bibkey: desogus2026threegates authors: Marco Desogus year: 2026 title: “The Three Gates: A Rooted-Operator Approach to Weil Positivity” doi: null url: https://arxiv.org/abs/2609.20367v3 claim: The preprint claims all-scale odd-channel Weil positivity via a common-cut Schur induction; this note records its actual-operator proof interfaces without adopting its claimed RH proof as a verified input. strata_touched: [] license: citation-only triage: anchor

Common-cut Schur induction: an unadopted all-scale claim

The source claims RH through all-scale positivity of the actual localized Weil operator on the real odd logarithmic channel. The claims below are version-specific: the current arXiv v3 is internally labelled V4; the older two-arm audit applies to arXiv v2. The full proof and supplementary certificates are not independently verified project premises, and no actual-arithmetic counterexample to either main claim is established here.

Current arXiv v3 interface: one exact source debit

The current primary is arXiv:2609.20367v3, revised 6 October 2026 and internally labelled V4, 69 pages. Its supplement is a different versioned deposit from the v2 archive. The paper still claims RH; the conclusion and the certificates are not independently verified project premises.

Audit statement 6.28 explicitly excludes the special multiplication-minus-rank-one folded Hessian from the active proof. Lemma 8.5 instead uses the exact reduced common-source metric and forcing, after one common-complement short, on the full-operator harmonic vector $F = H_{k} f$ . Its source row and single debit are

$C_{k}^{src} x_{k} = λ_{k} (c_{k}), D_{k}^{src} = ⟨ λ_{k} (c_{k}), (C_{k}^{src})^{- 1} λ_{k} (c_{k})⟩ .$

The square completion in MASTER-P3b is the standard exact Schur identity; (200) is its covariance under a unitary change of source coordinates. The source invokes the positive old block for the source metric’s positivity, and treats the endpoint fold solely as unitary transport. Thus the v2 two-arm formulas and their auxiliary physical-pivot condition are not active inputs to this version. The v2 scalar example below is not a counterexample to the v3 construction. No second debit should be imposed just because that older parametrization had two arms.

The reference term must be part of the actual reduced form

Lemma 8.1, (190)–(193), identifies the inherited row at its exact minimizer with the negative metric energy

$I_{k, m}^{act} = - S_{k, m}^{ex} [r_{m} c_{m}^{inh}] .$

Corollary 8.2 then asserts an aligned positive reference contribution on the same reduced row and uses

$S_{k, m}^{ex} [r_{m}^{0} c_{m}^{inh}] + I_{k, m}^{act} = ∣ c_{m}^{inh} ∣^{2} Δ S_{k, m}^{ex} \geq (1 - θ_{k, m}^{2}) Π_{m}^{(0)} ∣ X_{m} ∣^{2}, X_{m} = ε_{k, m} C_{m} c_{m}^{inh} .$

The equality is quadratic homogeneity once the reference term is present. The quantitative input not supplied by that equality is its provenance and allocation inside the actual full reduced Weil form. In (190) the frozen-coordinate remainder $Q_{k, m}^{rest}$ is independent of the varied inherited coordinate. Stationarity supplies the negative response contribution, not an extra positive reference energy. On the harmonic graph the inherited coordinate depends on the prescribed target, so this does not refute the possibility of a valid reference contribution. Its identification requires the actual remainder and target/source transport, with no reuse of the direct target diagonal $B_{k} ∥ c_{k} ∥^{2}$ , the same source metric, or a reserve already charged elsewhere.

Theorem 8.6 consumes that reference–actual splice, Lemma 8.3’s homogeneous parent estimate, the literal-slot true-ground leakage correction in Corollary 6.57/Certificate 6.58, and the complete one-debit source ledger. Theorem 6.53 states a joint operator lower form, stronger than testing pure directions separately; its statement can be cited as a source claim. Its active obligation is a lower bound on the whole same harmonic vector, including every source-row component and every mixed term. Separate bounds for pure target-ground and pure target-transverse data do not by themselves supply that combined lower bound. No additional ground/transverse covariance of the full source metric is inferred from branch ground-line invariance.

The complete archimedean/source correspondence remains necessary: Proposition 5.3 and the one-cell source metric used in Theorem 6.53/Lemma 6.61 must carry the full potential, regular kernel, pole and common-complement domain through to that same reduced vector. The normalization benchmarks below are reused, not recomputed. The v3 generic Schur and unitary identities do not establish this actual-family comparison.

The restricted odd Weil criterion and endpoint closure remain separate consumers; odd-only testing is not dismissed. The bounded primary check supplies no signed Robin main estimate, new positivity theorem, numerical certificate or RH proof. It also establishes no actual-arithmetic counterexample to v3. Its reusable source conclusion is that v2’s retired double-arm reduction must not be used to reject v3, while the positive reference contribution and full-form comparison still require verification before Theorem 8.6 can be adopted.

A full-pole bound for the literal physical collar

Use the physical form defined by (9)–(15) and (61)–(63) of arXiv v3. This calculation concerns its literal old/new compression, not the whole arithmetic target cell and not a free-complement source short. The source’s identification of that form with the global Weil form, and its subsequent common-cut transport, retain their separate verification obligations.

For every integer $k \geq 7$ , set

$a = \frac{1}{2} lo g k, b = \frac{1}{2} lo g (k + 1), h = b - a = \frac{w _{k}}{2}, w_{k} = lo g (1 + 1/ k) .$

Let $K_{k}$ be the odd functions supported in $(- b, - a) \cup (a, b)$ , initially on the smooth compactly supported core. Denote by $D_{k}^{A}$ the compression of the complete source-defined operator $A_{b, -}$ to this collar; thus this is the new diagonal in the full physical old/new block matrix. Write

$B_{k} = \frac{1}{2} + lo g (1/ w_{k}), τ_{k} = ∥ s^{ph} 1_{(- b, - a) \cup (a, b)} ∥_{2}^{2} = sinh b - sinh a - h .$

The pole quantity is the source’s exact restriction identity (114), not a profile replacement.

No prime-power term survives inside this new block

A same-side collar overlap would require a displacement smaller than $h < lo g 2$ . A cross-side overlap would require

$2 a < lo g n < 2 b, or equivalently k < n < k + 1.$

Neither can hold for an integer prime power $n \geq 2$ . Boundary contact at $n = k$ has zero measure. Consequently every prime-power translation in the complete new/new compression vanishes, including prime powers at the arithmetic endpoint. The old/new coupling remains present in the full operator.

Retain the reflected Gamma contribution and the true pole

Identify an odd collar vector isometrically with $u \in L^{2} (a, b)$ by taking $u (y) / 2$ on the positive interval and $- u (- y) / 2$ on the negative interval. Put $g (v) = h u (a + h v)$ for $0 < v < 1$ . Reuse the single-interval archimedean compression benchmark. Combining its two reflected copies gives the complete diagonal identity

$⟨ u, D_{k}^{A} u ⟩ = b [g] - [lo g (2 πh) + γ] ∥ u ∥_{2}^{2} - \iint_{(a, b)^{2}} ρ (∣ y - z ∣) u (y) \overline{u (z)} d y d z + \iint_{(a, b)^{2}} κ (y + z) u (y) \overline{u (z)} d y d z - 4 \int_{a}^{b} sinh (y /2) u (y) d y^{2}, (C1)$

where $ρ (r) = e^{- r /2} / (1 - e^{- 2 r}) - 1/ (2 r)$ and

$κ (r) = \frac{1}{2 r} + ρ (r) = \frac{e ^{- r /2}}{1 - e ^{- 2 r}} = j \geq 0 \sum e^{- (2 j + 1/2) r} .$

Thus the reflected term is positive as an operator, not merely pointwise:

$\iint κ (y + z) u (y) \overline{u (z)} d y d z = j \geq 0 \sum \int_{a}^{b} e^{- (2 j + 1/2) y} u (y) d y^{2} \geq 0. (C2)$

The sum converges because $a \geq \frac{1}{2} lo g 7 > 0$ . The reflected singular and regular kernels must be combined before using this sign.

For $0 \leq r \leq h < 1/2$ , the continuous extension of the regular kernel obeys

$0 \leq ρ (r) \leq \frac{1}{4} . (C3)$

For the upper bound, equivalently $(2 + r) sinh r \geq 2 r e^{r /2}$ , the odd power coefficients after the first are $2 [1 - (2 j + 1) / 4^{j}] / (2 j + 1)!$ and the even ones are $[1 - 1/ 4^{j}] / (2 j + 1)!$ , all nonnegative. For the lower bound, $sinh r / r \leq cosh r \leq e^{r^{2} /2} \leq 1 + r^{2} \leq 1 + r /2 \leq e^{r /2}$ for $r \leq 1/2$ . Schur’s kernel bound therefore gives an operator norm at most $h /4 = w_{k} /8$ on this interval.

The already supplied one-cell form has $b [g] \geq (lo g 2) ∥ g ∥_{2}^{2}$ . Its stronger spectral constant may be retained: the existing small-window source defines $μ_{1} > 0$ and gives $b [g] \geq (lo g 2 + μ_{1}) ∥ g ∥_{2}^{2}$ after the affine unitary. No local spectral theorem is reproved here. Finally Cauchy–Schwarz bounds the full negative pole by $2 τ_{k} ∥ u ∥_{2}^{2}$ . Hence (C1) yields the all- $k$ literal-collar lower bound

$D_{k}^{A} ⪰ [B_{k} - C_{k} + μ_{1}] I, C_{k} = \frac{1}{2} + lo g (π /2) + γ + \frac{w _{k}}{8} + 2 τ_{k} . (C4)$

Dropping only the known positive $μ_{1}$ gives the weaker explicit coefficient. This bound applies on the displayed smooth core and its Friedrichs form closure; it is not an assertion that an unproved transported source domain coincides with that closure.

There is a uniform elementary allowance:

$2 τ_{k} = k + 1 - k + \frac{1}{k} - \frac{1}{k + 1} - w_{k} < \frac{1}{2 k} + \frac{1}{2 k ^{3/2}} \leq \frac{4}{7 7} < \frac{20}{91}, \frac{w _{k}}{8} < \frac{1}{56} .$

Using the standard constant bounds $γ < 3/5$ , $lo g (π /2) < 23/50$ and $7 > 13/5$ , one obtains

$C_{k} < \frac{39}{25} + \frac{1}{56} + \frac{20}{91} = \frac{9}{5} - \frac{43}{18200} < \frac{9}{5} . (C5)$

Consequently $D_{k}^{A} ⪰ (B_{k} - 9/5) I$ in this realization. The Gamma reflection sign, vanished prime overlaps and true negative pole all refer to the same physical vector.

The constant target diagonal is not the whole physical compression

The existing small-window domain argument permits the constant profile in the closed form. Let $u_{k} = h^{- 1/2}$ on $(a, b)$ and take its odd pair as above. Then $b [1] = 1$ , the self-regular term is $O (h)$ , the reflected term is $O (h k^{- 1/2})$ , and the pole is $O (k^{- 1/2})$ . Since $h = w_{k} /2$ , (C1) gives

$⟨ u_{k}, D_{k}^{A} u_{k} ⟩ - B_{k} ⟶ \frac{1}{2} - lo g π - γ < 0. (C6)$

Therefore the scalar $B_{k} I$ is not equal to, or a lower bound for, the complete literal physical new diagonal. This does not refute a decomposition in which $B_{k}$ is one named primal term accompanied by further diagonal contributions. Any such decomposition must retain those contributions and the actual old/new transport before consuming MASTER-P3c. Nor is (C6) a negative test for the full Weil form: the actual diagonal grows like $lo g k$ .

The new interface is the complete prime-free odd collar comparison (C1)–(C5), with (C6) specifying its normalization. The one-cell floor, basic compression benchmark, Schur kernel estimate and exact pole restriction are reused. The global signed obligation remains the old/source inverse debit on the same harmonic vector; no positive aligned reference contribution is supplied by this diagonal bound, and no Robin or RH conclusion follows alone. The source’s finite MASTER margin must not be transported through (C5) without a paid common-source identification and a compatible all-scale tail.

A pole-free odd null family tests the raw old/source metric

The following calculation concerns the complete compact-test Weil form and its actual pole-free odd core. It tests a proposed identification of a raw old-block variational short with a uniformly positive one-cell source metric. It does not identify the source’s allocated common-cut component with that raw short, and does not assert that the source’s full induction is refuted.

Reuse the theta transform, all-order weighted derivative tails, compact weighted-two-jet bridge and multiplicity-weighted zero summability already recorded in the localization note. The published xi null-vector mechanism and the existing critical derivative family are not new results here. The additional interface is their quantitative comparison with a moving actual prime-power collar source slot, retaining the odd sector and annihilating the true pole.

An odd test family that also annihilates the pole

Let $Φ$ be the original positive even theta kernel, with

$Φ (z) = ξ (1/2 + i z), f (z) = \int_{R} f (y) e^{- i zy} d y .$

Define the real odd rapidly decaying function

$g (y) = Φ^{'''} (y) - \frac{1}{4} Φ^{'} (y) .$

Tilted integration by parts, justified by the retained all-order theta tails, gives

$\int_{R} g (y) e^{zy} d y = - z (z^{2} - 1/4) ξ (1/2 + z) . (O1)$

In particular this transform vanishes at every actual centered zero, with no assumption on its real part, and at both pole arguments $z = \pm 1/2$ . Thus

$\int_{R} g (y) sinh (y /2) d y = 0. (O2)$

The function $g$ is nonzero: its real Fourier transform is $(i t) ((i t)^{2} - 1/4) ξ (1/2 + i t)$ , which is nonzero for sufficiently small nonzero $t$ because $ξ (1/2) > 0$ . It follows by oddness and continuity that there is a positive point at which $g$ is nonzero. Fix a dyadic rational $M > 1$ such that

$y_{0} = \frac{1}{2} lo g M > 0, g (y_{0}) \neq = 0.$

Such an $M$ exists because dyadic rationals are dense in $(1, \infty)$ and the nonzero set is open. This choice requires no prime-distribution estimate.

Use a real fresh prime-power branch at arbitrarily large endpoints

Write $M = p / 2^{d}$ with positive integers $p, d$ , allowing $d = 0$ . For sufficiently large integers $j$ , put

$n_{j} = 2^{j}, k_{j} = M n_{j}^{2} + 1, a_{j} = \frac{1}{2} lo g k_{j}, b_{j} = \frac{1}{2} lo g (k_{j} + 1), h_{j} = b_{j} - a_{j} .$

For $j \geq d$ , $k_{j}$ is an integer, $M n_{j}$ is an integer, and $n_{j} ∤ k_{j}$ . Thus $n_{j}$ is an active fresh prime power in the source’s arithmetic routing, with $Λ (n_{j}) = lo g 2$ . Its actual translated positive physical collar source slot is

$J_{j} = (a_{j} - lo g n_{j}, b_{j} - lo g n_{j}) ⋐ (- a_{j}, a_{j}) . (O3)$

Both endpoints tend to $y_{0}$ , and $k_{j} h_{j} \to 1/2$ . Its reflected partner is $- J_{j}$ , disjoint from $J_{j}$ for sufficiently large $j$ .

These are literal prime-translation source intervals. They are also contained in the source’s full arithmetic slot: at endpoint $Y = k_{j} + 1$ , the exact Mellin coordinate is $t = e^{y + b_{j}}$ , so

$[\frac{k _{j}}{n _{j}}, \frac{k _{j} + 1}{n _{j}}) ⟷ [a_{j} - h_{j} - lo g n_{j}, b_{j} - lo g n_{j}) . (O4)$

The parent index is $m_{j} = ⌊ k_{j} / n_{j} ⌋ = M n_{j}$ and the non-divisorial residue is exactly $1$ . The Mellin substitution preserves $d y = d t / t$ , so a source-slot unitary preserves the norm used below. No abstract interval is substituted for this branch.

The complete old-core energy is smaller than every power of the slot width

Fix a real even $χ \in C_{c}^{\infty} (- 1, 1)$ , equal to one on $[- 1/2, 1/2]$ , and set

$u_{j} (y) = χ (y / a_{j}) g (y) .$

This is an actual real odd compact smooth test in the old physical support $(- a_{j}, a_{j})$ , and agrees with $g$ on $J_{j} \cup (- J_{j})$ for all sufficiently large $j$ . The reused weighted derivative tails, with the cutoff starting at $∣ y ∣ = a_{j} /2$ , give

$ϵ_{j} := \int e^{∣ y ∣/2} (∣ u_{j} - g ∣ + ∣ (u_{j} - g)^{''} ∣) d y = O_{R} (k_{j}^{- R}) for every R > 0. (O5)$

Indeed $e^{2∣ y ∣} \geq e^{a_{j}} = k_{j}$ on the error support; any polynomial factor from the fixed derivatives and the cutoff is absorbed by the original $e^{- c k_{j}}$ tail. The constants may depend on $R$ , $M$ , $g$ and the fixed cutoff, but not on $j$ .

Use the complete compact Weil explicit formula with

$z_{ρ} = ℑ ρ - i (ℜ ρ - 1/2), Q (u_{j}) = ρ \sum m_{ρ} u_{j} (z_{ρ}) \overline{u_{j} (\overline{z_{ρ}})} .$

Both limiting factors vanish by (O1), including the reflected zero $1 - \overline{ρ}$ . The weighted-two-jet estimate gives

$∣ u_{j} - g (t + i w) ∣ \leq \frac{2 ϵ _{j}}{1 + t ^{2}}, ∣ w ∣ \leq 1/2.$

Consequently the already supplied finite sum $S_{4} = \sum_{ρ} m_{ρ} (1 + (ℑ ρ)^{2})^{- 2}$ yields $∣ Q (u_{j}) ∣ \leq 4 S_{4} ϵ_{j}^{2}$ . The paired factors have not been replaced by modulus squares off the critical line.

Write the true polar-free old form, on this same compact core, as

$C_{a_{j}} [u_{j}] = Q (u_{j}) + 2 \int u_{j} (y) sinh (y /2) d y^{2} .$

Equation (O2) and $∣ sinh (y /2) ∣ \leq e^{∣ y ∣/2} /2$ bound the added term by $ϵ_{j}^{2} /2$ . Therefore

$∣ Q (u_{j}) ∣ + ∣ C_{a_{j}} [u_{j}] ∣ = O_{R} (k_{j}^{- R}) for every R > 0. (O6)$

Only compact tests are evaluated by the arithmetic form. No new form on noncompact functions, RH assumption or sign of $Q (u_{j})$ is asserted. If the source-defined $A_{a, -}$ and $C_{a}$ are identified with these complete Weil forms on their common core, the same estimate applies to those operators; that identification retains its independent obligation.

In contrast, their actual slot data have polynomially sized mass:

$ℓ_{j} := ∥ 1_{J_{j} \cup (- J_{j})} u_{j} ∥_{2}^{2}, k_{j} ℓ_{j} ⟶ ∣ g (y_{0}) ∣^{2} > 0. (O7)$

Thus neither the complete odd old form nor its true polar-free core admits, for any fixed $c > 0$ , the bound

$q_{a_{j}} [v] \geq c ∥ 1_{J_{j} \cup (- J_{j})} v ∥_{2}^{2} for every old compact odd test v (O8)$

at all sufficiently large $j$ . Taking $v = u_{j}$ gives the failure directly, independently of whether $q_{a_{j}}$ is positive.

A raw common-source short must retain these nearly null directions

Whenever the actual old form $q_{a_{j}}$ has a positive closed realization and the variational short under consideration, let $S_{j} = - S_{j} \subset (- a_{j}, a_{j})$ be either this reflected slot pair or a reflection-symmetric union of actual old source slots containing it. Work on $H_{j} = L_{odd}^{2} (S_{j}, d y)$ , with the inherited $L^{2}$ norm, and define

$q_{j}^{short} [r] = in f {q_{a_{j}} [v] : v \in D (q_{a_{j}}), v odd, 1_{S_{j}} v = r} .$

Assume the valid closed short on this data space when invoking its associated operator or inverse. Reflection symmetry makes the support restriction an orthogonal projection within the odd old space. Put $r_{j} = 1_{S_{j}} u_{j} \in H_{j}$ . By this definition and (O6)–(O7),

$0 \leq Short_{S_{j}} q_{a_{j}} [r_{j}] \leq q_{a_{j}} [u_{j}], \frac{Short _{S_{j}} q _{a_{j}} [ r _{j} ]}{∥ r _{j} ∥ _{2}^{2}} = O_{R} (k_{j}^{- R}) for every R > 0. (O9)$

This holds on any cofinal subsequence satisfying the stated positivity/shorting conditions, for the full odd form or the pole-free core. If the associated exact short operator on $H_{j}$ is additionally strictly positive with a bounded inverse, the usual positive-operator Cauchy–Schwarz inequality makes that inverse norm grow faster than every fixed power of $k_{j}$ along the subsequence. It supplies no lower bound for the actual debit $B_{A, k}^{*} A_{k}^{- 1} B_{A, k}$ , since that forcing need not lie in the normalized $r_{j}$ direction.

The one-cell source model and its CMC/ground lower bounds remain valid in their declared realization. Equations (O8)–(O9) exclude identifying them with a uniformly coercive lower bound for this whole raw actual old/source short by source-slot unitaries alone. A valid full-form allocation may instead separate an archimedean component, retain compensating arithmetic/pole/target terms, or estimate the actual forcing after a justified treatment of the critical family. Such an allocation must be proved on the same object; the known one-cell floor cannot be imported as a raw old inverse bound. This is a specific realization boundary, not a refutation of the source’s allocated-component claim or of its full induction.

The all-scale signed Robin estimate remains unproved. The application supplies a quantitative test for the missing raw metric correspondence and explains why an actual directional inverse estimate, including the nearly null family, is still required. Classical nullity, theta tails, zero summability and variational shorting are reused; the moving prime-power slot comparison (O3)–(O9) is the additional paper-level interface, without a new Lean certification or mathematical-priority claim.

Complete collar response versus one actual prime branch

Retain the actual odd family, integer endpoints and compact cutoffs from (O1)–(O9). This calculation gives the full old-to-new response of that family, uniformly against every $L^{2}$ odd collar probe. It complements the raw source-metric boundary; it does not estimate the old inverse debit on arbitrary forcing.

Reuse the complete mixed Weil pairing and compact-domain bridge, the general two-pole and prime formula, and the classical multiplicity-weighted inverse-square zero mass recorded in the resolvent supplier. No mixed-nullity or zero-summability theorem is reproved. The even theta-weighted commutator bounds in the exterior account are not assigned to the physical odd metric without a map.

Write

$E_{j} = (- b_{j}, - a_{j}) \cup (a_{j}, b_{j}), α_{j} = 2 h_{j} e^{b_{j} /2}, S_{2} = ρ \sum \frac{m _{ρ}}{1 + ( ℑ ρ ) ^{2}} < \infty, m_{j} = \int u_{j} (y) sinh (y /2) d y .$

All zeros in $S_{2}$ are actual nontrivial zeros, with both ordinate signs and every multiplicity. Put $B_{Q}$ for the full compact Weil pairing, linear in the first argument. For a compact smooth odd $v$ supported in $E_{j}$ , the disjoint supports eliminate multiplication terms. The same general explicit formula gives the physical response $Y_{j}^{Q}$ by

$Y_{j}^{Q} (y) = B_{Q} (u_{j}, v) = - \int_{- a_{j}}^{a_{j}} κ (∣ y - z ∣) u_{j} (z) d z - 2 sinh (y /2) m_{j} - 2 \leq n \leq k_{j} \sum \frac{Λ ( n )}{n} (u_{j} (y + lo g n) + u_{j} (y - lo g n)), y \in E_{j}, ⟨ Y_{j}^{Q}, v ⟩_{L^{2} (E_{j})}, κ (t) = \frac{e ^{- t /2}}{1 - e ^{- 2 t}} . (O10)$

The theta cutoff has compact support strictly inside the old interval, so this Gamma cross integral is nonsingular. Every prime-power translation that can reach this collar is retained; translations with $n \geq k_{j} + 1$ have no old/new overlap. The true negative odd pole has its coefficient $2$ . Formula (O10) refers to the actual compact Weil form. Identifying the source-defined $B_{A, k_{j}}^{*} u_{j}$ with it still requires the source’s complete common-core identification.

A bound uniform over all collar coefficients

For $∣ w ∣ \leq 1/2$ , Cauchy–Schwarz on the actual collar gives

$∣ v (t + i w) ∣ \leq e^{b_{j} /2} ∥ v ∥_{1} \leq α_{j} ∥ v ∥_{2} .$

The complete paired spectral formula is

$B_{Q} (u_{j}, v) = ρ \sum m_{ρ} u_{j} (z_{ρ}) \overline{v (\overline{z_{ρ}})}, z_{ρ} = ℑ ρ - i (ℜ ρ - 1/2) .$

Equation (O1) makes the limiting old factor zero at every $z_{ρ}$ ; (O5) bounds its actual error by $2 ϵ_{j} / (1 + (ℑ ρ)^{2})$ . The probe bound therefore pays the complete sum by $2 S_{2} α_{j} ϵ_{j} ∥ v ∥_{2}$ . No bound on the probe’s derivatives is required. Density of the smooth odd collar core and Riesz representation give

$∥ Y_{j}^{Q} ∥_{2} \leq 2 S_{2} α_{j} ϵ_{j} . (O11)$

For the true pole-free core, its response is $Y_{j}^{C} = Y_{j}^{Q} + 2 m_{j} sinh (y /2) ∣_{E_{j}}$ . Using $∣ m_{j} ∣ \leq ϵ_{j} /2$ from (O2) and $∥ sinh (y /2) ∥_{L^{2} (E_{j})} \leq α_{j} /2$ gives

$∥ Y_{j}^{C} ∥_{2} \leq (2 S_{2} + 1/2) α_{j} ϵ_{j} .$

Since $α_{j} ≍ k_{j}^{- 1/4}$ and $ϵ_{j} = O_{R} (k_{j}^{- R})$ for every $R > 0$ , both complete response norms are smaller than every fixed power of $k_{j}^{- 1}$ . This is a uniform $L^{2}$ collar estimate on the stated old family, not merely convergence for one fixed smooth probe. It uses the same actual zero real parts and paired factors; no critical-line or simple-zero assumption is made.

