bibkey: fibentropy2026weightedaggregates authors: trureturing research synthesis year: 2026 title: Fixed-modulus weighted aggregates, source entropy and moving product windows doi: null url: https://arxiv.org/abs/2607.00592v1 claim: “Smooth-number character estimates impose conductor, smoothness and coefficient conditions not established for the current Fibonacci target. Mellin inversion preserves moving product windows and exposes a weighted spectral interface, but supplies no bound at the Robin budget.” strata_touched: [] license: citation-only triage: anchor

Fixed-modulus weighted aggregates and the retained arithmetic target

This note distinguishes checked classical and recent source results from their proposed use in the FIB theory. Sections 223–224 there contain the full increment-law Rényi profile and its entropy and collision consequences. Section 225 combines the maximum source atom with short-factor multiplicative energy to obstruct every Hölder separation of the full-law residue majorant. Those are paper derivations without Lean verification or an originality claim. Their proofs are not duplicated here.

The Bohr phase-box note records the pointwise character-decay obstruction and exact moving-interval identity; the complementary-divisor note records the actual arithmetic kernel. This note specifies the conditions for importing aggregate literature and a Mellin interface that retains the product window. None of these interfaces establishes the needed actual FIB Robin budget.

1. The actual scale and the object requiring an estimate

Keep the prime-index family

$$V=F_r,I=[\lceil V/10\rceil ,\lfloor V/5\rfloor ]\cap \mathbb{Z},N_g=1+Vg,A=\min N_g,X=\max N_g,$$ $$y=\log A,\ell =\log y,s=y\ell ,R=y/{\ell }^2,H=e^{aR},D=X/H,Q=\phi (V).$$ Here $a>b_2={\pi }^2/6$ is fixed, $\log X=y+O(1)$, $\log Q=y/2+O(1)$, and $H^k<V$ eventually for every fixed $k$. The lower-cutoff candidate construction uses smoothness $w=s+1\asymp y\ell$. For every actual $1\le h<H$, the core scale $x\asymp X/h$ satisfies $\log x/\log V\to 2$, while $w\asymp \log x\log \log x$.

These smoothness parameters describe a proposed smooth-core decomposition. The full increment law below is not supported on smooth integers; applying a smooth-support theorem also requires a valid decomposition and control of its complementary contribution.

The actual increment law is

$$Z(n)=\sigma (n)/n,b_s(p^j)=Z(p^j{)}^s-Z(p^{j-1}{)}^s,b_s(1)=1,$$ $$U_p(s)=1+\sum _{j\ge 1}\frac{b_s(p^j)}{p^j},U(s)=\prod _p{U}_p(s),{\mu }_s(d)=\frac{b_s(d)}{dU(s)}.$$ Let ${\eta }_s$ be this probability conditioned on $(d,V)=1$, and ${\nu }_s$ its pushforward to $G=(\mathbb{Z}/V{)}^{\times }$. These are full, unbounded-source probabilities. A restriction to $d\le X$, an actual hit, or a cap on the same product $dh$ is a different law.

For nonnegative $a(d)$, the actual large-divisor sum is

$$S_a=\sum _{\begin{array}{c}d>D\\ \exists g\in I:d\mid N_g\end{array}}a(d)=\frac{1}{Q}\sum _{\chi \mathrm{mod}V}\sum _{\begin{array}{c}1\le h<H\\ (h,V)=1\end{array}}\chi (h)A_{\chi }(J_h),$$ $$J_h=(D,X]\cap [A/h,X/h]\cap \mathbb{N},A_{\chi }(J)=\sum _{d\in J,(d,V)=1}a(d)\chi (d).\text{\hspace{1em}(1)}$$ The actual kernel has $a(d)=(d/X){\mu }_s(d)=b_s(d)/(XU(s))$. A hit divisor is a unit, and $d>D>V>|I|-1$ eventually, so it hits at most one actual integer. This is why the pair identity counts the original sum exactly. The principal term and the signed nonprincipal aggregate must both be retained. No uniform pointwise character-decay claim is used below.

