trureturing — Discovering truth

bibkey: trureturing2026extrawindowpairing authors: trureturing research (derived proof; original supplier attributions in body) year: 2026 title: Ordinary supplier-relative forced-window, core and forward deductions doi: null url: https://raw.githubusercontent.com/the-omega-institute/trureturing/4a0698d331a1c734dbd978b52a833ce6116595bd/Library/ArithSums/trureturing2026extrawindowpairing.md claim: Ordinary supplier-relative forced-window, core and forward deductions under the hypotheses and scope stated in the body; no Lean verification, RH resolution, novelty or official acceptance is asserted. strata_touched: [] license: Apache-2.0 triage: anchor

Verdict: comment. N1–N5 are valid ordinary deductions with the supplier, tie and real-base qualifications below. The supplied answer is advisory data, not verification. This audit additionally consumes the completed two-price audit §9: N1 pays its cofinal-extra premise, and cofinal moderate distinct last-label edges now follow. For every fixed q>1, cofinally many such edges even satisfy C<=qA. This is separate from N4’s ratio-free argument. No bound on every edge, shrinking rate, nonzero core increment, or favorable signed Lambda estimate is obtained.

The original goal remains ACTIVE AND UNPROVED:

for EVERY actual integer n>5040,
I_psi(log n)+R(log n)+d_(log n)(n)>0.

This is ordinary fallible proof and finite execution evidence. No formal, Lean/kernel, official, consensus, sterile-context or RH acceptance is claimed. No role wrapper, new optimizer module or theorem assuming the missing sign was created. Assertion dispositions are in correction-register.json; exact quantifiers are recorded in quantifier-register.json and expanded below.

1. Complete intake and exact same-family source reuse

The entire 29,207-byte answer was read, including all five proofs, all H1–H5, plateau/tie/geometry statements, compensation/tent conclusions, remaining obligations, and literature limitations. Its SHA256 is 67eaa06e6d8437e39e0e92dd45f15294332faae39ab8502db5fda68fee07e1d2. Its completed intake receipt and all other manifest inputs match their exact lengths and hashes. A completed carrier is no mathematical approval. All foreign log_ref strings remain opaque; no pointed-to log was inspected.

The declared completed ordinary proofs were read IN FULL and reused:

• Two-price actual-tent audit, 39,317 bytes, §§1–11, especially the all-branch literal replacement in §3, balanced-Q proof in §7, real-base repair in §8, conditional cofinal geometry in §9, and convergent compensation in §10.
• Actual-value/envelope/tail-budget audit, 35,419 bytes, §§1–11: all-branch canonical order, last-label normalization, 9/A and 13/A budgets, exact arriving-clock Stieltjes indexing, corrected E7 and necessary orientation.
• Cross-prime source-family audit, 34,560 bytes, §§1–10: arbitrary-maximizer general-m characterization, legal replacements, canonical blocks and the full C0=120 discontinuous-psi estimate on ANY real base X>=1024.
• Forced-prime audit, 41,037 bytes, §§1–12: actual-G attainment, insertion and legal nonforced deletion, strict 5040 activation, all singleton ties, uniform V, and exact finite lcm identity.
• Full signed shorter-Q section, 30,012 bytes: all QH1–QH13, QQE/QQD and QG1–QG9. No higher-layer or reference-square term is omitted.

These are declared completed priors, not same-round peers. Their hashes appear in input-verification.json. Their ordinary analytic assumptions are retained, not silently promoted to compiled suppliers.

All new project intake used immutable objects at canonical Git common-dir /Users/macstudio/omega/trureturing/.git, exact commit 8629b924d8bd0e9d5156b933ee9947465c5aee06. No ref or live worktree supplied source bytes. source-search.json preserves four untruncated bounded searches, including private D5 constructions and the exact arithmetic family. Relevant source identities and snapshot scopes are in source-identity-manifest.json.

GoldenFutureExtensionMaximum already proves its private finite positive prefix and public golden_future_extension_maximum_attained for W-lambda log n on positive multiples, with lambda>0 and base>=1. GoldenColossalClosure already realizes max(base exponent, strict-positive prefix); its public closure price is the base threshold, not an arbitrary own-clock price. These full proofs were read. The public single-layer delta and existing first-layer prime antitonicity are also reused. Fixed-price optimization alone is not actual-G attainment. No duplicate construction is proposed.

The pinned NC object contains exactly once, byte for byte, the entire declared 30,012-byte full-Q section. The strict lcm identity, full Lambda kernel, QQE and unknown all-real starts thus already have their exact source. GAK40–GAK51 only give the separate conditional finite interval through 8e46; they do not supply a forced-family clock floor, unknown GM start or unbounded sign. The pinned CNS card exactly matches the supplied 53,774-byte snapshot. Its published unbounded GA2 supplier and distinct regular proper-GA1 scheme are existing content; neither is a selected forward sign theorem.

The searches identify no exact already-proved N1 window ceiling or N4 integer-rigidity package in the scoped pin. Broad N_p/R_X hits include unrelated prime and analytic notation; they are not such a supplier. This is a bounded search observation, with no priority or exhaustive absence claim. Completed Mathlib/OpenAI source searches in the full priors are reused: Chebyshev/VonMangoldt/factorization are helpers; the uninstantiated bounded multiplicative correlations and strict Re(s)>7/8 nonvanishing supply no selected full-Lambda sign here. Musin’s pinned contact card has different domains and explicit growth/concavity hypotheses; it supplies no N1–N5 premise. Its primary was not freshly retrieved or independently verified.

The actual CNS v2 PDF was extracted using installed pypdf. Printed pages 2, 13, 15, 16 were read, with printed page numbers checked against physical pages. cns-primary-excerpts.txt contains these extracts. The unavailable pdftotext executable was an extraction boundary, repaired with pypdf; it caused no source substitution or mathematical change.

2. Analytic quantifiers and exact arithmetic conventions

CNS arXiv:1112.6010v2 printed p.15 assumes RH failure, defines Theta as the supremum of real parts of nontrivial zeros, and has 1/2<Theta<=1. For ANY ONE FIXED b with 1-Theta<b<1/2, it states positive oscillation on CA integers, first in the cutoff x and then in log N using x~log N. Its explicit Omega-plus explanation means a positive power excess occurs infinitely often. Consequently there is a fixed a>0 and arbitrarily large actual integers m with X=log m and

G(m)>=exp(gamma)(1+a X^(-b)).

