Signed Type II keeper

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Examined on 15 October 2026. The assessed text is signed_type_ii.md, lines 9–481, compared throughout with Section 6 in the supplied source excerpts, lines 189–386. This is a written reconstruction under P and the stated elementary estimates, not a proof of P or an original-source fidelity check. The source method is OpenAI's; the supplied successor exposition is Plasma AI's, for the successor application attributed to Shisheng Li and Zhong Zhuang.

No numerical experiment is needed. Reproduction consists of checking the identities and inequalities below against the retained definitions and the explicit P statement. The variable called E in the splitting argument below is a scale, distinct from the hypothesis named E.

1. Weight bounds and the group-part tail

Use L=log x, T=log L and W=exp(sqrt L). Group counts are distinct-prime counts, even when the underlying integer has prime powers. Group primes are below exp(L^d), with d<0.47, hence below W eventually. The exceptional slot is an integer rough above W; it cannot contain a group prime.

Fixing this exceptional integer as a divisor of v leaves at most r_0! arrangements in each band. The normalization cancels those factors, so a_Q(v)<=tau(v). In each group, the sequence q^(t-ell)(t)_ell is bounded on nonnegative integers t: it is zero for t<ell and exponential damping dominates the fixed polynomial otherwise. Multiplication over O(T) groups gives W_ell<=L^C. Slotwise reciprocal summation gives H_Q<=L^C: each pure prime slot has bounded harmonic mass, the integer slot has O(L) mass, and there are O(T) slots. No divisor-moment theorem with a growing number of factors is being used here.

For theta=L^(-d), set a_p=(p^(1-theta)-1)^(-1). The inequality p^theta<=e gives a_p<=2e/p eventually. For one group the tilted marked sum is exactly

$$\frac1{V_g^\ell}\prod_{p\in\mathcal P_g}(1+qa_p) \sum_{p_1,\ldots,p_\ell\ {\rm distinct,ordered}} \prod_{i=1}^{\ell}\frac{a_{p_i}}{1+qa_{p_i}}.$$

The factors for marked primes sum their positive powers with weight a_p; all other present primes receive the additional factor q. This proves the identity including prime powers and the normalization q^(-ell). Its value is bounded in terms of the fixed group data. Taking the product over the groups and using Rankin's inequality yields

$$\sum_{v_*=1,\ v>x^{b_*/4}}\frac{W_\ell(v)}v \le L^{C_1}\exp\{-b_*L^{1-d}/4\}.$$

An integer N comparable to x has at most O(sqrt L) prime factors above W, counted with multiplicity. Its divisors rough above W consequently number at most exp(O(sqrt L)), since a+1<=2^a for every exponent a>=1. Multiplying harmonic mass by O(x) bounds the discarded u_P>x^(b_*/4) contribution by

$$xL^{C_2}\exp\{-b_*L^{1-d}/4+O(\sqrt L)\}.$$

This beats every logarithmic loss because 1-d>1/2. It uses N=2u-1 only through N comparable to x. It has no dependence on the later splitting precision.

2. The split and its costs before Cauchy's inequality

Put E=U L^(-K_0). On a good list assign u_P and the integer slot p to h. Their product is at most x^(b_/4+eta_)<x^(b_/2). Thus the pure slots still available have total logarithm at least (1-b_/2)L-O(1), whereas log E<=(1-b_)L-K_0 T. There is a positive initial margin for every fixed K_0 once x is large. Nonempty Type II scale ranges force b_<=1/2; larger b_* gives a vacuous eventual assertion.

Process pure slots from the largest band down, with logarithms v_i. The residual is R_i=log E-sum_(t<i) delta_t v_t. Taking a slot with positive probability requires R_i>v_i and reduces both residual and remaining total by v_i. Skipping with positive probability requires R_i<(c_2+1)s_(j(i)). If a later band remains, its at least r_0-1 pure slots have total logarithm at least (r_0-1)c_1 s_(j(i))/lambda >= (c_2+1)s_(j(i)). If no later band remains, the residual is at most (c_2+1)s_(j_x). These observations prove by a finite induction

$$0\le R_i\le\sum_{t\ge i}v_t+(c_2+1)s_{j_x}, \qquad E L^{-C_s}\le e\le E,\quad C_s=(c_2+1)c_+.$$

The number of slots is explicitly finite, O(T); the induction does not assume termination of a separate algorithm. The geometric tail also bounds R_i/s_(j(i)) by a fixed constant. Thus the fixed compact cutoff rho can be chosen independently of K_0. The compact transition products sum to one on every good list and to at most one on arbitrary lists. Adding the latter back costs only the already bounded group tail.