2. Primary-source results: what transfers and what does not

2.1 An arbitrary-weight smooth large sieve, but at much smaller conductors

S. Drappeau, A. Granville and X. Shao, Smooth-supported multiplicative functions in arithmetic progressions beyond the $x^{1/2}$-barrier, arXiv 1704.04831v2, updated 2017-06-15.

Original: arXiv text, version 2. Checked §5, Theorem 5.1, and §5.3, Theorem 5.6.

Theorem 5.1 gives absolute constants $C,c>0$ such that, for sufficiently large $(\log x{)}^C\le Y\le x$ and

$$Q_0=\min \{Y^c,\exp (c\log x/\log \log x)\},$$

one has, for arbitrary complex coefficients $a_n$,

$$\sum _{q\le Q_0}\sum _{\chi \mathrm{mod}q}^{\ast }{\left|\sum _{n\le x,P^{+}(n)\le Y}{a}_n\chi (n)\right|}^2\ll \Psi (x,Y)\sum _{n\le x,P^{+}(n)\le Y}|a_n{|}^2.\text{\hspace{1em}(2)}$$ The star means primitive characters. This arbitrary-coefficient clause is useful: growing coefficients are not excluded from (2) merely because they grow; their actual squared norm appears on the right. However, the FIB modulus $V=x^{1/2+o(1)}$ is much larger than $Y^c$ when $Y\asymp \log x\log \log x$. Moreover, the theorem does not promise $C\le 1$, so its smoothness hypothesis cannot simply be asserted at the current cutoff.

Theorem 5.6 permits $Q_0=x^c$, but inserts $q^{-1/2}$ before the character sum and still requires $Y\ge (\log x{)}^C$. Neither the unspecified small exponent $c$, nor the conductor weight, can be dropped. Extracting one fixed modulus from either nonnegative sum is legitimate if all its hypotheses hold; it does not eliminate these losses or provide cancellation between its characters. The target also includes imprimitive characters at $V$; using a primitive-character estimate for that target requires a conductor decomposition that retains the relevant nonunit restrictions.

For comparison, §1, the definition of class $\mathcal{C}$ requires $|{\Lambda }_f(n)|\le \Lambda (n)$ and implies $|f(n)|\le 1$. The progression theorem 1.2 concerns that class and averages moduli. It must not be confused with the arbitrary coefficients of Theorem 5.1. The unnormalised $b_s$ has $b_s(2)=(3/2{)}^s-1$, so it is outside this bounded class. Dividing by a global constant other than 1 does not preserve multiplicativity or $f(1)=1$.

The later paper I. E. Shparlinski, Character sums with smooth numbers, 1705.10148v2 (2017-06-11), original text, Lemma 2.2 and Theorem 1.1, uses this sieve and sharpens exceptional-pair counts. It retains the small-modulus and smoothness restrictions. It is not an estimate for all characters of the present fixed $V$.

2.2 A recent fixed-modulus low-moment theorem with explicitly incompatible scales

Seth Hardy and Max Wenqiang Xu, Character sums over smooth numbers, 2607.00592v1, 2026-07-01.

Original: arXiv text, version 1. Checked Theorems 1–3, the discussion following Theorem 1, and §4.1, equations (11)–(12).

Theorem 1 averages all characters at one fixed modulus, and permits composite moduli. Its assumptions include

$$2\le x\le q,(\log x{)}^6\le Y\le x^{1/(32\log \log x)}.$$ Writing $u=\log x/\log Y$ and $\alpha (x,Y)$ for the smooth-number saddle point, set

$$\mathcal{S}=(\log x{)}^{3/2}(\log Y{)}^{1/2}\left(Y^{1/2-\alpha (x,Y)}+\exp \left[-\frac{u}{4}{\left(\frac{\log \min (q,x^{3/2})}{\log x}-1\right)}^2\right]\right).$$ Then

$$\frac{1}{\phi (q)}\sum _{\chi \mathrm{mod}q}\left|\sum _{n\le x,P^{+}(n)\le Y}\chi (n)\right|\ll \sqrt{\Psi (x,Y)\mathcal{S}}.\text{\hspace{1em}(3)}$$ For every function $\epsilon (x)\to \infty$, the theorem gives $\mathcal{S}=o_{\epsilon }(1)$ uniformly when, in addition,

$$\frac{\log q}{\log x}\ge 1+\sqrt{\frac{6\log \log x+2\log \log Y+\epsilon (x)}{u}}.$$

Theorem 2 allows a completely multiplicative twist $h$ satisfying $|h(p)|\le 1$. It is not an arbitrary-weight theorem. Section 4.1 identifies the needed orthogonality/mean-square hypothesis at polynomial length at most the family-size parameter.