For large such X, a X^(-b)<=1 and log(1+u)>=u/2 on [0,1] gives

F(m)>=gamma+d X^(-b), with ONE fixed d=a/2>0.        (H1)

The precise value is immaterial; it is fixed before all windows. This deduction is stronger than bare GA2 infinitude. b>0 follows from Theta<=1. Theta need not be attained. This does not truncate zeros, choose an ordinate sign, or prove the cited Robin oscillation theorem anew. Printed p.16’s successive tail-maximizer argument supplies the global canonical family’s ordinary existence via the completed forced-source audit under RH failure.

CNS equation (2), printed p.2, has the unconditional upper envelope G(n)<=exp(gamma)+0.6482…/(log log n)^2. Printed p.15 explicitly uses the rounded larger 0.6483. Restrict to n>=3 so G(n)>0 and A=log n>1. With K_R=0.6483 exp(-gamma), log(1+u)<=u yields

F(n)-gamma<=K_R/(log A)^2.                         (H2)

This is an upper envelope with positive error, not Robin’s strict inequality. The n>1 wording of equation (2) is not used to take log G(2).

For x>1 set f(x)=log log x, lambda(x)=1/(x log x), w(x)=(log x+1)/(x^2(log x)^2)=-lambda’(x)>0. Then f’‘=-w<0 and w’=-x^-3[2/log x+3/(log x)^2+2/(log x)^3]<0. For n>=3, F(n)=log G(n)=W(n)-f(log n), W(n)=log(sigma(n)/n).

For ordinary prime q and k>=1, h_q=log q, ell_(q,k)=log(1+1/(q+…+q^k)), r_(q,k)=ell_(q,k)/h_q, c_q=ell_(q,1). Euler telescoping gives W as the finite occupied-layer sum. Stack prices strictly decrease; 0<r_(q,k)<1/(q^k log q), and r_(q,1)<lambda(q). Define the STRICT actual inventory N_t^str using r>lambda(t), K(t)=log N_t^str, and the WEAK reference psi(t) using q^k<=t. Q(t)=K(t)-psi(t). All powers and individual h_q weights are retained. Every compact positive-price range has one common finite actual/tie/reference support. A missing terminal equality layer is excluded, with zero pressure.

The reused actual characterization is for EVERY maximizing tie on positive multiples of m>=3. Insertion gives r_next<lambda(A), and deletion only above the forced exponent is legal with N/q>=m>=3, giving r_last>lambda(A). Thus N=lcm(m,N_A^str). For m=5040p, prime p>=29, A>11 and all 5040 layers are strictly active, so N=lcm(p,N_A^str) and extras are contained in {(p,1)}. The canonical source N_p is the LEAST actual maximizer. Global canonical clock order, value order, finite blocks and last label Pplus(N) are reused from the completed cross-prime and E1–E8 proofs. Arbitrary ties are not substituted in those global ordering assertions.

GM means exactly the ordinary input Corollary 1.3 of arXiv:2405.20552v2, fixed epsilon=1/60 and beta=7/12, with constants C_GM and Y0 such that for ALL real y>=Y0 and y^beta<=v<=y^.99,

|pi(y+v)-pi(y)-v/log y|<=C_GM v exp(-(log y)^(1/4)).  (GM)

One unknown enlarged start makes C_GM log y exp(-(log y)^(1/4))<=1/2. Then every permitted interval has a prime. In particular p_next<=p+p^beta eventually. This is ordinary label spacing, not a source-clock mesh or a favorable psi estimate. The full-Q prior supplies, for ALL sufficiently large real t, S_Q(t,t^.8)=(c_Q+o(1))t^1.1, c_Q=(sqrt2-1)/4, with one UNKNOWN common tolerance start and the seven retained errors

C[1/log t+t^(-1/5)+log t exp(-((log t)/3)^(1/4))
  +t^(-1/120)+t^(-3/10)log t+t^(-1/6)(log t)^(4/3)
  +t^(-19/180)(log t)^(43/36)].                    (QQE)

The absolute pi error, its log factor in theta, permitted enclosures below GM’s minimum, reference square coefficient 1/4, actual square sqrt2/4, growing-index support, strict endpoint atoms and all-real uniformity remain. No numerical common start is introduced.

3. N1: actual padding, full-domain attainment and ALL maximizing ties

Fix H1’s b,d once. For a qualifying witness clock X define EXACTLY

ell=(d/8) X^(1-b)log X,
R=exp(sqrt(2 K_R/d) X^(b/2)),
P_X={ordinary primes p: exp(ell)<=p<=exp(2ell)}.    (W)

R is this window ceiling, not the original residual R(t). H=X^.8 of an original consumer is also distinct from ell. Put m0=lcm(5040,m), d0=log(m0/m) in [0,log5040]. The legal comparator is the ACTUAL integer

D_p=p*m0,

even when p already divides m0. If k=v_p(m0), finite telescoping gives

F(D_p)-F(m)=W(m0)-W(m)+ell_(p,k+1)
            -[f(X+d0+log p)-f(X)].                (N1.1)

Both Euler increments are nonnegative. Since lambda decreases, the cost is at most (d0+log p)/(X log X). Uniformly for p in P_X it is at most (d/4)X^-b+log5040/(X log X), eventually <=(d/2)X^-b. Thus F(D_p)>=gamma+(d/2)X^-b uniformly. No absent-p assumption, missing old layer, or substitution D_p=lcm(p,m0) occurs.

For ANY integer n>=3 with A_n>R, H2 gives F(n)-gamma<K_R/(log R)^2=(d/2)X^-b. It therefore cannot attain the p-domain maximum. Also D_p itself has clock <=R, by this strict tail comparison. The nonempty finite set of multiples of 5040p with log n<=R attains the full infinite-domain maximum; every tie is in this finite set. For every such maximizing N,

ell<=log N<=R,  v(p)>=gamma+(d/2)X^-b.             (N1.2)

The lower bound uses N>=5040p. Equality at R is allowed. No comparison of sizes is inferred from an arbitrary competitor’s smaller objective.

Write a_R=sqrt(2 K_R/d)>0. Then log R=a_R X^(b/2), while

ell/log R=(d/(8a_R))X^(1-3b/2)log X -> infinity,

since 0<b<1/2 gives 1-3b/2>1/4. For EACH fixed M>0, ell-M log R->infinity, so exp(ell)/R^M->infinity. The eventual start may depend on M, b,d, analytic constants; it never depends on p or its tie. In particular p>2R eventually. For ANY maximizing tie, A=log N<=R<p, r_(p,1)<lambda(p)<lambda(A). Its forced layer is strictly inactive; the reused characterization yields

N=p N_A^str, with exactly extra {(p,1)}.            (N1.3)

Every prime in N_A^str is <A<=R, so p is Pplus(N). If this SAME integer were maximizing for another ordinary label q>=29, its nonempty own extra would have to be contained in {(q,1)} as well. Therefore q=p, including q outside the window. This proves label uniqueness for every tie and last-label fidelity for the canonical integer. Plateaus and distinct-clock equal-value transitions outside these windows are not excluded.

