The shift sign in the weighted-dilation analytic core

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The smooth-successor route uses Sections 3–5 of the weighted-dilation manuscript only through the statement of its Theorem 5.1, premise (P) of the Type II estimate for 2u-1. That statement allows every integer shift $0<|k|\le L^C$, and the manuscript's own Type II estimate for $2u+1$ already applies it at both signs, since its $m'-m=2ek$ takes either sign. So the change from $2u+1$ to $2u-1$ needs no new estimate from Sections 3–5. This page records, as a check of that statement against its proof, every place where $k$ enters the proof and why none depends on its sign. It then records how far this reading checked the proof of (P), re-derives in outline its most intricate step, and states what remains. The observation that Sections 2–5 are used as stated is Li and Zhuang's; the inventory, the checks and the re-derivation below are Plasma AI's, author-recorded.

Notation follows the manuscript: $D$ is the product of the shared and free labels of an edge, $\sigma_i=\pm1$ marks forward and adjoint edges, $N=2R$ with $R=\lceil L^{0.52}\rceil$ is the length of the moment, and $P_*=\exp(L^c)$ is the smallest big-group prime.

Where k enters

1. The physical operator (Definition 3.3). An edge moves the position $n$ to $n+kD$ with $0<|k|\le L^{C_2}$, and its adjoint moves it by $-kD$. Positions range over all of $\mathbb Z$ with counting measure, and the absolute bounds of Lemma 3.4 use only divisibility and translation.
2. Path offsets (3.8). A path's offsets satisfy $h_{i+1}-h_i=\sigma_ikD_i$, so the sign of $k$ enters only through the signs of these steps. The paths of $(\mathcal A\mathcal A^*)^R$ already contain steps of both signs, from forward and adjoint edges, and every later use of the offsets is through their residues, translations and the phase $e(\theta h_N)$.
3. Residue replacement (Lemma 3.6). The average over the root uses the offsets only through their residues modulo group primes and through counts of Chinese-remainder classes. The return $h_N=0$ is imposed by $\int_0^1e(\theta h_N)\,d\theta$.
4. The memory construction (Lemma 3.7 and the edge operation). Residues are translated by $\Delta=\sigma_ikD$. Compatibility uses that every group prime exceeds $|k|$: a prime active at both ends of an edge divides $kD$, hence $D$.
5. Rare transfers (Lemma 3.9). A promotion requires $r+\sigma_ikD_{\mathcal B}D_{\mathcal S}\equiv0\pmod p$ for a big prime $p>|k|$. Since $F$, $k$ and $D_{\mathcal B}$ are units modulo $p$ and the $\mathcal D$ products of omitted small labels are distinct integers below $p$, at most one omission choice allows a given promotion. Only the invertibility of $k$ modulo $p$ is used.
6. Clean edges (Lemma 3.10). After Fourier transformation in the small residue torus and the pending residues, translation by $\sigma_ikd'D_{\mathcal B}$ becomes the phase $e(\sigma_i\Theta D_{\mathcal B})$, with a real $\Theta$ depending on $k$, $d'$, the memory primes, the frequency and $\theta$. Hypothesis (3.6) is a supremum over all real $\Theta$, so $k$, whatever its sign, is absorbed into $\Theta$.
7. Endpoint pairing (Corollary 3.11). The block length $H$ bounds $|kD|$.
8. Ideal kernels (Section 4). The shift does not occur; the norm bound (4.1) is uniform in $\Theta\in\mathbb R$ and $|\zeta|\le L^{C_4}$.
9. The physical lift (Section 5). The sum (5.5) runs over $z>0$ with $z+k>0$. In the lift (5.7), $w'=w+kD_C$ is positive because $|kD_C|=x^{o(1)}$ while $w=x^{1+o(1)}$. The free-label exclusion writes $w'=p'(v+kD_C/p')$ with $p'\mid D_C$, again for either sign.
10. Comparison shifts (5.9)–(5.10). The shift is $kD_AD_B$ and the multiplier is $M(\theta)=\mathbb E\,b_1(D_A)b_2(D_B)e(-k\theta D_AD_B)$. Its minor-arc bound applies Dirichlet's approximation to $k\theta$ and the bilinear Lemma 4.2. Dividing by $k$ places $\theta$ within $L^{O(1)}/H'$ of a fraction with denominator at most $|k|r'$; the sign of $k$ changes only the numerator.
11. Major arcs and the Mellin argument (5.11)–(5.17), Lemmas 5.2–5.4. These concern the first endpoint alone; $k$ does not occur.