The present core scale has $x=V^{2-o(1)}>V$, while its cutoff $Y\asymp \log x\log \log x$ is below $(\log x{)}^6$. Its increment weight is also not the allowed twist. Thus three independent conditions fail. Theorem 3’s average over primitive characters and varying $q\le Q_0$ has a family-size parameter $Q_0^2$; this is not a replacement for the fixed-modulus character family.

The paper itself identifies the small-smoothness regime $Y\le (\log x{)}^6$ as a further problem after Theorem 1. Its methods therefore suggest a direction but do not close the present interface.

2.3 Older smooth-number mean squares do not silently become fixed-modulus estimates

Adam J. Harper, Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers, 1208.5992v1, 2012-08-29.

Original: arXiv text, version 1. Checked Theorems 1–3.

Theorem 2, for an absolute large $K$, $(\log x{)}^K\le Y\le x$, and $Q_0\le \Psi (x,Y)$, gives

$$\sum _{q\le Q_0}\sum _{(b,q)=1}{\left|\Psi (x,Y;q,b)-\frac{{\Psi }_q(x,Y)}{\phi (q)}\right|}^2\ll \Psi (x,Y{)}^2\left(e^{-cu/{\log }^2(u+1)}+Y^{-c}\right)+\Psi (x,Y)Q_0.\text{\hspace{1em}(4)}$$ It is an actual aggregate estimate, but averages moduli and has an unproved smoothness premise at the present cutoff. Theorem 3 instead controls individual characters with conductor at most $Y^b$, alongside a specified zero-free condition; that does not cover the current large conductors. These are accurate theorem boundaries, not a claim that no suitable fixed-modulus result exists anywhere.

2.4 Weighted multiplicative energy is a useful norm, not freely chosen weights

Régis de la Bretèche, Marc Munsch and Gérald Tenenbaum, Small Gál sums and applications, 1906.12203v4, 2020-07-10.

Original: arXiv text, version 4. Checked §1.2, equations (1.3)–(1.6), Theorem 1.2.

For nonnegative $c_n$, ${\sum }_{n\le N}{c}_n=1$, define

$$\mathcal{E}(c;N)=\sum _{m\le N^2}{\left(\sum _{dt=m,d,t\le N}{c}_d{c}_t\right)}^2,{\mathcal{E}}_N=\underset{c}{\inf }{N}^2\mathcal{E}(c;N).$$ With ${\delta }_{\mathrm{Gal}}=1-(1+\log \log 2)/\log 2\approx 0.08607$, their theorem states

$$(\log N{)}^{{\delta }_{\mathrm{Gal}}}(\log \log N{)}^{3/2}\ll {\mathcal{E}}_N\ll (\log N{)}^{{\delta }_{\mathrm{Gal}}}(\log \log N{)}^6.\text{\hspace{1em}(5)}$$ Thus carefully chosen nonnegative short-factor weights can improve on the all-ones energy, but not remove all logarithmic cost. The upper bound asserts existence of good weights, not small energy of the actual core law or every prescribed set. Replacing the target’s counting weight by a favourable $c_h$ requires a domination, a partition, or an amplification identity and must pay for the compensating factor. With $N^2<V$, integer energy becomes an exact fixed-modulus fourth moment, as in §3 below; that transfer itself does not need $V$ prime.

A nearby composite-modulus result is Bryce Kerr, Moments of character sums to composite modulus, 1904.04578v1, 2019-04-09, original text, §2, Theorem 2. It averages the shift $\lambda \mathrm{mod}q$ of ${\sum }_{v\le N}\chi (\lambda +v)$ for one primitive character, with explicit square/cubefull-part factors. That is a different average from the current sum over all characters of one fixed modulus; its shift variable cannot be replaced by the moving source interval $J_h$.