The selected prime-power response alone is polynomially sized

Keep the literal contribution of $n_{j} = 2^{j}$ in (O10):

$P_{j} (y) = - \frac{lo g 2}{n _{j}} (u_{j} (y + lo g n_{j}) + u_{j} (y - lo g n_{j})) .$

On the positive collar only $u_{j} (y - lo g n_{j})$ is present, and on the negative collar only its odd reflected partner is present. The source slots are exactly $J_{j}$ and $- J_{j}$ from (O3); the two translations preserve Lebesgue measure. Therefore

$∥ P_{j} ∥_{2}^{2} = \frac{( lo g 2 ) ^{2}}{n _{j}} ℓ_{j}, k_{j}^{3/2} ∥ P_{j} ∥_{2}^{2} ⟶ (lo g 2)^{2} M ∣ g (y_{0}) ∣^{2} > 0. (O12)$

The selected response is thus of order $k_{j}^{- 3/4}$ in norm. It is not a hypothetical edge or a prime sample: $n_{j}$ is the fresh branch with parent $M n_{j}$ and residue one, at the same integer endpoint used in (O3)–(O9).

Let $R_{j}^{Q}$ consist of the Gamma and true pole terms in (O10), together with all the displayed prime-power terms except $n_{j}$ . This is an independently specified sum of the actual remaining terms. Equations (O10)–(O12) give

$Y_{j}^{Q} = P_{j} + R_{j}^{Q}, \frac{∥ P _{j} + R _{j}^{Q} ∥ _{2}}{∥ P _{j} ∥ _{2}} ⟶ 0. (O13)$

Thus the remaining actual response cancels the polynomially sized selected prime response in relative $L^{2}$ norm on these collars. The same statement holds for the pole-free core with its actual pole-removal term included in $R_{j}^{C}$ . Independent absolute estimates for these components would miss this joint relation. The cancellation is quantitative and on one actual source; it does not follow from independently achievable component bounds or from replacing a signed response by its norm.

The inverse budget remains a separate estimate

Equations (O9) and (O11) show that a nearly null raw source metric and a small complete collar response can coexist, even though a single real prime branch remains polynomially large. They do not show that $B_{A, k}^{*} A_{k}^{- 1} B_{A, k}$ is small. In particular, an upper bound on the coupling numerator and an upper bound on the old energy do not bound their quotient; a matching directed estimate and control of all other old directions remain required. No whole-form positivity, Robin main upper bound or RH conclusion is inferred.

The additional interface is the complete physical collar norm (O11) and its same-source comparison with the isolated actual prime branch (O12)–(O13). Classical mixed nullity, support Cauchy–Schwarz, Riesz representation, zero summability and the accepted moving-slot construction are reused. This is a paper-level application, without Lean certification or a mathematical-priority claim.

A natural-tail cutoff pays one actual old direction

The small response in (O11) cannot be divided by the old-energy upper bound in (O6). The following chooses a different cutoff of the same odd null function and proves a matching lower denominator. It pays only the resulting one-dimensional old direction; the full inverse supremum remains separate.

Reuse the original theta series and derivative bounds in the derivative-family account, the compact weighted-jet/domain bridge and complete two-pole formula, and the Gamma scaling calculation underlying the physical compression benchmark. The even theta-weighted exterior and commutator estimates in Lenz’s application have a different metric and are not used as odd physical estimates.

Retain $g$ , the fixed $M$ , and the actual endpoints $k_{j}, a_{j}, b_{j}, h_{j}, E_{j}$ from (O1)–(O3) and (O10). Put

$R_{j} = a_{j} /2 = \frac{1}{4} lo g k_{j}, δ_{j} = e^{- 2 R_{j}} = k_{j}^{- 1/2} .$

Fix a real smooth function $0 \leq η \leq 1$ , zero on $(- \infty, 0]$ and one on $[1, \infty)$ . Define a new compact odd old test and its actual error tail by

$u_{j} (y) = (1 - η ((∣ y ∣ - R_{j}) / δ_{j})) g (y), e_{j} (y) = g (y) - u_{j} (y) . (T1)$

The multiplier is constant near zero, so the use of $∣ y ∣$ does not impair smoothness. The support of $u_{j}$ lies in $[- R_{j} - δ_{j}, R_{j} + δ_{j}] ⋐ (- a_{j}, a_{j})$ for large $j$ . It still agrees with $g$ on the actual reflected source slots. This is a change of the chosen test family, not a change of any arithmetic operator, prime branch or earlier cutoff statement.

The actual tail has one natural width

Suppress $j$ temporarily and write $R = R_{j}$ , $δ = δ_{j}$ . Directly differentiating the reused theta series gives

$g (R) = - 32 π^{5} e^{21 R /2} e^{- π e^{2 R}} (1 + O (e^{- 2 R})) .$

Thus $A_{R} = - g (R) > 0$ for large $R$ . Let

$H_{R} (s) = - e_{j} (R + δs) / A_{R}, H (s) = η (s) e^{- 2 π s}, c_{η} = \int_{0}^{\infty} H (s)^{2} d s > 0,$

with both profiles zero for $s \leq 0$ . The same first theta term and its derivatives, with the remaining normally convergent series exponentially smaller, show

$H_{R} ⟶ H in L^{1} \cap H^{2}, ∣ H_{R}^{(m)} (s) ∣ \leq C_{m} e^{- cs} (s \geq 0, 0 \leq m \leq 2), (T2)$

for fixed $c > 0$ and constants independent of large $R$ . Indeed $e^{2 R} (e^{2 δs} - 1) \geq 2 s$ and $δ e^{2 R} = 1$ ; the polynomial factors from each fixed derivative are absorbed by this exponential. On bounded $s$ the leading term tends to $e^{- 2 π s}$ , giving the stated dominated convergence. Consequently the actual odd error mass is

$M_{R} := ∥ e_{j} ∥_{2}^{2} = 2 A_{R}^{2} δ (c_{η} + o (1)) . (T3)$

These theta consequences are intermediate applications, not independent new theta identities.

A complete-form bridge for this noncompact error

Only $u_{j}$ is an old compact arithmetic test. To calculate its energy, approximate the fixed- $j$ error $e_{j}$ by real even compact cutoffs times $e_{j}$ . The reused all-order theta tails pay the weighted two-jet error and both pole integrals. They also supply the Gamma small-shift bound by the first derivative and an integrable large-shift majorant.

For the complete prime sum, the same two-tail estimate used in the derivative-family domain passage applies:

$e^{2∣ y ∣} + e^{2∣ y - l o g n ∣} \geq 2 n .$

Products of fixed theta derivatives therefore have integrated majorant $C_{j} n^{d_{j}} e^{- c_{j} n}$ , including every prime power and independent of the outer approximation cutoff. This proves absolute convergence and dominated passage for the prime correlations. The full paired zero sum passes by the weighted-jet bridge and multiplicity-weighted inverse-fourth zero mass.

Write $Q^{tail} (e_{j})$ for this particular limit of the complete arithmetic expressions. This defines neither a new closed operator nor positivity on an arbitrary noncompact domain. At every actual zero, (O1) gives $e_{j} (z_{ρ}) = - u_{j} (z_{ρ})$ , and likewise for the reflected factor. Thus, with actual zero real parts, both signs and multiplicities unchanged,

$Q (u_{j}) = Q^{tail} (e_{j}), B_{Q} (u_{j}, v) = - B_{Q}^{tail} (e_{j}, v) (T4)$

for every compact smooth odd collar probe $v$ . The same domain passage supplies the mixed equality. This reuses full mixed nullity rather than replacing the off-line pair by a modulus square.

On this tail the complete expression is

$Q^{tail} (e_{j}) = c_{Γ} ∥ e_{j} ∥_{2}^{2} + \int_{0}^{\infty} κ (t) ∥ e_{j} - τ_{t} e_{j} ∥_{2}^{2} d t - 2 n \geq 2 \sum \frac{Λ ( n )}{n} ℜ ⟨ e_{j}, τ_{l o g n} e_{j} ⟩ - 2 \int e_{j} (y) sinh (y /2) d y^{2}, (T5)$

where $c_{Γ} = ℜ digamma (1/4) - lo g π$ and $κ$ is exactly (O10). The unbounded prime sum is paid by the domain bridge; it is not truncated at the compact old endpoint.

The lower denominator retains Gamma, every prime and the true pole

For one positive-tail copy $f_{R} (y) = - A_{R} H_{R} ((y - R) / δ)$ , the reused Gamma dilation calculation has the explicit form

$\int_{0}^{\infty} κ (t) ∥ f_{R} - τ_{t} f_{R} ∥_{2}^{2} d t = A_{R}^{2} δ \int_{0}^{\infty} δ κ (δs) ∥ H_{R} - τ_{s} H_{R} ∥_{2}^{2} d s = (lo g (1/ δ) + O (1)) ∥ f_{R} ∥_{2}^{2} .$

For completeness, the profile bounds in (T2) pay the uniform remainder: below $s = 1$ use the derivative bound; above it write the squared increment as $2∥ H_{R} ∥_{2}^{2} - 2ℜ ⟨ H_{R}, τ_{s} H_{R} ⟩$ , with exponentially decaying correlation. The scalar integral obeys $2 \int_{δ}^{\infty} κ (t) d t = lo g (1/ δ) + O (1)$ . This is the existing kernel scaling mechanism, consumed here with a noncompact but controlled profile.

The reflected tail cross term is at separation at least $2 R$ , so its absolute value is at most $C A_{R}^{2} δ^{2} e^{- R} = O (e^{- 3 R} M_{R})$ . Combining the two tails and the actual $c_{Γ}$ gives the Gamma part $(2 R + O (1)) M_{R}$ .

The same-side prime correlations satisfy $C A_{R}^{2} δ e^{- c t / δ}$ . Their sum over $t = lo g n \geq lo g 2$ , using only $Λ (n) \leq lo g n$ , is exponentially small in $1/ δ$ relative to $M_{R}$ . Opposite tails only correlate for $t \geq 2 R$ ; putting $v = (t - 2 R) / δ$ bounds the correlation by $C A_{R}^{2} δ (1 + v) e^{- c v}$ . Absorb the polynomial into a smaller exponential. For $T = e^{2 R} = 1/ δ$ , elementary integral comparison then gives

$n \geq ⌈ T ⌉ \sum \frac{lo g n}{n} e^{- c l o g (n / T) / δ} \leq CR e^{- R} .$

Hence the absolute value of the full prime correction is at most $CR e^{- R} M_{R}$ . No prime-distribution hypothesis or finite prime sample is used. The true odd pole is also controlled on this same error:

$\int e_{j} (y) sinh (y /2) d y \leq C A_{R} δ e^{R /2}, 2 \int e_{j} sinh (y /2)^{2} \leq C e^{- R} M_{R} .$

Equations (T3)–(T5) therefore give the matching complete lower energy

$Q (u_{j}) = (2 R_{j} + O (1)) M_{R_{j}} > 0 (j large) . (T6)$

The actual pole-free old energy has the same asymptotic, since it adds precisely the displayed pole square. This establishes positivity only for this specified compact family.

The complete physical collar row at the same scale

Let $Y_{j}^{Q}$ be the literal response (O10) with $u_{j}$ replaced by $u_{j}$ . Use (T4) to estimate it from the tail. The whole-line Gamma row of $e_{j}$ has its local multiplier and symmetric difference integral retained. On $y \in (a_{j}, b_{j}) = (2 R_{j}, 2 R_{j} + h_{j})$ its local small-shift part is negligible by the original theta tails, while the remote part is bounded by

$C A_{R} δ e^{- R /2} .$

One can split at $t = R /2$ : below it, $y \pm t \geq 3 R /2$ and the symmetric small-shift difference is paid by the second derivative; above it, (T2) and $κ (t) \leq C e^{- t /2}$ bound the near- $R$ tail by $A_{R} δ e^{- (y - R) /2}$ and its reflected partner by $A_{R} δ e^{- (y + R) /2}$ . The remaining theta tail is smaller than these budgets.

The full prime row is estimated without deleting the translations beyond the old compact threshold. For $T_{-} = e^{y - R}$ and $T_{+} = e^{y + R}$ , the two source regions of $e_{j} (y - lo g n)$ give

$2 \leq n \leq T_{-} \sum \frac{Λ ( n )}{n} ∣ e_{j} (y - lo g n) ∣ n \geq T_{+} \sum \frac{Λ ( n )}{n} ∣ e_{j} (y - lo g n) ∣ \leq C A_{R} lo g T_{-} (T_{-}^{- 1/2} + δ T_{-}^{1/2}), \leq C A_{R} lo g T_{+} (T_{+}^{- 1/2} + δ T_{+}^{1/2}) .$

These follow by summing the bounds $(n / T_{-})^{c / δ}$ and $(n / T_{+})^{- c / δ}$ from (T2), respectively, and using $Λ (n) \leq lo g n$ . Endpoint terms are retained in the integral comparison, including when $δ T_{-} < 1$ . Since $T_{-} ≍ e^{R}$ , $T_{+} ≍ e^{3 R}$ and $δ = e^{- 2 R}$ , their total is $C A_{R} R e^{- R /2}$ . The other orientation $e_{j} (y + lo g n)$ is smaller than every exponential in $R$ relative to $A_{R}$ by the same bound. Odd reflection supplies the negative collar, with no omitted orientation.

Finally the true pole row is bounded by $2∣ \int e_{j} sinh (y /2) ∣ sinh (y /2) \leq C A_{R} e^{- R /2}$ on the positive collar. Thus (T4), with all local, Gamma, prime and pole terms retained, gives

$∥ Y_{j}^{Q} ∥_{2}^{2} \leq C h_{j} A_{R_{j}}^{2} R_{j}^{2} e^{- R_{j}} . (T7)$

This is a full physical $L^{2} (E_{j})$ row bound, uniform in every collar coefficient. The true pole-free core response satisfies the same bound, since its actual pole-removal term has the just-displayed size. Its identification with source-defined operators still requires the complete common-core map.

A correctly directed rank-one debit, with the other directions unpaid

Since $h_{j} ≍ e^{- 4 R_{j}}$ , (T3), (T6) and (T7) now supply a lower denominator and upper numerator on the same actual test:

$\frac{∣ B _{Q} ( u _{j} , v ) ∣ ^{2}}{Q ( u _{j} )} \leq C R_{j} e^{- 3 R_{j}} ∥ v ∥_{2}^{2} \leq C (lo g k_{j}) k_{j}^{- 3/4} ∥ v ∥_{2}^{2} . (T8)$

It holds for every odd $L^{2}$ collar $v$ , by the response representation and density; the denominator is positive for large $j$ by (T6), without an all-support positivity assumption. The same statement holds with the true pole-free form in both numerator and denominator. Constants may depend on the fixed smooth transition and theta kernel, but not on $j$ or $v$ .

Equation (T8) pays the variational debit restricted to the line $C u_{j}$ . It is not the supremum over the full old space, does not remove cross terms with its old complement, and does not bound the entire operator $B_{A, k}^{*} A_{k}^{- 1} B_{A, k}$ . Applying it to that source’s restricted line remains conditional on the source’s complete form and norm identification. The original signed Robin main estimate and RH remain open.

The additional interface is the matched complete physical old-energy/collar-response quotient (T6)–(T8) at the natural theta-tail width. Theta nullity and tails, Gamma dilation, the full explicit formula and standard comparison/domain arguments are reused as inputs. The calculation is paper-level, without Lean certification or a mathematical-priority claim.

Joint control of a fixed derivative space at actual collars

Fix an integer $d \geq 1$ . A separate bound for each of $d + 1$ old lines does not control their joint variational debit: their actual energy Gram can have much smaller directions. The following controls every complex combination, including coefficients depending on the endpoint, using the same natural-tail cutoff and physical form as (T1)–(T8).

Reuse the full theta derivative family and its compact-domain bridge, Romik’s original series and natural tail scale, and the complete Gamma, prime and two-pole bounds in (T4)–(T7). The common negative-edge Gram in the even critical-space account has a different measure, parity and form; its invertibility is not an inverse bound for the present odd physical energy. Classical polynomial interpolation and finite-dimensional Gram stability are used inside the joint estimate, not presented as new supplier theorems.

Let $D = d / d y$ and define

$g_{m} = D^{2 m + 1} (D^{2} - 1/4) Φ, 0 \leq m \leq d .$

The existing theta transform and tilted integration by parts give

$\int g_{m} (y) e^{zy} d y = - z^{2 m + 1} (z^{2} - 1/4) ξ (1/2 + z) .$

Every $g_{m}$ is odd, vanishes at all actual zeros in this transform, and annihilates both pole arguments. Retain the actual $k_{j}, a_{j}, b_{j}, h_{j}, E_{j}$ , $R_{j} = lo g (k_{j}) /4$ , $δ_{j} = e^{- 2 R_{j}}$ and the one fixed smooth transition $η$ from (T1). Write

$χ_{j} (y) = 1 - η ((∣ y ∣ - R_{j}) / δ_{j}), U_{j, d} = span_{C} {χ_{j} g_{0}, \dots, χ_{j} g_{d}} . (T9)$

All these tests have the same compact old support. The coefficients in $U_{j, d}$ are unrestricted at each $j$ ; no fixed-vector limit is substituted for a uniform estimate.

Resolve coalescing tails before taking an inverse

Suppress $j$ and put $R = R_{j}$ , $δ = e^{- 2 R}$ , $T = e^{2 R} = 1/ δ$ . In the original theta series the $n = 1$ term is

$Φ_{1} (y) = e^{y /2} P (X) e^{- π X}, X = e^{2 y}, P (X) = 4 π^{2} X^{2} - 6 π X .$

Differentiation acts on its polynomial by

$D = \frac{1}{2} + 2 X \frac{d}{d X} - 2 π X .$

Thus the first-term polynomial for $g_{m}$ is

$p_{m} = (D^{2 m + 3} - \frac{1}{4} D^{2 m + 1}) P, ν_{m} = de g p_{m} = 2 m + 5, a_{m} = 4 π^{2} (- 2 π)^{2 m + 3}$

with leading coefficient $a_{m}$ . Introduce the scaled jet matrix

$J (T)_{s, m} = \frac{T ^{s} p _{m}^{(s)} ( T )}{s ! a _{m} T ^{ν_{m}}}, 0 \leq s, m \leq d . (T10)$

Since every lower polynomial coefficient is fixed, $J (T)_{s, m} = (s ν _{m}) + O_{d} (T^{- 1})$ . The limiting matrix is invertible: the standard polynomial Vandermonde determinant is

$det [(s ν _{m})]_{s, m = 0}^{d} = \frac{\prod _{m < n} ( ν _{n} - ν _{m} )}{\prod _{s = 0}^{d} s !} = 2^{d (d + 1) /2} \neq = 0.$

For large $R$ , define a real change of basis by

$β_{m r} (T) = \frac{T ^{r - ν_{m}}}{a _{m}} [J (T)^{- 1}]_{m r}, G_{r, R} = m = 0 \sum d β_{m r} (T) g_{m}, 0 \leq r \leq d .$

Its first-term polynomial $q_{r, T} = \sum_{m} β_{m r} p_{m}$ has exact jets

$\frac{q _{r, T}^{(s)} ( T )}{s !} = 1_{{s = r}} (0 \leq s \leq d), ∣ β_{m r} (T) ∣ \leq C_{d} T^{r - ν_{m}} . (T11)$

The higher jets obey $q_{r, T}^{(s)} (T) / s! = O_{d} (T^{r - s})$ for $d < s \leq 2 d + 5$ . Consequently its exact Taylor polynomial is

$q_{r, T} (X) = (X - T)^{r} + s = d + 1 \sum 2 d + 5 O_{d} (T^{r - s}) (X - T)^{s} .$

This change of basis is invertible and depends on the same endpoint as the tests. It explicitly retains combinations that cancel the common leading tail; it does not assume the raw derivative Gram is uniformly conditioned.

A stable common profile Gram

Put $B_{R} = e^{R /2} e^{- π T}$ and define the positive-tail error profiles

$F_{r, R} (s) = η (s) G_{r, R} (R + δs) / B_{R}, F_{r} (s) = η (s) (2 s)^{r} e^{- 2 π s},$

zero for $s \leq 0$ . The first theta term is exactly

$η (s) e^{δs /2} q_{r, T} (T e^{2 δs}) e^{- π T (e^{2 δs} - 1)} .$

Here $T e^{2 δs} - T \to 2 s$ on bounded $s$ . Equations (T10)–(T11) give $q_{r, T} (T e^{2 δs}) \to (2 s)^{r}$ . For a uniform majorant use $T (e^{2 δs} - 1) \geq 2 s$ and $T (e^{2 δs} - 1) \leq 2 s e^{2 δs}$ . The finite polynomial degree, the higher-jet bounds and two scaled derivatives are absorbed by the exponential. For the remaining theta terms $n \geq 2$ , the coefficients in (T11) and the normally convergent differentiated series give a bound $C_{d} T^{d} e^{- 3 π T} e^{- cs}$ , also through two scaled derivatives. Thus

$F_{r, R} ⟶ F_{r} in L^{1} \cap H^{2}, ∣ F_{r, R}^{(l)} (s) ∣ \leq C_{d} e^{- cs} (0 \leq l \leq 2), (T12)$

uniformly in large $R$ , with $c > 0$ fixed. These are applications of the original series and the new finite jet coordinates; no differentiated asymptotic remainder is used.

The limiting Gram is

$K_{d} (r, t) = \int_{0}^{\infty} η (s)^{2} (2 s)^{r + t} e^{- 4 π s} d s .$

It is positive definite: a nonzero polynomial cannot vanish almost everywhere on $s \geq 1$ , where $η = 1$ . Strong convergence in (T12) makes the finite Gram converge in operator norm. Hence there are constants $0 < c_{d} < C_{d}$ and a sufficiently large $R_{d}$ such that for every $a \in C^{d + 1}$ and $R \geq R_{d}$ ,

$c_{d} ∣ a ∣^{2} \leq r = 0 \sum d a_{r} F_{r, R}_{2}^{2} \leq C_{d} ∣ a ∣^{2} .$

Writing $F_{a, R} = \sum_{r} a_{r} F_{r, R}$ , the pointwise bounds, first two derivative norms and $L^{1}$ norm are also at most $C_{d} ∣ a ∣$ . In particular they are controlled by $C_{d} ∥ F_{a, R} ∥_{2}$ . This is uniform over coefficients depending on $R$ ; separate convergence of fixed derivative vectors would not provide it.

Let

$g_{a, R} = r \sum a_{r} G_{r, R}, u_{a, R} = χ_{j} g_{a, R}, e_{a, R} = g_{a, R} - u_{a, R} .$

Odd reflection gives the exact mass and its two-sided coefficient comparison

$M_{a, R} := ∥ e_{a, R} ∥_{2}^{2} = 2 B_{R}^{2} δ ∥ F_{a, R} ∥_{2}^{2} ≍_{d} B_{R}^{2} δ ∣ a ∣^{2} . (T13)$

Joint complete energy and response bounds

For fixed $j, d, a$ , the compact outer-cutoff passage in (T4)–(T5) applies to this finite linear combination of theta derivatives. Each transform and both pole moments vanish for the uncut combination. Therefore its compact old energy equals the complete tail energy, and its mixed collar pairing is the negative tail pairing. The full paired zero multiset, every prime power and true pole are unchanged.

All estimates in (T6)–(T7) depend on a scaled profile only through its norm, the first two derivative budgets and an exponential envelope. Equations (T12)–(T13) provide those budgets uniformly in $a$ , relative to its actual tail mass. In the energy calculation the Gamma remainder is $C_{d} M_{a, R}$ ; the reflected Gamma term is $C_{d} e^{- 3 R} M_{a, R}$ ; same-side prime correlations are exponentially small in $1/ δ$ ; the full opposite-tail prime correction is $C_{d} R e^{- R} M_{a, R}$ ; and the actual negative pole is $C_{d} e^{- R} M_{a, R}$ . The discrete endpoint term in the prime sum is retained. Thus, uniformly for every coefficient vector,

$∣ Q (u_{a, R}) - 2 R M_{a, R} ∣ \leq C_{d} M_{a, R} . (T14)$

The same estimate holds for the true pole-free core. It pays all cross terms in the complete old Gram, and makes it positive definite for sufficiently large $j$ .

For the collar row use the same two source thresholds $e^{y - R}$ and $e^{y + R}$ as (T7), with both translation orientations and their infinite tail. Replace the profile amplitude there by $B_{R} ∣ a ∣$ , justified by (T12); the singular Gamma neighborhood and local multiplier are paid by the uniform second-derivative tail. The true pole moment is at most $C_{d} B_{R} ∣ a ∣ δ e^{R /2}$ . With $Y_{j}^{Q} (u)$ denoting the full physical $L^{2} (E_{j})$ response,

$∥ Y_{j}^{Q} (u_{a, R}) ∥_{2}^{2} \leq C_{d} h_{j} B_{R}^{2} ∣ a ∣^{2} R^{2} e^{- R} \leq C_{d} h_{j} R^{2} e^{- R} δ^{- 1} M_{a, R} . (T15)$

This is uniform over all old coefficients and all odd collar coefficients. The core response has the same bound after its actual pole-removal term. Neither weighted even commutators nor an independently optimized component budget is substituted for this physical row.

The actual restricted inverse is paid jointly

Equations (T13)–(T15), $h_{j} ≍ e^{- 4 R_{j}}$ and $δ_{j} = e^{- 2 R_{j}}$ give, for each fixed $d \geq 1$ and all sufficiently large $j$ ,

$0 \neq = u \in U_{j, d} sup \frac{∣ B _{Q} ( u , v ) ∣ ^{2}}{Q ( u )} \leq C_{d} R_{j} e^{- 3 R_{j}} ∥ v ∥_{2}^{2} \leq C_{d} (lo g k_{j}) k_{j}^{- 3/4} ∥ v ∥_{2}^{2} . (T16)$

It holds for every odd $L^{2}$ collar $v$ by the complete response representation. In any basis $u_{r}$ of this actual compact old space, put $G_{r t} = B_{Q} (u_{t}, u_{r})$ and $b_{r} (v) = \overline{B_{Q} (u_{r}, v)}$ . Then $G ≻ 0$ and the standard finite-dimensional variational identity identifies the left side with $b (v)^{*} G^{- 1} b (v)$ . Consequently (T16) is a bound for the complete restricted inverse, including its smallest-energy combinations, rather than the sum of $d + 1$ rank-one bounds. It also holds with the true pole-free form in both Gram and forcing.