Every step is therefore either independent of $k$ or uses only $|k|\le L^{O(1)}$, translation by $kD$, positivity of shifted positions that are $x^{1+o(1)}$ in size, or invertibility of $k$ modulo primes above $|k|$. These transformations preserve the proposed proof for both signs. No additional sign-specific estimate was identified in Sections 3--5. This does not establish the estimates themselves by sign symmetry.

What this reading checked

The author examination of 2026-10-15 (UTC) read Sections 3–5 in full in the linked source's library transcription, at proof depth. The following steps were checked in detail, and no error was found in them.

• Corollary 3.11 from Theorem 3.5. Spectral Hölder for $\mathcal A\mathcal A^*$, the trace bound for the positive semidefinite block matrix that uses only mark independence of the first vector, the block length $\log H=O_J(TL^d)$, and the Hölder summation with exponents $2$, $2R/(R-1)$ and $2R$.
• Lemma 3.6. The Bonferroni truncation at degree of order $NT$, the count $\exp(O(h(L^d+\log N+\log s)))=\exp(o(L))$ of Chinese-remainder terms, which needs $0.52+d<1$, and the relative comparison with $Q_{\rm act}^{-1}\exp(-O(NT))$.
• The lifespan expansion (3.12)–(3.14). Each represented big prime has one lifespan and is charged $V_g^{-r}/(pb_p)$ for $r$ active runs, matching the original label normalization and the reciprocal-prime cost paid once.
• The memory charges, prime by prime (Lemma 3.7). An active birth costs $\widetilde\mu_g(p)$ and each later promotion $V_g^{-1}$; a ghost birth costs $V_g\widetilde\mu_g(p)=1/(pb_p)$ with the factor $-\eta_j$; a drop and a store followed by a ghost termination reproduce the two terms of the suffix identity (3.13), and a ghost birth followed by a promotion the prefix term; the splitting $\rho\,(q_i/\rho^2)\,\rho$ gives $q_i$ at each internal hit of a pending particle, and no damping to a particle that is active, stored or promoted at that visit, including at the boundary visits where $q_0=q_N=\sqrt q$; the exclusion $p\nmid D$ for stored and promoted particles follows from the bans; orphans reproduce (3.14).
• The factorial measures and absolute bounds. The ghost adjoint (3.19), with the selection of $t$ old zeros among $k+t$; the edge accounting (3.23)–(3.25), $(j)_a/j!=1/(j-a)!=(j-a+b)_b/(j-a+b)!$; the weighted Schur test (3.18) with $w=v^{j_{\rm tot}}$, the ghost row and column ratios (3.20) under the condition (3.21), and the edge bounds (3.26) of Lemma 3.8 in both directions; and the memory truncation (3.27)–(3.28), where doubling every ghost birth keeps (3.21) and charges $2^{-B+O((\ell+m)sN)}$ to the paths that exceed the truncation.
• Lemma 3.9 and (3.32). The uniform omission choice under the small-group symmetrization, at most $B$ of the $\mathcal D$ choices allowing a promotion, the reversal argument for stores, and the saving $L^{-(c_0/2)\log\binom{J+\ell}\ell}$.
• The reduction in Lemma 3.10. The maps $P_{\mathbf d}$ and (3.35), the diagonalization of the translations, the identification of the remaining operator with $\mathcal T(\Theta)$, and the negligible cost of replacing $\mu_g$ by $\widetilde\mu_g$ and of removing the free-label ban.
• The distinct-birth restoration (3.40)–(3.47). The cycle-sign identity (3.41), the point-mass saving $O(P_*^{-1})$ per dependent birth at large rank, and the Fourier representation of the equalities at small rank, which keeps the signed contraction on all but $2r=o(N)$ edges.
• Section 4. The collapse (4.4), Lemma 4.2 and (4.6), the kernel bounds (4.8)–(4.11), Lemma 4.3 in both ranges of the denominator, and the cancellation in the proof of Theorem 4.1: the Efron–Stein components, the symmetric-probe bound (4.22)–(4.24), which uses $\ell\ge1$, and the sign identity (4.26).
• Section 5. The endpoint norm bound independent of $J$, the Mellin decoupling of $\Psi(n/(ZD))$ into the ideal frequency $\zeta$, the raw-pattern identity (5.7), the minor-arc claim, the reduction to local Fourier energy (5.11), the small-mark identity (5.12), the Gallagher-type conversion (5.14)–(5.16), and Lemmas 5.2–5.4.