2.5 Standard entropy results and exact migration conditions

Tim van Erven and Peter Harremoës, Rényi Divergence and Kullback–Leibler Divergence, 1206.2459v2, 2014-04-24.

Original: arXiv text, version 2. Checked Introduction equations (2), (4), Theorem 9 (data processing), and Theorem 28, equations (42)–(43) (finite/countable additivity, with the stated exception at order zero).

For a probability on a finite group of order $Q$, $D_{\alpha }(\nu \Vert \mathrm{Unif})=\log Q-H_{\alpha }(\nu )$. This reference supplies the standard entropy/divergence framework; it does not itself estimate the present prime-valuation law. A hypothetical “uniform probability on all integers” is not a valid reference law. The nonnegative Euler sums and the order-dependent valuation tails in theory §223 justify the infinite-source limit without such an object.

Jonathan Hermon and Sam Olesker-Taylor, Cutoff for Almost All Random Walks on Abelian Groups, 2102.02809v2, 2025-10-10, original text, §2.6, lower-bound proof of Theorem 2.5, supplies the finite-image entropy mechanism with generators held fixed. If a source law $\rho$ puts most probability on $E_L=\{w:\rho (w)\ge e^L/Q\}$, then $|E_L|\le Q{e}^{-L}$ and every image has size at most this, hence

$$\Vert {\pi }_{\ast }\rho -\mathrm{Unif}{\Vert }_{\mathrm{TV}}\ge \rho (E_L)-e^{-L}.\text{\hspace{1em}(6)}$$ Theory §§223–224 supply source power-sum and entropy estimates for the specific increment law. This closes the source-concentration premise left open in the earlier Bohr note, without importing a random-generator mixing upper bound. The source remains the full unit-conditioned law; actual-hit conditioning or a product cap is a different operation.

3. An exact way to keep moving product windows while using short-factor energy

The existing fixed-rectangle fourth moment extends uniformly to a continuous Mellin twist. For a unit subset $\mathcal{H}\subseteq [1,H]$ and real $t$, define

$$B_{\chi }(t)=\sum _{h\in \mathcal{H}}\chi (h)h^{-it}.$$ If $H^2<V$, then character orthogonality gives

$$\frac{1}{Q}\sum _{\chi }|B_{\chi }(t){|}^4=\#\{h_1{h}_2=h_3{h}_4:h_i\in \mathcal{H}\}\le 2H^2(1+\log H).\text{\hspace{1em}(7)}$$ Indeed modular product equality becomes integer equality, and its continuous twist is then exactly one. The gcd parametrization $h_1=gu,h_3=gv,h_2=kv,h_4=ku$, $(u,v)=1$, proves the last bound. Nonnegative coefficients $c_h\le 1$ can be inserted without increasing it. For arbitrary complex coefficients, the exact fourth moment is instead

$$\frac{1}{Q}\sum _{\chi }{\left|\sum _{h\in \mathcal{H}}{c}_h\chi (h)h^{-it}\right|}^4=\sum _m{\left|\sum _{h_1{h}_2=m}{c}_{h_1}{c}_{h_2}\right|}^2,$$

with coefficients zero outside $\mathcal{H}$. This identity also needs only $H^2<V$, not a prime modulus. It does not supply a small energy for prescribed coefficients.

Take a nonnegative $W\in C_c^{\infty }((0,\infty ))$ with $W(u)\ge 1$ on $[A/X,1]$, and let

$$\tilde{W}(it)={\int }_0^{\infty }W(u)u^{it-1}du,A_{\chi }(t)=\sum _{D<d\le X,(d,V)=1}a(d)\chi (d)d^{-it}.$$ The original moving-window sum has the legitimate positive majorant

$$S_a\le S_W:=\sum _{\begin{array}{c}D<d\le X,1\le h<H\\ dh\equiv 1(\mathrm{mod}V)\end{array}}a(d)W(dh/X).$$ Mellin inversion gives the exact identity for this majorant