The quantifiers are $\forall d \exists C_{d}, j_{d} \forall j \geq j_{d}$ : no degree-uniform constant, growing-degree rate or certified numerical starting endpoint is supplied. The actual full old inverse can have additional directions outside $U_{j, d}$ , and its interaction with their complement remains unpaid. Source-operator application still requires the full common-core and physical norm identification. The original selected-integer signed Robin estimate and RH remain unproved.

The added interface is the coefficient-uniform complete physical Gram/forcing bound (T13)–(T16), made possible by the endpoint-dependent tail-jet construction (T10)–(T12). The individual cutoff estimate, theta nullity and tails, kernel scaling and classical interpolation/Gram identities are reused. This is a paper-level joint estimate, without Lean certification or a mathematical-priority claim.

Quantified degree growth in the physical inverse budget

The fixed- $d$ statement (T16) does not control a prescribed degree growing with the endpoint. A qualitative choice of an unspecified slow sequence would not pay this parameter gap. Here the original polynomial construction is used to bound its constants and thresholds together, on the same actual collars and in the same physical odd metric.

Retain the fixed smooth transition $η$ , the actual endpoints and all objects from (T9)–(T16). The original theta series, derivative nullity and compact-domain passage from the derivative-family account, Gamma scaling and complete prime/pole estimates remain suppliers. Classical interpolation, adjugate bounds and elementary factorial estimates below are intermediate tools; no separate new polynomial theorem is asserted.

One budget for the degree-sensitive construction

Put $N = d + 1$ , $D = 2 d + 5$ and retain $T = e^{2 R}$ , $δ = T^{- 1}$ . There is a constant $C_{0} \geq 1$ , depending on the fixed transition and normalization but not on $d$ or $R$ , such that with

$K_{d} = exp (C_{0} N^{2}), H_{d} = K_{d}^{10}, (T17)$

the following construction estimates hold whenever $R \geq H_{d}$ . The exponential budget is derived as follows, rather than substituted for the unspecified constants in (T16).

For the coefficient $ℓ^{1}$ norm of a polynomial of degree at most $D$ , the original operator satisfies

$∥ D p ∥_{coeff, 1} \leq (1/2 + 2 D + 2 π) ∥ p ∥_{coeff, 1} .$

At every stage constructing $p_{m}$ , its degree is at most $D$ . Applying this inequality at most $2 d + 3$ times gives $max_{m} ∥ p_{m} / a_{m} ∥_{coeff, 1} \leq exp (O (N lo g (N + 1)))$ , with a constant independent of $d$ . The entries of $V = [(s ν _{m})]$ are at most $2^{D}$ and its determinant is at least one by (T10). The adjugate formula therefore gives

$∥ V^{- 1} ∥_{2} \leq N d! (2^{D})^{d} \leq exp (O (N^{2})) .$

The lower polynomial coefficients similarly pay $∥ J (T) - V ∥_{2} \leq exp (O (N lo g (N + 1))) / T$ . The ordinary inverse perturbation estimate, at the stated $R \geq H_{d}$ , bounds $J (T)^{- 1}$ by $exp (O (N^{2}))$ . The coefficients $β_{m r}$ and higher Taylor coefficients in (T11) consequently have the same budget, after the displayed powers $T^{r - ν_{m}}$ and $T^{r - s}$ are factored out. All the constants implicit in these bounds are independent of the degree. Enlarging one $C_{0}$ absorbs their finite products.

The first-term profile error can also be quantified. For $s \geq 0$ ,

$0 \leq T (e^{2 s / T} - 1) - 2 s \leq \frac{2 s ^{2}}{T} e^{2 s / T} .$

Use this in the exact Taylor formula following (T11). For the exponential factor, $∣ e^{- a} - e^{- b} ∣ \leq (a - b) e^{- b}$ when $a \geq b \geq 0$ ; its first two derivatives use the same $T^{- 1}$ remainder and finite polynomial factors. The higher jets have $T^{r - s} \leq T^{- 1}$ for $r \leq d < s$ . Thus, through two scaled derivatives, the errors are bounded by $T^{- 1} exp (O (N^{2}))$ times a polynomial of degree at most $D + 6$ and a fixed decreasing exponential. The bound $sup_{s \geq 0} (1 + s)^{D + 6} e^{- cs} \leq exp (O (N lo g (N + 1)))$ and the corresponding integrals pay that polynomial.

For every remaining theta term, its polynomial coefficients contain only powers $n^{O (N)}$ and the same degree- $D$ recurrence. After the jet change its profile contribution is bounded by $exp (O (N^{2})) T^{d} e^{- 3 π T} e^{- cs}$ through two scaled derivatives. The elementary Gaussian moment bound $\sum_{n \geq 2} n^{O (N)} e^{- c (n^{2} - 4)} \leq exp (O (N lo g (N + 1)))$ pays the series uniformly in degree. For $T \geq N^{2}$ , $T^{d} e^{- 3 π T} \leq e^{- 2 π T}$ , so this also lies in the $T^{- 1}$ error budget. These estimates differentiate the original series and exact polynomial expressions, not an asymptotic remainder.

After increasing $C_{0}$ once, for every $0 \leq r \leq d$ and $0 \leq l \leq 2$ the construction therefore gives, with a fixed $c > 0$ ,

$∣ F_{r, R}^{(l)} (s) ∣ \leq K_{d} e^{- cs} (s \geq 0), ∥ F_{r, R} - F_{r} ∥_{L^{1}} + ∥ F_{r, R} - F_{r} ∥_{H^{2}} \leq K_{d} / T . (T18)$

The threshold $R \geq H_{d}$ pays every inverse-perturbation and exponential-absorption condition above. No unspecified $R_{d}$ from the fixed-degree argument is retained as an extra hypothesis.

A quantitative lower Gram on the same profiles

Let $p (x) = \sum_{r = 0}^{d} a_{r} x^{r}$ . Partition $[1, 2]$ into $N$ cells and, in the first third of each cell, choose $s_{m}$ with

$∣ p (2 s_{m}) ∣ \leq 3 N ∥ p (2 \cdot) ∥_{L^{2} (1, 2)} .$

Such a point exists by the integral bound on that subinterval, also for complex $p$ . The nodes $x_{m} = 2 s_{m}$ lie in $[2, 4]$ and have pairwise separation at least $4∣ m - n ∣/ (3 N)$ . In the Lagrange basis, the coefficient $ℓ^{1}$ norm of each numerator is at most $5^{d}$ , while the denominator is at least

$(\frac{4}{3 N})^{d} m! (d - m)! .$

Using $(m d) \leq 2^{d}$ , $d! \geq (d / e)^{d}$ and $N / d \leq 2$ gives a coefficient norm at most $(15 e)^{d}$ for each Lagrange polynomial. Consequently

$∣ a ∣ \leq N 3 N (15 e)^{d} ∥ p (2 \cdot) ∥_{L^{2} (1, 2)} .$

Since $η = 1$ on this interval, the actual limiting profile Gram obeys the degree-explicit bound

$λ_{m i n} (K_{d}) \geq \frac{e ^{- 8 π}}{3 N ^{3} ( 15 e ) ^{2 d}} \geq K_{d}^{- 1} . (T19)$

Here $C_{0}$ has been chosen large enough to include the last inequality. The finite profile Gram has error at most $K_{d}^{4} / T$ by (T18), Cauchy–Schwarz and $N \leq K_{d}$ . For $R \geq K_{d}^{10}$ this is less than $(2 K_{d})^{- 1}$ . Thus, uniformly over every complex coefficient vector,

$∥ F_{a, R} ∥_{2}^{2} \geq (2 K_{d})^{- 1} ∣ a ∣^{2} .$

The exponential envelope, first two derivative norms and $L^{1}$ norm of the combination are at most $K_{d}^{2} ∣ a ∣$ , after absorbing their fixed constants into $C_{0}$ . Relative to the actual combination norm they are at most $K_{d}^{3} ∥ F_{a, R} ∥_{2}$ . This explicitly pays endpoint-dependent cancellation directions before taking any inverse.

The complete physical remainder fits the same budget

For each parameter pair $d, R$ and coefficient vector, use the fixed-parameter compact outer-cutoff passage from (T4)–(T5). It applies to the finite derivative combination at that endpoint, irrespective of how $d$ or its coefficients are selected at other endpoints. Every actual zero, paired off-line factor and multiplicity is retained; there is no interchange of an infinite-degree limit with the explicit formula.

Insert the just-proved relative profile budgets into the same Gamma and full prime/pole estimates used in (T14)–(T15). The Gamma scaling remainder and singular small-shift derivative budget are at most a fixed constant times $K_{d}^{6} M_{a, R}$ . Its reflected cross term has the additional $e^{- 3 R}$ factor. The same-side prime correlations have the additional exponential in $- 1/ δ$ , the complete opposite-tail prime correction has $R e^{- R}$ , and the actual pole square has $e^{- R}$ . The prime sums still include their discrete endpoint terms; only $Λ (n) \leq lo g n$ is used. These scalar integrals and sums have constants independent of the degree because the profile exponential rate $c$ is fixed.

For the collar row, both threshold sums at $e^{y - R}$ and $e^{y + R}$ , the other translation orientation, the Gamma local and remote pieces and the true pole use the same relative envelope and two-jet budgets. Squaring its bound costs at most $K_{d}^{6}$ times the common scalar constant. Choosing $C_{0}$ to contain those fixed scalar constants makes $H_{d} = K_{d}^{10}$ an upper budget for both complete estimates:

$∣ Q (u_{a, R}) - 2 R M_{a, R} ∣ ∥ Y_{j}^{Q} (u_{a, R}) ∥_{2}^{2} \leq H_{d} M_{a, R}, \leq H_{d} h_{j} R^{2} e^{- R} δ^{- 1} M_{a, R} . (T20)$

Both inequalities hold for all $d \geq 1$ , every complex coefficient vector and each actual endpoint with $R_{j} \geq H_{d}$ . The true pole-free form satisfies the same estimates. In particular $Q (u) \geq R_{j} M_{a, R} > 0$ for $u \neq = 0$ in that subspace. The error budget is compared to the main term before division; it is not an upper denominator used as a lower one.

A paid increasing number of old directions

Put $C = 10 C_{0}$ and, for sufficiently large actual endpoints, choose

$d_{j} + 1 = ⌊ \frac{lo g R _{j}}{4 C} ⌋ . (T21)$

Then $d_{j} \geq 1$ eventually, $d_{j} \to \infty$ and $H_{d_{j}} = exp (C (d_{j} + 1)^{2}) \leq R_{j}^{1/4} \leq R_{j}$ . All thresholds and both estimates in (T20) are therefore paid by this same choice. Since $h_{j} = \frac{1}{2} lo g (1 + e^{- 4 R_{j}}) \leq \frac{1}{2} e^{- 4 R_{j}}$ , the complete restricted variational debit obeys

$0 \neq = u \in U_{j, d_{j}} sup \frac{∣ B _{Q} ( u , v ) ∣ ^{2}}{Q ( u )} \leq \frac{1}{2} R_{j}^{5/4} e^{- 3 R_{j}} ∥ v ∥_{2}^{2} = O ((lo g k_{j})^{5/4} k_{j}^{- 3/4}) ∥ v ∥_{2}^{2} . (T22)$

This holds for every odd $L^{2}$ collar $v$ , with the same core version. The actual space dimension is $d_{j} + 1 ≍ lo g lo g k_{j}$ , and all its endpoint-dependent complex combinations are covered. The constants in the chosen schedule and its numerical starting endpoint are not certified numbers. The quantitative degree rate and vanishing debit follow from the paid exponential construction budget, not from a qualitative diagonalization of (T16).

An increasing derivative space does not by itself identify or estimate the complete old complement, nor prove approximation in the original energy norm. The full old-space inverse and the source’s common-core/norm map remain separate obligations. The original selected-integer signed Robin main estimate, its coefficients and strict core are unchanged; RH remains unproved.

The added interface is the simultaneous degree/threshold control (T17)–(T20) consumed by the growing-space physical inverse bound (T21)–(T22). The fixed-degree estimate, theta representations and nullity, interpolation and Gamma/prime/pole mechanisms are reused. This is paper-level work without Lean certification or a mathematical-priority claim.

Move the theta cutoff toward the actual old endpoint

The growing spaces in (T21) still use a cutoff at half of the actual old radius. Increasing their degree does not include the outer half of that radius. The following pays a different physical collar row when the theta cutoff and the arithmetic endpoint are independent. It keeps the same integers and collars, and permits a cutoff whose ratio to the old radius tends to one. This is a spatial extension of the specified derivative family, not a density or full-old-space assertion.

Retain the actual $k_{j}$ , $a_{j} = \frac{1}{2} lo g k_{j}$ , $h_{j} = \frac{1}{2} lo g (1 + e^{- 2 a_{j}})$ and reflected collar $E_{j}$ from (O10). Suppress $j$ and write $a = a_{j}$ , $h = h_{j}$ . Choose

$\frac{a}{2} \leq r \leq a - 1, g = a - r, δ = e^{- 2 r}, B_{r} = e^{r /2} e^{- π e^{2 r}}, r \geq H_{d} . (T23)$

Here $g$ is the physical gap, and is unrelated to the previously named null function $g_{0}$ . Use the same transition $η$ , derivative family $g_{m}$ , and jet basis at the new cutoff $r$ . Define

$χ_{r} (y) = 1 - η ((∣ y ∣ - r) / δ), U_{j, d}^{(r)} = span_{C} {χ_{r} g_{m} : 0 \leq m \leq d} .$

Its tests are compact inside the same old interval $(- a, a)$ since $r + δ < a$ . Coefficients remain unrestricted and may depend on both parameters.

Reuse the tail coordinates and complete energy

The construction in (T17)–(T19) depends on the theta cutoff alone; neither the arithmetic endpoint nor the collar occurs in its polynomial, profile or Gram calculation. Apply it at $r$ . For a coefficient vector $c$ in that jet basis, write $g_{c, r}$ for its uncut null function, $u = χ_{r} g_{c, r}$ and $e = g_{c, r} - u$ . With $F = F_{c, r}$ and $P = ∥ F ∥_{2}$ , the reused budgets give

$M := ∥ e ∥_{2}^{2} = 2 B_{r}^{2} δ P^{2}, ∣ F^{(ℓ)} (s) ∣ \leq K_{d}^{3} P e^{- cs} (0 \leq ℓ \leq 2), ∥ F ∥_{1} \leq K_{d}^{3} P, (T24)$

for a fixed $c > 0$ independent of $d, r, a$ . The lower Gram ensures $P > 0$ for nonzero coefficients. The fixed-parameter compact outer-cutoff passage and mixed nullity remain (T4): all actual zero real parts, both signs, heights and multiplicities are unchanged.

The complete tail-energy proof in (T20) also contains no collar parameter. Its Gamma scale is $lo g (1/ δ) = 2 r$ , the two tails are separated by $2 r$ , and its full opposite-tail prime threshold is $e^{2 r}$ . Thus the existing proof directly supplies

$∣ Q (u) - 2 r M ∣ \leq H_{d} M, Q (u) \geq r M > 0 (u \neq = 0) . (T25)$

The true pole-free core satisfies the same bounds. Choose the admissible $C_{0}$ in (T17) large enough once to absorb the additional fixed scalar constants below; $K_{d} = e^{C_{0} (d + 1)^{2}}$ and $H_{d} = K_{d}^{10}$ still pay all earlier construction conditions. No new unquantified degree-dependent threshold is introduced.

Pay the new physical row, including its discrete endpoints

On the positive collar $a < y < a + h$ , put $T_{-} = e^{y - r}$ and $T_{+} = e^{y + r}$ . The same complete prime-row comparison used in (T7), now at these two distinct thresholds, gives

$2 \leq n \leq T_{-} \sum \frac{Λ ( n )}{n} ∣ e (y - lo g n) ∣ n \geq T_{+} \sum \frac{Λ ( n )}{n} ∣ e (y - lo g n) ∣ \leq C B_{r} K_{d}^{3} P lo g T_{-} (T_{-}^{- 1/2} + δ T_{-}^{1/2}), \leq C B_{r} K_{d}^{3} P lo g T_{+} (T_{+}^{- 1/2} + δ T_{+}^{1/2}) . (T26)$

Indeed, (T24) bounds the two tails by $(n / T_{-})^{c / δ}$ and $(n / T_{+})^{- c / δ}$ respectively. Elementary increasing/decreasing integral comparison and $Λ (n) \leq lo g n$ supply the displayed estimates, including their first discrete endpoint terms. There is no contribution from $T_{-} < n < T_{+}$ because that argument lies in $(- r, r)$ , where $e = 0$ . The noncompact tail sum is not stopped at $k_{j}$ .

The other orientation $e (y + lo g n)$ is bounded by $B_{r} K_{d}^{3} P e^{- c g / δ} n^{- c / δ}$ ; its entire weighted sum is smaller than $C B_{r} K_{d}^{3} P e^{- g /2}$ for the paid small $δ$ . Since $h \leq 1$ , $r \geq a /2$ and $g \geq 1$ , the logarithms in (T26) are at most $C a$ . Moreover

$T_{-}^{- 1/2} ≍ e^{- g /2}, δ T_{+}^{1/2} ≍ e^{(a - 3 r) /2},$

while $δ T_{-}^{1/2}$ and $T_{+}^{- 1/2}$ are bounded by these two budgets. The complete prime row is therefore at most $C B_{r} K_{d}^{3} P a (e^{- g /2} + e^{(a - 3 r) /2})$ .

The complete Gamma row retains its local multiplier and symmetric difference integral. Split its shifts at $g /2$ . For the smaller shifts, $y \pm t \geq r + g /2$ and (T24) gives

$∣ e^{''} (y \pm t) ∣ \leq B_{r} K_{d}^{3} P δ^{- 2} e^{- c g / (2 δ)} .$

The fixed scalar integral $\int_{0}^{\infty} t^{2} κ (t) d t < \infty$ pays the local singularity. Uniformly for $g \geq 1$ and the paid small $δ$ , $δ^{- 2} e^{- c g / (2 δ)} \leq C δ e^{- g /2}$ . The local multiplier has a smaller bound. For shifts at least $g /2$ , use $κ (t) \leq C e^{- t /2}$ and the two tail envelopes. Changing to their scaled coordinates bounds the near- $r$ tail by $C B_{r} K_{d}^{3} P δ e^{- (y - r) /2}$ and the reflected tail by $C B_{r} K_{d}^{3} P δ e^{- (y + r) /2}$ . The remaining untranslated and far positive-tail terms are smaller. This gives a full Gamma-row bound $C B_{r} K_{d}^{3} P δ e^{- g /2}$ ; no nonsingular separated-kernel formula is substituted for the local part.

Finally, the actual odd pole satisfies

$\int e (x) sinh (x /2) d x \leq C B_{r} K_{d}^{3} P δ e^{r /2} .$

Multiplication by the true collar factor $2 sinh (y /2)$ therefore bounds its row by $C B_{r} K_{d}^{3} P e^{(a - 3 r) /2}$ . Odd reflection supplies the negative collar. These estimates include every prime power and both translation orientations, and the actual pole coefficient. The fixed-parameter mixed identity identifies the compact test’s physical row with the negative tail row. Squaring the complete row, integrating over both collars and absorbing only fixed constants in the same $H_{d}$ yields

$∥ Y_{j}^{Q} (u) ∥_{L^{2} (E_{j})}^{2} \leq H_{d} h a^{2} δ^{- 1} M (e^{- (a - r)} + e^{a - 3 r}) . (T27)$

The true pole-free core has the same estimate after its actual pole-removal term. This bound is uniform over every old coefficient vector at the specified $d, r, a$ ; the profile-to-mass comparison pays cancellations before the square is taken.

A vanishing debit with a cutoff approaching the old radius

Divide (T27) by the independently proved lower energy (T25), and use the exact width inequality $h \leq \frac{1}{2} e^{- 2 a}$ . For every odd physical $L^{2}$ collar $v$ ,

$0 \neq = u \in U_{j, d}^{(r)} sup \frac{∣ B _{Q} ( u , v ) ∣ ^{2}}{Q ( u )} \leq \frac{H _{d} a ^{2}}{2 r} (e^{- 3 (a - r)} + e^{- (a + r)}) ∥ v ∥_{2}^{2} . (T28)$

This is the complete restricted inverse, not a sum over individual derivative lines. The same core version holds. At the earlier midpoint it has the same exponential order as (T22); changing the cutoff changes both threshold contributions, so the midpoint collar estimate cannot simply be reused with $r$ substituted for $a /2$ .

At sufficiently large actual endpoints choose

$r_{j} = a_{j} - lo g a_{j}, d_{j} + 1 = ⌊ \frac{lo g r _{j}}{4 C} ⌋, C = 10 C_{0} . (T29)$

Then $a_{j} /2 \leq r_{j} \leq a_{j} - 1$ , $H_{d_{j}} \leq r_{j}^{1/4} \leq r_{j}$ , and all conditions are paid. Equation (T28) becomes

$0 \neq = u \in U_{j, d_{j}}^{(r_{j})} sup \frac{∣ B _{Q} ( u , v ) ∣ ^{2}}{Q ( u )} \leq C_{1} a_{j}^{- 7/4} ∥ v ∥_{2}^{2} = O ((lo g k_{j})^{- 7/4}) ∥ v ∥_{2}^{2} . (T30)$

Here $C_{1}$ is fixed, and the schedule’s constant and numerical start are not certified numbers. Indeed, $a_{j}^{2} / r_{j} \leq 2 a_{j}$ , $e^{- 3 (a_{j} - r_{j})} = a_{j}^{- 3}$ and the other exponential is smaller. The actual dimension remains $d_{j} + 1 ≍ lo g lo g k_{j}$ . The new tests agree with their uncut null functions on $[- r_{j}, r_{j}]$ and have support inside $[- r_{j} - δ_{j}, r_{j} + δ_{j}] ⋐ (- a_{j}, a_{j})$ , with $r_{j} / a_{j} \to 1$ .

The weaker decay in (T30) pays a different, larger spatial reach than the midpoint family. It does not establish that this finite derivative family approximates arbitrary old tests, even in the newly reached interior. The full complementary directions, their cross terms and the energy-norm approximation cost remain unestimated; they cannot be described as only a logarithmic boundary layer. The source common-core/physical-norm map and original selected-integer signed Robin estimate also remain unproved, and no RH conclusion follows.

The additional interface is the independent-cutoff, same-collar response estimate (T26)–(T28), consumed by the spatial schedule (T29)–(T30). Tail coordinates, degree thresholds, complete energy, mixed nullity and elementary prime comparisons are reused suppliers. This is paper-level work without Lean certification or a mathematical-priority claim.

Pay the actual cross Gram of both cutoff families

The midpoint and near-endpoint estimates cannot be added on their joint span without a lower bound for its whole Gram. They are different cutoff families, even though their uncut functions belong to the same null derivative family. The following estimates their actual mixed energy and then combines the already paid forcing bounds.

Retain the same actual $k_{j}, a_{j}, h_{j}, E_{j}$ , complete physical odd form $Q$ and true pole-free core. Suppress $j$ and take, for sufficiently large actual $a = a_{j}$ ,

$r_{0} = a /2, r_{1} = a - lo g a, Δ = r_{1} - r_{0}, Σ = r_{1} + r_{0}, δ_{i} = e^{- 2 r_{i}} .$

Use both accepted degree schedules, without replacing either family by a single line:

$d_{i} + 1 = ⌊ \frac{lo g r _{i}}{4 C} ⌋, H_{d_{i}} \leq r_{i}^{1/4}, C = 10 C_{0} . (T31)$

The corresponding actual spaces are $U_{i} = U_{j, d_{i}}^{(r_{i})}$ . For $u_{i} \in U_{i}$ , write its uncut null function as $g_{i}$ , its error as $e_{i} = g_{i} - u_{i}$ , and its profile as $F_{i}$ . This $g_{i}$ is a finite derivative combination, not a new source function. Reuse (T24)–(T25): with $P_{i} = ∥ F_{i} ∥_{2}$ , $B_{i} = e^{r_{i} /2} e^{- π e^{2 r_{i}}}$ and $M_{i} = ∥ e_{i} ∥_{2}^{2}$ ,

$M_{i} = 2 B_{i}^{2} δ_{i} P_{i}^{2}, Q (u_{i}) \geq r_{i} M_{i}, ∣ F_{i}^{(ℓ)} (s) ∣ \leq K_{d_{i}}^{3} P_{i} e^{- cs} (0 \leq ℓ \leq 2) .$

Put $W = K_{d_{0}}^{3} K_{d_{1}}^{3}$ . The existing degree budget gives $W \leq (r_{0} r_{1})^{3/40} \leq a^{3/20}$ . All constants below are independent of the degree, endpoint and complex coefficient vectors. At each fixed parameter pair the mixed nullity and compact outer-cutoff bridge give the exact equality $B_{Q} (u_{0}, u_{1}) = B_{Q}^{tail} (e_{0}, e_{1})$ , with the actual zero pairing, heights and multiplicities unchanged. The same core equality retains its true pole correction.

Unequal-width correlations retain both prime-shift centers

Write the positive tails as $f_{i} (y) = B_{i} F_{i} ((y - r_{i}) / δ_{i})$ , extended by zero below $r_{i}$ ; oddness gives $e_{i} (y) = f_{i} (y) - f_{i} (- y)$ . Eventually $δ_{1} \leq δ_{0} /2$ . The profile envelopes then bound the complete two-orientation correlation by

$∣ ⟨ e_{0}, τ_{t} e_{1} ⟩ ∣ + ∣ ⟨ e_{1}, τ_{t} e_{0} ⟩ ∣ \leq C B_{0} B_{1} W P_{0} P_{1} δ_{1} (e^{- c ∣ t - Δ∣/ δ_{0}} + 1_{{t \geq Σ}} e^{- c (t - Σ) / δ_{0}}), t \geq 0. (T32)$

For like-sign tails the two possible centers are $Δ$ and $- Δ$ . Integrating the narrower profile gives the factor $δ_{1}$ ; the term centered at $- Δ$ is bounded by the displayed one. For opposite signs the convolution is zero below $Σ$ . Above it, putting $w = t - Σ$ gives the explicit envelope integral

$\int_{0}^{w} e^{- c x / δ_{0}} e^{- c (w - x) / δ_{1}} d x \leq \frac{2 δ _{1}}{c} e^{- c w / δ_{0}} .$

Thus (T32) covers all reflected pairs and both translations for complex profiles, without a phase or sign assumption.