The explicit parameter inequalities were checked line by line: in (3.37), $E_{\rm id}>G+C_s+2$ makes the clean part at most $\frac12L^{-G}$ and the choice of $J$ does the same for (3.32), and $G>E+C_v+3$ beats the ghost operations; in (4.20), the five lines control the dyadic sum, (4.14), (4.17), the square root of (4.16), and (4.11) with (4.19), in that order. The order of choices stated in the manuscript (the transfer target, then the ideal family, then $J$; the analytic precisions; the discrepancy exponent last) was followed through these steps without finding a circular dependence.

The memory identity as a bijection

The most intricate step is the exact memory identity, Lemma 3.7, which the manuscript proves in a few paragraphs. It was re-derived here in outline as a bijection of two expansions.

Work in the independent-residue model of Lemma 3.6 with the phase $e(\theta h_N)$. A skeleton fixes the patterns, the endpoint permutations, the auxiliary free labels and the active big and small lists at every visit, with their prime values; it determines the offsets $h_i$ and a weight (the label normalizations $V_g^{-M}$ at visit $0$ and $V_g^{-t_g}$ at each edge, the free-label probabilities, the coefficients and the phases). The residue expectation factors over primes. For a big prime $p$ with no active visit, the factor is $b_p$ plus one orphan term for each pair $j<i$ of hits, by (3.14). For a big prime with active visits whose offsets agree modulo $p$, the factor is $p^{-1}\prod_{i\text{ inactive}}(1-\eta_iH_i)$, and the identities (3.13) expand it into a start (the first active visit, or a ghost birth at an earlier hit) and an end (the last active visit, or a ghost termination at a later hit), leaving $q_i$ at the internal hits. After division by $\prod_pb_p$, the path sum is a sum over histories: a skeleton, a finite set of never-active big primes each with an orphan interval, and a start and an end for each active big prime.

The operator product $\langle\mathbf 1_\varnothing,G_0\mathcal E_0\cdots \mathcal E_{N-1}G_N\mathbf 1_\varnothing\rangle$ expands into a sum over sequences of local choices: at each edge the pattern, the permutations, the free labels, which unshared source slots are stored and which dropped, which target slots are promoted and the injection into old pending particles, and the fresh target values; at each visit the terminated pending zeros and the ordered batch of ghost births with its values. Under the distinct-birth rule, sending a history to the local choices it dictates is a bijection. An unshared source label is stored when its prime has a later active run or a ghost termination, and dropped otherwise; an unshared target label is promoted when its prime is pending, and freshly born otherwise; ghost births and terminations sit at the start and end hits, and an orphan is a ghost birth followed by a ghost termination. Conversely, distinct births give each prime value a single birth, so the local choices determine every prime's active visits, start and end.

The constraints agree on the two sides. A promotion checks displacement zero, which is the congruence of the offsets of consecutive uses. The operator's exclusion $p\nmid D$ for stored and promoted particles removes no history, since an unshared label is neither a shared nor a free label and so does not divide $D$. The same fact, with $p>|k|$, makes the displacement of a newly stored particle nonzero after the step, so it cannot be promoted across the edge that stored it. A pending prime cannot equal an active label, because that would be a second birth of the same value, while it may equal a free label, which is not a birth and whose divisibility of $D$ leaves its residue unchanged. The weights agree prime by prime, as listed in the previous section, with $V_g^{-1}$ per active run and $1/(pb_p)$ once per prime. The factor $1/b!$ of a ghost batch and the factorial memory measure count each set of distinct particles once, by (3.19) and (3.25). The symmetrizations of two adjacent edges merge into the single projection between the factors of $(\mathcal A\mathcal A^*)^R$, and the ghost operations commute with them. The small groups keep their Haar coordinate and physical lists on both sides.