$$S_W=\frac{1}{2\pi Q}{\int }_{\mathbb{R}}\tilde{W}(it)X^{it}\sum _{\chi \mathrm{mod}V}{A}_{\chi }(t)B_{\chi }(t)dt,\text{\hspace{1em}(8)}$$ where $\mathcal{H}=\{1\le h<H:(h,V)=1\}$. The sums are finite and $\tilde{W}$ decays rapidly, so the interchange is justified. Let $M_W$ be its principal-character term. Hölder and (7) yield

$$|S_W-M_W|\le \frac{[2H^2(1+\log H){]}^{1/4}}{2\pi }{\int }_{\mathbb{R}}|\tilde{W}(it)|{\left(\frac{1}{Q}\sum _{\chi \ne 1}|A_{\chi }(t){|}^{4/3}\right)}^{3/4}dt.\text{\hspace{1em}(9)}$$

This explicitly accommodates the moving inequalities $A\le dh\le X$; it does not replace them by an unrelated rectangle. It exposes a concrete weighted $L^{4/3}$ spectral quantity, averaged also against a controlled Mellin kernel. To use a sharper signed correlation one can work with (8) before taking absolute values.

The price of smoothing remains real: $S_W-S_a$ is the sum, over the same congruence pairs, of the nonnegative terms

$$a(d)\left[W(dh/X)-1_{[A/X,1]}(dh/X)\right].$$

If $W$ is chosen equal to one on the original interval, this excess is supported outside it. Narrowing the edge region requires quantitative control of the resulting Mellin norm; no uniform bound under such sharpening is assumed. Conditions such as a joint cap on $dh$ still cannot be factorized into an independent $A_{\chi }{B}_{\chi }$ without an additional representation. Dropping them supplies an upper bound, which may be too large. No bound at the Robin budget is proved by (9).

4. What remains to be estimated

The full-law Rényi asymptotic and near-maximal total variation in theory §§223–224 concern the complete source or its unit-conditioned residue image. They do not locate its large atoms relative to the prescribed inverse residues. Unit conditioning deletes the local coordinates for primes dividing $V$; conditioning on an actual hit does not have that product form. The raw and unit-conditioned transforms are related by

$${\hat{\mu }}_s(\chi )=c_V{\hat{\nu }}_s(\chi ),c_V={\mu }_s((d,V)=1),$$

and the mass factor must be retained when returning to the raw target.

Theory §225 uses the maximum source atom and a short-factor $L^1$ lower bound, obtained from second and fourth moments through multiplicative energy, to show that even the exact full-law Hölder norm product, optimized over every exponent, exceeds the needed budget exponentially in $R$. That is a lower bound on an upper-bound certificate, not on the actual correlation. It does not automatically apply to the truncated, Mellin-twisted $A_{\chi }(t)$ in (8)–(9), whose coefficients retain different information.

For the actual application one must control the principal term $M_W$, the nonprincipal contribution in (8), and any smoothing or discarded-filter cost after multiplication by

$$\Lambda =\frac{XU(s)}{(e^{\gamma }\ell {)}^s}.$$

Equation (9) is a sufficient route only if its specific weighted spectral integral and these costs are bounded at that scale. The references above do not supply that bound under the current simultaneous parameter conditions. The signed identity (8), a direct weighted intersection estimate, or an additional valid representation of product-dependent filters remains an open interface.

The source locators above refer to the specified arXiv versions. They support the displayed theorem scopes; they do not establish exhaustive prior-art coverage or originality of the arithmetic application. The Mellin and energy formulas are standard mathematical operations applied to the stated finite sums. No full-FIB Robin estimate or unrestricted RH criterion is settled here.

5. Exact arithmetic smoothing, size tilts and the remaining endpoint

Theory §§226–228 develop the same weighted kernel further. The arithmetic endpoints satisfy $A\equiv X\equiv 1(\mathrm{mod}V)$. A logarithmic cutoff can put its two smooth transitions entirely in the adjacent gaps of that residue class. With $\eta =V/(8X)$, the specified nonnegative cutoff $W_{\eta }$ equals the hard indicator on every actual product $dh\equiv 1(\mathrm{mod}V)$. This exact equality is specific to these arithmetic gaps; it is not a general assertion that smoothing arbitrary windows is free. Its Fourier one-norm is $O(y)$, and the associated frequency scale is ${\eta }^{-1}\asymp V$.