The elementary threshold comparison in (T26), applied on both sides of $e^{b}$ , supplies for $b \geq 1$ and the paid small $δ_{0}$

$n \geq 2 \sum \frac{lo g n}{n} e^{- c ∣ l o g n - b ∣/ δ_{0}} \leq C (1 + b) (e^{- b /2} + δ_{0} e^{b /2}) .$

The first term retains the discrete endpoint, including when the local expected count is below one. Apply this at $b = Δ$ and $b = Σ$ , use $Λ (n) \leq lo g n$ , and normalize by the actual masses. The complete prime part of the mixed energy satisfies

$\frac{∣ B _{prime} ( e _{0} , e _{1} ) ∣}{M _{0} M _{1}} \leq C Wa (e^{- 3Δ/2} + δ_{0} e^{- Δ/2} + e^{- Δ - Σ/2} + e^{- Σ/2}) . (T33)$

This notation denotes the prime part of the full polarized tail expression; if one mass is zero the equivalent multiplicative inequality is used. No prime power, endpoint or translation orientation is omitted. In particular the discrete shift near $e^{Δ}$ , rather than the direct tail overlap, supplies the first term in (T33).

The singular Gamma part and true pole also fit

The direct overlap is exponentially small in $Δ/ δ_{0}$ , but this alone does not bound the Gamma energy. For its symmetric increment pairing split the shifts at $Δ/2$ . Below that point, expressing both increments as integrals of their first derivatives gives

$∣ ⟨ e_{0} - τ_{t} e_{0}, e_{1} - τ_{t} e_{1} ⟩ ∣ \leq C B_{0} B_{1} W P_{0} P_{1} δ_{0}^{- 1} t^{2} e^{- c Δ/ (2 δ_{0})} .$

Indeed, wherever a translated $e_{1}^{'}$ is nonzero, the argument of the corresponding $e_{0}^{'}$ has absolute value at least $r_{0} + Δ/2$ . Its supremum is at most $B_{0} K_{d_{0}}^{3} P_{0} δ_{0}^{- 1} e^{- c Δ/ (2 δ_{0})}$ , while the full $L^{1}$ norm of $e_{1}^{'}$ is at most $C B_{1} K_{d_{1}}^{3} P_{1}$ . The finite scalar integral $\int t^{2} κ (t) d t$ therefore pays the small-shift singularity. No divergent total jump rate is separated out.

For shifts at least $Δ/2$ , use (T32) and $κ (t) \leq C e^{- t /2}$ . Its translated terms integrate to at most $C B_{0} B_{1} W P_{0} P_{1} δ_{0} δ_{1} (e^{- Δ/2} + e^{- Σ/2})$ . The untranslated overlap and actual $c_{Γ}$ multiplier fit the small-overlap budget. Consequently the complete Gamma pairing obeys

$\frac{∣ B _{Γ} ( e _{0} , e _{1} ) ∣}{M _{0} M _{1}} \leq C W [e^{3 r_{0} + r_{1}} e^{- c Δ/ (2 δ_{0})} + δ_{0} δ_{1} (e^{- Δ/2} + e^{- Σ/2})] . (T34)$

The actual pole moments, from the same error profiles, satisfy $∣ \int e_{i} (y) sinh (y /2) d y ∣ \leq C B_{i} K_{d_{i}}^{3} P_{i} δ_{i} e^{r_{i} /2}$ . Their polarized term, with true coefficient $2$ , is therefore at most $C W e^{- Σ/2} M_{0} M_{1}$ . The pole-free core removes precisely this actual term and has the same mixed upper budget.

A coefficient-uniform mixed Gram lower bound

For the actual choices in (T31), $Δ = a /2 - lo g a$ and $Σ = 3 a /2 - lo g a$ . The first term in (T34) is at most $e^{- a}$ once $a$ exceeds a fixed threshold independent of the degree: use $Δ \geq a /4$ , $δ_{0} = e^{- a}$ and exponential domination. All other Gamma and pole terms, and every term in (T33), can be compared explicitly with $a^{3/2} e^{- 3 a /4}$ after division by $r_{0} r_{1} \geq a /2$ . For example the discrete difference-shift term is $e^{- 3Δ/2} = a^{3/2} e^{- 3 a /4}$ , whereas the opposite-shift integral term is $e^{- Σ/2} = a^{1/2} e^{- 3 a /4}$ .

Together with $W \leq a^{3/20}$ , this gives a fixed $C_{*} > 0$ such that, for all sufficiently large actual endpoints and every coefficient pair,

$∣ B_{Q} (u_{0}, u_{1}) ∣ \leq ε_{a} r_{0} M_{0} r_{1} M_{1}, ε_{a} = C_{*} a^{2} e^{- 3 a /4} ⟶ 0. (T35)$

The same estimate holds for the true pole-free core. The remaining starting threshold is fixed by scalar constants and $c$ , independently of the degrees; no unspecified $d$ -dependent threshold is imported. Its numerical value and $C_{*}$ are not certified numbers.

Choose a sufficiently large endpoint with $ε_{a} < 1/2$ . Write $Q_{i} = Q (u_{i})$ . The existing lower energies imply $r_{0} M_{0} r_{1} M_{1} \leq Q_{0} Q_{1}$ . Therefore the actual full joint Gram satisfies

$Q (u_{0} + u_{1}) \geq (1 - ε_{a}) (Q_{0} + Q_{1}) \geq \frac{1}{2} (r_{0} M_{0} + r_{1} M_{1}) . (T36)$

This proves positivity on the whole joint space without assuming positivity on an arbitrary old complement. It also makes the two spaces a direct sum: a zero $u_{0} + u_{1}$ forces both actual masses, and hence both coefficient vectors, to vanish. Its dimension is $(d_{0} + 1) + (d_{1} + 1) ≍ lo g lo g k_{j}$ . The separate spaces alone would not establish (T36).

The complete joint inverse is now paid

Let $D_{0} (a)$ and $D_{1} (a)$ be the already proved uniform upper allowances in (T22) and (T30) for these two actual spaces; thus $D_{0} (a) = O (a^{5/4} e^{- 3 a /2})$ and $D_{1} (a) = O (a^{- 7/4})$ . For the same physical odd collar $v$ and any $u_{i} \in U_{i}$ , those bounds give $∣ B_{Q} (u_{i}, v) ∣ \leq D_{i} (a) ∥ v ∥_{2} Q_{i}$ . Ordinary two-coordinate Cauchy–Schwarz, now consumed with the actual joint denominator (T36), gives

$0 \neq = u \in U_{0} \oplus U_{1} sup \frac{∣ B _{Q} ( u , v ) ∣ ^{2}}{Q ( u )} \leq \frac{D _{0} ( a ) + D _{1} ( a )}{1 - ε _{a}} ∥ v ∥_{2}^{2} = O (a^{- 7/4}) ∥ v ∥_{2}^{2} . (T37)$

The same statement holds with the true pole-free form in both Gram and forcing. The existing finite variational identity identifies this quotient with the whole joint Gram inverse, including coefficients depending on the endpoint and near-cancelling directions across the two cutoffs. The sum in (T37) is justified by the new cross-Gram estimate; it is not obtained by assuming independent optimizers or adding rank-one inverses.

The added interface is the complete unequal-cutoff mixed estimate (T32)–(T35), consumed by joint positivity and inverse control in (T36)–(T37). Tail coordinates, degree and single-space energy/forcing budgets, mixed nullity/domain passage, Gamma kernel estimates, prime threshold comparisons and finite Gram algebra are reused suppliers. The full old complement and its interaction with this joint family, energy-norm approximation cost, source common-core/physical-norm map and original selected-integer signed Robin estimate remain unpaid. This paper-level joint estimate has no Lean certification, full-old-space or RH conclusion, or mathematical-priority claim.

Pay the complete physical collar forcing on an infinite exterior

The preceding theta spaces leave arbitrary old directions unestimated. Here the old-to-new pairing is bounded on the whole physical odd $L^{2}$ space, and the existing Fourier-exterior floor supplies a denominator on an infinite old subspace. The retained low-frequency Schur block remains separate.

Reuse the complete compact-test row (O10), the classical disjoint-source/squared-weight mechanism in arXiv v3 under “Ambient cross- $n$ structure” and “Fixed-target gauge and simultaneous source shorting”, especially the corollary “Source-resolved transverse Feshbach bound” (source-resolved-transverse), the weighted Schur criterion, classical Chebyshev bounds, and the existing infinite-exterior allowance (16). The source’s positive normalized pre-short metric is not identified with the full physical old metric. The additional interface below is the complete adjacent physical row, including its shared-boundary Gamma singularity, consumed by that independently supplied exterior denominator. No generic Carleman, prime-weight or Fourier-leakage theorem is reproved.

For an integer $k \geq 7$ set

$a = \frac{1}{2} lo g k, b = \frac{1}{2} lo g (k + 1), h = b - a \leq \frac{1}{2 k}, ℓ = 2 a = lo g k,$

and put $H_{a} = L_{odd}^{2} (- a, a)$ and $K_{k} = L_{odd}^{2} ((- b, - a) \cup (a, b))$ with the physical Lebesgue norms. Write $R_{k} : H_{a} \to K_{k}$ for the cross row in (O10), initially on compact smooth old tests, and $b_{k} (v) = R_{k}^{*} v$ for its old forcing. Thus $B_{Q} (u, v) = ⟨ R_{k} u, v ⟩$ ; no positivity of the whole old form is assumed. For a collar vector outside the form domain, this pairing denotes the bounded cross-row extension established below.

Retain the reflected integer source slots

On the positive collar the prime row is $- \sum_{2 \leq n \leq k} Λ (n) n^{- 1/2} u (y - lo g n)$ ; the other translation is zero there. Its source slots are

$J_{n} = (a - lo g n, b - lo g n) \subset (- a, a), 2 \leq n \leq k .$

Distinct $J_{n}$ are disjoint: consecutive logarithmic gaps exceed $2 h$ . The reflected slots also remain disjoint from them. Indeed $J_{n} \cap (- J_{m})$ could have positive length only if $2 a < lo g (nm) < 2 b$ , or $k < nm < k + 1$ , which no integer product satisfies. Products $nm = k$ or $k + 1$ give only touching endpoints. This verifies the needed reflection interface in the actual thin collar; it does not replace the source metric by a normalized one-cell form.

The two reflected copies and physical odd normalization therefore give, for every complex odd collar vector,

$∥ R_{k, p}^{*} v ∥_{2}^{2} = S_{k} ∥ v ∥_{2}^{2}, S_{k} = 2 \leq n \leq k \sum \frac{Λ ( n ) ^{2}}{n} . (T38)$

This is the disjoint-source squared-weight mechanism with its physical reflection map checked, not a new principle. The endpoint $n = k$ is retained: its old/new slot has positive length. In contrast, the old self-form has only $n < k$ .

Fix a classical Chebyshev constant $B \geq 1$ such that $Ψ (t) = \sum_{n \leq t} Λ (n) \leq Bt$ for $t \geq 1$ . Directly reuse partial summation and $Λ (n) \leq lo g k$ to obtain

$S_{k} \leq B ℓ (1 + ℓ), W (k) := 2 \leq n < k \sum \frac{Λ ( n )}{n} \leq 2 B k . (T39)$

Pay the Gamma singularity at the common boundary

Use the isometry $u \mapsto 2 u ∣_{(0, a)}$ and its collar counterpart. The negative odd Gamma row on $(0, a) \times (a, b)$ has absolute kernel

$κ (y - x) - κ (y + x), κ (t) = \frac{e ^{- t /2}}{1 - e ^{- 2 t}} .$

Both reflected terms have been kept. The kernel $κ$ is decreasing, so this difference is nonnegative and bounded above by $κ (y - x)$ . Put $C_{κ} = (1 - e^{- 2})^{- 1}$ . For $0 < t \leq 1$ , concavity gives $1 - e^{- 2 t} \geq t (1 - e^{- 2})$ ; for $t \geq 1$ use $t e^{- t /2} \leq 2/ e < 1$ . Hence $κ (t) \leq C_{κ} / t$ for all $t > 0$ .

After $s = y - a$ and $r = a - x$ , the dominating kernel is $C_{κ} / (s + r)$ on $(0, h) \times (0, a)$ . The classical Carleman bound, equivalently the cited weighted Schur criterion with weight $r^{- 1/2}$ and its standard scalar integral, gives

$∥ R_{k, Γ} ∥ = ∥ R_{k, Γ}^{*} ∥ \leq π C_{κ} . (T40)$

The bound is independent of $k, h$ . The row is not treated as a separated smooth kernel or claimed to have norm $O (h)$ : its $1/ (2 (y - x))$ singularity reaches the shared endpoint. Compact-core density gives the bounded $L^{2}$ cross extension used here, without splitting off a divergent diagonal jump rate.

Keep the true pole in the same row

Let $s_{a} = sinh (y /2) ∣_{(- a, a)}$ and $t_{k} = sinh (y /2) ∣_{(- b, - a) \cup (a, b)}$ . The actual pole row is $- 2 t_{k} s_{a}^{*}$ , and its exact norm is

$∥ R_{k, pole} ∥ = 2 (sinh a - a) (sinh b - sinh a - h) \leq e^{h /2} \leq e^{1/8} . (T41)$

For the inequality use $sinh a - a \leq e^{a} /2$ , $sinh b - sinh a - h = \int_{a}^{b} (cosh t - 1) d t \leq h e^{b}$ and $2 h e^{2 a} \leq 1$ . This retains the true coefficient $2$ and both physical norms. Multiplication terms have no old/new cross part. Combining (T38)–(T41) gives the complete all-old forcing allowance

$∥ b_{k} (v) ∥_{2} \leq K (k) ∥ v ∥_{2}, K (k) = π C_{κ} + e^{1/8} + B ℓ (1 + ℓ) = O (1 + ℓ) . (T42)$

The true pole-free core has the same bound with the $e^{1/8}$ term omitted. These statements cover every complex odd $L^{2}$ collar vector and every old direction, without a theta-family, derivative or positivity restriction. They are $L^{2}$ forcing bounds, not yet inverse-energy bounds.

Consume the existing floor on the whole infinite Fourier exterior

Use the same Fourier/Plancherel identification and actual Weil-form domain bridge as in the cited exterior account. Let $P_{> M}$ be the projection in $H_{a}$ onto the Fourier modes $∣ m ∣ > M$ on the length- $ℓ$ old interval. Oddness is preserved. On the closed-form restriction to this infinite exterior, the already supplied lower allowance is

$d (k, M) = η = (1 - η) [lo g \frac{M}{4 ℓ} - \frac{2 ℓ}{π M}] + η h_{0} - 2 W (k) - 4 sinh (ℓ /2), \frac{4}{3 π ^{2}}, h_{0} = - γ - \frac{π}{2} - 3 lo g 2 - lo g π .$

Its conditions include $π M / (2 ℓ) \geq 15/4$ . Neither the floor nor the restriction’s positivity requires RH or positivity of the full old form. The underlying analytic/Fourier/domain identification remains a paper-level bridge; it is not certified merely by citing the existing Lean leakage statement.

Choose

$β = \frac{4 B + 4}{1 - η}, M = ⌈ e^{β k} ⌉ .$

Reuse (T39) and $4 sinh (ℓ /2) \leq 2 k$ . Then

$d (k, M) \geq 2 k - (1 - η) [lo g (4 ℓ) + \frac{2 ℓ}{π M}] + η h_{0} \geq k > 0 for all sufficiently large integers k . (T43)$

The last step needs only the displayed scalar frequency condition and $(1 - η) [lo g (4 ℓ) + 2 ℓ / (π M)] - η h_{0} \leq k$ . No numerical starting endpoint is certified.

Let $A_{k, > M}$ be the operator of this restricted closed form. The standard positive-form variational identity, now with an independently paid denominator, consumes the same complete forcing in (T42):

$0 \neq = u \in D (Q) P_{> M} u = u sup \frac{∣ B _{Q} ( u , v ) ∣ ^{2}}{Q ( u )} = ⟨ P_{> M} b_{k} (v), A_{k, > M}^{- 1} P_{> M} b_{k} (v)⟩ \leq \frac{K ( k ) ^{2}}{d ( k , M )} ∥ v ∥_{2}^{2} \leq \frac{K ( k ) ^{2}}{k} ∥ v ∥_{2}^{2} = O (\frac{( lo g k ) ^{2}}{k}) ∥ v ∥_{2}^{2} . (T44)$

The full true pole-free form and its own forcing obey the same estimate, since removing the negative pole only increases the exterior energy and removes its row term. Each quotient uses its corresponding complete form and row. This covers every coefficient combination in the entire infinite exterior; it is not a sum of finite or rank-one inverse budgets.

The added interface is (T38)–(T42) for the complete physical adjacent forcing, consumed by the reused exterior floor in (T43)–(T44). The disjoint-slot principle, classical Carleman/Schur and prime-weight estimates, Fourier leakage, scalar digamma bound and positive-form inverse algebra are reused. The retained space has dimension of order $exp (β k)$ : this conservative cutoff is not an efficient certification scheme. Its full retained Schur sign and coupling, and a paid relative-energy identification of the joint theta space (T31) with that retained Fourier space, remain unproved. The theta and exterior debit estimates cannot simply be added as a bound on the whole old inverse.

Applying this physical estimate to the preprint’s allocated source debit still requires its common-core, norm and full-form transport identification. The original selected-integer signed Robin estimate, its original coefficients, actual zero real parts, both signs, all heights and multiplicities, elementary correction and strict core are unchanged and unproved. This paper-level interface is neither a Lean result nor an RH proof, and carries no mathematical-priority claim.

Join the theta directions to the entire infinite exterior

The theta debit (T37) and infinite-exterior debit (T44) cannot be added without paying their actual mixed energy. A state-projection approximation is not needed for this particular joint estimate: mixed nullity expresses its forcing through the small original theta error. The following pays that error’s complete old response, including its endpoint trace, and consumes it on the whole exterior.

Reuse (T24), (T31)–(T37), the complete tail expression (T5), the mixed-nullity/domain bridge, (T42)–(T44), and the classical $H^{1}$ Fourier integration-by-parts, trace and Parseval estimates. The weighted even approximation and spectral-projector statements in the weighted exterior account have a different physical norm and are not transported here. The additional interface is the complete physical old error response and its mixed bound with the infinite exterior; no generic Fourier approximation or Schur theorem is new.

Keep the same integer $k$ , $a = \frac{1}{2} lo g k$ , $ℓ = 2 a$ , both cutoffs $r_{i}$ and degree schedules from (T31), and the same exterior cutoff $M = ⌈ e^{β k} ⌉$ from (T43). Put $U = U_{0} \oplus U_{1}$ and $E = P_{> M} H_{a}$ . Every $u = u_{0} + u_{1} \in U$ retains its actual null functions $g_{i}$ , errors $e_{i} = g_{i} - u_{i}$ , profiles and masses from (T24); all complex coefficients may depend on the endpoint. The constants below are independent of these coefficients and degrees. Assume the already stated Fourier/Plancherel and closed physical form-domain bridge when using $E$ and its positive restricted operator $A_{E}$ ; it does not imply positivity of the complete old form.

The actual errors have a paid two-derivative norm

Write $K_{i} = K_{d_{i}}$ , $δ_{i} = e^{- 2 r_{i}}$ and $M_{i} = ∥ e_{i} ∥_{2}^{2}$ . From the same two profiles in (T24), for $0 \leq j \leq 2$ ,

$∥ e_{i}^{(j)} ∥_{2} \leq C K_{i}^{3} δ_{i}^{- j} M_{i}, ∣ e_{i}^{(j)} (y) ∣ \leq C B_{i} K_{i}^{3} P_{i} δ_{i}^{- j} e^{- c (∣ y ∣ - r_{i}) / δ_{i}} (∣ y ∣ \geq r_{i}) . (T45)$

The errors and their derivatives vanish inside $(- r_{i}, r_{i})$ , with no boundary distribution. This is direct parameter reuse, not a new theta estimate. The paid schedule gives $K_{i}^{3} \leq a^{3/40}$ and $δ_{i}^{- 1} \leq k$ .

The full tail response is bounded in physical old $H^{1}$

For the fixed actual tail define its arithmetic response on the whole old interval by

$Z_{i} (y) = m_{i} = c_{Γ} e_{i} (y) + \int_{0}^{\infty} κ (t) [2 e_{i} (y) - e_{i} (y + t) - e_{i} (y - t)] d t - n \geq 2 \sum \frac{Λ ( n )}{n} [e_{i} (y + lo g n) + e_{i} (y - lo g n)] - 2 sinh (y /2) m_{i}, \int_{R} e_{i} (y) sinh (y /2) d y, - a < y < a . (T46)$

Here $c_{Γ}$ and $κ$ are exactly (T5). The sum is global: the error is noncompact, so even $n = k$ and all $n > k$ remain present. This row is distinct from the disjoint-support collar row (O10), which has no multiplication term.

The Gamma integral is a convergent $H^{1}$ Bochner integral. Reuse the translation estimate $∥ e_{i} - τ_{t} e_{i} ∥_{H^{1}} \leq t ∥ e_{i}^{'} ∥_{H^{1}}$ and $\int_{0}^{\infty} t κ (t) d t < \infty$ to bound it by $C ∥ e_{i} ∥_{H^{2}}$ . This keeps the local singularity in its increment; no divergent diagonal rate is isolated.

For $2 \leq n \leq k$ , restrict translated functions only after taking their full $H^{1} (R)$ norm. The reused Chebyshev allowance $\sum_{n \leq k} Λ (n) / n \leq 2 B k$ pays this complete finite part by $C k K_{i}^{3} δ_{i}^{- 1} M_{i}$ , with both orientations.

For $n > k$ and $∣ y ∣ \leq a$ , one has $∣ y \pm lo g n ∣ - r_{i} \geq (a - r_{i}) + lo g (n / k)$ . The exact profile envelope in (T45), through the first derivative, therefore bounds the remaining $H^{1}$ norm by

$C ℓ B_{i} K_{i}^{3} P_{i} δ_{i}^{- 1} e^{- c (a - r_{i}) / δ_{i}} n > k \sum \frac{lo g n}{n} (n / k)^{- c / δ_{i}} .$

Once $c / δ_{i} \geq 2$ , the complete discrete sum is at most $C k (1 + ℓ)$ by integral comparison, including its endpoint term. Divide by $M_{i} = 2 B_{i} P_{i} δ_{i}$ ; this costs at most $C ℓ K_{i}^{3} δ_{i}^{- 3/2} k (1 + ℓ)$ , independently of the degree. The exponentially smaller factor may be discarded in this conservative upper bound; the prime tail itself is not discarded.

The same envelope gives the true polar moment $∣ m_{i} ∣ \leq C B_{i} K_{i}^{3} P_{i} δ_{i} e^{r_{i} /2}$ , while $∥ sinh (y /2) ∥_{H^{1} (- a, a)} \leq C e^{a /2}$ . Its coefficient- $2$ row is consequently at most $C K_{i}^{3} e^{(a - r_{i}) /2} M_{i}$ . Since $a /2 \leq r_{i} < a$ , every preceding budget is bounded by a fixed multiple of $k^{3} M_{i}$ at sufficiently large endpoints. Thus the full same tail satisfies

$∥ Z_{i} ∥_{H^{1} (- a, a)} \leq C k^{3} M_{i} . (T47)$

This is uniform for the complete actual coefficient combination, not only for the basis vectors. It uses no old positivity and no weight change. The pole-free response omits only the displayed true pole and obeys the same bound.

Mixed nullity exposes a small high-frequency forcing

The existing fixed-parameter outer-cutoff passage applies to every compact smooth old probe, not only a disjoint collar probe: the uncut $g_{i}$ has zero transform at every actual zero, and its true polar moments vanish. The complete prime/Gamma/pole expression passes by the same original theta-tail and weighted-jet majorants. Hence

$B_{Q} (u, f) = - ⟨ Z_{0} + Z_{1}, f ⟩ (f \in C_{c}^{\infty} (- a, a), f odd) .$

On the closed physical form realization this also holds for every admitted odd old form vector $f$ . Indeed the compact smooth $u$ has its actual operator row in $L^{2}$ ; core density and (T47) identify that row with $- Z$ , where $Z = Z_{0} + Z_{1}$ . This is an operator-row extension under the existing form bridge, not an assumption that a zero quadratic value implies mixed nullity.

For this actual $H^{1}$ row the ordinary Fourier integration-by-parts estimate retains its endpoint jump:

$∥ P_{> M} Z ∥_{2}^{2} \leq C [\frac{ℓ ∣ Z ( a ) - Z ( - a ) ∣ ^{2}}{M} + \frac{ℓ ^{2} ∥ Z ^{'} ∥ _{2}^{2}}{M ^{2}}] \leq \frac{C ( 1 + ℓ ^{2} )}{M} ∥ Z ∥_{H^{1} (- a, a)}^{2} \leq \frac{C k ^{6} ( 1 + ℓ ^{2} )}{M} (M_{0} + M_{1}) . (T48)$

Traces are not assumed to vanish or match. In the coefficient formula their contribution is $(- 1)^{m} [Z (a) - Z (- a)] / (i (2 πm / ℓ) ℓ)$ up to the irrelevant overall sign; the derivative part uses Parseval. This is a normalized application of the classical trace/Fourier estimate, rather than a new Fourier theorem.