The truncation. Memory grows only by stores, at most $(\ell+m)|\mathcal B|$ per edge, and by ghost births. A history whose memory ever exceeds $B$ thus has at least $B-O((\ell+m)sN)$ ghost births, and since doubling every ghost birth keeps (3.21), such histories weigh at most $2^{-B+O((\ell+m)sN)}L^{O(N)}$ in absolute value, with or without the distinct-birth indicator. In the truncated evolution, Lemma 3.9 uses $B$ as the bound on pending particles, Lemma 3.10 absorbs the projections into its test vectors as contractions, Lemma 3.8 persists under them, and the rank expansion is applied to the truncated evolution, its large-rank terms using only absolute bounds and its small-rank terms using (3.38) for the unaffected truncated edges. So the truncation is consistent with every later step.

Quantitative transference obligations

The following calculation explains why the separate source estimates are strong enough, and which uniformities are essential. It is a reconstruction of the deductions in Sections 3--4, not an additional assumption that their unrefereed lemmas have been independently established.

For the memory identity, put

$$b_p=1-\frac1p\sum_{i=0}^N(1-q_i),\qquad \mu_g(p)=\frac1{V_gp},\qquad \widetilde\mu_g(p)=\frac1{V_gpb_p},\qquad q_0=q_N=\sqrt q,\quad q_i=q\ (0<i<N).$$

All big primes exceed $P_*=\exp(L^c)$, whereas $N=O(L^{0.52})$. Thus $b_p>0$ and $\widetilde\mu_g(p)/\mu_g(p)=1+O(N/P_*)$. Counting measure on each pending residue is essential: the test $r=0$ selects a point and supplies no further $1/p$. A lifespan with $r$ active runs must carry exactly $V_g^{-r}/(pb_p)$. A probability measure on that residue coordinate would charge an extra reciprocal prime at later returns and would not give the identity.

With memory cutoff $B_{\rm mem}=\lceil L^2\rceil$ and $\mathcal D=\binom{J+\ell}{\ell}^{|\mathcal S|}$, the two edge estimates needed from Lemmas 3.9 and 3.10 are

$$\|\mathcal E_{\rm dirty}\| \le 2L^{C_a}\sqrt{B_{\rm mem}/\mathcal D}, \qquad \|\mathcal E_{\rm clean}\|\le2L^{-E_{\rm id}+C_s}.$$

For the first inequality, fix a pending big prime $p$ and all big-label choices. If $F$ is the product of the small source labels and $U$ the product of the omitted ones, promotion requires $r+\sigma_i kD_{\mathcal B}F/U=0$ modulo $p$, with all denominators units. The $\mathcal D$ omission products are distinct positive integers smaller than $p$. Therefore a fixed pending particle permits at most one omission choice, and at most $B_{\rm mem}$ choices permit any promotion. The absolute row bound gains $B_{\rm mem}/\mathcal D$; the column bound does not need that gain. Schur's inequality gives its square root. Reversal treats stores. This reasoning is unchanged when $k<0$.

For the clean edge, the small-state maps in (3.34) obey $\sum_{\mathbf d}\|P_{\mathbf d}w\|^2\le L^{C_s}\|w\|^2$. In one small group the multiplier is bounded by

$$\sup_{y\ge M}q^{y-M}\frac{(y-M+\ell)_\ell}{V_g^\ell}, \qquad M=J+\ell,$$

which is independent of $J$. With memory prime lists fixed, translations of the small and pending residues have a finite Fourier decomposition. Each fiber sees the ideal operator at some real frequency $\Theta$, even for negative $k$. The density perturbation from $\mu_g$ to $\widetilde\mu_g$, and the cost of removing the free-label collision ban, are each $L^{O_J(1)}/P_*$ in norm. These are smaller than any fixed negative power of $L$. The stated clean bound follows from the uniform ideal bound.