Keeping the existing weight $d/X$ also changes the complementary transform: the identity

$$\frac{d}{X}{W}_{\eta }(\log (dh/X))=\frac{1}{h}{k}_{\eta }(\log (dh/X)),k_{\eta }(u)=e^u{W}_{\eta }(u),$$

places the reciprocal weight $1/h$ on the short factor. The resulting transform

$$C_H(\chi ,\tau )=\sum _{1\le h<H,(h,V)=1}{h}^{-1+i\tau }\chi (h)$$

has fourth character moment at most $\zeta (2{)}^4/\zeta (4)$ for every real $\tau$ when $H^2<V$. This is the classical equal-product energy calculation with reciprocal coefficients and the convergent divisor-square Dirichlet series. Neither operation is a new general Fourier or zeta-function theorem. Their combined application to the actual weighted kernel is proved in §226. The hard principal term is at most $c_D/Q$, and the smoothed principal term is at most $e^{2\eta }{c}_D/Q$, where $c_D={\mu }_s(D<d\le X,(d,V)=1)$. The two principal terms need not equal each other even though the full actual sums do.

The source estimate in §227 concerns a different object from a uniform random integer or an ordinary smooth-number count. For each fixed $q\ge 1$, it explicitly uses

$$P_q(d)=\frac{{\mu }_s(d{)}^q}{{\sum }_n{\mu }_s(n{)}^q}$$

and, when specified, conditions that source on $(d,V)=1$. Under small real size tilts, the local exponent weights are proportional to $(b_s(p^j)/p^j{)}^q{p}^{vj}$. The manuscript estimates all actual integer prime-power exponents before reducing the principal prime range to a logistic kernel. At zero tilt, the mean and variance of $P_q$ and its unit-conditioned version are

$$\mathbb{E}\log d=y+\frac{2b_2}{q^2}\frac{y}{{\ell }^2}+O_q(y/{\ell }^3),\mathrm{Var}(\log d)=\frac{y}{q}+O_q(y/\ell ).$$

Here $\ell =\log y$. The strong prime-density input is the one already located in Weingartner’s versioned author manuscript, equation (9), and used in theory §223. Weingartner’s stated moment expansion is not itself a theorem about this power-tilted increment source. The source cumulants and moving-window rates are separate repository paper deductions; differentiating a prior asymptotic remainder would not justify them.

For fixed $q\ge 1$, $\beta \ge 0$, and $a>0$, §227 obtains

$$\log {\mathbb{E}}_{P_q}\left[(d/X{)}^{\beta }{1}_{D<d\le X}{1}_{(d,V)=1}\right]=-\frac{2b_2^2}{q^3}\frac{y}{{\ell }^4}+O_{q,\beta ,a}(y/{\ell }^5).$$

The negative exponential tilt supplies the upper estimate. The lower estimate uses a second, explicitly chosen tilt whose mean lies inside the actual integer window, followed by Chebyshev concentration and the exact change-of-measure factor. No central or local limit theorem is assumed. Removing the unit restriction, or working under the explicitly unit-conditioned source, gives the same displayed rate. This is a source-size statement; an additional actual-hit condition or a cap on the product $dh$ needs its own analysis.

Thus the size-window cost is $o(R)$, where $R=y/{\ell }^2$. It preserves the first two logarithmic terms of each fixed $q$-power sum. The $y/{\ell }^4$ coefficient of the size loss does not determine a full third coefficient for that sum, since the prior complete-source expansion still has an error of that order.

Section 228 applies the standard finite inverse Hausdorff–Young inequality with normalized counting on the characters and unnormalized counting on $G$. An author-hosted source is Terence Tao, The Fourier transform (6 April 2009), equation (6): on a compact abelian group with probability Haar measure, the transform maps $L^p$ to counting-measure ${\ell }^{p^{\prime }}$ with norm at most one for $1\le p\le 2$. The preceding paragraph explicitly derives this by Riesz–Thorin interpolation from the $L^1\to {\ell }^{\infty }$ bound and Parseval. Apply it to the finite character group, whose dual identifies with $G$; reversing the Fourier sign only permutes the dual coordinates. This supplies the required normalization and exponent range, not an arithmetic estimate.