At sufficiently large endpoints (T36) gives $Q (u) \geq a (M_{0} + M_{1}) /4$ , and (T43) gives $Q (f) \geq k ∥ f ∥_{2}^{2}$ on the whole $E$ . Consume these independent denominators in the exact mixed identity. Increasing a fixed $C_{*} > 0$ if needed, put $ζ_{k} = C_{*} k^{4} / M \to 0$ . Then for every coefficient combination and every exterior form vector,

$∣ B_{Q} (u, f) ∣ \leq ζ_{k} Q (u) Q (f), Q (u + f) \geq (1 - ζ_{k}) [Q (u) + Q (f)] . (T49)$

The scalar threshold is independent of the degree; no numerical starting endpoint is certified. The proof has used positivity only on $U$ and $E$ , separately supplied by (T36) and (T43). In particular $U \cap E = {0}$ , so this is a genuine joint positive space with all its near-cancelling coefficient combinations controlled. The true pole-free form has its corresponding row identity and the same estimate.

Pay the full joint inverse and its retained theta short

Let $D_{U} (k) = [D_{0} (a) + D_{1} (a)] / (1 - ε_{a}) = O (a^{- 7/4})$ from (T37), and $D_{E} (k) = K (k)^{2} / d (k, M) = O ((lo g k)^{2} / k)$ from (T44). The two estimates concern the same collar vector $v$ . Now (T49), followed by ordinary two-coordinate Cauchy–Schwarz, gives

$0 \neq = x \in U + E x \in D (Q) sup \frac{∣ B _{Q} ( x , v ) ∣ ^{2}}{Q ( x )} \leq \frac{D _{U} ( k ) + D _{E} ( k )}{1 - ζ _{k}} ∥ v ∥_{2}^{2} = O ((lo g k)^{- 7/4}) ∥ v ∥_{2}^{2} . (T50)$

This covers the entire infinite exterior plus both original theta spaces, not a finite collection of exterior modes. Their inverse budgets can be combined because the actual mixed term has now been paid. The standard positive-form variational identity supplies the complete restricted inverse interpretation.

There is also an exact finite retained consumer. The projection $u \mapsto l = P_{\leq M} u$ is injective on $U$ because $U \cap E = 0$ . Finite odd Fourier polynomials belong to $H_{0}^{1} (- a, a)$ and hence the admitted physical form domain, so $l$ and $u - l$ are legitimate form vectors. For its image define the actual exterior short, using the same closed form, $Q_{short} (l) = in f_{f \in E \cap D (Q)} Q (l + f)$ . Since $l + E = u + E$ , completing the already positive exterior square gives

$Q_{short} (P_{\leq M} u) (1 - ζ_{k}^{2}) Q (u) = Q (u) - ⟨ P_{> M} Z, A_{E}^{- 1} P_{> M} Z ⟩, \leq Q_{short} (P_{\leq M} u) \leq Q (u) . (T51)$

Thus the exact high-frequency elimination retains a positive theta image of dimension $(d_{0} + 1) + (d_{1} + 1)$ inside the retained Fourier space. This pays its shorted metric without assuming raw Fourier projection is a small relative-energy perturbation. It does not establish the sign or conditioning of the rest of that finite space.

The added interface is the coefficient-uniform complete old error row (T46)–(T48), consumed by the actual theta/exterior mixed estimate (T49), whole joint inverse (T50) and exterior-shorted retained theta metric (T51). Tail profiles, nullity/domain passage, Fourier trace algebra, positive block inversion and the two prior restricted budgets are reused. The full retained complement and its mixed interaction with this positive theta image remain unpaid; they cannot be inferred from the image dimension or from an independently positive diagonal.

The original selected-integer signed Robin estimate, all its original coefficients, actual zero real parts, both signs, heights and multiplicities, elementary correction and strict core remain unchanged and unproved. Source common-core/norm/full-form transport is still required before consuming this physical result as the preprint’s allocated source metric. This is a paper-level mixed interface, without Lean certification, a numerical endpoint certificate, an all-old inverse or RH conclusion, or a mathematical-priority claim.

The remaining finite short still has nearly null directions

The positive space in (T49) includes both growing theta families and the entire infinite exterior. Removing it does not leave a uniformly polynomially coercive finite block. The following concerns the actual remaining Schur form after both eliminations, not the raw old/source metric already tested in (O8)–(O9). Its additional requirement is a normalized witness surviving a growing number of removed theta directions.

Reuse the original null functions $g_{m}$ from (T9), the original theta-series recurrence in (T17), the polynomial $L^{2}$ interpolation bound used in (T19), the full weighted-jet/nullity passage (T4)–(T5), and the exact positive joint/exterior shorts (T49)–(T51). Classical Plancherel, derivative Fourier tails, finite-dimensional kernels and variational shorting are inputs, not new general theorems. The fixed-window full-form source does not supply a cofinal polynomial gap on the different block considered here.

Keep the same actual $k, a, ℓ, M, U, E$ as above and put

$W_{M} = P_{\leq M} H_{a}, L_{M} = P_{\leq M} U, V_{M} = W_{M} \cap L_{M}^{⊥} .$

Orthogonality here uses the physical $L^{2}$ norm. Under the same physical/Fourier/closed-form bridge, the exact exterior short $S_{M}$ is defined on $W_{M}$ . Its restriction to $L_{M}$ is positive by (T51). Thus the remaining finite form is the legitimate second short

$S_{M} (l) S_{M}^{rem} (w) = h \in E \cap D (Q) in f Q (l + h), = l \in L_{M} in f S_{M} (w + l) = u \in U h \in E \cap D (Q) in f Q (w + u + h), w \in V_{M} . (T52)$

The last equality uses $P_{\leq M} U = L_{M}$ and the same fibres as (T51). Only the eliminated blocks are required positive; the remaining form is not presumed nonnegative. Let $D = dim L_{M} = (d_{0} + 1) + (d_{1} + 1) = O (lo g lo g k)$ .

Pay raw null-family normalization at growing degree

For $c = (c_{0}, \dots, c_{D})$ put $G_{c} = \sum_{m = 0}^{D} c_{m} g_{m}$ and $p_{c} (x) = \sum_{m = 0}^{D} c_{m} x^{m}$ . The reused Fourier identity gives

$G_{c} (t) = (i t) ((i t)^{2} - 1/4) ξ (1/2 + i t) p_{c} (- t^{2}) .$

Fix a sufficiently small $ϵ > 0$ . The known $ξ (1/2) > 0$ and continuity make the prefactor’s modulus uniformly positive on $[ϵ, 2 ϵ]$ . Plancherel and the already used polynomial interpolation inequality, rescaled to the fixed interval $[- 4 ϵ^{2}, - ϵ^{2}]$ , therefore supply constants $c_{g}, C_{g} > 0$ , independent of the degree, such that

$∥ G_{c} ∥_{2}^{2} \geq c_{g} e^{- C_{g} (D + 1)} ∣ c ∣^{2} . (T53)$

This is a raw global $L^{2}$ bound, not a conditioning claim for the original energy. No endpoint-dependent tail-jet inverse is used here.

The same original derivative recurrence also pays a global upper budget $A_{D} = exp (C_{g}^{'} (D + 1) lo g (D + 2))$ after increasing a fixed $C_{g}^{'}$ :

$∥ G_{c} ∥_{H^{2} (R)} \leq A_{D} ∣ c ∣, ∣ G_{c}^{(j)} (y) ∣ \leq A_{D} ∣ c ∣ e^{- c_{0} e^{2∣ y ∣}} (0 \leq j \leq 2) . (T54)$

For clarity about uniformity, each original $n$ -th theta term on $y \geq 0$ has the form $e^{y /2} P (n^{2} e^{2 y}) e^{- π n^{2} e^{2 y}}$ . At most $2 D + 5$ derivatives and the polynomial coefficient bound in (T17) give coefficient size $exp (O ((D + 1) lo g (D + 2)))$ and degree $O (D + 1)$ . Absorb its polynomial into half the same exponential; use $n^{2} e^{2 y} \geq (n^{2} + e^{2 y}) /2$ and sum the remaining Gaussian in $n$ . A further fixed fraction pays $e^{y /2}$ and integration. Odd reflection controls the other half-line. This derives (T54) from the normally convergent original series, with no unspecified fixed-order constants.

A compact witness survives all the removed directions

Fix one smooth even $χ$ , equal to one on $[- 1, 1]$ and zero outside $(- 2, 2)$ , and set

$R = a /4, t_{c} (y) = χ (y / R) G_{c} (y), e_{c} = G_{c} - t_{c} .$

The compact test $t_{c}$ is supported inside $(- a /2, a /2)$ , hence is a genuine old vector. Choose a physical orthonormal basis $l_{1}, \dots, l_{D}$ of $L_{M}$ . The $D$ complex linear conditions

$⟨ P_{\leq M} t_{c}, l_{j} ⟩ = 0, 1 \leq j \leq D,$

on $D + 1$ coefficients have a nonzero solution; scale it so $∣ c ∣ = 1$ . The coefficients may depend on every actual endpoint and on the removed space. Put $t = t_{c}$ and $w = P_{\leq M} t \in V_{M}$ .

The original tails (T54) pay $∥ e_{c} ∥_{2} \leq C A_{D} e^{- c_{1} e^{2 R}}$ . Meanwhile $∥ t^{''} ∥_{2} \leq C A_{D}$ for $R \geq 1$ . Since $t$ vanishes near both old endpoints, ordinary derivative Parseval gives

$∥ P_{> M} t ∥_{2} \leq (\frac{ℓ}{2 π ( M + 1 )})^{2} ∥ t^{''} ∥_{2} .$

There is no endpoint-trace assumption on a general row here: this particular compact test has zero traces. At the actual $D = O (lo g lo g k)$ and $M = ⌈ e^{β k} ⌉$ , each loss is smaller than one quarter of the lower norm in (T53). Consequently, uniformly over the selected constraint solution,

$∥ w ∥_{2}^{2} \geq c_{2} e^{- C_{2} (D + 1)} > 0 (T55)$

for all sufficiently large actual endpoints. This pays the growing-codimension normalization rather than assuming the raw derivative Gram is stable.

The complete arithmetic energy remains small

At each finite parameter pair, $G_{c}$ has the original mixed nullity and zero polar moments. The same fixed-parameter outer-cutoff passage yields $Q (t) = Q^{tail} (e_{c})$ , with all actual zero real parts, both signs, heights and multiplicities retained. No infinite-degree limit is interchanged with the explicit formula.

The differentiated cutoff tails and (T54) give $|e_{\mathbf c}|{H^2}+\int|e{\mathbf c}(y)|e^{|y|/2}dy \le C A_De^{-c_3e^{2R}}$. The Gamma increment bound and finite $\int t κ (t) d t$ pay the singular part by the squared $H^{1}$ tail norm. The true coefficient- $2$ pole uses the displayed exponential moment.

The full noncompact prime sum is also retained. Whenever both error factors contribute, put $S = e^{2∣ y ∣} + e^{2∣ y - l o g n ∣}$ . Then $S \geq 2 e^{2 R}$ , $S \geq 2 n$ and $S \geq e^{2∣ y ∣}$ . Splitting its exponential into three fixed parts in the actual product envelope gives

$n \geq 2 \sum \frac{Λ ( n )}{n} ∣ ⟨ e_{c}, τ_{l o g n} e_{c} ⟩ ∣ \leq C A_{D}^{2} e^{- c_{4} e^{2 R}} n \geq 2 \sum \frac{lo g n}{n} e^{- c_{4} n} \leq C A_{D}^{2} e^{- c_{4} e^{2 R}} .$

No $n \geq k$ term is dropped. These estimates include all terms of (T5), so, decreasing a fixed $c_{5} > 0$ if necessary,

$∣ Q (t) ∣ \leq C A_{D}^{2} e^{- c_{5} e^{2 R}} = C A_{D}^{2} e^{- c_{5} k^{1/4}} . (T56)$

The true pole-free form satisfies the same absolute estimate with its own complete expression and removed pole. No sign of either compact energy is inferred.

A polynomial floor fails on the actual remaining short

Let $v_{k} = w /∥ w ∥_{2}$ and use the same normalization for $t$ . Since $t - w \in E \cap D (Q)$ , it is an admissible competitor in (T52), with the theta coordinate zero. Thus

$S_{M}^{rem} (v_{k}) \leq \frac{Q ( t )}{∥ w ∥ _{2}^{2}} \leq C exp (C_{3} (D + 1) lo g (D + 2) - c_{5} k^{1/4}) \leq C_{*} e^{- c_{*} k^{1/4}}, ∥ v_{k} ∥_{2} = 1, (T57)$

for all sufficiently large actual integers $k$ , with fixed positive constants and no certified numerical start. In particular, for any fixed $q > 0$ and $c > 0$ , the uniform lower bound $S_{M}^{rem} ⪰ c k^{- q} I$ cannot hold on this remaining finite space at every sufficiently large endpoint. If the remaining matrix is positive definite, its inverse norm is correspondingly at least $C_{*}^{- 1} e^{c_{*} k^{1/4}}$ . This conditional statement concerns its operator norm only.

The argument applies separately to the true pole-free form: use its own positive $U + E$ elimination and the corresponding constraints and row. It does not assert that one vector is an eigenvector for both remaining matrices.

The added interface is the normalized, degree-uniform remaining-short witness (T53)–(T57), which survives the actual growing theta removal and the complete exterior elimination. The fixed raw-source obstruction (O8)–(O9), original nullity and theta recurrences, polynomial Gram inequality, Fourier derivative tail and variational algebra are reused. A small remaining energy is not a negative test, an RH counterexample, or a lower bound on the actual collar inverse debit. The actual forcing can be correspondingly small or correlated. This Schur debit route still needs control of the actual forcing against the nearly null directions; that comparison is unpaid. In particular the complete physical $L^{2}$ forcing norm cannot be turned into a polynomial inverse budget by a presumed polynomial floor on this remainder.

The original selected-integer signed Robin target, all original coefficients and actual zero data, elementary correction and strict core are unchanged and unproved. Source common-core/norm/full-form transport still requires its separate identification. This is a paper-level obstruction to a specific coercivity route, without Lean certification, a numerical endpoint certificate, a proof or refutation of RH, or a mathematical-priority claim.

Pay the actual forcing on a growing part of the remaining short

The small-energy witness in (T57) does not determine its actual forced debit. The next estimate pays a growing subspace of the same remaining finite short, including the change to the collar forcing caused by the elimination of $X = U + E$ . It does not establish the sign or inverse budget of the whole remaining block.

Reuse the original recurrence and tail-jet construction (T17)–(T19), complete energy and response calculations (T20), (T24)–(T28), unequal-cutoff correlations (T32)–(T34), full error row (T46)–(T48), joint positive space and debit (T49)–(T51), actual remaining short (T52), and raw normalization (T53)–(T56). Their original theta/nullity, Gamma, prime, pole, Fourier and variational suppliers remain inputs. The added interface is the coefficient-uniform residual forcing quotient on an actual growing-dimensional remaining space. No new general interpolation, shorting or Fourier theorem is asserted.

Keep $k, a, h, M, U, E, W_{M}, L_{M}, V_{M}$ and $C_{0}$ as above. All statements use the same physical/Fourier/closed-form bridge, and all sufficiently large actual integer endpoints; no fresh-prime branch of (O3) is needed. Put $X = U + E$ and let $D_{X} (k) = O (a^{- 7/4})$ denote the paid complete restricted debit in (T50).

A sufficient degree threshold leaves more directions than were removed

The convenient condition $r \geq H_{d} = K_{d}^{10}$ in (T17) is stronger than the construction and complete energy require. Inspect its original proof with $T = e^{2 r}$ and $K_{d} = e^{C_{0} (d + 1)^{2}}$ . The coefficient and Vandermonde inverse bounds are $O (K_{d})$ ; their perturbation product is $O (K_{d}^{2} / T)$ . The original profile error is $O (K_{d} / T)$ and its finite Gram error is $O (K_{d}^{4} / T)$ . The higher theta terms use $T \geq (d + 1)^{2}$ and the same $T^{- 1}$ remainder. Thus $T \geq C_{*} K_{d}^{10}$ pays these requirements directly, including the lower profile Gram $c_{*} K_{d}^{- 1}$ and the relative two-derivative/envelope budget $C_{*} K_{d}^{3}$ .

After that construction, the complete energy remainder in the paragraph preceding (T20) is $C_{*} K_{d}^{6}$ times the actual tail mass. The reflected Gamma, same-side prime, opposite-tail prime and true pole terms carry only the already displayed decreasing scalar factors. In particular the opposite-tail factor $r e^{- r}$ is bounded. Therefore sufficient conditions, with fixed constants independent of the degree, are

$e^{2 r} \geq C_{*} K_{d}^{10}, r \geq C_{*} K_{d}^{6}, ∣ Q (t) - 2 r m (t) ∣ \leq C_{*} K_{d}^{6} m (t), Q (t) \geq r m (t) . (T58)$

Here $t = χ_{r} g$ is any nonzero compact theta-derivative combination, $e = g - t$ , and $m (t) = ∥ e ∥_{2}^{2} > 0$ . Increase the fixed $C_{*}$ as needed; do not change the original $C_{0}$ or either previously selected degree schedule. This is a check of the sufficient thresholds in the existing construction, not an application of its stronger displayed hypothesis to parameters that fail that hypothesis.

Let $D = dim L_{M}$ as in (T52). The original schedules give $D \leq lo g a / (10 C_{0})$ . Choose

$r_{*} = a /4, N_{*} = ⌊ \frac{lo g a}{9 C _{0}} ⌋, d_{*} = N_{*} - 1, K_{d_{*}} \leq a^{1/9} . (T59)$

Both conditions in (T58) hold eventually: $e^{2 r_{*}} = e^{a /2}$ exceeds $C_{*} a^{10/9}$ , and $r_{*} = a /4$ exceeds $C_{*} a^{2/3}$ . Let $T_{*}$ be the $N_{*}$ -dimensional space of these naturally cut off derivatives, with the same smooth transition at width $δ_{*} = e^{- 2 r_{*}}$ . Its dimension follows either from the lower profile Gram or from the original raw independence and analyticity on the interior where the cutoff equals one. Moreover $N_{*} - D \to \infty$ , because $1/3 > 1/ 10$ . All coefficients can depend on the actual endpoint.

Keep the whole collar row at this smaller radius

The response proof in (T26)–(T27) also works at $r = a /4$ . Its lower-cutoff restriction $r \geq a /2$ is not needed for this particular estimate: $g = a - r \geq 1$ , both thresholds $T_{\pm} = e^{y \pm r}$ have logarithm at most $C a$ , and $δ T_{-}^{1/2} \leq δ T_{+}^{1/2}$ and $T_{+}^{- 1/2} \leq T_{-}^{- 1/2}$ for every $r > 0$ . Their two leading row budgets remain $e^{- g /2}$ and $e^{(a - 3 r) /2}$ . The latter is allowed to grow; the actual collar width is retained before division.

The other translation orientation, singular local Gamma increment and remote reflected Gamma terms use only $r \to \infty$ , $g \geq 1$ and the paid relative two-derivative envelope. The actual coefficient- $2$ pole is still bounded by $C K_{d_{*}}^{3}$ times the same polar moment and collar factor. Thus, with $R_{k}$ the complete physical cross row from (T38), every $t \in T_{*}$ satisfies

$\begin{aligned} \|R_kt\|_2^2 &\le C K_{d_*}^6h a^2\delta_*^{-1}m(t) \left(e^{-(a-r_*)}+e^{a-3r_*}\right),\\ \boxed{ \frac{|B_Q(t,v)|^2}{Q(t)} &\le D_*(a)\|v\|_2^2,\qquad D_*(a)\le C a^{5/3}e^{-5a/4} }\qquad(v\in\mathcal K_k). \end{aligned} \tag{T60}$

Use $Q (t) \geq r_{*} m (t)$ from (T58), $h \leq e^{- 2 a} /2$ , and $K_{d_{*}}^{6} \leq a^{2/3}$ to obtain the second line. The two exponential debit terms before comparison are $e^{- 3 (a - r_{*})}$ and $e^{- (a + r_{*})}$ . The complete noncompact prime sum is not cut at $n = k$ ; it enters through the same nullity/tail bridge as before. For collar vectors outside the form domain the mixed expression means its bounded cross-row extension.

The eliminated space is nearly orthogonal in the actual energy

For the new cutoff and the two old ones, the separation and sum radii are $(Δ_{0}, Σ_{0}) = (a /4, 3 a /4)$ and $(Δ_{1}, Σ_{1}) = (3 a /4 - lo g a, 5 a /4 - lo g a)$ . The narrower-tail condition in (T32) holds. Its mixed profile factor is at most $K_{d_{*}}^{3} K_{d_{i}}^{3} \leq a^{1/3 + 3/40}$ . Apply the complete two-center prime and singular Gamma estimates (T33)–(T34), with the actual true pole, to these radii. Dividing by the independently paid energies leaves a bound $C a^{2} e^{- 3 a /8} Q (t) Q (u)$ for every $u \in U$ ; the first center $Δ_{0}$ and the pole sum $Σ_{0}$ both fit that allowance. The small-overlap Gamma term is exponentially smaller. This comparison covers all complex coefficient combinations, not only pairs of basis vectors.

The actual old error row for $t$ satisfies $∥ Z_{*} ∥_{H^{1} (- a, a)} \leq C k^{3} m (t)$ . To see that (T47)’s argument still pays it, replace its old profile budget by $K_{d_{*}}^{3} \leq a^{1/3}$ and use $δ_{*}^{- 1} = k^{1/4}$ . The finite prime norm, Gamma two-derivative norm, complete $n > k$ tail and pole are still bounded by $C k^{3} m (t)$ . Keep the endpoint jump in (T48). Its exterior mixed allowance is consequently $ν_{k} = C k^{4} / M$ as before.

Using the already proved joint lower bound (T49), define

$∣ B_{Q} (t, x) ∣ ρ_{k}^{2} \leq ρ_{k} Q (t) Q (x) = \frac{( C a ^{2} e ^{- 3 a /8} ) ^{2} + ν _{k}^{2}}{1 - ζ _{k}} \leq C a^{4} e^{- 3 a /4} + C k^{8} / M ⟶ 0. (t \in T_{*}, x \in X \cap D (Q)), (T61)$

Only $T_{*}$ and the independently paid $X$ are positive here. No Cauchy–Schwarz inequality on the unproved whole old or remaining form is used.

Pay the Schur-corrected forcing on its actual fibre

Impose the $D$ linear conditions $P_{\leq M} t ⊥ L_{M}$ and let $Z_{*} = {P_{\leq M} t : t \in T_{*}, P_{\leq M} t ⊥ L_{M}} \subset V_{M}$ . The projection is injective on $T_{*}$ : if $P_{\leq M} t = 0$ , then $t \in E \subset X$ , and (T61) with $x = t$ contradicts $ρ_{k} < 1$ unless $t = 0$ . Hence $dim Z_{*} \geq N_{*} - D \to \infty$ .

Reuse the positive restricted form inverse on $X$ from (T50). For $t \in T_{*}$ let $x_{t} \in X \cap D (Q)$ solve $B_{Q} (x_{t}, x) = B_{Q} (t, x)$ for every $x \in X \cap D (Q)$ . The paid mixed functional has a Riesz representative in the energy domain of that same positive closed-form restriction. Equation (T61) gives $Q (x_{t}) \leq ρ_{k}^{2} Q (t)$ . For $w = P_{\leq M} t \in Z_{*}$ , the exact minimizing lift of its fibre is $t - x_{t}$ , because $t - w \in E \subset X$ . Thus the actual remaining energy and residual collar row are

$S_{M}^{rem} (w) b_{k}^{rem} (w, v) ∣ b_{k}^{rem} (w, v) ∣ = Q (t) - Q (x_{t}) \geq (1 - ρ_{k}^{2}) Q (t) > 0 (w \neq = 0), = B_{Q} (t, v) - B_{Q} (x_{t}, v), \leq (D_{*} (a) + ρ_{k} D_{X} (k)) Q (t) ∥ v ∥_{2} . (T62)$

The second forcing term cannot be omitted. Its bound uses (T50) on the same actual $x_{t}$ and collar vector $v$ , rather than an uncorrected numerator divided by the shorted denominator. Consume the independently proved lower denominator in the first line. The entire growing-dimensional residual Gram, with every coefficient combination, therefore satisfies

$0 \neq = w \in Z_{*} sup \frac{∣ b _{k}^{rem} ( w , v ) ∣ ^{2}}{S _{M}^{rem} ( w )} \leq \frac{( D _{*} ( a ) + ρ _{k} D _{X} ( k ) ) ^{2}}{1 - ρ _{k}^{2}} ∥ v ∥_{2}^{2} \leq C a^{9/4} e^{- 3 a /4} ∥ v ∥_{2}^{2} . (T63)$

The $k^{8} / M$ term is smaller than the displayed allowance eventually. Positivity on $Z_{*}$ gives the standard restricted inverse interpretation of this supremum. It is not an estimate of the inverse or sign of the full remaining matrix.

Small remaining energy and small forced debit coexist on this space

The raw norm and global derivative budgets (T53)–(T54) apply at degree $d_{*} = N_{*} - 1$ . For unit raw coefficients they give $∥ g ∥_{2}^{2} \geq c e^{- C N_{*}}$ and $A_{*} = exp (C N_{*} lo g (N_{*} + 1))$ . Cutting at $r_{*} = a /4$ loses at most $C A_{*} e^{- c k^{1/4}}$ in $L^{2}$ . The cutoff derivatives cost only powers of $e^{2 r_{*}}$ ; the same Gaussian tail absorbs them. In particular $∥ t^{''} ∥_{2} \leq C A_{*}$ eventually. The zero-trace Fourier tail from (T55) is at most $C (ℓ / M)^{2} A_{*}$ .

Each loss is less than one quarter of the original lower norm, uniformly over the constraint kernel. Therefore $∥ w ∥_{2}^{2} \geq c^{'} e^{- C^{'} N_{*}}$ for unit raw coefficients. The full energy tail estimate (T56), including all infinite prime powers and the true pole, applies to the same natural-width cutoff after its polynomial derivative factors are absorbed. It gives $∣ Q (t) ∣ \leq C A_{*}^{2} e^{- c k^{1/4}}$ . As $N_{*} = O (lo g lo g k)$ , (T62) supplies

$0 < S_{M}^{rem} (w) \leq C e^{- c k^{1/4}} ∥ w ∥_{2}^{2} (0 \neq = w \in Z_{*}), dim Z_{*} ⟶ \infty. (T64)$

Equations (T63) and (T64) concern the same actual remaining subspace and the same corrected forcing. They show that its nearly null metric does not create a large actual forced debit: the restricted debit instead tends to zero. This is stronger than giving separate upper bounds for numerator and denominator; (T62) has independently paid their relative denominator before division.