4. N2: the exact common finite core and all choices of ties

Let C_X be the finite set of literal positive u with 5040|u, log u<=R, and every prime divisor <=R. It contains 5040 eventually. For real h in [ell,2ell], put V_X(h)=max_(u in C_X)[W(u)-f(log u+h)]. Each summand has derivative -lambda(log u+h)<0 and second derivative w(log u+h)>0. Finite maxima give a decreasing convex function (indeed strictly decreasing here); no off-prime arithmetic source is asserted.

For actual p in P_X, every full-domain tie is pu with u=N_A^str in C_X by N1. Conversely each u in C_X makes pu a legal p-domain integer and p>R keeps p absent from u. Hence

F(pu)=c_p+W(u)-f(log u+log p),
v(p)=c_p+V_X(log p).                              (N2.1)

Every core-envelope maximizer at log p is a full-domain maximizing tie, and every full-domain tie is obtained this way. The least integer pu corresponds to the least u. The common finite set is exact, not a relaxed real core optimization or a restricted substitute for the original domain.

If log u>log v, the difference of their summands has strictly positive derivative lambda(log v+h)-lambda(log u+h). If u maximizes at h1, then at every h2>h1 that difference is strictly positive, so v cannot maximize there. Thus EVERY maximizing core clock at h1 is <= EVERY maximizing core clock at h2. At actual labels p<r this gives, for ANY choices of ties,

N=pU, N_r=rV,  A=log N<C=log N_r,
C-A=log V-log U+log(r/p)>0.

Now U=N_A^str, V=N_C^str, so the increasing strict inventories prove U|V. This ordering of arbitrary ties is paid inside the supplied window; it is not a blanket arbitrary-tie ordering outside it.

At a real h, if active core clocks have minimum a_min and maximum a_max, the finite convex maximum has left derivative -lambda(h+a_min), right derivative -lambda(h+a_max), and subgradient interval between them. Actual ties form a finite set of integer cores; convex intermediate slopes are not necessarily attainable clocks. Even at h1=h2, no chosen arbitrary tie is declared equal to the canonical least one. Envelope convexity alone yields no C/A bound, uniform value gap or favorable Lambda mean.

5. N3: SAME-source replacement and expanded arriving-clock errors

Take consecutive ordinary primes p<r both in P_X. Use canonical sources for the Stieltjes conclusions below; the finite comparisons in this section also hold for arbitrary chosen window ties. Define

N=pU, N_r=rV, A=log N, C=log N_r,
delta=log(r/p)>0, D=log(V/U)>=0,
Z=pV, z=log Z=A+D=C-delta,
epsilon=F(N)-F(Z), beta0=F(N_r)-F(rU).

p,r>R exclude them from both cores. Thus p does not divide N_r, r does not divide N: the sources are distinct and nonnested. U|V makes Z an actual multiple of the SAME N=pU. Z is legal in its p-domain and rU is legal in the r-domain, preserving all of 5040. Maximality gives epsilon, beta0>=0. Both Euler c_p,c_r cancel in their sum:

epsilon+beta0=f(A+D)-f(A)-f(A+D+delta)+f(A+delta)
  =int_0^delta int_0^D w(A+s+t)dt ds
  <=int_0^delta lambda(A+s)ds<=delta lambda(A).    (N3.1)

No deletion of an intermediate integer was compared. If D=0, Z=N and both deficiencies vanish. If D>0, a p-domain tie can still give epsilon=0; canonical least arrival implies beta0>0 but provides no quantitative margin.

The literal comparison F(pV)-F(rV)=c_p-c_r+f(C)-f(C-delta)>0 proves v(p)>v(r). The other legal competitor rU gives the upper drop bound

0<v(p)-v(r)<=c_p-c_r+f(A+delta)-f(A)
  <=delta[1/p+lambda(A)].                         (N3.2)

Here c_p-c_r=int_(log p)^(log r)du/(exp u+1)<=delta/p. Strictness is cross-label order, not a usable positive scale margin.

Let E=exp(-5ell/12). Applying GM only to the permitted interval (p,p+p^beta] gives r<=p+p^beta and

0<delta<=p^(-5/12)<=E,
0<=epsilon<=E/(ell log ell).                      (N3.3)

For each fixed M>0, delta R^M and epsilon R^M tend uniformly to zero over all such pairs and all choices of window ties, since log R=o(ell). The following estimates are at the ARRIVING scale, without C/A control. For EVERY t in [ell,R],

w(t)>=1/(R^2 log R),
w(t)t^1.1>=(R^(.9) log R)^(-1).

Consequently

epsilon/w(C)<= E R^2 log R/(ell log ell) ->0,
epsilon/[w(C)C^1.1]
   <= E R^(.9)log R/(ell log ell) ->0,             (N3.4)

and the identical bounds hold with z in place of C. These are uniform ceiling estimates; no epsilon=O(1/A) argument is transported to C without paying its actual denominator. The absolute cross-label drop has bound E[exp(-ell)+(ell log ell)^(-1)]; divided by w(C)C^1.1 its bound is E R^(.9)log R[exp(-ell)+(ell log ell)^(-1)]->0 as well. This explicitly explains why cross-label strictness supplies no fixed positive margin. R may be arbitrarily large relative to a departure A.

6. N4: exact integer rigidity, closed strict inventory, signed contact

Let M_z=lcm(N,N_z^str), y=log M_z. Since A<=z<C, monotonicity gives U|N_z^str|V, while p>C keeps p absent. Thus

M_z=p N_z^str | Z=pV, and A<=y<=z.                (N4.1)

Reuse the original finite positive-prefix optimizer over literal multiples of N at price lambda(z). Its deficiency against the SAME Z is

Delta_z=[W(M_z)-lambda(z)y]-[W(Z)-lambda(z)z]>=0.

Direct algebra, with C_z(u)=f(z)+lambda(z)(u-z)-f(u), gives

epsilon=[F(N)-F(M_z)]+C_z(y)+Delta_z.              (N4.2)

The first term is >=0 by actual GA2 of N; the other two are >=0 by concavity and fixed-price optimality. For y<=z,

C_z(y)=int_y^z(t-y)w(t)dt>=w(z)(z-y)^2/2.