The ghost operations have norm at most $L^{C_v}$, with $C_v$ independent of $J$ and of the comparison complexity. Choose

$$G>E+C_v+3,\qquad E_{\rm id}>G+C_s+2,$$

then construct the ideal family, and finally choose $J$ large enough that $2L^{C_a}\sqrt{B_{\rm mem}/\mathcal D}\le\frac12L^{-G}$. The clean term is also at most $\frac12L^{-G}$ eventually. This gives $\|\mathcal E_i\|\le L^{-G}$ while different births are still permitted to use the same prime. In particular, $C_a$ may depend on the already chosen comparison complexity; the target $E_{\rm id}$ must not depend on that complexity. This is the order supplied by the source.

Global birth distinctness cannot be restored by a single absolute collision error. For any fixed positive $C_h$,

$$L^{C_hN}P_*^{-1} =\exp(C_hN\log L-L^c)$$

does not even tend to zero, since $c<0.47<0.52$. Thus that proposed shortcut would lose the required saving. The source instead expands distinctness by equality rank. At rank $r$ the total absolute inclusion-exclusion coefficient is at most $L^{C'r}$. The two bounds are

$$L^{C_h(N+1)}(C/P_*)^r \quad\hbox{and}\quad O\!\left(L^{(-G+C_v)N+C_v+2r(G+C_a)}\right).$$

The first comes from one fresh-label atom bound per dependent birth, keeping the earlier pivot value in the conditioning even if that particle has died. The second inserts equality by Fourier phases at the birth operations only. At most $2r$ edges lose their signed norm bound; all other edges keep their complete pattern sum and both symmetrizations. A multiplier on a particular ghost-batch address is averaged over that batch before applying the symmetric memory operator, so its modulus remains at most one.

Split at $r_0=\lceil C''N\log L/\log P_*\rceil$. For large $C''$, the first bound summed over $r\ge r_0$ is $o(L^{-EN})$. Since $r_0/N=O(\log L/L^c)+O(1/N)=o(1)$, the second bound and the coefficient count sum to $L^{(-G+C_v+o(1))N}=o(L^{-EN})$ for $r<r_0$. The memory tail is $\exp(-\Omega(L^2))$, and the residue replacement has arbitrarily strong relative precision after fixing its Bonferroni order. This explains the proposed root-moment estimate (3.7). The single-collision calculation is an obstruction to that shortcut, not a counterexample to the equality-rank argument or the theorem.

The required ideal bound is also uniform over the entire coefficient use. For a selected product of labels of size $U$, its probability at a particular integer is at most $L^\kappa/U$. Fiberwise Cauchy--Schwarz therefore converts an arbitrary tuple test to an integer coefficient sequence of squared norm at most $L^\kappa/U$ times its original squared norm. The integer bilinear estimate handles rational denominators larger than a fixed power of $L$. For smaller denominators, the cellwise character Gram matrix is $I+O(Q^2\xi^{-1}L^{-D_*})$ in operator norm and hence has norm at most two after the precision is chosen. Matching cells control the Mellin and remaining additive phase. This supplies (4.12) for arbitrary tuple tests, including the conditional tests used in Theorem 4.1.

For $M\ge(m+\ell)(m+\ell-1)$, the remaining ideal cancellation uses the fraction

$$\frac{\binom{M-m-\ell}{j-m}}{\binom Mj} =\frac{(j)_m(M-j)_\ell}{(M)_{m+\ell}} \le2\ell^\ell m^{-\ell}$$

of symmetric coordinate components containing every probe and avoiding the last $\ell$ slots. This is small by choosing $m$ large, precisely because $\ell\ge1$. Across a block of $\gg\log L$ groups it gives any prescribed logarithmic saving. Alternating nonempty probe subsets cancel the other components. The $3^P$ approximation errors are paid by the precision in (4.12), before choosing $J$. The application takes $\ell=1$ throughout.

The local Fourier-energy estimate

This is the remaining analytic use of the endpoint hypotheses after the residual physical pairing. The norm calculation for that pairing uses $\tau(n_*)^4$, a fixed divisor moment. Summing a prescribed physical list first contributes its reciprocal product, whose total normalized mass is at most one. Thus the endpoint norm exponent is independent of $J$. Corollary 3.11 and the ideal estimate can be chosen in the required order. Removing shared labels then gives the raw correlation plus comparison shifts $kD_AD_B$, as in (5.7).