The true pole-free form admits the same construction with its own positive $X$ , minimizing lifts and residual row. It removes only the actual pole contribution and does not identify the two remaining matrices or their minimizing vectors.

The added result is the growing-dimensional actual remaining-short consumer (T59)–(T64), with its paid degree threshold and Schur correction. The previously supplied raw near-null obstruction, theta/nullity/Gram/row estimates and positive short algebra are reused. The whole remaining complement, its interactions outside $Z_{*}$ and the full old inverse are still unpaid; adding this family does not prove that these spaces exhaust the old form domain or that a repeated elimination converges in the original energy. Source common-core/norm/full-form transport remains a separate identification. The original selected-integer signed Robin target, all coefficients and actual zero data, elementary correction and strict core remain unchanged and unproved. These are paper estimates without Lean certification, a numerical starting certificate, a proof or refutation of RH, or a mathematical-priority claim.

A separated radius mesh pays a larger actual remaining space

The remaining space paid in (T63) comes from one extra cutoff. The following uses a growing number of cutoff radii on the same old interval. Its full mixed Gram and collective Fourier row must be estimated before their debit allowances can be combined. The resulting actual remaining image has dimension of order $a / lo g a$ , and its corrected debit still tends to zero.

Reuse (T58)’s sufficient construction thresholds, (T59)’s $T_{*}$ , the complete response calculation (T60), the unequal-width Gamma/prime/pole estimates (T32)–(T34), complete error-row/Fourier-tail bounds (T46)–(T48), exterior floor (T43), actual short (T52), and residual-fibre construction (T62). Classical matrix row-sum and positive-form variational identities are intermediate tools. The added interface is their uniform consumer with a growing number of actual theta cutoffs, followed by the exact remaining quotient.

Keep the original $k, a, h, M, C_{0}, U, E, X = U + E$ and physical/Fourier/closed-form bridge. Only form-domain vectors enter any expression $Q (x)$ . All constants below are independent of the number of radii, selected degrees, endpoint and complex coefficients.

Choose the mesh while retaining both original families

Put $w_{a} = 8 lo g a$ and start with the radii $a - lo g a - j w_{a}$ in $[a /4, a - lo g a]$ . Remove any of these within distance strictly less than $w_{a}$ of $a /4$ or $a /2$ , and adjoin $a /4$ and $a /2$ . For all sufficiently large $a$ , the top radius $a - lo g a$ is retained. Denote this set by $R_{a}$ and its size by $J_{a}$ . Its increasing enumeration is separated by at least $w_{a}$ , and

$J_{a} ≍ \frac{a}{lo g a} .$

Removing at most a fixed number of grid points does not change this order. Use

$N_{g} N_{a /4} G_{a} = ⌊ \frac{lo g a}{30 C _{0}} ⌋, = N_{*} from (T59), = r \in R_{a} \sum G_{r} . N_{r} G_{r} = N_{g} (r \neq = a /4), = span {χ_{r} g_{m} : 0 \leq m < N_{r}}, (T65)$

The cutoff $χ_{r}$ is the original smooth natural-width cutoff at $δ_{r} = e^{- 2 r}$ . For regular radii, $K_{N_{r} - 1}^{6} \leq a^{1/5}$ ; for $r = a /4$ , it is at most $a^{2/3}$ . Since every $r \geq a /4$ , both sufficient conditions in (T58) hold uniformly. For $t_{r} \in G_{r}$ let $m_{r} = ∥ g_{r} - t_{r} ∥_{2}^{2}$ be its actual tail mass. Then $Q (t_{r}) \geq r m_{r} > 0$ for $t_{r} \neq = 0$ .

At $r = a /2$ and $r = a - lo g a$ , $N_{g}$ is at least the corresponding original degree count in (T31), because $30 < 40$ and $lo g r \leq lo g a$ . Therefore $U \subset G_{a}$ , and $T_{*} = G_{a /4} \subset G_{a}$ . The original $C_{0}$ , degrees and coefficients are retained.

Pay every mesh cross term with one row-sum bound

For $r < s$ in the mesh put $Δ = s - r$ and $Σ = s + r$ . The narrower-width condition in (T32) holds since $Δ \geq w_{a} \to \infty$ . The profile product is at most $a^{2/3}$ , using the conservative maximum at the special radius. The complete prime estimate (T33), divided by the independent lower energies, gives the difference-center term $C a^{2/3} e^{- 3Δ/2}$ . Its other terms are at most $C a^{2/3} e^{- a /4}$ , since $δ_{r} \leq e^{- a /2}$ and $Σ \geq a /2$ .

The singular Gamma estimate (T34) retains its small-shift term. Uniformly over this mesh, that term is at most $C a^{2/3} e^{4 a} e^{- c w_{a} e^{a /2} /2}$ before harmless energy-normalization factors; it is eventually smaller than $C a^{2/3} e^{- a /4}$ . The regular shifted terms and the true coefficient- $2$ pole also fit this latter allowance. Thus all actual reflected orientations, prime centers, discrete endpoints and Gamma terms give

$∣ B_{Q} (t_{r}, t_{s}) ∣ \leq η_{rs} Q (t_{r}) Q (t_{s}), η_{rs} = C a^{2/3} (e^{- 3∣ r - s ∣/2} + e^{- a /4}) .$

Separation bounds the whole coefficient-matrix row sum:

$μ_{a} := r max s \neq = r \sum η_{rs} Q (r \sum t_{r}) \leq C a^{2/3} (\frac{2 a ^{- 12}}{1 - a ^{- 12}} + J_{a} e^{- a /4}) \leq C^{'} a^{- 10} ⟶ 0, \geq (1 - μ_{a}) r \sum Q (t_{r}) \geq \frac{a}{4} (1 - μ_{a}) r \sum m_{r} . (T66)$

Apply $2 Q (t_{r}) Q (t_{s}) \leq Q (t_{r}) + Q (t_{s})$ to obtain the second line. This proves independence of the actual cutoff spaces as well as positivity of their full joint Gram; no constants fixed in $J_{a}$ are used. Consequently $dim G_{a} = \sum_{r} N_{r}$ .

The complete collar forcing is concentrated near the top radius

For every mesh radius the full response proof in (T60) applies with $a /4 \leq r \leq a - lo g a$ . Its two thresholds and local/remote Gamma split require $r > 0$ , $a - r \geq 1$ and the paid profile budgets. The actual collar width is $h \leq e^{- 2 a} /2$ . Thus its independently divided allowance is

$\frac{∣ B _{Q} ( t _{r} , v ) ∣ ^{2}}{Q ( t _{r} )} \leq D_{r} (a) ∥ v ∥_{2}^{2}, D_{r} (a) \leq C K_{N_{r} - 1}^{6} \frac{a ^{2}}{r} (e^{- 3 (a - r)} + e^{- (a + r)}) .$

This retains the full noncompact prime sum, both orientations and the true pole. The special $r = a /4$ term is the exponentially small allowance already paid in (T60). For the regular grid, $K^{6} a^{2} / r \leq C a^{6/5}$ and

$regular r \sum e^{- 3 (a - r)} \leq \frac{a ^{- 3}}{1 - a ^{- 24}} + e^{- 3 a /2} .$

The added $a /2$ radius contributes the final term. The second exponential contributes at most $J_{a} e^{- 5 a /4}$ . Therefore

$r \in R_{a} \sum D_{r} (a) \leq C a^{- 9/5}, 0 \neq = g \in G_{a} sup \frac{∣ B _{Q} ( g , v ) ∣ ^{2}}{Q ( g )} \leq \frac{\sum _{r} D _{r} ( a )}{1 - μ _{a}} ∥ v ∥_{2}^{2} = O (a^{- 9/5}) ∥ v ∥_{2}^{2} . (T67)$

The last inequality uses coefficient Cauchy–Schwarz with the already paid joint denominator (T66). It is not a sum of separately optimized inverse values without their cross Gram.

Pay the growing mesh’s collective row against the entire exterior

For each actual error the complete physical old response from (T46) has $∥ Z_{r} ∥_{H^{1} (- a, a)} \leq C k^{3} m_{r}$ , uniformly over this mesh. The proof in (T61) permits the maximum $K^{3} \leq a^{1/3}$ , while $δ_{r}^{- 1} \leq k$ , so all finite and infinite prime parts, singular Gamma and actual pole are still paid by that same conservative polynomial. The errors’ actual multiplication terms are included.

For the sum row, retain the growing-count cost:

$r \sum Z_{r}_{H^{1}}^{2} \leq C k^{6} J_{a} r \sum m_{r} .$

Use the endpoint-jump Fourier estimate (T48), the joint lower energy (T66), and the whole exterior floor $Q (f) \geq k ∥ f ∥_{2}^{2}$ . Set $σ_{k} = C k^{4} J_{a} / M \to 0$ . Then for $g \in G_{a}$ and $f \in E \cap D (Q)$ ,

$∣ B_{Q} (g, f) ∣ \leq σ_{k} Q (g) Q (f), Q (g + f) \geq (1 - σ_{k}) [Q (g) + Q (f)] .$

In particular $G_{a} \cap E = {0}$ . Put $H_{a}^{mesh} = G_{a} + E$ , with its same positive closed-form restriction. The whole infinite-exterior debit (T44) and (T67) now combine:

$D_{mesh} (v) := 0 \neq = z \in H_{a}^{mesh} \cap D (Q) sup \frac{∣ B _{Q} ( z , v ) ∣ ^{2}}{Q ( z )} \leq \frac{C a ^{- 9/5} + K ( k ) ^{2} / d ( k , M )}{1 - σ _{k}} ∥ v ∥_{2}^{2} = O (a^{- 9/5}) ∥ v ∥_{2}^{2} . (T68)$

This includes all mesh coefficients and the entire infinite exterior. Its positive restriction is complete: (T66)–(T68) give energy-norm equivalence with the finite direct sum of the cutoff energies and the same closed exterior energy space. No positivity on arbitrary old vectors is inferred.

Transport that paid space to the actual remaining quotient

Let $P_{V}$ be physical $L^{2}$ projection from $W_{M}$ onto $V_{M}$ in (T52), and define

$Y_{a} = P_{V} P_{\leq M} G_{a} \subset V_{M} .$

The kernel of this map on $G_{a}$ is exactly $U$ . If $P_{\leq M} g \in L_{M}$ , choose the unique $u \in U$ with the same retained projection. Then $g - u \in G_{a} \cap E = {0}$ . The reverse inclusion follows from $P_{\leq M} U = L_{M}$ . Thus

$dim Y_{a} = r \sum N_{r} - D ≍ \frac{a}{lo g a} ⟶ \infty. (T69)$

For $y = P_{V} P_{\leq M} g$ , one has $g - y \in X \cap D (Q)$ after selecting the corresponding $u \in U$ as above and retaining the exterior part. Let $x_{g} \in X \cap D (Q)$ be the energy Riesz representative with $B_{Q} (x_{g}, x) = B_{Q} (g, x)$ for every $x \in X \cap D (Q)$ . Positivity on the already paid $H_{a}^{mesh}$ gives the energy-orthogonal decomposition

$H_{a}^{mesh} \cap D (Q) = (X \cap D (Q)) \oplus_{Q} {g - x_{g} : g \in G_{a}} .$

The finite second summand maps isomorphically to $Y_{a}$ and is the exact minimizing lift of its fibres. Hence $S_{M}^{rem} > 0$ on $Y_{a} ∖ {0}$ , and its actual residual collar functional is $b_{k}^{rem} (y, v) = B_{Q} (g - x_{g}, v)$ , independent of the representative $g$ .

Let $D_{X} (v)$ mean the actual restricted debit on $X$ , not merely its known upper allowance. The standard positive inverse identity, applied to this same-source orthogonal decomposition, gives

$D_{mesh} (v) 0 \leq 0 \neq = y \in Y_{a} sup \frac{∣ b _{k}^{rem} ( y , v ) ∣ ^{2}}{S _{M}^{rem} ( y )} = D_{X} (v) + 0 \neq = y \in Y_{a} sup \frac{∣ b _{k}^{rem} ( y , v ) ∣ ^{2}}{S _{M}^{rem} ( y )}, \leq C a^{- 9/5} ∥ v ∥_{2}^{2} . (T70)$

The last bound uses $D_{X} (v) \geq 0$ and (T68); it does not subtract an upper bound for $D_{X}$ from another upper bound. All minimizers and quantified energy arguments belong to their positive form restrictions. The actual Schur correction is included throughout.

Since $T_{*} \subset G_{a}$ , the previously paid nearly null subspace $Z_{*}$ is contained in $Y_{a}$ , and (T64) remains its $L^{2}$ energy estimate. No such metric upper estimate is supplied for the other vectors of $Y_{a}$ . The true pole-free form has the same estimates with its own positive restrictions, minimizing lifts and residual row.

The added result is the growing-radius, complete joint consumer (T65)–(T70), giving the larger actual residual space and its corrected inverse debit. All theta, complete arithmetic cross/row, Fourier/exterior and variational suppliers are reused. The rest of the retained finite space, its interactions with $Y_{a}$ , full old inverse and energy-norm exhaustion remain unestimated. Source common-core/norm/full-form transport is still separate. The original selected-integer signed Robin target, all coefficients and actual zero data, elementary correction and strict core are unchanged and unproved. These paper estimates have no Lean certification, numerical starting certificate, RH proof or refutation, or mathematical-priority claim.

The paid mesh does not remove the fixed compact-test problem

The radius mesh pays a growing part of the actual remaining space. Its increasing rank does not establish that it exhausts the original energy. The following gives a quantitative test: eliminating the whole paid mesh and infinite exterior changes the complete Weil energy of any fixed-support compact test by a quantity tending to zero. This identifies an actual remaining obligation rather than adding another cutoff family.

Reuse the actual paired-zero explicit formula and finite zero mass $S_{2}$ from (O10)–(O11), mixed nullity and tail passage (T4), relative profiles (T24), sufficient degree thresholds (T58), complete physical response formula (T46), endpoint-trace Fourier estimate (T48), exterior floor (T43), and complete positive mesh/quotient (T65)–(T70). Classical weighted integration by parts, Riesz representation and variational shorting are tools. The new consumer is the uniform compact-test dual estimate and its exact remaining-form comparison, not a new general shorting identity.

Keep the same $k, a, M, R_{a}, G_{a}, E$ and shared physical/Fourier/closed-form bridge. Put $H_{a}^{mesh} = G_{a} + E$ as in (T68). Fix $R > 0$ , independent of $a$ , and let $ϕ$ be any smooth compactly supported complex odd function in $(- R, R)$ , zero extended to $R$ . Assume $a > R + 1$ . All estimates below are uniform in this test with its actual $H^{2} (R)$ norm. They neither require nor assert $Q (ϕ) \geq 0$ .

The compact probe, rather than the tiny tail, pays zero summability

For the actual zero coordinate $z_{ρ}$ from (O10), compact support and two integrations by parts supply

$∣ ϕ (\overline{z_{ρ}}) ∣ \leq \frac{C _{R} ∥ ϕ ∥ _{H^{2}}}{1 + ( ℑ ρ ) ^{2}} .$

The bound is uniform for the actual real parts $0 < ℜ ρ < 1$ ; boundary terms vanish because $ϕ$ is a compact smooth test. Thus the complete sum of these absolute probe values is bounded by $C_{R} S_{2} ∥ ϕ ∥_{H^{2}}$ , retaining both ordinate signs and every multiplicity.

For a mesh direction $t_{r} = χ_{r} g_{r}$ write its actual null error as $e_{r} = g_{r} - t_{r}$ , with the same $B_{r}, P_{r}, δ_{r}, m_{r}$ as in (T24). Its relative profile envelope gives

$\int_{R} ∣ e_{r} (y) ∣ e^{∣ y ∣/2} d y \leq C K_{N_{r} - 1}^{3} B_{r} P_{r} δ_{r} e^{r /2} .$

Indeed the positive tail has coordinate $y = r + δ_{r} s$ ; the fixed profile decay absorbs $e^{δ_{r} s /2}$ eventually, and reflection pays the other tail. No derivative of $e_{r}$ is needed in this numerator. Mixed nullity gives $t_{r} (z_{ρ}) = - e_{r} (z_{ρ})$ at every actual zero, and the original paired explicit formula then yields

$\begin{aligned} |B_Q(t_r,\phi)| &\le C_RK_{N_r-1}^3B_rP_r\delta_r e^{r/2} \|\phi\|_{H^2},\\ \boxed{ \frac{|B_Q(t_r,\phi)|^2}{Q(t_r)} &\le C_RK_{N_r-1}^6\frac{e^{-r}}r\, \|\phi\|_{H^2}^2 . } \end{aligned} \tag{T71}$

The second line uses the independent lower energy $Q (t_{r}) \geq r m_{r}$ and $m_{r} = 2 B_{r}^{2} δ_{r} P_{r}^{2}$ . The actual off-line pair is kept; it is not replaced by a modulus square.

Separation of the mesh gives $\sum_{r \in R_{a}} e^{- r} \leq e^{- a /4} / (1 - a^{- 8})$ . With $K_{N_{r} - 1}^{6} \leq a^{2/3}$ and $r \geq a /4$ , the whole joint lower Gram (T66), followed by coefficient Cauchy–Schwarz, therefore pays

$0 \neq = g \in G_{a} sup \frac{∣ B _{Q} ( g , ϕ ) ∣ ^{2}}{Q ( g )} \leq C_{R} a^{- 1/3} e^{- a /4} ∥ ϕ ∥_{H^{2}}^{2} . (T72)$

This is a coefficient-uniform dual energy bound against a fixed-support test, distinct from the adjacent collar bound (T67). A weighted two-derivative norm of the theta error would introduce unnecessary inverse-width factors; here the compact probe pays the summable zero count.

The complete compact operator row pays the entire exterior

On $(- a, a)$ the actual arithmetic row of $ϕ$ is

$A_{a} ϕ (y) = c_{Γ} ϕ (y) + \int_{0}^{\infty} κ (t) [2 ϕ (y) - ϕ (y + t) - ϕ (y - t)] d t - n \geq 2 \sum \frac{Λ ( n )}{n} [ϕ (y + lo g n) + ϕ (y - lo g n)] - 2 sinh (y /2) \int_{R} ϕ (x) sinh (x /2) d x .$

This is the same complete row as (T46), applied to an actual compact test. The Gamma integral is a convergent $H^{1}$ Bochner integral, bounded by $C ∥ ϕ ∥_{H^{2}}$ using $\int_{0}^{\infty} t κ (t) d t < \infty$ . Its local singularity remains in the increment.

For $∣ y ∣ \leq a$ , compact support makes both translated prime terms vanish if $lo g n > a + R$ . All prime powers up to that actual support threshold remain, with both orientations. The existing Chebyshev allowance gives $\sum_{n \leq e^{a + R}} Λ (n) / n \leq C_{R} e^{a /2}$ . Full translation invariance of the $H^{1} (R)$ norm pays the finite prime part after restriction. The true coefficient- $2$ pole has its actual moment bounded by $C_{R} ∥ ϕ ∥_{2}$ and its old $H^{1}$ factor bounded by $C e^{a /2}$ . Thus

$∥ A_{a} ϕ ∥_{H^{1} (- a, a)} \leq C_{R} e^{a /2} ∥ ϕ ∥_{H^{2}} .$

The smooth compact test belongs to the operator row on this physical realization, so $B_{Q} (ϕ, f) = ⟨ A_{a} ϕ, f ⟩$ for every admitted old form vector $f$ . Its row generally has a nonzero endpoint jump; retain that trace in (T48). As $d (k, M) \geq k = e^{a}$ on the entire $E$ , this gives

$0 \neq = f \in E \cap D (Q) sup \frac{∣ B _{Q} ( f , ϕ ) ∣ ^{2}}{Q ( f )} \leq \frac{∥ P _{> M} A _{a} ϕ ∥ _{2}^{2}}{d ( k , M )} \leq C_{R} \frac{1 + a ^{2}}{M} ∥ ϕ ∥_{H^{2}}^{2} . (T73)$

The support argument makes the remaining prime terms exactly zero for this compact row; it does not truncate the noncompact theta-error sum used elsewhere. Multiplication, singular Gamma, pole and Fourier trace belong to the same actual row.

Eliminate the whole paid space on its actual fibre

Use the joint mesh/exterior lower estimate in (T68) before combining the two dual bounds. For the actual compact-test functional put

$D_{a}^{comp} (ϕ) ϵ_{R} (a) = 0 \neq = z \in H_{a}^{mesh} \cap D (Q) sup \frac{∣ B _{Q} ( z , ϕ ) ∣ ^{2}}{Q ( z )}, = C_{R} (a^{- 1/3} e^{- a /4} + \frac{1 + a ^{2}}{M}) ⟶ 0.$

Since $σ_{k} \to 0$ , increasing the fixed $C_{R}$ once gives the uniform bound

$0 \leq D_{a}^{comp} (ϕ) \leq ϵ_{R} (a) ∥ ϕ ∥_{H^{2}}^{2} . (T74)$

This divides by a independently paid positive restriction, never by $Q (ϕ)$ .

Let $L_{a}^{mesh} = P_{\leq M} G_{a} \subset W_{M}$ and $V_{a}^{bulk} = W_{M} \cap (L_{a}^{mesh})^{⊥}$ , using physical $L^{2}$ orthogonality. Set $b_{a} (ϕ) = P_{V_{a}^{bulk}} P_{\leq M} ϕ$ . Projection is injective on $G_{a}$ by (T68), so the retained mesh part has a unique original mesh representative. Accordingly $ϕ - b_{a} (ϕ) \in H_{a}^{mesh} \cap D (Q)$ . Define the actual remaining short after this whole elimination by

$S_{a}^{bulk} (b) = z \in H_{a}^{mesh} \cap D (Q) in f Q (b + z), b \in V_{a}^{bulk} .$

This is the same successive short obtained by removing $X$ and then the paid image $Y_{a}$ ; only the eliminated positive restrictions are assumed positive. Let $x_{ϕ} \in H_{a}^{mesh} \cap D (Q)$ be its positive energy Riesz representative, satisfying $B_{Q} (x_{ϕ}, z) = B_{Q} (ϕ, z)$ for every $z$ in that same energy domain. Then $Q (x_{ϕ}) = D_{a}^{comp} (ϕ)$ , and $ϕ - x_{ϕ}$ is the exact minimizing lift of $b_{a} (ϕ)$ . Therefore

$S_{a}^{bulk} (b_{a} (ϕ)) 0 \leq Q (ϕ) - S_{a}^{bulk} (b_{a} (ϕ)) = Q (ϕ) - D_{a}^{comp} (ϕ), \leq ϵ_{R} (a) ∥ ϕ ∥_{H^{2}}^{2} ⟶ 0. (T75)$

The complete $Q (ϕ)$ is independent of the enlarged old endpoint once $a > R$ , by the same zero-extension/core identification. No $L^{2}$ convergence of the projected state $b_{a} (ϕ)$ is asserted. The comparison is a uniform scalar form estimate on the exact actual fibres.

There is also a same-source mixed consumer. For any physical odd collar vector $v$ , let $x_{v}$ represent its bounded functional on the paid positive mesh space. Equation (T68) gives $Q (x_{v}) \leq C a^{- 9/5} ∥ v ∥_{2}^{2}$ . The fully corrected bulk/collar pairing differs from $B_{Q} (ϕ, v)$ by $B_{Q} (x_{ϕ}, v) = B_{Q} (x_{ϕ}, x_{v})$ . Cauchy–Schwarz on this positive restriction, with both actual representatives, pays

$∣ B_{Q} (ϕ - x_{ϕ}, v) - B_{Q} (ϕ, v) ∣ \leq C a^{- 9/10} ϵ_{R} (a) ∥ ϕ ∥_{H^{2}} ∥ v ∥_{2} . (T76)$

This retains the Schur correction and makes no positivity claim for the complete bulk or original old form.

A nonvanishing remaining test survives the growing elimination

The known literal-collar lower bound (C4)–(C5) supplies a fixed compact smooth odd unit test $ϕ_{0}$ supported in the collar for $k = 7$ , with $Q (ϕ_{0}) \geq q_{0} := B_{7} - 9/5 > 0$ in the same physical realization. Choose $R$ enclosing that fixed support. Equation (T75) then gives, for all sufficiently large endpoints,

$S_{a}^{bulk} (b_{a} (ϕ_{0})) \geq q_{0} /2 > 0.$

In particular this actual remaining fibre is nonzero and its energy does not vanish. Conversely, if a compact odd test $ϕ$ with $Q (ϕ) < 0$ existed, its actual remaining fibre would have $S_{a}^{bulk} (b_{a} (ϕ)) \leq Q (ϕ) < 0$ at every sufficiently large endpoint where the paid elimination is defined. This is a conditional statement about a hypothetical test, not a constructed negative test or an RH refutation.

The new comparison (T71)–(T76) shows that the paid mesh becomes weak in the dual energy seen by fixed-support compact tests. Its growing rank and vanishing collar debit do not settle the original compact-core sign: the remaining form converges to the full original $Q$ on every such test. Further enlargement by directions of this same type cannot be declared an exhaustion without a separate approximation or remaining-form estimate.

The true pole-free form uses its own complete compact row, positive mesh restriction, Riesz representatives and remaining short. The absolute weighted polar moment of a theta error has the same budget as (T71), so omitting the actual pole does not change the displayed rates. Its scalar comparison is with its own original complete core energy, not the full $Q$ .