Using N3’s actual common ceiling,

(z-y)^2<=2epsilon/w(z)
   <=2 E R^2 log R/(ell log ell) ->0.              (N4.3)

Eventually z-y<log2 UNIFORMLY, with no z<=2A or C<=2A premise. Z/M_z is a positive integer by (N4.1). If it exceeded one it would be >=2, contradicting (N4.3). Therefore

M_z=Z EXACTLY, y=z, N_z^str=N_C^str=V.            (N4.4)

Cancellation of the literal common factor p is legal. Every strict inventory on the CLOSED interval [z,C] equals V by monotonicity. A missing layer tied at z would become strictly active before C and contradict this equality; at C a missing tie remains excluded, with zero terminal pressure. No neutral layer was silently inserted. In (N4.2), Delta_z and C_z(y) are now zero.

The paired canonical arriving clock and literal optimizer clock obey C-y=delta<=E. This exact pairing does not declare Z GA2 or canonical. N3.4 proves epsilon=o(w(z)z^1.1) and o(w(C)C^1.1) uniformly, including arrivals with large C/A. For clarity the small remaining concavity cost is

C_C(z)=int_z^C(t-z)w(t)dt<=w(z)delta^2/2.

The exact bound |w’/w|<=4/t gives w(z)/w(C)<=exp(4delta/z)->1 uniformly. Thus C_C(z)/[w(C)C^1.1]<=delta^2/C^1.1 eventually, tending uniformly to zero, with absolute bound w(ell)E^2/2. This cost is kept with its sign.

Use the completed E1–E8 definition beginning at p’s LAST label:

J_p(t)=sum_(i>=j, A_(p_(i+1))<=t) log(p_(i+1)/p_i).

These are ARRIVING-clock atoms, weak/right-continuous. N1 proves p’s label is unique. p,r are consecutive labels, and N_r also has a unique producing label, including outside the window. Canonical global clock order therefore gives no atom in [A,C), and exactly the single mass delta at C. A later equal clock would be the SAME N_r with another label, contradicting uniqueness. Hence J_p(A)=0, J_p(t)=0 for A<=t<C, and J_p(C)=delta. No future plateau mass is omitted by this argument; the completed general family aggregates such masses outside this window.

On [z,C], Y_N(t)=log lcm(N,N_t^str)=z. Consequently

Y_N(t)-t+J_p(t)=z-t for z<=t<C,
Y_N(C)-C+J_p(C)=z-C+delta=0,
int_z^C[Y_N(t)-t+J_p(t)]w(t)dt=-C_C(z)<=0.        (N4.5)

Here delta>0 makes the integral strictly negative. The endpoint jump has zero ordinary integration measure, and cannot reverse this sign. The same finite envelope E_N(t)=F(M_t)-F(N)-C_t(Y_N(t)) has derivative (Y_N(t)-t)w(t) almost everywhere, E_N(A)=0. Therefore its finite integrals from A to z and A to C are respectively -epsilon and -epsilon-C_C(z). These are finite exact comparisons, with no subtraction of divergent infinite integrals. Compensated endpoint zero supplies no favorable component.

7. N5: cofinal literal forward integer, full Lambda and REAL base transfer

Let T be ANY required finite common start. Along H1 witnesses, ell->infinity. GM at the REAL start exp(ell), length exp(beta ell), gives an ordinary prime p just above exp(ell). GM at p gives its next prime r. Eventually r<4 exp(ell)<exp(2ell), so both are in P_X. Their canonical clocks A,C and z are >=ell. Choosing a sufficiently large witness passes T and all required starts together. These are actual original N_p,N_r, not the H1 seed m or padding D_p. There are cofinally many distinct such integers. Indeed (N1.3) supplies cofinal extra-bearing canonical source blocks.

At Z, let h=z^.8, B_z=z+h, L=p N_(B_z)^str, Y=log L. N4 gives N_z^str=V, so Z=pV divides L and the SAME original N divides Z. Since p>2R and B_z<2z<=2R, r_(p,1)<lambda(p)<lambda(B_z). Its unique extra stays strictly inactive on [z,B_z]. Thus the occupied loss V_Z(z,h) is EXACTLY zero, including all endpoint conventions. At the arriving original N_r, the identical argument with r and C+C^.8 proves V_(N_r)(C,C^.8)=0. This zero-loss conclusion is limited to the supplied branch, not all forced-prime sources.

The finite lcm identity requires N_z^str|Z and literal successors, not GA2 of Z. For completeness, put M_t=lcm(Z,N_t^str). Its clock is z+K(t)-K(z)-P_Z(t). The common support is finite; integrating the missing layer steps and subtracting the occupied activations gives

F(L)-F(Z)=int_z^B_z[log M_t-t]w(t)dt+C_(B_z)(Y)
         =w(z)[T_psi(z,h)+S_Q(z,h)]+C_(B_z)(Y).

No K(z)=z or Y=B_z is assumed. Against the actual original GA2 source,

F(L)-F(N)=w(z)[T_psi(z,h)+S_Q(z,h)]
              +C_(B_z)(Y)-epsilon.               (N5.1)

The strict terminal inventory is unchanged. If a subset of missing tied terminal layers of mass J were deliberately installed, its reward would be lambda(B_z)J and its clock Y+J. The pressure integral would be unchanged but its cost would become C_(B_z)(Y+J). Dropping that change is invalid.

The COMPLETE Lambda identity follows by finite integration of psi(t)-psi(z)=sum_(z<n<=t)Lambda(n):

w(z)T_psi(z,h)
  =sum_(z<n<=B_z)Lambda(n)[lambda(n)-lambda(B_z)]
   -[f(B_z)-f(z)-h lambda(B_z)].                  (N5.2)

Every higher prime power appears with Lambda(q^k)=log q. The initial atom is excluded; the terminal atom is included with ZERO weight. This is the original forward kernel, not a prime-only or balanced-tent kernel.

L>Z is proved independently of any sign. Apply GM to real y=z+h/3 and length h/3. Uniformly as z>=ell->infinity, h/3 lies between y^beta and y^.99 because beta<4/5<.99. A prime q satisfies z+h/3<q<=z+2h/3. Since q>z, its first layer is absent from N_z^str. Eventually q<B_z-1<q_1(B_z), using the reused strict cutoff enclosure B_z-1<q_1(B_z)<B_z, so q is strictly in N_(B_z)^str. Also q<p. Thus q divides L/Z and L>Z>=N, a nontrivial literal integer multiple. No full-domain maximality or GA2 status is claimed for L or Z.