In the common frequency interval $|\tau |\le 1/(2aR)$, each residue’s positive divisor masses lie in one short phase arc. The actual truncated source power sum then lower-bounds its Fourier norm, with a separately controlled principal-coordinate removal. The arithmetic smoothing kernel has positive mass at frequency zero uniformly in the shrinking transition width.

For every fixed $\epsilon >0$, these deductions yield a positive-norm certificate obstruction uniformly over $p\ge 1+\epsilon$: put

$$q_{\epsilon }=\max \{2,(1+\epsilon )/\epsilon \}.$$

The normalized logarithmic lower limit of the separated certificate is at least $b_2/q_{\epsilon }^2$. This includes every fixed $p>1$ and choices varying with scale while staying that fixed distance from one. It does not cover $p=1$ or $p(y)\downarrow 1$, because the source estimates are not uniform for unbounded $q$. The $4/3$–$4$ instance gives $b_2/16$.

This obstruction retains the exact arithmetic product window and reciprocal cofactor; it does not simply transfer the full-law result from §225. It still lower-bounds an upper-bound expression formed after absolute values and norm separation, not the signed source/cofactor correlation or the actual Robin kernel. The signed representation, the one-norm endpoint, and joint loss counts remain unresolved routes. None of the new estimates settles the complete FIB family or RH. These are paper deductions, with no new Lean verification or claim of exhaustive prior-art coverage.

6. Small rational ratios, the Fourier endpoint and one possible exception

Theory §§230–231 use a uniform constraint on integers with small loss to extend the preceding analysis. Keep the actual prime-index FIB family and source of §1, and put

$$m_s=\underset{d\ge 1}{\max }{\mu }_s(d),J_s(d)=\log \frac{m_s}{{\mu }_s(d)}\ge 0.$$

The maximum exists by theory §218. Its reference integer is

$$n_y=\prod _{p\le y}{p}^{a_p},a_p=\{\begin{array}{ll}\lceil \log s/\log p\rceil ,& p\le \sqrt{s},\\ 1,& \sqrt{s}<p\le y.\end{array}$$

For every fixed $K>0$, theory §230 proves the uniform implication

$$J_s(d)\le KR,\frac{d}{n_y}=\frac{u}{v}\text{ in lowest positive terms}⟹\log u+\log v=O_K(y/\sqrt{\ell })=o(y).$$

This holds for every integer in the specified loss set. The proof treats all small-prime valuations, changes of occupancy in the central prime range, and the entire large-prime tail. It retains the same reference $n_y$ for all integers. Since every prime factor of $n_y$ is at most $y$ and every prime factor of $V$ is eventually greater than $2y$, this reference is a unit modulo $V$.

For the actual truncated transform

$$F_D(\chi ,\tau )=\sum _{\begin{array}{c}D<d\le X\\ (d,V)=1\end{array}}{\mu }_s(d)\chi (d)d^{i\tau },$$

let $\Vert \cdot {\Vert }_{p,\ast }$ use the $Q$-point character probability measure with the principal coordinate set to zero. Fix $a>0$, $H=e^{aR}$ and $D=X/H$. With

$$\kappa =\frac{1}{10},g_{\kappa }=\frac{1}{2}+\frac{2}{3}{e}^{-\kappa }>1,c_{\kappa }=2\kappa \log g_{\kappa }>0,$$

theory §230 obtains, for every fixed $0<c<c_{\kappa }$, a threshold independent of $\tau$ such that

$$\Vert F_D(\cdot ,\tau ){\Vert }_{1,\ast }\ge m_s{e}^{cR}(\tau \in \mathbb{R}).$$

The new argument constructs a bounded test on the actual character group, prime by prime from degree-eight polynomials. Its constant and linear coefficients create a positive gain on a finite integer cube. Disjoint buffer primes place that cube’s mean inside the actual size window, and a variance bound retains asymptotically all of its auxiliary positive weight. The small rational ratios turn the relevant modular matches into integer equalities, cancelling the continuous phases exactly. Higher coefficients have signs; their complete absolute contribution is bounded before deriving the norm lower bound. This supplies uniformity for all real frequencies without assuming that prime character phases are independent.