For a comparison, Fourier inversion in the integer position leaves the multiplier

$$M(\theta)=\mathbb E\, b_1(D_A)b_2(D_B)e(-k\theta D_AD_B),\qquad |b_1|,|b_2|\le1.$$

Both free products are nonempty. The same integer bilinear estimate makes $M$ smaller than any required negative log power outside arcs of width $L^{C_{\rm arc}}/H'$ around rationals of denominator at most $L^{C_{\rm arc}}$. Here $D_AD_B\asymp H'$ and $\exp(L^c)\le H'\le\exp(O(\log L\,L^d))$. Dirichlet approximation is applied to $k\theta$; dividing by negative $k$ only changes a numerator sign. Set $Y=ZH'$ and $H=H'/L^{C_{\rm arc}}$.

On the remaining arcs, the needed estimate is the first endpoint's local energy

$$\int_{|\beta|\le1/H}|\widehat{\mathcal U}(a/r+\beta)|^2\,d\beta \ll_A YL^{-A}.$$

One small marked prime can be removed because $\ell=1$; the square-divisible exception costs $YL^{O(1)}\exp(-L^a)$ in squared norm. The factors $m$, the small prime, and the long rough integer are units modulo the arc denominator. Splitting the residual factor by its gcd with that denominator and using characters on the resulting quotient retains principal and imprimitive characters. These have exactly the modulus bounds allowed by the discrepancy hypothesis.

The additive-to-Mellin conversion (5.14)--(5.16), followed by dyadic separation, reduces the estimate to

$$\int_{|t|\le 2LY/H}|P_s(t)P_l(t)\mathcal R(t)|^2\,dt\ll L^{-A'} \tag{*}$$

with

$$\begin{aligned} P_s(t)&=\sum_{p_s\asymp P}b_s(p_s)p_s^{-1+it},\\ P_l(t)&=\sum_{\substack{p\asymp P',\ p\in I_{**}\\P^-(p)>W}} \chi_p(p)p^{-1+i\sigma_1+it},\\ \mathcal R(t)&= \left(\sum_{m\asymp M_0}\alpha_m\chi_m(m)m^{-1+i\sigma_0+it}\right) \left(\sum_{r\asymp R_0}d_rr^{-1+it}\right). \end{aligned}$$

The variable $p$ is an integer, not necessarily prime. The parameters satisfy $\exp(L^a)/2\le P\le\exp(L^b)$, $x^\tau/2\le P'\le x^\eta$ with $\eta<1/4$, and $PM_0P'R_0\asymp Y$. Every fixed subproduct, collapsed to a polynomial of size $U$, has coefficient-square sum $L^{O(1)}/U$. The remaining mark weight has already been bounded by a fixed log power; it does not increase the order of the divisor moment.

Here is how the three time regimes prove $(*)$.

• Where $|P_s(t)|\le L^{-A_s}$, ordinary mean square for $P_l\mathcal R$ gives $O(L^{-2A_s+C})$. Its size is $Y/P$, which exceeds the time range: $H/(2LP)\to\infty$ because $b<c$.
• The set where $|P_s(t)|>L^{-A_s}$ meets only $Y^{o(1)}$ unit intervals. Indeed, for $j=\lfloor\log Y/\log(2P)\rfloor$, the squared coefficients of $P_s^j$ sum to at most $j!(C/P)^j$. A unit-separated mean square gives an interval count at most $YL^{O(1)}j!(CL^{2A_s}/P)^j$. Its logarithm is $O(L^b+L^{1-a}\log L)=o(\log Y)$.
• On these exceptional intervals, $\mathcal R$ has a sparse mean square bounded by $L^{O(1)}$. Its size is $U\asymp Y/(PP')\ge Y^{1-\eta-o(1)}>Y^{0.6}$. For unit-separated sample times, the Gram kernel satisfies $|\sum_{n\asymp U}n^{iz}|\ll U/(1+|z|)+Y^{1/2}$ for $|z|\ll Y$, by the first and second derivative tests. With $R=Y^{o(1)}$ samples, the absolute row sum is $O(U\log(2Y)+RY^{1/2})=O(U\log(2Y))$. Multiplying by the coefficient-square sum proves $\sum_j\sup_{t\in I_j}|\mathcal R(t)|^2\ll L^{O(1)}$.