Pro’s phrase “dyadic base z” must be replaced with the actual real-base application, not an assumption that log Z is a power of two. Set X=z, centers a=z and c=C=z+delta. Eventually X>=1024, a,c in [X,2X], delta<=1<=X^.8/2. The completed cross-prime audit §8 proves, for ANY real X satisfying these hypotheses,

|T_psi(c,c^.8)-T_psi(a,a^.8)|
  <=120 X^.8(|c-a|+1)log X.                      (N5.3)

The proof does not use a power-of-two grid. To retain every discontinuity cost explicitly, write L_X=X^.8, H_s=s^.8, B_s=s+H_s, Q_s(t)=psi(t)-psi(s)-(t-s), R_s(t)=w(t)/w(s). On [X,3X], 0<=psi(v)-psi(u)<=(v-u+1)log(3X), counting actual integers in (u,v]. Also H_s<=2L_X, |Q_s|<=5L_X log(3X), R_s<=1. For d=|c-a|<=L_X/2, the symmetric difference of the two moving intervals has length <=3d, costing <=15L_X d log(3X). On the common interval the exact integrand change is d-[psi(c)-psi(a)], costing <=4L_X(d+1)log(3X). The normalization change satisfies 0<=R_c-R_a<=4d/X by |w’/w|<=4/t, costing <=40d L_X^2 log(3X)/X. Add these, use L_X/X<=1 and log(3X)<=2log X for X>=1024, to obtain C0=120. The upper endpoint movement is d+(H_c-H_a), not merely H_c-H_a. The indispensable +1 accounts for an initial prime-power atom crossed by an arbitrarily small shift. At such q the exact jump is

Lambda(q)[lambda(q)-lambda(q+q^.8)]/w(q)>0.

Thus a uniform Lipschitz claim for psi or T_psi with only d is not licensed.

In this application the absolute transfer error is at most 120 z^.8(delta+1)log z. Dividing by C^1.1 gives <=240 z^(-.3)log z eventually. Since z>=ell and t^(-.3)log t decreases eventually, this is uniformly <=240 ell^(-.3)log ell ->0. C^1.1/z^1.1->1 uniformly because delta/z<=E/ell->0. All actual real and comparable-base conditions are therefore paid; no common unknown analytic start was replaced numerically.

QQE, epsilon=o(w(z)z^1.1), exact zero occupied loss and the actual nonnegative C_(B_z)(Y) are the paid analytic/accounting terms in N5.1. Its sign remains unpaid. A fixed strict favorable Lambda margin would transfer from these partners to the arriving ORIGINAL canonical consumer at C by N5.3, with a fixed amount of slack; no such margin is inferred.

8. ACTUAL consumption of completed two-price audit §9

The completed §9 had a proved conditional consequence: cofinal extras imply cofinal moderate last-label transitions, and its alternative eventual zero-extra doubling branch remains possible until extras are supplied. N1 and §7 now supply those extras. Here is the entire consumption proof, also giving the optional fixed-ratio generalization.

Enumerate DISTINCT canonical source blocks N_j in prime-label order. Clock order makes repeated labels consecutive; equal clocks mean the same integer; labels divide their source so each block is finite. Unbounded primes force infinitely many blocks. Let A_j=log N_j, last label p_j=Pplus(N_j), next ordinary prime r_j, arrival C_j=A_(j+1)=A_(r_j). The completed SAME-source replacement Z_j=lcm(N_j,N_(C_j)^str) has A_j<=z_j=log Z_j<=C_j, with all three branch formulas retained in the two-price §3 proof. A_j,p_j->infinity. The LAST-label bounds are

A_j<3p_j always; if departure has zero extra, p_j<A_j. (G1)

They are never applied to a small arrival label of a zero-extra plateau.

Fix ANY q>1 once, and choose rho=(1+q)/2 in (1,q). A zero-extra departure followed by an extra arrival has r_j as that arrival’s UNIQUE label. An extra r_j cannot be <=C_j-1, by the strict first cutoff, hence C_j<r_j+1. Actual GM label spacing and (G1) give

C_j<r_j+1<=p_j+p_j^beta+1
   <A_j+A_j^beta+1<q A_j eventually.              (G2)

This uses the DEPARTURE’s last label and the ARRIVAL’s singleton property. It does not use A_(r_j)<3r_j for a zero-extra arrival.

Suppose moderate edges C_j<=qA_j were not cofinal. Then beyond some index every C_j>qA_j. By (G2), no zero-extra source in this tail can be followed by an extra source. If one zero-extra source occurred there, induction would make every later source zero-extra, contradicting N1’s cofinal extras. Therefore this tail consists entirely of extra sources. Each such arrival has unique label r_j, so p_(j+1)=r_j, consecutive ordinary primes. From the ACTUAL p_next/p<=1+p^(-5/12)->1, enlarge the tail so p_(j+1)<=rho p_j. Meanwhile A_(j+1)>q A_j. Iterating and using (G1) at EACH departure LAST label yields

q^k A_j < A_(j+k) < 3p_(j+k) <=3rho^k p_j.

Thus (q/rho)^k<3p_j/A_j for all k, impossible. Hence, under the same ordinary RH-false suppliers,

for EACH FIXED q>1 and EVERY T,
some distinct LAST-label edge has A_j>=T and
A_j<C_j<=q A_j, and A_j<=z_j<=C_j.                (G3)

In particular q=2 consumes the previously conditional §9 moderate geometry. This actually proves the slightly stronger arriving-clock bound C_j<=q A_j, not merely its needed z_j<=q A_j. Thresholds depend on q; the statement does NOT say every edge ratio tends to one and asserts no shrinking rate. Z_j may equal N_j, and D may be zero. A block transition is nontrivial because its sources are distinct, not because Z_j>N_j.

The completed two-price §8 consumer is consequently available cofinally: for q=2 its all-branch projection y_j=log lcm(N_j,N_(z_j)^str) obeys

0<=C_j-y_j<=8*3^(5/24) A_j^(7/24)sqrt(log A_j)
                 +log(8A_j)
            =o(A_j^(.3)/log A_j).

With real comparable base X=y_j, its C0=120 transfer is o(C_j^1.1). This pays that former geometry premise, but a favorable full Lambda quantity at those projected clocks is still unpaid. This moderate-family consumer is separate from N4’s ratio-free exact rigidity.

One can also obtain moderate edges INSIDE each supplied window, so the same-window quantifiers need not be conflated with the block proof. Fix q>1 and put k_X=floor(log(R/ell)/log q)+1; then q^(k_X)ell>R and k_X=O(log R+log ell) for large witnesses. Choose p_0 just above exp(ell) as in §7, so log p_0<=ell+log2. Iterating actual successor primes gives delta_i<=E as long as labels are in P_X. Inductively

log p_i<=ell+log2+iE<2ell for 0<=i<=k_X,

because k_X E->0. This also justifies the induction’s membership, using GM for all p_i>=exp(ell). N1 gives k_X+1 distinct canonical sources with clocks between ell and R. If every one of these k_X edges had C>qA, the last clock would exceed q^(k_X)ell>R, a contradiction. Thus every sufficiently large qualifying window contains a N3 pair with C<=qA. Their A>=ell makes this cofinal. No upper clock-ratio premise was used to prove N4; this is an additional supply consequence, with q fixed first.