There is a potential dimension trap here: the number of slot variables grows. The proof avoids it by multiplying one-dimensional Fourier norms, rather than using the fixed-dimensional separation lemma in dimension O(T). For N_s slots, the sum of total variations over all patterns is at most (2C_F)^(N_s)=L^(O(1)). Truncating each input frequency at L has error at most 2^(N_s) N_s C_M C_F^(N_s-1) L^(-M). The constants in this estimate precede K_0. The separated slot frequencies are exactly

$$\sigma_t=-\frac{\xi_t}{s_{j(t)}} -\delta_t\sum_{i>t}\frac{\xi_i}{s_{j(i)}}.$$

Since sum_i 1/s_(j(i))=O(1/T), these frequencies are O(L/T). The only remaining dependence on E in the Fourier exponent is a scalar phase, of absolute value one.

The smooth cutoff Psi(eh/x) has a fixed one-dimensional Mellin L1 norm. Its separation contributes common powers e^(i rho) and h^(i rho). After separation, fixed pure-slot products have only L^(O(1)) ordered assignments: each band contributes at most r_0! and the number of bands is O(T). Consequently

$$|a_e(e)|\le L^{C_a},\qquad a_h(h)=h^{i\varrho}W_\ell(h) \sum_{pr=h_*}\mathbf1_{p\in I_{**},\ P^-(p)>W}p^{i\sigma}c_r, \qquad |c_r|\le L^{C_a}.$$

All of C_a and the expansion-mass exponent C_f are independent of K_0. The identity (eh)*=e h* and W_ell(eh)=W_ell(h) holds because e is group-free. The integer slot remains in h, with its complete roughness and interval restrictions.

Uniform truncation errors are bounded against

$$L^{O(1)}\sum_{u\asymp x}\tau(u)\tau(2u-1)\ll xL^{O(1)}.$$

Cauchy's inequality and fixed second divisor moments prove this estimate; 2u-1 is an injective selection of positive integers of size O(x). Removing the compact Psi support during separation does not create an unbounded size range: m and n remain on their original dyads and force u comparable to x. Taking the Fourier accuracy large enough therefore gives the stated B_0 expansion with error O(x L^(-D_*-1)).

3. Cauchy, the diagonal, and the order of K_0

The smooth majorants of e and n equal one on their original ranges. Applying Cauchy's inequality in those two variables gives

$$|B_0|^2\le EVL^{C_3}\mathcal N.$$

The outside factor follows from the number of e,n pairs and the bounds on a_e and beta. Enlarging the range in the nonnegative square norm is legitimate. Only the lower edge of the e cutoff moves by L^(-C_s); its derivative cost depends on C_s, not K_0.

On the diagonal m=m', the equations force h=h'. For fixed m,n, e divides (mn+1)/2, and

$$\sum_{e\mid(mn+1)/2}\tau((mn+1)/(2e))^2\le\tau(mn+1)^3.$$

There are at most tau(v) pairs m,n with mn=v. Cauchy's inequality and the second and sixth fixed divisor moments therefore give

$$\mathcal N_{\rm diag} \ll L^{O(1)}\sum_{v\ll x}\tau(v)\tau(v+1)^3 \ll xL^{C_5}.$$

No estimate for a shifted divisor correlation is assumed. Ordinary unshifted moments suffice. Because EV is comparable to xL^(-K_0), choosing K_0>=C_3+C_5+2C_f+2D_*+2 controls this contribution even after multiplying the square root by the whole pre-Cauchy Fourier mass. The quantities on the right have already been fixed.

4. The signed bijection

Suppose

$$mn=2eh-1,\qquad m'n=2eh'-1,\qquad m\ne m'.$$

The first congruence implies gcd(n,2e)=1. Subtraction then gives 2e dividing m'-m. Define k=(m-m')/(2e). Direct calculation gives

$$m'-m=-2ek,\qquad h'-h=-kn,\qquad mh'-m'h=k.$$

Thus z=m'h and z+k=mh' are both positive. The enlarged e range gives 0<|k|=O(L^(K_0+C_s)); eventually |k|<W.

Conversely, suppose m,m',h,h' are positive, m and m' are rough above W, 0<|k|<W, m'h=z, mh'=z+k, and m'=m modulo 2|k|. If e=(m-m')/(2k)>0, then e is an integer and

$$m(h'-h)=-k(2eh-1).$$

Every prime factor of m exceeds |k|, so gcd(m,k)=1, including the case m=1. It follows that n=(2eh-1)/m is a positive integer. Substitution recovers h'-h=-kn and m'n=2eh'-1. All variables of the original pair are recovered uniquely. This proves the converse, including divisibility and positivity; it does not merely count a superset.