The same comparison covers a growing inner region

The fixed- $R$ limit in (T75) must not be applied to moving supports by silently treating $C_{R}$ as uniform. Its actual support cost can be paid explicitly. For $1 \leq R < a - 1$ and the same compact odd $ϕ$ , weighted two-fold integration by parts gives

$∣ ϕ (\overline{z_{ρ}}) ∣ \leq \frac{C 1 + R e ^{R /2}}{1 + ( ℑ ρ ) ^{2}} ∥ ϕ ∥_{H^{2}} .$

Here multiplication by $e^{w y}$ , $∣ w ∣ \leq 1/2$ , costs $e^{R /2}$ , its first two derivatives have uniformly bounded coefficients, and the $L^{1}$ conversion costs $2 R$ . The finite actual zero mass $S_{2}$ is independent of $R$ . Therefore the squared compact-probe constant in (T71)–(T72) is at most $C (1 + R) e^{R}$ .

The complete physical row has the same support cost. Its prime threshold is $e^{a + R}$ and the weighted sum is $C e^{(a + R) /2}$ . The true polar moment has bound $C 1 + R e^{R /2} ∥ ϕ ∥_{2}$ , and the old polar factor costs $e^{a /2}$ . Gamma and multiplication require no growing support constant. Consequently

$∥ A_{a} ϕ ∥_{H^{1} (- a, a)}^{2} \leq C (1 + R) e^{a + R} ∥ ϕ ∥_{H^{2}}^{2} .$

Apply the actual endpoint-trace/exterior estimate and the whole mesh lower Gram exactly as in (T73)–(T75). With fixed constants independent of $a, R, ϕ$ , the resulting same actual compact debit and remaining form satisfy

$0 \leq Q (ϕ) - S_{a}^{bulk} (b_{a} (ϕ)) = D_{a}^{comp} (ϕ) \leq C (1 + R) e^{R} (a^{- 1/3} e^{- a /4} + \frac{1 + a ^{2}}{M}) ∥ ϕ ∥_{H^{2}}^{2} . (T77)$

In particular choose $R (a) = a /4 - 2 lo g a$ . For every sufficiently large actual endpoint and every smooth compact odd test supported in $(- R (a), R (a))$ , even when the test depends on that endpoint,

$0 \leq Q (ϕ) - S_{a}^{bulk} (b_{a} (ϕ)) \leq C a^{- 4/3} ∥ ϕ ∥_{H^{2}}^{2} .$

Indeed $(1 + R (a)) e^{R (a)} a^{- 1/3} e^{- a /4} \leq C a^{- 4/3}$ ; the $M$ term is smaller by its already paid exponential growth. Thus the remaining fibre keeps the complete original compact energy across an inner region whose radius tends to infinity. The comparison is an absolute $H^{2}$ -controlled error, not a relative energy bound on nearly null tests, a lower bound for the entire bulk, or $L^{2}$ convergence of the projected state. The separately defined pole-free comparison retains the same explicit support cost.

All original coefficients, actual zero real parts, both signs, all heights and multiplicities, the selected global Robin-ratio maximizing integer, elementary correction and strict core are preserved. The original signed Robin estimate and RH remain unproved. Source common-core/norm/full-form transport is still a separate identification. These are paper-level complete-row and remaining-form estimates, without Lean certification, a numerical starting certificate, energy-norm exhaustion, an all-old inverse or a mathematical-priority claim.

A polar-containing partition pays the whole actual tail

The cached primary’s named subsection True-core finite polar-cyclic endpoint criterion supplies the finite Schur implication. Its partition-tail remark supplies a harmonic half-unit floor and a regular-Gamma allowance, not a positive tail floor after all prime-power translations have been included. Reuse that criterion, the complete physical row (T46), its compact-support bound (T77), classical Dirichlet/Neumann Poincare, Plancherel and Jensen. The consumer below is a complete actual partition-tail estimate with an explicit finite-rank cost and compact-probe recovery over the growing physical interval. It is a paper application, not a new general Schur or Poincare theorem or a mathematical-priority claim.

Use the same actual odd physical space, zero extension, common closed-form bridge and full coefficient- $2$ pole as before. Write $Q_{a} (f)$ for the complete Weil form on $(- a, a)$ , and $C_{a} (f)$ for its true polar-free core. In this paragraph $Π_{a}$ is a projection; it is not a quadratic form. Put

$s_{a}^{ph} (y) W_{a} c_{Γ} = sinh (y /2) 1_{(- a, a)} (y), = 2 \leq n < e^{2 a} \sum \frac{Λ ( n )}{n}, = ℜ digamma (1/4) - lo g π = - γ - π /2 - 3 lo g 2 - lo g π .$

Every prime power below the actual overlap threshold appears in $W_{a}$ . Boundary contact at $n = e^{2 a}$ gives a zero-measure overlap, so the strict cutoff is unchanged. The same core relation is $Q_{a} (f) = C_{a} (f) - 2∣ ⟨ s_{a}^{ph}, f ⟩ ∣^{2}$ .

Cell means force an actual logarithmic archimedean floor

Let $P$ be a finite partition of $(- a, a)$ into intervals of length at most $h$ . Take an odd $f$ in the actual form domain with $\int_{I} f = 0$ on every cell. Its zero-extended primitive $F (y) = \int_{- a}^{y} f (x) d x$ is zero at every cell endpoint and outside $(- a, a)$ . Dirichlet Poincare on each cell, followed by Plancherel, gives

$∥ F ∥_{2}^{2} \leq \frac{h ^{2}}{π ^{2}} ∥ f ∥_{2}^{2}, \frac{1}{2 π} \int_{R} \frac{∣ f ( t ) ∣ ^{2}}{t ^{2}} d t = ∥ F ∥_{2}^{2} . (T78)$

The Fourier convention is still $f (t) = \int e^{- i t y} f (y) d y$ . The apparent singularity at zero is controlled by that genuine compactly supported primitive; no inverse-frequency estimate is imposed on arbitrary mean-nonzero vectors.

The actual archimedean multiplier, including its scalar and zero-extension boundary interaction, is

$A_{Γ} (t) = c_{Γ} + 2 \int_{0}^{\infty} κ (u) (1 - cos (t u)) d u, κ (u) = \frac{e ^{- u /2}}{1 - e ^{- 2 u}} .$

Since $1 - e^{- 2 u} \leq 2 u$ , the elementary Laplace integral gives $A_{Γ} (t) \geq c_{Γ} + \frac{1}{2} lo g (1 + 4 t^{2})$ . This bound concerns the complete archimedean form, not just the singular harmonic difference energy with its endpoint potential removed. For $f \neq = 0$ normalize the measure $∣ f (t) ∣^{2} d t / (2 π ∥ f ∥_{2}^{2})$ . Equation (T78) bounds its mean of $(1 + 4 t^{2})^{- 1}$ by $h^{2} / (4 π^{2})$ . Jensen for the convex function $- lo g$ therefore yields

$a_{Γ, a} (f) \geq [c_{Γ} + lo g \frac{2 π}{h}] ∥ f ∥_{2}^{2} . (T79)$

The extended Jensen inequality also applies when the positive logarithmic moment is infinite; the retained form vectors have the required finite moment. The multiplier description is the same physical/closed-form identification used in the preceding estimates. It neither relies on nor supplies the source’s separate Cauchy–Carleman collar correspondence.

Include the true pole and every prime-power orientation

Fix $λ > 0$ . Define

$h_{a} = 2 π exp (c_{Γ} - 2 W_{a} - λ), r_{a} = max {1, ⌈ a / h_{a} ⌉} .$

Partition each half-interval into $r_{a}$ equal cells of length $ℓ_{a} = a / r_{a} \leq h_{a}$ , and reflect the partition. Let $V_{a}$ be the odd piecewise-constant cell space together with the actual vector $s_{a}^{ph}$ . Let $Π_{a}$ be its physical $L^{2}$ orthogonal projection and $Θ_{a} = I - Π_{a}$ . Every vector in its orthogonal complement has zero mean on each cell and is orthogonal to the true pole. Each zero-extension prime translation is a contraction. Consequently the two actual orientations give the lower allowance $- 2 W_{a} ∥ f ∥_{2}^{2}$ , and (T79) gives

$C_{a} (f) = Q_{a} (f) \geq λ ∥ f ∥_{2}^{2} for every f \in Θ_{a} H_{a} \cap D (Q_{a}) . (T80)$

This is positivity of the whole infinite partition tail, without positivity of the old full form. The pole is zero on this tail because it was included in the actual finite space; it has not been estimated away or assigned another coefficient. No prime phase, zero real part or sign hypothesis is used.

The odd cell space has dimension $r_{a}$ , and the nonconstant pole adds one independent vector. Thus

$dim V_{a} = r_{a} + 1 \leq 2 + \frac{a}{2 π} exp (2 W_{a} + λ - c_{Γ}) . (T81)$

For fixed $λ$ , the existing unconditional half-weighted Mangoldt asymptotic $W_{a} \sim 2 e^{a}$ gives $lo g dim V_{a} \sim 4 e^{a}$ . At the actual arithmetic endpoints $a = \frac{1}{2} lo g k$ this is $lo g dim V_{a} \sim 4 k$ . The construction pays a finite but rapidly growing rank. This is an upper-cost statement for this construction, not a necessary cost for other methods or an efficiency claim.

The finite barrier is well defined, but its sign is unpaid

The coarse vectors belong to the actual closed form and operator domains. A finite step function has Fourier decay $O_{a} (∣ t ∣^{- 1})$ , hence finite logarithmic form norm. Its archimedean row is bounded by a finite sum of $1 + ∣ lo g ∣ y - b ∣∣$ near the finitely many cell and support endpoints $b$ ; these singularities are locally square integrable. The truncated smooth pole has the same endpoint behavior. The finite prime-power translation row is $L^{2}$ . Inward dilation followed by smooth odd mollification supplies the common compact-core approximation in logarithmic form norm. Thus $C_{a} V_{a} \subset L^{2} (- a, a)$ in this realization, where $C_{a}$ is the operator of the true core. These are domain checks; constants here may depend on the finite partition and are not uniform rank estimates.

The cached source’s true-core finite Schur lemma now applies directly with $P = Π_{a}$ , $Q = Θ_{a}$ and the paid $λ$ . One sufficient remaining finite certificate is

$Π_{a} C_{a} Π_{a} - 2∣ s_{a}^{ph} ⟩ ⟨ s_{a}^{ph} ∣ - λ^{- 1} (Θ_{a} C_{a} Π_{a})^{*} (Θ_{a} C_{a} Π_{a}) ⪰ 0. (T82)$

All factors belong to the same actual core and the same partition. Its last term is the finite Gram of the complete $L^{2}$ rows $Θ_{a} C_{a} v$ ; there is no omitted intermediate band, prime-power tail, Gamma trace or pole normalization. The source supplies the implication from (T80) and (T82) to full positivity. Equation (T80) does not establish (T82), and no cofinal family of such finite certificates is supplied. A Lanczos space starting only at the pole need not contain $V_{a}$ ; its different orthogonal tail cannot be assigned this $λ$ without an inclusion or an independent bound. The source’s one-link Krylov–Radau criterion is reused only under its own tail hypothesis.

The entire compact row reaches the actual finite quotient

There is a quantitative consumer before any finite matrix sign is known. For $1 \leq R < a - 1$ , take any smooth compact odd $ϕ$ supported in $(- R, R)$ , including endpoint-dependent tests. Reuse the complete row bound (T77), and write $A_{a}$ for the full physical operator. The projection $Θ_{a}$ lands in the cellwise-zero-mean tail. Neumann Poincare for the cell averages of the actual row, followed by (T80), gives

$D_{a}^{part} (ϕ) := 0 \neq = f \in Θ_{a} H_{a} \cap D (Q_{a}) sup \frac{∣ B _{Q_{a}} ( f , ϕ ) ∣ ^{2}}{Q _{a} ( f )} \leq λ^{- 1} ∥ Θ_{a} A_{a} ϕ ∥_{2}^{2} \leq \frac{ℓ _{a}^{2}}{π ^{2} λ} ∥ (A_{a} ϕ)^{'} ∥_{L^{2} (- a, a)}^{2} \leq \frac{C}{λ} (1 + R) e^{a + R} h_{a}^{2} ∥ ϕ ∥_{H^{2}}^{2} .$

The extra pole direction in $V_{a}$ can only decrease the $L^{2}$ projection error from the cell-average space. Its full coupling is retained in $A_{a} ϕ$ ; both prime orientations and all terms with $lo g n \leq a + R$ remain in that row. Primes above this compact-support threshold vanish exactly, as in (T73), whereas (T80) retains all possible tail prime powers.

Apply the already used actual positive-tail Riesz/short identity to the exact fibre $b = Π_{a} ϕ$ , not to an auxiliary fixed matrix. For $S_{a}^{part} (b) = in f_{f \in Θ_{a} H_{a} \cap D (Q_{a})} Q_{a} (b + f)$ ,

$S_{a}^{part} (Π_{a} ϕ) 0 \leq Q_{a} (ϕ) - S_{a}^{part} (Π_{a} ϕ) ∥ ϕ - Π_{a} ϕ ∥_{2} = Q_{a} (ϕ) - D_{a}^{part} (ϕ), \leq \frac{C}{λ} (1 + R) e^{a + R} h_{a}^{2} ∥ ϕ ∥_{H^{2}}^{2}, \leq \frac{h _{a}}{π} ∥ ϕ^{'} ∥_{2} . (T83)$

For fixed $λ$ , the coefficient of $∥ ϕ ∥_{H^{2}}^{2}$ tends to zero even on the growing region $R = a - 2$ . Its logarithm is at most $- 4 W_{a} + 2 a + O (lo g (a + 2))$ , which tends to $- \infty$ by the same unconditional supplier. These are absolute norm-controlled bounds, not a relative energy estimate or a sign certificate for arbitrary endpoint-dependent tests. A strict negative compact test, if one existed, would remain negative on its exact finite fibre. A zero-energy test need not retain zero energy. The finite problem is a short, rather than just the positive-looking compression with the coupling discarded.

The explicit mesh cost and full-row comparison supply the missing all-prime partition-tail interface to the source criterion. They do not supply its finite matrix sign or the source’s full collar transport. All original selected global Robin-ratio maximizer, signed target, coefficients, actual zero real parts, both signs, heights, multiplicities, elementary correction and strict core remain unchanged. RH and the original signed Robin estimate remain unproved. No Lean certification, numerical certificate, whole-old inverse or mathematical-priority claim is made.

Pay the whole coarse-to-tail row, including the almost aligned pole

The complete positive tail in (T80) does not bound the finite coupling Gram in (T82). The compact-probe row estimate (T83) also does not apply directly to every coarse vector: a step function’s row has logarithmic endpoint singularities and need not be $H^{1}$ . The following pays the whole polar-containing coarse space in its actual $L^{2}$ norm. Reuse the same physical realization, multiplier, zero extension, domains and exact positive-tail short. Classical Fourier series Parseval, continuous Plancherel, translation moments and the source’s finite Schur implication are inputs. The consumer is a width-uniform actual Gamma coupling, followed by the complete arithmetic finite-short comparison; neither a general Fourier alias theorem nor a new Schur criterion is claimed.

Let $P_{a}^{0}$ denote the odd cell-average projection used in (T80), with actual equal cell width $ℓ = ℓ_{a} \leq h_{a} < 1$ . Let $P_{ℓ}$ denote cell averaging on the whole lattice of intervals $[j ℓ, (j + 1) ℓ)$ , $j \in Z$ . The physical endpoints $\pm a$ are lattice endpoints. Continue to use $Π_{a}$ for the projection onto $V_{a}$ , and $Θ_{a} = I - Π_{a}$ . Write $Γ$ for the whole-line archimedean operator with multiplier $A_{Γ} (t)$ from (T79); the physical row is its restriction after zero extension. Its scalar and the entire regular Gamma part are retained.

The large scalar Gamma row cancels in the cell tail

Split the actual kernel as

$κ (u) = \frac{e ^{- u /2}}{2 u} + b (u), b (u) \geq 0, \int_{0}^{\infty} b (u) d u < \infty.$

The nonnegativity uses $1 - e^{- 2 u} \leq 2 u$ . At zero $b (u) = 1/2 + O (u)$ , and at infinity it decays exponentially. Thus

$A_{Γ} (t) = c_{Γ} + L (t) + B (t), L (t) = \frac{1}{2} lo g (1 + 4 t^{2}), ∣ B (t) ∣ \leq 4∥ b ∥_{1} .$

For a finitely supported cell-constant function $p (y) = ℓ^{- 1/2} \sum_{j} d_{j} 1_{[j ℓ, (j + 1) ℓ)} (y)$ , put $D (θ) = \sum_{j} d_{j} e^{- ij θ}$ and

$c_{m} (θ) = \int_{0}^{1} e^{- i (θ + 2 πm) v} d v, - π \leq θ \leq π .$

Its exact Fourier values at the common aliases are $p ((θ + 2 πm) / ℓ) = ℓ D (θ) c_{m} (θ)$ . Fourier series Parseval of $e^{- i θ v}$ on $(0, 1)$ gives $\sum_{m} ∣ c_{m} (θ) ∣^{2} = 1$ . Cell averaging is the orthogonal projection onto the line with components $c_{m} (θ)$ in each such Fourier fibre. Hence the squared row norm of $(I - P_{ℓ}) L (D) p$ is the integral of $∣ D (θ) ∣^{2}$ times the variance of the real numbers $L_{m} = L ((θ + 2 πm) / ℓ)$ under the weights $∣ c_{m} ∣^{2}$ , with the common measure $d θ / (2 π)$ .

For $m \neq = 0$ and $0 < ∣ θ ∣ \leq π$ ,

$∣ c_{m} (θ) ∣^{2} \leq \frac{θ ^{2}}{π ^{2} m ^{2}}, 0 \leq L_{m} - L_{0} \leq C + lo g (1 + ∣ m ∣) + lo g^{+} (1/∣ θ ∣),$

uniformly for $0 < ℓ \leq 1$ . Indeed $π ∣ m ∣ \leq ∣ θ + 2 πm ∣ \leq 3 π ∣ m ∣$ and $L_{m} - L_{0} = \frac{1}{2} lo g [(ℓ^{2} + 4 (θ + 2 πm)^{2}) / (ℓ^{2} + 4 θ^{2})]$ . Bounding the variance by the second moment centered at $L_{0}$ gives a uniform constant: the sum of $lo g^{2} (1 + ∣ m ∣) / m^{2}$ is finite and $θ^{2} [lo g^{+} (1/∣ θ ∣)]^{2}$ is bounded. At $θ = 0$ all noncentral weights vanish, so the same statement holds. The term $lo g (1/ ℓ)$ cancels in these differences; a full row bound before this cancellation would lose uniformity.

The scalar $c_{Γ}$ has no cell-tail component, and the bounded multiplier $B$ supplies only a fixed allowance. Therefore a fixed finite $G_{0}$ , independent of $a$ , $ℓ$ , the cell count and the coefficients, satisfies

$∥ (I - P_{ℓ}) Γ p ∥_{L^{2} (R)} \leq G_{0} ∥ p ∥_{2}, ∥ Θ_{a} (Γ p) ∣_{(- a, a)} ∥_{2} \leq G_{0} ∥ p ∥_{2} . (T84)$

The second estimate follows because the restriction of $P_{ℓ} Γ p$ is an odd physical cell-constant vector, killed by $Θ_{a}$ . It includes the physical endpoint jumps rather than discarding them. The finite step functions are in the operator domain by the same logarithmic Fourier decay used in (T82).

Orthogonalize the actual pole before estimating it

The actual pole is almost cell-constant as $ℓ$ decreases. Bounds on arbitrary coefficients in the raw basis $(cell constants, s_{a}^{ph})$ would not pay that conditioning. Instead use its exact orthogonal remainder

$w_{a} = s_{a}^{ph} - P_{a}^{0} s_{a}^{ph} .$

It is nonzero, is odd, has zero integral on each cell, and lies in $V_{a}$ . Each $u \in V_{a}$ has the unique orthogonal splitting $u = p + z w_{a}$ with $p = P_{a}^{0} u$ , and $∥ u ∥_{2}^{2} = ∥ p ∥_{2}^{2} + ∣ z ∣^{2} ∥ w_{a} ∥_{2}^{2}$ .

On every cell the derivative of $s (y) = sinh (y /2)$ is $s^{'} (y) = \frac{1}{2} cosh (y /2) > 0$ , with ratio of maximum to minimum at most $e^{ℓ /2}$ . The variance identity $\int_{I} ∣ s - s_{I} ∣^{2} = (2 ℓ)^{- 1} \iint_{I^{2}} ∣ s (y) - s (x) ∣^{2} d x d y$ therefore gives fixed constants $c, C > 0$ such that

$c ℓ^{2} ∥ s^{'} ∥_{L^{2} (- a, a)}^{2} \leq ∥ w_{a} ∥_{2}^{2} \leq C ℓ^{2} ∥ s^{'} ∥_{L^{2} (- a, a)}^{2} .$

Let $J_{b}$ be every jump of the zero-extended $w_{a}$ , including those at $\pm a$ . Interior jumps are differences of neighboring cell averages of $s$ . The endpoint deviations have the same cellwise derivative allowance. Summing these estimates gives

$ℓ b \sum ∣ J_{b} ∣^{2} + ℓ^{2} ∥ (w_{a})_{pw}^{'} ∥_{2}^{2} \leq C ∥ w_{a} ∥_{2}^{2} .$

For $0 < t < ℓ$ , a path of length $t$ meets at most one lattice jump. The integrated derivative part has squared norm at most $t^{2} ∥ (w_{a})_{pw}^{'} ∥^{2}$ , and the integrated jump part at most $t \sum_{b} ∣ J_{b} ∣^{2}$ . Larger shifts use the usual $4∥ w_{a} ∥^{2}$ bound. Together with the genuine primitive estimate (T78), this supplies

$∥ τ_{t} w_{a} - w_{a} ∥_{2}^{2} \frac{1}{2 π} \int \frac{∣ w _{a} ( t ) ∣ ^{2}}{( ℓ t ) ^{2}} d t \frac{1}{2 π} \int ∣ ℓ t ∣^{1/2} ∣ w_{a} (t) ∣^{2} d t \leq C min (t / ℓ, 1) ∥ w_{a} ∥_{2}^{2}, t > 0, \leq π^{- 2} ∥ w_{a} ∥_{2}^{2}, \leq C ∥ w_{a} ∥_{2}^{2} . (T85)$

For the last line use the classical translation-moment identity: integrating $∥ τ_{t} w - w ∥^{2}$ against $t^{- 3/2} d t$ is a fixed positive multiple of its Fourier $∣ t ∣^{1/2}$ moment. The short-shift bound makes that integral $O (ℓ^{- 1/2} ∥ w ∥^{2})$ . No smoothness across a jump, endpoint-free approximation or constant uniform in an unnormalized polar coefficient is assumed.

The pole remainder also has a width-uniform Gamma row

For $0 < ℓ \leq 1$ and $η \neq = 0$ ,

$\frac{1}{2} lo g (ℓ^{2} + 4 η^{2})^{2} \leq C [1_{{∣ η ∣ \leq 1}} ∣ η ∣^{- 2} + 1_{{∣ η ∣ > 1}} (1 + ∣ η ∣^{1/2})] .$

Apply this with $η = ℓ t$ , and use (T85), the multiplier split and Plancherel. It gives $∥ [Γ - lo g (1/ ℓ) I] w_{a} ∥_{2} \leq G_{1} ∥ w_{a} ∥_{2}$ on the whole line, for a fixed $G_{1}$ . Since $Θ_{a} w_{a} = 0$ , restriction and projection remove the subtracted scalar exactly. Combining this with (T84) on the orthogonal splitting of $u$ gives a fixed $G_{*}$ such that

$∥ Θ_{a} (Γ u) ∣_{(- a, a)} ∥_{2} \leq G_{*} ∥ u ∥_{2} for every u \in V_{a} . (T86)$

All constants are independent of the endpoint, the width, the finite rank and the choice of $λ$ in (T80). This is a uniform row estimate, not positivity of the Gamma-minus-prime core. The scalar cancellation has been paid separately for both actual orthogonal components, including the near-alignment of the original pole and cell constants.

The complete finite coupling is now paid in $L^{2}$

The actual core row retains every $n < e^{2 a}$ and both zero-extension prime translations. Reuse their full norm allowance $2 W_{a}$ . The rank-one pole has zero tail component because $Θ_{a} s_{a}^{ph} = 0$ . Thus the same coupling for the true core and full form satisfies

$∥ K_{a} u ∥_{2} \leq (G_{*} + 2 W_{a}) ∥ u ∥_{2}, K_{a} = Θ_{a} C_{a} Π_{a}, K_{a}^{*} K_{a} ⪯ (G_{*} + 2 W_{a})^{2} Π_{a} . (T87)$

This bound concerns one complete row with its common physical projection. It uses a conservative triangle allowance for the prime terms; it neither independently adjusts their phases nor claims a cancellation between them. It loses their possible favorable common-source interference, but omits none of them. The true coefficient- $2$ pole remains in the finite diagonal below.

Let $H_{a}$ be the positive operator associated with the closed form restriction to the actual tail $Θ_{a} H_{a}$ , using (T80), and let $D_{a} = Π_{a} C_{a} Π_{a} - 2∣ s_{a}^{ph} ⟩ ⟨ s_{a}^{ph} ∣$ be the full finite compression. Reuse the exact positive-tail Schur identity, with $S_{a}^{part}$ the actual finite short from (T83). Equations (T80) and (T87) supply the full finite-operator comparison

$S_{a}^{part} 0 ⪯ D_{a} - S_{a}^{part} = D_{a} - K_{a}^{*} H_{a}^{- 1} K_{a}, ⪯ δ_{a} Π_{a}, δ_{a} = \frac{( G _{*} + 2 W _{a} ) ^{2}}{λ} . (T88)$

The inverse is the positive operator of this tail restriction, not an inverse of an unproved positive full old operator. The Gram is the one defined by the complete $L^{2}$ rows, as in (T82). Unlike the compact-only $H^{2}$ allowance (T83), (T88) covers every vector in the actual finite space with its $L^{2}$ norm, including arbitrary coefficients and the normalized orthogonal polar remainder. No finite diagonal or short sign has been assumed.