9. Compensation, tents, signs and the complete remaining conclusion

The completed same-LAST-label tail budgets are retained: v(s)-gamma=L_s+u_s, |u_s|<=9/A, sum eta_i<=13/A. The series L_s and its Stieltjes integral are proved convergent BEFORE subtraction. These errors are o(w(A)A^1.1), but neither v(s)-gamma nor L_s individually is declared that small. The corrected E7 identity is exactly

w(A)T_psi=I_psi(A)-I_psi(B)
              -(psi(A)-A)(lambda(A)-lambda(B)).

The original wrong literal E7 line in a prior Pro answer is not restored. With the original residual and terminal occupied defect,

Xi_s(B)+u_s=F(N)-F(M_B)+C_B(Y)>=0.

The convergent compensation int_A^infinity[Y_N(t)-t+J_s(t)]w(t)dt=-u_s is reused with its actual boundary proof. It is not assembled by subtracting divergent int Yw and int tw. N4’s local zero contact and negative preceding integral do not localize a positive component of that global compensation. No new monotonicity, one-sided variation, Tauberian condition or correlation with the full von Mangoldt measure has been supplied.

The completed balanced-tent identity retains FULL Lambda, Q_tent, occupied crossings and the fixed-price cost. Neither a finite nonzero favorable same-family average nor the required balanced-left placement z-z^.8>=A is proved here. Moderate geometry alone does not prove that placement, and D=0 plainly leaves it unavailable. If the tent extends left of A, its whole occupied crossing sum remains; it is not a singleton loss. N5 is the valid forward consumer that avoids this placement requirement.

From GA2 of the SAME original N and the literal L, N5.1 yields only the necessary comparison

w(z)[T_psi(z,z^.8)+S_Q(z,z^.8)]
   <=epsilon-C_(B_z)(Y).                          (SIGN)

Equivalently T_psi<=-S_Q+[epsilon-C_(B_z)(Y)]/w(z). This direction cannot produce a favorable LOWER estimate. QQE and arrival loss control leave the critical coefficient c_Q, not any fixed kappa<c_Q. The original arriving GA2 source also obeys its necessary upper comparison with V=0. Positive Q drift, strict cross-label order, least attainment, integer optimizer contact and C0 transfer create no strict favorable coefficient. The ordinary GM allowance at main prime scale, after normalization by the gain, still has size sqrt(z)log z exp(-c(log z)^(1/4)), which does not tend to zero. This diagnoses that allowance, not the actual error’s sign.

The original unpaid selected-source quantifier can be written without changing its objects:

ONE fixed kappa<c_Q must work beyond EVERY common start T:
there is an ORIGINAL canonical N_p with A_p>=T and
T_psi(A_p,A_p^.8)>=-kappa A_p^1.1.

For this window route, its support can be confined to the now-proved cofinal window sources; one still owes that favorable estimate there. Alternatively one owes a fixed positive normalized lower margin for the COMPLETE w(z)(T_psi+S_Q)-epsilon on cofinal proved partners. The finite consumer, its actual terminal cost, cofinal same-family support and every error are already paid. In prose, if that still-missing fixed margin were established it would make the actual L outperform the same original N; this is an obligation analysis, not a new theorem assuming a sign.

A fixed favorable partner bound with coefficient kappa0<c_Q could be transferred using N5.3 to an arriving coefficient kappa1 with kappa0<kappa1<c_Q, after every UNKNOWN QQE/transfer start. Its original successor is r N_(C+C^.8)^str, with occupied loss zero. No two different integers are identified. Finite weighted variants must retain finite nonnegative weights, nonzero total weight, original-family support and tail starts; none of their favorable inequalities is proved.

Cofinal moderate geometry is now paid, even at each fixed ratio q>1; the answer’s blanket statement that cofinal C<=2A is unpaid is superseded by §8. Cofinal Z>N, positive core increments, every-edge ratio convergence, shrinking-rate coverage, balanced-left placement and the favorable sign remain unpaid. No least-counterexample or least-global-maximizer finite bound pays an UNKNOWN analytic start. No zero is computed, no full RH criterion weakened, and the full original goal remains ACTIVE/unproved.

10. Executed meaningful controls and artifact scope

python3 exact_checks.py exited 0: 103 exact finite checks and 18 deliberate error controls. It uses integer/Fraction arithmetic, 160-bit outward dyadic intervals, 96 positive range-reduced atanh terms with proved geometric tail, and integer fifth-root brackets. No float supplies a proof decision. It checks actual padding when p is absent/present/repeated, exact Euler contributions, scalar core single crossing, actual integer replacement rectangles, strict/missing marginal ties and retained terminal clock, optimizer cost/Delta decomposition and the integer log2 gap, right-continuous arriving Stieltjes atoms including plateau aggregation, negative compensated integral, full Lambda versus step integration, initial/terminal atoms, higher powers, the original lcm identity with occupied loss, exact zero inactive-singleton loss, and the C0=120 bound at real base 2047.75 across the actual initial 2048 prime-power atom. It rejects the no-jump, prime-only, wrong-sign, early-jump, omitted-Delta and omitted-occupied-loss alternatives. Finite samples are not asserted GA2 or full-domain maximizers. The general proofs above supply all uniformity, asymptotics and cofinality.

python3 intake_sources.py exited 0: eight immutable mathematical objects, four complete searches, pinned CNS snapshot equality, exact full-Q byte reuse, and actual printed CNS page extraction. Input hashes and final proof/register/manifest consistency are checked in the final owned check. execution-record.json records actual terminal outcomes and extraction repair; artifact-manifest.json covers all mathematical source/check/proof outputs and the runner result/sentinel publication scope.

All writes are confined to this own persistent target and the runner-derived attempt directory. No repository/worktree/Git/GitHub lifecycle or mutation, clone, build, tool/judge/config/auth change, Terminal/osascript action, Pro, subagent, live geometry82344, other review seat, actualW77605, or ongoing native job was inspected or operated. All foreign and own log_ref pointers remain opaque. This audit iteration preserves substantive ordinary proof progress while the full original RH research goal stays ACTIVE and unproved.

Project application: two legal directions at the same source

This is a paper application of the full-domain sources in N1–N2 and the complete Q increment suppliers. It introduces no Lean declaration and no literature priority claim. The classical explicit formula and finite-envelope selector are reused. The new interface is the legal backward deletion comparison at the same original source, combined with its forward comparison; its favorable prime-error input remains unproved.