For completeness, the long factor is uniformly small on those intervals when $L^{B_0}\le|t|$ and $|t+\sigma_1|\le x^2$. A block-sieve upper polynomial with $h_s=2\lceil(\log L)^2\rceil$ and $D=W^{4h_s+2}=x^{o(1)}$ replaces roughness with absolute harmonic error $O(e^{-h_s}+D/P')$. Its coefficients obey $\sum_d|\lambda_d|/d\ll\sqrt L$. For each $d$, split into character residue classes and apply the derivative tests to $\exp(iu\log(q'y+a))$, where $u=t+\sigma_1$ and $N=P'/(dq')\ge x^{\tau/2}$. For $|u|\le cN$ the normalized bound is $O(|u|^{-1})$. Otherwise finitely many ranges $N^{j-2+1/10}\le|u|\le N^{j-1/10}$ cover $cN<|u|\le x^2$. After $j-2$ differencing steps with shifts at most $N^{1/(20j)}$, the second derivative test saves a fixed power $N^{-\delta_\tau}$. Partial summation and the sum over $d$ give

$$|P_l(t)|\ll e^{-h_s}+D/P' +\sqrt L\bigl(L^{-B_0}+x^{-\tau\delta_\tau/2}\bigr).$$

Thus $|P_l|\ll L^{-A_l}$ for any fixed $A_l$ by choosing $B_0$ large enough after the fixed frequency bound. Combining this with the sparse mean square bounds the high-time exceptional contribution to $(*)$ by $L^{-2A_l+C}$, with no factor counting the exceptional intervals.

Finally, for $|t|\le L^{B_0}$ the $m$-polynomial is at most $L^{-B}$ by the original discrepancy hypothesis, after $B$ exceeds all character, frequency and interval requirements. The other factors have absolute log-power bounds. This part is therefore $O(L^{-2B+B_0+C})$. Choose $A_s$, then $A_l$ and $B_0$, and finally $B$. All three contributions have the saving required in $(*)$. The Fourier tails are controlled by ordinary mean square; their fixed cutoff exponents do not grow when their accuracy is increased. This completes the proposed comparison cancellation at the uniformity required in (P), without a new restriction on the sign.

What remains

No failing estimate or counter-configuration was found. The memory argument and the quantitative deductions above are author examinations of OpenAI's source proof. They supply no independent acceptance. The classical prime number theorem, Siegel--Walfisz theorem, multiplicative large sieve and derivative tests are used in the forms stated in Section 2.

The remaining mathematical dependence is (P), together with the elementary source inputs (E) specified in the combined sketch. It is not merely a missing label on an established result: the unrefereed analytic proof remains the premise being consumed. The absence of a detected error does not prove its uniform assertion.

The full Theorem 5.1 is stronger than the application needs. A sufficient replacement is this precise restriction. For every fixed group and band data and every fixed $D_0,C>0$, there must be a fixed $B$ such that the harmonic correlation displayed as (P) is $O(L^{-D_0})$, uniformly for all $0<|k|\le L^C$, $xL^{-C}\le Z\le xL^C$, the allowed cutoffs and frequencies, and the two endpoint formulas in step 5 of the Type II page. Here $\alpha$ is the original restricted $W$-rough sequence satisfying $(\mathrm D_B)$, the common residual coefficient $c_r$ and the common rough interval $I_{**}$ arise from the band partition, and the character modulus is $2|k|$. The range includes both signs of $k$. It is enough to prove this for the prime and roughness proxy differences used in the extraction, with uniformity in their interval restrictions and their final cell precision. The required $B$ is chosen before that precision.

This restriction states exactly what could replace (P) if its general form failed; no such failure is asserted here. A proof for a common uniform endpoint class would apply to both sign choices. A proof for selected calls from the predecessor calculation need not apply to all successor calls, whose separated cutoff coefficients differ.