The existing Fejér card records the historical attribution of the classical positive kernel. Section 230 proves the needed normalized-square and convolution identities directly. This citation does not assert that the original article was inspected or that it contains the new arithmetic application.

For the reciprocal cofactor $C_H$ defined in §5, the same character probability measure gives $\Vert C_H(\cdot ,\tau ){\Vert }_{1,\ast }\ge 1/2$ eventually, uniformly in $\tau$. Norm monotonicity therefore yields

$$\Vert F_D(\cdot ,\tau ){\Vert }_{p,\ast }\Vert C_H(\cdot ,\tau ){\Vert }_{p^{\prime },\ast }\ge \frac{1}{2}{m}_s{e}^{cR}(1\le p\le \infty ),$$

where $p^{\prime }$ is conjugate to $p$. One threshold covers all $p$ and $\tau$, including choices of $p$ depending on scale or frequency. This separate construction covers the endpoint left outside the fixed-$\epsilon$ argument in §5; it does not take an uncontrolled $q\to \infty$ limit. Since $\log (\Lambda m_s)=o(R)$, the normalized positive norm product grows exponentially. The same conclusion applies to a nonnegative shared weight $w_y$ when an interval $I_y$ has length $\asymp 1/R$ and ${\int }_{I_y}{w}_y(\tau )d\tau \ge e^{-o(R)}$, as for the specified arithmetic-gap kernel. These are lower bounds on separated positive upper-bound certificates. They do not lower-bound the signed arithmetic correlation or an additional joint product filter.

Theory §231 uses the same uniform rational-ratio bound to compare actual hits. Fix $a>b_2$, $K=8b_2$ and define

$$E_y=\{g\in I:\exists d\mid N_g,d>D,J_s(d)\le KR\}.$$

If $d_i{h}_i=N_{g_i}$ are two such hits and $d_i/n_y=u_i/v_i$, their strict cofactor bounds are $1\le h_i<H$. Cancelling the same unit $n_y$ in the two actual congruences gives

$$u_1{h}_1{v}_2\equiv u_2{h}_2{v}_1(\mathrm{mod}V).$$

Both positive sides have uniformly $o(y)$ logarithm and hence are strictly below $V$, whose logarithm is $y/2+O(1)$. They are equal as integers, so $N_{g_1}=N_{g_2}$. Consequently $|E_y|\le 1$ eventually. The possible single integer may carry many small-loss large divisors; this conclusion counts actual integers, not divisors. It uses the uniform ratio bound and unit reference, without depending on the Fourier endpoint theorem.

The complement is controlled by its full positive moment. Write $T_I=|I|$ and $t=e^{\gamma }\ell$, so $\Lambda =XU(s)/t^s$. The exact divisor-increment expansion and the unique-hit property for $d>D$ give

$$\sum _{g\in I\setminus E_y}{\left(\frac{Z(N_g)}{t}\right)}^s\le \Lambda \left(\frac{T_I}{X}+\frac{D}{X}+{\mu }_s\{J_s>KR\}\right).$$

The fixed $q=3/4$ complete-source power sum, through the exact identity ${\mathbb{E}}_{{\mu }_s}{e}^{J_s/4}=m_s^{1/4}{S}_{3/4}({\mu }_s)$, gives

$${\mu }_s\{J_s>8b_{2}R\}\le e^{-(5b_2/3)R+o(R)}.$$

Thus the complement moment is at most $e^{-c_{a}R+o(R)}\to 0$, where $c_a=\min (a-b_2,2b_2/3)>0$. For each integer in that complement, this yields $Z(N_g)<t\le e^{\gamma }\log \log N_g$. The resulting statement includes equality among the possible violations:

$$\#\{g\in I:Z(N_g)\ge e^{\gamma }\log \log N_g\}\le 1$$

for every sufficiently large prime-index window specified in §1.

These are repository paper deductions, with no new Lean or kernel verification and no verdict of literature originality. No effective numerical starting index is supplied. The possible exceptional integer has not been excluded or proved safe; there can be a different candidate in each window. The result does not establish Robin for the complete FIB family, and no bridge capturing every arbitrary-integer Robin counterexample has been proved. RH remains unresolved.