Fix an original canonical source in a qualifying window,

$$N=N_p^{\mathrm{mult}}=p{N}_A^{\mathrm{str}},A=\log N,h=A^{4/5},a=A-h,b=A+h.$$

Keep $f(t)=\log \log t$, $\lambda =f^{\prime }$, $w=-{\lambda }^{\prime }$, $K(t)=\log N_t^{\mathrm{str}}$, and $Q(t)=K(t)-\psi (t)$ from the existing supplier. Require $a>11$, reusing the existing AVB07 activation calculation for $5040\mid N_a^{\mathrm{str}}$. This enlarges only the common unknown start by a fixed elementary activation threshold. N1 gives $A\le R$ and $p>2R$; thus $p$ is absent from both strict cores at $a,b$. The literal integers

$$M_{-}=p{N}_a^{\mathrm{str}},M_{+}=p{N}_b^{\mathrm{str}},Y_{\pm }=\log M_{\pm }$$

satisfy $5040p\mid M_{\pm }$ and $M_{-}\mid N\mid M_{+}$. Full-domain maximality gives $F(M_{\pm })-F(N)\le 0$, for $F(n)=W(n)-f(\log n)$ and $W(n)=\log (\sigma (n)/n)$. The backward comparison uses this complete domain; GA2 alone does not provide it. Neither competitor has to be a canonical maximizer.

Set $m_{-}=\lambda (a)-\lambda (A)>0$ and $m_{+}=\lambda (A)-\lambda (b)>0$. Define the continuous price tent

$$k_A(t)=\{\begin{array}{ll}[\lambda (a)-\lambda (t)]/m_{-},& a\le t\le A,\\ [\lambda (t)-\lambda (b)]/m_{+},& A\le t\le b,\\ 0,& \text{otherwise}.\end{array}\text{\hspace{1em}(PT1)}$$

It is nonnegative, has height $k_A(A)=1$, vanishes at $a,b$, and has $\int |k_A^{\prime }|=2$. Keep both actual terminal concavity costs,

$$C_t(y)=f(t)+\lambda (t)(y-t)-f(y)\ge 0,{\mathcal{C}}_A=C_a(Y_{-})/m_{-}+C_b(Y_{+})/m_{+}.$$

Define the full prime-power and layer responses

$${\mathcal{B}}_{\psi }(A)=\sum _{n\ge 1}\Lambda (n)k_A(n)-{\int }_a^b{k}_A(t)dt,{\mathcal{B}}_Q(A)={\int }_a^b{k}_A(t)dQ(t).$$

All Stieltjes terms are sums on the common finite activation/prime-power support. The exact same-source identity is

$$\boxed{{\mathcal{B}}_{\psi }(A)+{\mathcal{B}}_Q(A)+{\mathcal{C}}_A=\frac{F(M_{-})-F(N)}{m_{-}}+\frac{F(M_{+})-F(N)}{m_{+}}\le 0.}\text{\hspace{1em}(PT2)}$$

To verify the equality, reuse the finite layer-pressure calculation $H(t)=W(N_t^{\mathrm{str}})-\lambda (t)K(t)$. Activation reward equals activation price times layer mass, so $H$ is continuous at each event and $H(t)-H(A)={\int }_A^{t}K(v)w(v)dv$, including $t<A$. With $Y(t)=\log p+K(t)$ and $Y(A)=A$, this gives

$$F(p{N}_t^{\mathrm{str}})-F(N)={\int }_A^t[Y(v)-v]w(v)dv+C_t(Y(t)).$$

The two oriented integrals are

$$I_{-}={\int }_a^A[K(A)-K(v)-(A-v)]w(v)dv,I_{+}={\int }_A^b[K(v)-K(A)-(v-A)]w(v)dv.$$

Finite event-by-event integration gives $I_{-}/m_{-}+I_{+}/m_{+}={\int }_a^b{k}_A(t)[dK(t)-dt]$. Using $dK=d\psi +dQ$ proves PT2. Strict inventory conventions are unchanged. A layer exactly at $A$ is supplied by the forward integral with weight one; atoms at $a,b$ have zero tent weight. The reference prime-power atom at $A$, if present, also has its full $\Lambda$ weight. The forced $(p,1)$ reward cancels between the three literal integers. This cancellation does not assert zero own-price defect for $N$.

Reusing the full Q estimates on the new consumer

Let $d_0=(\sqrt{2}-1)/2=2c_Q$ and let ${\mathcal{F}}_Q$ be the existing seven-term error function in QQE. The pointwise ingredients of that supplier, rather than its forward integrated conclusion alone, give

$$Q(t)-Q(a)=\frac{d_0}{\sqrt{A}}(t-a)+E_A(t),\underset{a\le t\le b}{\sup }|E_A(t)|\le C\frac{h}{\sqrt{A}}{\mathcal{F}}_Q(A).\text{\hspace{1em}(PT3)}$$

Here is the parameter correspondence. Start with $c_0=a$, set $c_{j+1}=c_j+c_j^{4/5}$, and truncate the last interval at $b$. At most three such windows cover $[a,b]$ eventually, with all $c_j\asymp A$; telescope their pointwise increments. For partial windows, retain the supplier’s common enclosing square length $O(A^{7/24})$, the actual square derivative $1/\sqrt{2A}$ and the reference derivative $1/(2\sqrt{A})$. Their difference is $d_0/\sqrt{A}$; derivative variation costs $O(h^2/A^{3/2})$. The first-layer and strict endpoint atoms cost $O(\log A)$. QH8 bounds the absolute signed contribution of every higher layer by the sum of both nonnegative inventories. Relative to $h/\sqrt{A}$, these are exactly the retained QQE errors: square asymptotics $1/\log A$, weight/derivative variation $A^{-1/5}$, GM error, enclosing-length $A^{-1/120}$, atom $A^{-3/10}\log A$, and the two growing-layer errors. Fixed coverage changes only the constant and the common unknown start; no pointwise PNT remainder is subtracted and no layer is omitted.

Integration by parts in PT3, with the two zero tent endpoints, and the existing $|w^{\prime }/w|\le 4/t$ give

$${\int }_a^b{k}_A(t)dt=h[1+O(h/A)],{\mathcal{B}}_Q(A)=[d_0+O({\mathcal{F}}_Q(A))]A^{3/10}.\text{\hspace{1em}(PT4)}$$

Consequently every such original source satisfies the necessary upper comparison

$$\frac{{\mathcal{B}}_{\psi }(A)}{A^{3/10}}\le -d_0+O({\mathcal{F}}_Q(A))-\frac{{\mathcal{C}}_A}{A^{3/10}}.\text{\hspace{1em}(PT5)}$$

A fixed $\epsilon >0$ and a source beyond every required common start with ${\mathcal{B}}_{\psi }(A)\ge -(d_0-\epsilon )A^{3/10}$ would contradict PT5. That favorable lower estimate is not supplied here. In particular, PT5 is an upper comparison, with the same direction issue as SIGN.