Moving the tail floor pays a smaller error at an explicit rank cost

The width-uniform proof allows $λ$ to grow with the endpoint. Take

$λ_{a} = (1 + a) (1 + W_{a})^{2}, h_{a} = 2 π e^{c_{Γ} - 2 W_{a} - λ_{a}}$

and use the same reflected partition and actual pole. For a fixed $C_{*}$ independent of all parameters,

$0 ⪯ D_{a} - S_{a}^{part} ⪯ \frac{C _{*}}{1 + a} Π_{a}, lo g dim V_{a} \sim 4 a e^{2 a} . (T89)$

The rank statement follows from (T81) and the same unconditional $W_{a} \sim 2 e^{a}$ . At $a = \frac{1}{2} lo g k$ its logarithm is asymptotic to $2 k lo g k$ . This is a larger construction than the fixed- $λ$ partition with log rank $\sim 4 k$ ; the increased resolution pays the uniformly smaller whole-coarse coupling debit. It is not an efficient finite certificate, a necessary lower cost for RH, or a sign bound. A large rank by itself proves no exhaustion.

There is a direct consumer for the remaining sign obligation. Suppose on an unbounded cofinal family of these actual endpoints one could establish

$D_{a} ⪰ - ϵ_{a} Π_{a}, ϵ_{a} \geq 0, ϵ_{a} ⟶ 0. (T90)$

Then (T88)–(T89) give $S_{a}^{part} ⪰ - (ϵ_{a} + C_{*} / (1 + a)) Π_{a}$ . Completing the same positive tail square implies $Q_{a} (f) \geq - (ϵ_{a} + C_{*} / (1 + a)) ∥ f ∥_{2}^{2}$ for every admitted full vector at those endpoints, since $∥ Π_{a} f ∥ \leq ∥ f ∥$ . Every fixed compact odd test is eventually admitted, with its full energy unchanged by zero extension. Letting the endpoint tend to infinity would therefore give its nonnegative complete Weil energy. The source’s restricted odd criterion and exact support closure are separate existing suppliers for the final RH implication under the stated realization bridge.

Thus an asymptotically nonnegative full finite compression, allowing small negative finite eigenvalues, is a sufficient input once this complete vanishing coupling debit is paid. It is not necessary to certify a fixed positive gap or exact nonnegative finite barrier at every finite endpoint. No bound (T90) has been established here. The unsigned coupling estimate does not imply it, and a failed conservative finite barrier would not refute the complete form.

The actual finite compression must still retain its complete prime translations, Gamma potential and coefficient- $2$ pole on the common coarse space. Its cofinal negative-eigenvalue control, the source’s full collar correspondence, the original selected-maximizer signed Robin estimate and RH remain unproved. All original coefficients, actual zero real parts, both signs, all heights and multiplicities, elementary correction and strict core are preserved. These are conditional paper-level actual-row and finite-short estimates, without Lean, numerical certification, a full old inverse or a mathematical-priority claim.

An offcritical zero forces exponential descent of the actual compression

Reuse the compact odd zero-isolation argument in the primary’s proof of its restricted Weil criterion, together with the same complete zero-side identity and physical realization used above. The extra estimate here concerns the physical $L^{2}$ normalization of those witnesses and their transfer to the actual finite compression (T88)–(T89). It keeps every actual zero, both ordinate signs and all multiplicities. It does not establish a lower bound for that compression or adopt the source’s full RH argument.

Normalize the isolating bump with a contracting convolution norm

Suppose one actual centered zero is $z_{0} = ρ_{0} - 1/2 = δ + iγ$ , $δ > 0$ . There are no real nontrivial zeta zeros, so $γ \neq = 0$ . Choose a real even nonnegative $η \in C_{c}^{\infty} (- r_{0}, r_{0})$ , $\int η = 1$ , whose bilateral Laplace transform $E (z) = \int η (x) e^{z x} d x$ has $d = ∣ E (z_{0}) ∣ > 0$ . A sufficiently narrow unit bump supplies this nonvanishing.

For fixed sufficiently large $b$ put

$η_{b} (x) R_{b} = \frac{1}{2} [η (x - b) + η (x + b)], = ∣ E (z_{0}) cosh (b z_{0}) ∣, ψ_{b} = η_{b} / R_{b}, r = b + r_{0} .$

Since $∣ cosh (b z_{0}) ∣^{2} = sinh^{2} (b δ) + cos^{2} (bγ)$ , one has

$∣ Ψ_{b} (z_{0}) ∣ = 1, q_{b} := ∥ ψ_{b} ∥_{1} = R_{b}^{- 1} < 1, κ_{b} := \frac{2 lo g R _{b}}{r} ⟶ 2 δ (b \to \infty) . (T91)$

The limit uses $lo g R_{b} = δ b + lo g (d /2) + o (1)$ ; it requires no choice of an ordinate phase or a largest zero real part.

Write $Ψ_{b}$ for the bilateral Laplace transform of $ψ_{b}$ . For this one fixed bump, reuse the source’s fixed height cut, even zero-killing polynomial $P$ , and real even polynomials $U_{M} (z) = u_{M} + v_{M} z^{2}$ with uniformly bounded coefficients. Its real odd compact tests have transforms

$F_{M} (z) = z P (z) U_{M} (z) Ψ_{b} (z)^{M}, F_{M} (z_{0}) = 1.$

The coefficient bound follows from the fixed invertible real-linear map $(u, v) \mapsto u + v z_{0}^{2}$ , since $ℑ (z_{0}^{2}) = 2 δ γ \neq = 0$ . The existing isolation step includes all other zeros: the polynomial removes the finite head, and a fixed rapid-decay majorant times $q_{tail}^{2 (M - M_{0})}$ , $q_{tail} < 1$ , pays the whole infinite-height remainder. Thus $Q (f_{M}) \leq - 2 m_{ρ_{0}}$ for sufficiently large $M$ .

Put the fixed-order differential operator corresponding to $z P U_{M}$ on just one convolution factor. Its $L^{2}$ norm has an $M$ -independent bound, because the bump and polynomial are fixed and the two coefficients of $U_{M}$ are bounded. Young’s inequality then gives fixed $C_{b} < \infty$ with

$supp f_{M} \subset [- M r, M r], ∥ f_{M} ∥_{2} \leq C_{b} q_{b}^{M - 1}, Q (f_{M}) \leq - 2 m_{ρ_{0}} < 0. (T92)$

No estimate for a growing derivative order is used. Constants and the starting index may depend on the chosen zero, bump and fixed height cut; uniformity over zero heights is not asserted. The same test has a fixed negative zero response while its physical norm decreases geometrically.

Transfer the witness to the same full finite matrix

Let $m_{a}^{odd}$ denote the infimum of $Q_{a} (f) /∥ f ∥_{2}^{2}$ over nonzero admitted real odd tests; no existence of an extremizing vector is needed. Fix $0 < κ < 2 δ$ and choose $b$ with $κ_{b} > κ$ . At an endpoint $a$ , take $M = ⌊ a / r ⌋ - 1$ , so that the corresponding test is strictly inside the admitted interval. Equation (T92) gives its physical Rayleigh quotient at most $- c_{κ} e^{κa}$ for every sufficiently large endpoint.

Let $u = Π_{a} f_{M}$ . It is nonzero, because a vector wholly in the actual positive tail cannot have negative energy. Exact tail completion gives $S_{a}^{part} (u) \leq Q (f_{M})$ , and $∥ u ∥_{2} \leq ∥ f_{M} ∥_{2}$ . Dividing the negative energy by the smaller norm strengthens the inequality. The whole-finite comparison (T89), with its original pole and common metric, therefore yields

$λ_{m i n} (D_{a}) \leq - c_{κ} e^{κa} + \frac{C _{*}}{1 + a} \leq - \frac{1}{2} c_{κ} e^{κa} for all sufficiently large a . (T93)$

The full infinite-height contribution was paid before this projection. No shifted metric, replacement zero spectrum or whole-old inverse appears in this transfer.

Consequently an actual lower allowance on an unbounded cofinal sequence,

$D_{a_{j}} ⪰ - M_{j} Π_{a_{j}}, M_{j} \geq 0, \frac{lo g ( 1 + M _{j} )}{a _{j}} ⟶ 0, (T94)$

would exclude every offcritical zero. This sufficient input allows growing negative finite eigenvalues; it need not tend to zero as in (T90). More generally a logarithmic allowance rate at most $ν$ would exclude zeros with $ℜ ρ > 1/2 + ν /2$ , and reflection gives the corresponding left bound. With the arithmetic clock $k = e^{2 a}$ , an offcritical displacement $δ$ forces descent at least $k^{η}$ for every $0 < η < δ$ ; (T94) asks for a $k^{o (1)}$ negative allowance. Selecting a cofinal FIB schedule supplies none of this actual joint estimate.

The available conservative allowance still has rate one

The existing whole Gamma floor, complete prime norm bound and true pole give

$∥ s_{a}^{ph} ∥_{2}^{2} D_{a} = sinh a - a, ⪰ [c_{Γ} - 2 W_{a} - 2 (sinh a - a)] Π_{a} .$

Using the already-retained unconditional $W_{a} \sim 2 e^{a}$ , the corresponding nonnegative lower allowance satisfies

$M_{a}^{raw} := 2 W_{a} + 2 (sinh a - a) + ∣ c_{Γ} ∣ \sim 5 e^{a}, \frac{lo g ( 1 + M _{a}^{raw} )}{a} ⟶ 1. (T95)$

It cannot contradict the forced rate $κ < 2 δ < 1$ . The vanishing coupling debit pays the finite-short transfer but does not change this finite diagonal allowance. A joint arithmetic estimate reaching (T94), the source’s independent full-form correspondence, and the original selected-maximizer signed Robin budget remain unproved. The witness rate and actual-matrix transfer are conditional paper deductions, without mathematical-priority or Lean-certification claims; the sufficient growth interface is not an achieved lower estimate.

v2: the claimed induction step

The following induction and fold formulas are those of arXiv:2609.20367v2, revised 20 September 2026, 69 pages. At the arithmetic endpoint $a_{k} = (lo g k) /2$ , the source retains the pole term in $A_{k} = C_{k} - 2∣ s_{k} ⟩ ⟨ s_{k} ∣$ . Lemma 6.27, under the old endpoint’s positivity and inverse hypotheses, forms the actual harmonic extension

$H_{k} f = (f - A _{k}^{- 1} B _{A, k} f), B_{A, k} = B_{k} - 2 s_{k} t_{k}^{*}, T_{k} = H_{k}^{*} A_{k + 1} H_{k} .$

Theorem 8.5, printed pp.58–59, claims for every integer $k \geq 7$ , assuming $A_{k, -} > 0$ , a lower bound on this same extension by a strictly positive ground-coordinate coefficient plus a nonnegative transverse remainder, strictly positive when the transverse component is nonzero. This is the claimed estimate yielding $T_{k} > 0$ . The old-block positivity is an induction hypothesis, not an assumption of all-scale positivity. Theorem 1.2’s restricted odd Weil criterion and the endpoint-to-all-support closure are separate consumers of the induction.

The load-bearing comparisons to inspect before any reuse are:

Theorem 6.53 and Theorem 6.55: simultaneous ground/transverse budgets on the same common complement and the required transverse reserve.
Lemma 8.1: placement of the full actual Schur response, with its positive saving and negative reference charge attached to the same contribution.
Lemma 8.2: the homogeneous mixed and aligned-forcing estimate for all complex coefficients, not only a normalized scalar case.
Lemma 8.4: the common-cut, transported form and endpoint-fold identities for the same harmonic vector, including the pole term, actual forcing and physical-pivot positivity.
Certificate 6.62 and Certificate 8.3: the base at $Y = 7$ and the all- $k$ ground margin, whose stated finite interval verification is paired with an analytic tail.

These are propositions that the source claims to prove. They are not merely extra conjectural hypotheses declared by its author; they also have not become independently verified project premises by being listed here. Checking scalar certificates alone would leave the actual-operator and function-space correspondence obligations untouched.

v2: debit multiplicity and physical-pivot inputs

The displayed local formulas require a specific reconciliation before they can supply an all-scale estimate. In Lemma 8.4, equations (207)–(212) give two endpoint arms with the same positive target block $P$ and coupling $b$ . Completing both squares deducts

$2 D = 2∥ P^{- 1/2} b x ∥^{2} .$

Equation (212) explicitly calls $D = ∣ h_{0} ∣^{2} / Π^{phys}$ the one-arm debit. In contrast, (205) and MASTER-P3b use the once-deducted remainder $Q - D = E (1 - u)$ . At these displayed coefficients, that remainder alone does not bound the two-arm remainder. A reuse must identify the additional payment, or an explicit normalization or allocation that reconciles the same $Q$ , $D$ and direct target diagonal. An overall change of units must transport all three together.

A scalar check isolates this issue without claiming to realize an arithmetic endpoint. In the one-defect model of Lemma 6.28, take both pieces to have measure one, $a_{-} = a_{+} = 1$ , $β = 1/4$ , $x_{-} = 1/3$ , $x_{+} = 1$ . Then the left pivot is $3/4 > 0$ , $γ^{c} = 1/3$ , $g_{-} = 0$ , $h = Π^{phys} = 2/3 > 0$ , and

$Q = D = \frac{2}{3}, Q - D = 0, Q - 2 D = - \frac{2}{3} .$

Equivalently, the two displayed positive arm blocks $P = 2/3$ and couplings $b = 2/3$ , minimized at $t_{+} = t_{-} = - 1$ , give $- 2/3$ after subtracting the separately retained target diagonal. These are exact rational values. The example shows that the once-deducted estimate is insufficient for those local displayed formulas; it is not a counterexample to the actual Weil operator, nor does it show that all the other reserves in the complete argument fail.

The sign needed for the physical pivot is a separate input. With the notation of (124)–(135), positive $∣ R ∣, J$ and $1 - β I > 0$ give

$Π^{phys} = \frac{∣ R ∣}{J} \frac{1 - β ( I + J )}{1 - β I} .$

Thus the old left-block condition alone does not supply the numerator’s positivity. The algebra in Lemma 6.28 is valid under its stated positive-pivot condition; the relation $g = A^{pre} x$ transports the forcing but does not establish that sign. The actual compression/shorting map must identify this pivot as one whose positivity follows from the permitted induction inputs. The review has not completed that identification. This is an unclosed proof input, not a claim that an old-block induction hypothesis is inherently circular.

v2: a same-vector payment interface

The original square completion (209)–(212) retains two nonnegative arm squares. Write their sum as

$S_{k} [F] = \pm \sum ∥ P_{k}^{1/2} t_{\pm} + P_{k}^{- 1/2} b_{k} x_{k} ∥^{2} .$

At the displayed coefficients, its exact accounting is

$Φ_{k} = B_{k} ∣ γ_{k} (f) ∣^{2} + S_{k} [F] + Q_{k} - 2 D_{k} .$

Lemma 6.61 states the one-debit identity on the identified normalized vector $F_{k}^{#} = H_{k} f_{k}^{♮}$ . Lemma 8.4 asserts the same identity on its general harmonic vector. Write $C_{k} [F] = Q_{k} [F] - D_{k} [F]$ . The displayed fold remainder is $S_{k} [F] + C_{k} [F] - D_{k} [F]$ . Granting the source identification on the same $F$ , it writes $C_{k} = E_{k} (1 - u_{k})$ when $E_{k} > 0$ . The interface below uses $Q_{k} - D_{k}$ directly, so it remains meaningful at $E_{k} = 0$ without dividing by $E_{k}$ . This is a conditional accounting of the displayed form, not a verified transport identity for the actual Weil operator.

There is further coefficient slack in the source’s parent-loss estimate. With $w_{k} = lo g (1 + 1/ k)$ , put

$L_{k} = \frac{439}{250} m \sum χ_{m} + \frac{5}{2} W_{k}, δ_{k} = - lo g (k w_{k}) + \frac{439}{250} (\frac{\sum _{m} V _{k, m}}{lo g 2} - m \sum χ_{m}) .$

The coefficients in (203), (204) and MASTER-P3c give

$B_{k} - L_{k} = g_{k}^{M +} + δ_{k} .$

The source bounds $0 < k w_{k} < 1$ and $χ_{m} \leq V_{k, m} / lo g 2$ make $δ_{k}$ nonnegative. Nonnegative unused terms also arise from the two Young inequalities in Lemma 8.2 and from any excess of the exact inherited pivot over its certified floor. None of these sign statements supplies a comparison with the remaining $D_{k}$ .

Throughout the proposed interface, $k$ is an integer at least seven, $A_{k, -} ≻ 0$ , $f \in K_{k}$ , and $F = H_{k} f$ is the exact full-operator harmonic extension with the pole retained. Retain the order in Lemma 8.4: complete full-form transport, fixed-target gauge, simultaneous common-complement source short, then root-adapted endpoint fold. The one-defect and forcing identifications of Lemma 6.28, its positive multiplication coefficients, $1 - β_{k} I_{k} > 0$ , $J_{k} > 0$ , $Π_{k}^{phys} > 0$ , and the arm block $P_{k} ≻ 0$ must all hold in those coordinates. These correspondence and positive-pivot inputs remain unverified here; none is inferred from positivity of the future block.

A proposed unused remainder $U_{k}$ is admissible only after establishing the common lower-form estimate

$⟨ F, A_{k + 1, -} F ⟩ \geq (g_{k}^{M +} + δ_{k}) ∣ γ_{k} (f) ∣^{2} + R_{k}^{⊥} [f] + S_{k} [F] + U_{k} [F] + Q_{k} [F] - 2 D_{k} [F] .$

This lower bound is itself an unverified actual-family obligation. Conditional on it, a sufficient payment preserving the advertised ground coefficient and transverse reserve is

$D_{k} [F] \leq C_{k} [F] + δ_{k} ∣ γ_{k} (f) ∣^{2} + S_{k} [F] + U_{k} [F] for every f \in K_{k} .$

Here $U_{k}$ may contain only explicitly identified, proved nonnegative Young-residual or pivot-excess terms retained in that lower bound. Each must be disjoint from the expenditures already made in MASTER-P2 and the $3/5 + 2/5$ parent allocation, and from $S_{k}$ , the coefficient slack $δ_{k} ∣ γ_{k} (f) ∣^{2}$ , and the transverse reserve $R_{k}^{⊥}$ supplied by Theorems 6.53 and 6.55. It is not defined as the unknown difference. Taking $U_{k} = 0$ introduces no additional reserve; a nonzero choice needs the displayed lower-form proof. This interface is sufficient for the stated allocation scheme; positivity could also follow from a different allocation that spends part of the claimed ground margin. It is not asserted to be necessary for RH or for operator positivity.

The source reserves have distinct existing uses:

Source input	Existing allocation	Additional obligation before using it for $D_{k}$
Lemma 8.1, MASTER-P2	Comparator saving and its negative reference charge are one inherited-response contribution.	Retain both terms; the saving is not an independent positive summand.
Lemma 8.2	The certified inherited surplus is split $3/5 + 2/5$ to pay mixed and aligned forcing.	Identify the unused remainder and compare it with $D_{k}$ on the actual harmonic response.
Theorems 6.53 and 6.55	The transverse budget supplies the reserved transverse term.	Prove any proposed reallocation while preserving the claimed transverse conclusion.
Lemma 8.4, (213)	One target diagonal splits into ground and transverse parts.	Keep the split and both expenditures in the same coordinates.

The source does not identify an extra factor-of-two normalization in (209)–(213): the symmetric coupling is explicitly $2 b_{k}$ , and the diagonal is retained once. No same-vector payment satisfying the interface above has been verified here. Harmonic restrictions could make the arm squares or parent remainders large enough; their quantitative consequence remains unverified. This narrows the reuse obligation without producing an actual-arithmetic counterexample or deciding the source’s claimed RH conclusion. The coefficient calculation and conditional scalar accounting do not constitute new mathematical estimates.

The v2 supplementary archive, file The_Three_Gates_Supplementary_V2.zip, contains a Gate-II audit of full-comb scalarization. That audit explicitly limits its verdict to that particular risk and says it does not independently reprove every theorem. Its reported PASS therefore does not verify this second-arm payment. The archive describes its master/ programs as finite scalar sweeps and its Y7_end_to_end/ calculation as a conservative base-endpoint replication; those stated scopes do not supply the missing all-step, same-vector lower bound. These computations have not been rerun here.

Full-form transport benchmarks and the v2 correspondence input

Proposition 5.3 must be read as an obligation about the whole form, including its diagonal. For a change of variables $x = ϕ (u)$ with $J = ϕ^{'} > 0$ and $g (u) = J (u) f (ϕ (u))$ , substitution in the singular difference expression produces

$K (ϕ (u), ϕ (v)) J (v) g (u) - J (u) g (v)^{2} .$

Using only the transported off-diagonal kernel $J (u) J (v) K (ϕ (u), ϕ (v))$ in an ordinary $∣ g (u) - g (v) ∣^{2}$ form leaves a multiplication term to account for, together with the transported endpoint potential. The exponential kernel identities in Lemma 5.1 do not by themselves perform this diagonal comparison. Before applying the later one-cell lower form, its full transported potential must be identified or bounded in the same coordinates. The bounded review has not completed the identification of the source’s $c_{A}$ with all the later collar and fixed-target forms. No failure of the entire RH claim follows merely from this outstanding correspondence.

The source formulas give a specific normalization benchmark for this interface. Denote the archimedean form in (61), with the pole and prime terms excluded, by $a_{arch, a}$ . For $a > 0$ , use the affine unitary

$(T f) (t) = 2 f (2 t - 1), 0 < t < 1.$

On the smooth compactly supported core, substitution in (12)–(15), (50) and (62) gives

$a_{arch, a} [T^{- 1} g] = b [g] + (c_{0} (a) - lo g 2) ∥ g ∥_{2}^{2} - ⟨ g, K_{γ, 2 a}^{(0, 1)} g ⟩, c_{0} (a) - lo g 2 = - lo g (4 πa) - γ,$

where $K_{γ, 2 a}^{(0, 1)}$ has kernel $2 a ρ (2 a ∣ t - s ∣)$ . The singular difference energy retains its coefficient, while

$V (2 t - 1) = - lo g 2 - \frac{1}{2} lo g (t (1 - t)) .$

The original odd domain becomes $g (1 - t) = - g (t)$ . Extension of this core calculation requires the actual transported closed-form domain and common form-core contract. This affine map is not identified with the exponential $U_{A}$ or the paper’s complete collar gauge. The benchmark states which scalar and regular-kernel terms occur before that identification; it neither supplies a new lower bound nor contradicts Proposition 5.3. Reusing the later $b$ estimates still requires the complete realization and allocation of these terms in the same direct/source forms. The displayed calculation is paper-level source bookkeeping, without a Lean validation of the integral or domain transport.

There is also a benchmark in the actual parent-cell coordinate of §6.9. Transport the same archimedean formula by $R_{a}$ from (8), before odd restriction. Its physical gamma kernel is $ρ (∣ y - z ∣)$ , and its multiplication coefficient is

$c_{0} (a) + V (y / a) = - lo g (2 π) - γ - \frac{1}{2} lo g (a^{2} - y^{2}) .$

Write $h_{J}$ for the singular quarter difference form on an interval $J$ . Let $I = (ℓ, r) \subset (- a, a)$ , $w = r - ℓ$ , $f \in C_{c}^{\infty} (I)$ , and let $E_{I} f$ be its zero extension. The exterior strips give

$h_{(- a, a)} [E_{I} f] - h_{I} [f] = \frac{1}{2} \int_{I} lo g \frac{a ^{2} - y ^{2}}{( y - ℓ ) ( r - y )} ∣ f (y) ∣^{2} d y .$

The physical endpoint potential is $- \frac{1}{2} lo g (a^{2} - y^{2})$ ; it cancels the numerator in this strip contribution. With $g (u) = w f (ℓ + w u)$ , the resulting compression is

$a_{arch, a}^{ph} [E_{I} f] = b [g] - [lo g (2 π w) + γ] ∥ g ∥_{2}^{2} - ⟨ g, K_{γ, w}^{(0, 1)} g ⟩, K_{γ, w}^{(0, 1)} (u, v) = wρ (w ∣ u - v ∣) .$

For a Mellin parent $C_{m} = [m, m + 1)$ within $[1, e^{2 a}]$ , use $t = e^{y + a}$ . Its physical interval is $I_{m} = (lo g m - a, lo g (m + 1) - a)$ , with $w = w_{m}$ . The coordinate $u = lo g (t / m) / w_{m}$ is the one used in (103)–(104). This calculation is a compression, whereas Theorem 6.53 takes an infimum over a free common complement. For the same form, prescribed data and admissible complement, zero extension is only one competitor and gives an upper comparison for that infimum. The physical compression has not been identified with the theorem’s short of $b$ . A nonsymmetric single-parent test also needs its reflected component and all cross terms to become an odd test. The pole and prime terms are excluded from these archimedean benchmarks; no counterexample to the actual odd Weil form is asserted.

The remaining realization must specify its full potential, starting interval, preceding map from the odd physical space, and relation between $A$ , $a$ and $w_{m}$ . The displayed proof of Proposition 5.3 does not identify these data. Lemma 6.16 preserves the target ground line and its orthogonal splitting; that alone is not a transformed-potential identity. Lemma 6.51 and Theorem 6.53 calculate on the already-specified $b$ , and Lemma 8.4 invokes Proposition 5.3 again. These source dependencies locate the missing correspondence without proving that no such correspondence can exist.

The existing small-support spectral supplier and localization account already cover the corresponding basic boundary energy and its small-window use. They should be reused; neither supplies this paper’s all-scale collar correspondence. The supplementary scalarization audit limits its PASS to the stated scalarization risk and does not independently establish the complete diagonal identification.

Relation to the FIB research gap

The retained-old-block, mixed-coupling and Schur-induction architecture is standard and is explicitly attempted at all scales in this source. Naming the support schedule after Fibonacci therefore supplies no architectural novelty. The project’s exact block reduction and golden positivity induction remain reusable under their own assumptions; this review did not rebuild them.

For an actual finite positive old block $H$ , the additional estimate is $B^{*} H^{- 1} B ⪯ D$ for the matching new block and coupling; a semidefinite old block also needs the appropriate range condition. This source’s claimed budget bridges are relevant candidates for detailed comparison with that obligation. They are not adopted as a supplier that has already closed it. The local scalar audit does not settle the full comparison. No external certificates, prime or zero samples, or Lean declarations were produced for this review.

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