Both m and m' are odd and coprime to k. Therefore they are units modulo q_k=2|k|, and all Dirichlet characters modulo q_k impose the congruence exactly:

$$\mathbf1_{m'=m\ ({\rm mod}\ q_k)} =\frac1{\varphi(q_k)}\sum_{\chi\bmod q_k} \chi(m)\overline{\chi(m')}.$$

The pulled-back cutoff arguments are e=(m-m')/(2k) and

$$n=\frac{(m-m')z}{kmm'}-\frac1m.$$

The sign of the last term is essential. The two possible signs of k are both retained. Extending the smooth cutoffs by zero to nonpositive arguments imposes the original positivity conditions; it changes no arithmetic restriction.

5. Separation, conjugations, and P's normalization

The support gives z comparable to x times a fixed logarithmic power. There are O(T) dyads Z needed. In t_1=log(m/U), t_2=log(m'/U), t_3=log(z/Z), the derivatives of e/E are O(U/(|k|E)), and those of n/V are O(Z/(|k|UV)+(UV)^(-1)). Both are fixed powers of L after K_0 is chosen. Fixed smooth localizations give a compact three-dimensional box. The fixed-dimensional Fourier lemma now applies legitimately, with polynomial total variation and a frequency exponent fixed before the desired tail saving.

Absolute endpoint convolutions have bound L^(O(1)) tau(z_*)^2; with W_ell, their square sum on a Z dyad is O(Z L^(O(1))). The fourth divisor moment proves this. Cauchy's inequality at z and z+k bounds the absolute correlation by Z L^(O(1)); |k| is negligible relative to Z. This controls tails after all polynomially many shifts and character terms. No cancellation is used to justify the Fourier truncation itself.

For fixed chi and separated frequencies xi_1,xi_2,xi_3, define

$$F_0(v)=\sum_{mpr=v_*}\alpha_m\chi(m)m^{i(\varrho-\xi_2)} \mathbf1_{p\in I_{**},\ P^-(p)>W}p^{-i\sigma}\overline{c_r},$$ $$G_0(v)=\sum_{mpr=v_*}\alpha_m\chi(m)m^{i(\xi_1+\varrho)} \mathbf1_{p\in I_{**},\ P^-(p)>W}p^{-i\sigma}\overline{c_r}.$$

Conjugating F_0 at z=m'h gives exactly the factors involving conjugate alpha_(m'), conjugate chi(m'), p^(i sigma)c_r and (m')^(i xi_2-i varrho). Multiplying by z^(i varrho) restores h^(i varrho). At z+k=mh', G_0 and (z+k)^(-i varrho) give the other factor alpha_m chi(m) m^(i xi_1) conjugate(a_h(h')) after its group weight is restored. Since m,m' are group-free, z_=m'h_, (z+k)*=mh'*, W_ell(z)=W_ell(h), and W_ell(z+k)=W_ell(h'). The separated correlation is consequently

$$\sum_z\psi_Z(z/Z)z^{i(\varrho+\xi_3)}(z+k)^{-i\varrho} \overline{F_0(z)}G_0(z+k)W_\ell(z)W_\ell(z+k).$$

The same original alpha with its actual m restrictions is inside F_0. Character and power twists are explicit parts of the allowed endpoint, not a replacement sequence on which discrepancy is being assumed without proof. Its modulus is 2|k|, its frequencies and coefficients have fixed logarithmic power bounds, and 0<tau_<eta_<b_*/4<=1/8<1/4. The integer slot and the unrestricted residual coefficient meet P exactly.

With tilde_psi(t)=t psi_Z(t), the displayed unweighted sum is Z times P's harmonically normalized sum: Z tilde_psi(z/Z)/z=psi_Z(z/Z). P applies to both signs of k and the actual conjugation. It supplies O(Z L^(-D_0)). Summing the shifts, dyads, averaged characters and Fourier variation gives O(x L^(-D_0+C_9)) for the off-diagonal norm. The total character weight at each k is one; even polynomial overestimates are absorbed in C_9.

Choose positive D_0 at least C_3+C_9-K_0+2C_f+2D_+2, then choose the discrepancy exponent B supplied by P. The diagonal and off-diagonal estimates give |B_0|=O(x L^(-D_-C_f-1)). Integrating the pre-Cauchy expansion and adding its errors proves the signed Type II estimate at the claimed strength.

Result of the attacks

The finite allocation, the two signs, the determinant converse, the character units, the conjugations, the integer slot, the arbitrary beta sequence and the factor Z all survive the checks above. The order of choices is acyclic: fixed geometry and requested saving; K_0; post-Cauchy separation bounds and D_0; B; then any proxy precision whose coefficient bound is independent of that precision. No extra analytic hypothesis beyond the stated P/E interfaces was found in this deduction. This finding supplies no proof of P.