The continuous kernel preserves the complete zero response

For ${\hat{k}}_A(s)={\int }_a^b{k}_A(t)t^{s-1}dt$, two integrations by parts give

$$\begin{array}{rl}s(s+1){\hat{k}}_A(s)=& w(a)a^{s+1}/m_{-}+w(b)b^{s+1}/m_{+}\\ & -w(A)A^{s+1}(1/m_{-}+1/m_{+})\\ & +m_{-}^{-1}{\int }_a^A{w}^{\prime }(t)t^{s+1}dt-m_{+}^{-1}{\int }_A^b{w}^{\prime }(t)t^{s+1}dt.\end{array}\text{\hspace{1em}(PT6)}$$

Thus for an actual nontrivial zero $\rho =\sigma +i\gamma$,

$$|{\hat{k}}_A(\rho )|\ll \min \{h{A}^{\sigma -1},A^{\sigma +1}/[h(1+{\gamma }^2)]\}.$$

The classical full-$\Lambda$ explicit formula therefore yields the absolutely convergent, multiplicity-preserving response

$${\mathcal{B}}_{\psi }(A)=-\sum _{\rho }{m}_{\rho }{\hat{k}}_A(\rho )-{\int }_a^b\frac{k_A(t)}{t(t^2-1)}dt.\text{\hspace{1em}(PT7)}$$

The sum runs over distinct nontrivial zeros with their multiplicities $m_{\rho }$. The last term includes all trivial zeros and is $O(h/A^3)$. No real part is replaced by $1/2$, no simplicity or independence assumption is made, and no favorable sign follows from convergence. The analytic gain is elimination of the original $\psi (A)$ starting term and second-order frequency decay; it is not a new proof of the classical explicit formula.

A joint estimate on the original selector, with gaps retained

Take all ordinary prime labels $p_0<\cdots <p_m$ in a qualifying window, with $m\ge 1$, $t_i=\log p_i$, ${\delta }_i=t_{i+1}-t_i$, and the canonical $A_i=\log N_{p_i}^{\mathrm{mult}}$. Put

$${\mathcal{W}}_X(v)=\sum _{i=0}^{m-1}{\delta }_i{A}_i^{-3/10}{k}_{A_i}(v),L_X=\sum _{i=0}^{m-1}{\delta }_i>0.$$

A sufficient, still unproved, joint estimate is: one fixed $\epsilon >0$ has, beyond every common finite start, a qualifying original window for which

$$\sum _{n\ge 1}\Lambda (n){\mathcal{W}}_X(n)-\int {\mathcal{W}}_X(v)dv\ge -(d_0-\epsilon )L_X.\text{\hspace{1em}(PT8)}$$

Indeed the left side equals the nonnegative weighted sum of PT5’s normalized responses. Its upper bound is $[-d_0+O(\sup_{\ell\le A\le R}\mathcal F_Q(A))]L_X -\sum_i\delta_i\mathcal C_{A_i}/A_i^{3/10}$, contradicting PT8 at sufficiently large $\ell$. The nonzero total weight and every same-source terminal cost remain. No marginal optimum is substituted for a joint one.

The existing finite envelope and tie law in N2 determine the monotone selector ${\mathcal{A}}_X(t)=t+\log u_{\min }(t)$, where $u_{\min }$ is the least active core. In the interior it equals ${\lambda }^{-1}(-V_{X,-}^{\prime }(t))$, and ${\mathcal{A}}_X(t_i)=A_i$. The direct core definition also fixes the outer endpoints. Monotonicity sandwiches its whole range between $A_0$ and $A_m$, inside $[\ell ,R]$. Real intervening labels are used for integration only and are not new integer sources. On a fixed-core interval ${\mathcal{A}}_X(t)=t+\log u$. Set $g(A)=A^{-3/10}{\mathcal{B}}_{\psi }(A)$. Kernel continuity and the bound $|{\partial }_A{k}_A(n)|\ll h^{-1}$ on its support, together with $\Lambda (n)\le \log n$, give the coarse sufficient variation bound ${\mathrm{Var}}_{[\ell ,R]}g\ll R\log R$. Monotone selector composition cannot increase this variation. Reusing N1–N2 and GM label spacing, $\max {\delta }_i\le E=e^{-5\ell /12}$ and $\log R=o(\ell )$, the existing Dragomir Corollary 7, equations (5.16)–(5.17), with left tags $t_i$ and integrand $g\circ {\mathcal{A}}_X$, gives

$$\sum _{i=0}^{m-1}{\delta }_{i}g(A_i)={\int }_{t_0}^{t_m}g({\mathcal{A}}_X(t))dt+O(ER\log R).\text{\hspace{1em}(PT9)}$$

The existing label-spacing supplier gives $L_X\sim \ell$ for the full window. Thus the necessary normalized condition $ER\log R=o(L_X)$ holds, and fixed slack can absorb this error. An arbitrary short subblock would require its own relative-error check. No prime-error independence or independent-sample model is used. The variation calculation differentiates the locally finite arithmetic sum, not the infinite zero series.

At each envelope switch $s_j$, write $U_j,V_j$ for the smallest-clock left and largest-clock right cores. The skipped interval is $G_j=(s_j+\log U_j,s_j+\log V_j)$. Include a switch at $t_0$, if present, and exclude a switch at $t_m$, consistently with the canonical left derivative. Piecewise change of variables gives

$${\int }_{t_0}^{t_m}g({\mathcal{A}}_X(t))dt={\int }_{{\mathcal{A}}_X(t_0)}^{{\mathcal{A}}_X(t_m)}g(v)dv-\sum _j{\int }_{G_j}g(v)dv.\text{\hspace{1em}(PT10)}$$

The common envelope imposes the exact switch constraint

$$W(V_j)-W(U_j)=f(s_j+\log V_j)-f(s_j+\log U_j).\text{\hspace{1em}(PT11)}$$

Retain, in addition to PT11, dominance over every core in ${\mathcal{C}}_X$ at each actual switch. Pairwise equality alone does not identify an envelope switch. A normalization check is $A_m-A_0=L_X+{\sum }_j|G_j|$: dense prime labels do not imply dense source clocks. The first integral in PT10 has selected endpoints, and the second retains signed responses over the skipped intervals. A generic free-center mean does not control either selection issue automatically. Their joint estimate, with full envelope dominance, remains the route to the unproved PT8. The completed far tails, existing source construction, and FIB fiber transport are reused unchanged. Neither PT8 nor the original signed finite-head lower bound nor RH is proved by this application.

The actual price-gap application retains whole-switch moments, the original tent gap response and the exact exterior-flux debt, with both original terminal costs separate.