工程紀錄 · 第五弧線 v3.8 · 2026-09-03 · RAMANUJAN_AXIS_CANCELLATION · LIMIT_PERIODIC_SUBPOWER_CLOSURE · HIGH_CONDUCTOR_QUANTITATIVE_GAP · RH_CLAIM_FALSE # Parallelogram Ramanujan Tail：整條 Spectral Axis 消失、Limit-Periodic Subpower Closure 與 High-Conductor Quantitative Gap **RH-Parallelogram-RamanujanTail v3.8** 本節點承接： - `RH-RefinedSingularSeries-Parallelogram v3.7` - `RH-CenteredShift-FourierGap v3.6` - Montgomery–Soundararajan refined singular-series rational-frequency expansion - Bloom–Kuperberg adding-fractions framework - limit-periodic harmonic analysis v3.7 已證： $$ \widehat{\mathfrak K}_{4,q}(0,0)=0 $$ for every finite squarefree modulus $q$。 v3.8 將這個結果大幅加強： $$ \boxed{ \widehat{\mathfrak K}_{4,q}(\alpha,0)=0 \qquad \forall\alpha\in\frac1q\mathbb Z/\mathbb Z. } $$ 也就是整條： $$ \boxed{\beta=0} $$ spectral axis完全消失。 等價地，對每個 fixed： $$ h\bmod q, $$ 都有： $$ \boxed{ \sum_{d\bmod q} \mathfrak K_{4,q}(h,d)=0. } $$ 這不是 asymptotic。 這是 coefficient-level identity。 本輪第二個結果是 full deterministic singular-series model的 **subpower closure**： $$ \boxed{ \mathfrak M_{4,N}=o(N^5). } $$ 因此 high-conductor tail不能恢復一個非零 $cN^5$ deterministic main term。 但是： $$ o(N^5) $$ 仍可能是： $$ \frac{N^5}{\log N}, \qquad N^{5-o(1)}, $$ 所以 fixed： $$ \eta_M>0 $$ 仍未證明。 本輪第三個結果是 exact two-dimensional weight transform： $$ \boxed{ \widehat\Omega_N(\alpha,\beta) = \sum_{2\le n1, \\ 0, & r(\theta)=1. \end{cases} } $$ This is the finite-modulus coefficient appearing in the refined singular-series expansion with all individual denominators $>1$。 --- # 2. Montgomery–Soundararajan / Kuperberg expansion The standard finite refined singular series has the form： $$ \boxed{ \begin{aligned} \mathfrak S_{0,q}(d_1,\ldots,d_k) &= \sum_{ \substack{ q_1,\ldots,q_k>1\\ q_i\mid q } } \prod_{i=1}^k \frac{\mu(q_i)}{\phi(q_i)} \\ &\quad\times \sum_{ \substack{ (a_i,q_i)=1\\ \sum_i a_i/q_i\in\mathbb Z } } e\left( \sum_i \frac{d_i a_i}{q_i} \right). \end{aligned} } $$ For the parallelogram： $$ (d_1,d_2,d_3,d_4) = (0,h,d,h+d), $$ define： $$ \theta_i = \frac{a_i}{q_i}. $$ The constraint is： $$ \theta_1+\theta_2+\theta_3+\theta_4=0. $$ The phase becomes： $$ \boxed{ h\alpha+d\beta, } $$ where： $$ \boxed{ \alpha=\theta_2+\theta_4, } $$ $$ \boxed{ \beta=\theta_3+\theta_4. } $$ This is the canonical two-dimensional rational spectrum。 --- # 3. Four-point spectral coefficient Fix： $$ \alpha,\beta\in G_q. $$ Set： $$ t=\theta_4. $$ Then： $$ \theta_2=\alpha-t, $$ $$ \theta_3=\beta-t, $$ $$ \theta_1=t-\alpha-\beta. $$ Therefore the finite four-point coefficient is： ## Theorem 3.1 $$ \boxed{ C_q(\alpha,\beta) = \sum_{t\in G_q} \rho_q(t) \rho_q(\alpha-t) \rho_q(\beta-t) \rho_q(t-\alpha-\beta). } $$ --- # 4. Pair spectral coefficient For the pair： $$ \{0,h\}, $$ the constraint gives： $$ \theta_1=-\theta_2. $$ Since： $$ \rho_q(-\theta)=\rho_q(\theta), $$ the centered pair singular series is： $$ \boxed{ \mu_q(h) = \sum_{\theta\in G_q} \rho_q(\theta)^2e(h\theta). } $$ Thus its square has spectral coefficient： $$ \boxed{ P_q(\alpha) = \sum_{t\in G_q} \rho_q(t)^2 \rho_q(\alpha-t)^2. } $$ and is independent of $d$。 --- # 5. Finite covariance spectrum Define： $$ \boxed{ \mathfrak K_{4,q}(h,d) = \mathfrak S_{0,q}(0,h,d,h+d) - \mu_q(h)^2. } $$ Then： $$ \boxed{ \widehat{\mathfrak K}_{4,q}(\alpha,\beta) = C_q(\alpha,\beta) - \mathbf1_{\beta=0} P_q(\alpha). } $$ The normalization is： $$ \mathfrak K_{4,q}(h,d) = \sum_{\alpha,\beta\in G_q} \widehat{\mathfrak K}_{4,q}(\alpha,\beta) e(h\alpha+d\beta). $$ --- # 6. Entire $\beta=0$ axis cancellation Set： $$ \beta=0. $$ Then： $$ \begin{aligned} C_q(\alpha,0) &= \sum_t \rho_q(t) \rho_q(\alpha-t) \rho_q(-t) \rho_q(t-\alpha) \\ &= \sum_t \rho_q(t)^2 \rho_q(\alpha-t)^2 \\ &= P_q(\alpha). \end{aligned} $$ Therefore： ## Theorem 6.1 · Spectral Axis Cancellation For every finite squarefree： $$ q, $$ and every： $$ \alpha\in G_q, $$ $$ \boxed{ \widehat{\mathfrak K}_{4,q}(\alpha,0)=0. } $$ This strictly strengthens the v3.7 zero-mode theorem。 --- # 7. Rowwise mean-zero theorem By Fourier orthogonality： ## Corollary 7.1 For every： $$ h\bmod q, $$ $$ \boxed{ \frac1q \sum_{d\bmod q} \mathfrak K_{4,q}(h,d)=0. } $$ So pair centering removes every component that is constant in the second parallelogram gap $d$。 --- # 8. The surviving $\alpha=0$ axis Now set： $$ \alpha=0, $$ and： $$ \beta\ne0. $$ Then： $$ \boxed{ \widehat{\mathfrak K}_{4,q}(0,\beta) = \sum_t \rho_q(t)^2 \rho_q(\beta-t)^2. } $$ This is the spectral form of the second Wick-like pair channel： $$ (0,d)(h,h+d). $$ Thus v3.8 does **not** say that all pairing structure disappears。 It says specifically： ```text h-pair square is removed completely d-pair channel survives as beta != 0 spectrum ``` --- # 9. Endpoint-Cesàro parallelogram weight Recall： $$ \boxed{ \Omega_N(h,d) = \sum_{ r=1 }^{2N-1-h-d} w_N(r+h) w_N(r+h+d), } $$ with support： $$ h,d\ge1, \qquad h+d\le2N-2. $$ --- # 10. Exact two-dimensional Fourier transform Define： $$ \boxed{ \widehat\Omega_N(\alpha,\beta) = \sum_{h,d} \Omega_N(h,d) e(h\alpha+d\beta). } $$ Set： $$ n=r+h, $$ $$ m=n+d. $$ Then： $$ 1\le hn} w_N(m)e(m\beta) \right| \ll N M_N(\beta). } $$ This includes the trivial： $$ N^2 $$ bound when： $$ \|\beta\|\lesssim N^{-1}. $$ --- # 13. Weight spectral bound Use Theorem 10.1 and sum first over $m$。 Since： $$ \sum_nw_N(n)\ll N^2, $$ Sections 11–12 give： ## Theorem 13.1 · Parallelogram Weight Multiplier $$ \boxed{ |\widehat\Omega_N(\alpha,\beta)| \ll N^3 M_N(\alpha) M_N(\beta). } $$ In particular： $$ \boxed{ |\widehat\Omega_N(\alpha,\beta)| \ll N^4 M_N(\beta). } $$ --- # 14. Why the spectral-axis theorem matters Without centering, a： $$ \beta=0 $$ mode receives no $d$-oscillation and can see： $$ |\widehat\Omega_N| \asymp N^5. $$ Theorem 6.1 removes every such mode。 Every surviving deterministic covariance frequency therefore has： $$ \beta\ne0. $$ So high-conductor analysis always has at least one genuine rational oscillation direction available。 This is stronger than merely knowing the total mean is zero。 --- # 15. Finite-modulus spectral formula For a finite squarefree： $$ q, $$ $$ \boxed{ \mathfrak M_{4,N}^{(q)} = \sum_{ \substack{ \alpha,\beta\in G_q\\ \beta\ne0 } } \widehat{\mathfrak K}_{4,q}(\alpha,\beta) \widehat\Omega_N(-\alpha,-\beta), } $$ up to the already-lower-order collision lines removed in v3.6。 Hence： $$ \boxed{ |\mathfrak M_{4,N}^{(q)}| \ll N^3 \sum_{ \beta\ne0 } | \widehat{\mathfrak K}_{4,q}(\alpha,\beta) | M_N(\alpha) M_N(\beta). } $$ This is the exact spectral high-conductor interface。 --- # 16. Finite-$q$ $d$-Sobolev diagnostic Because： $$ \beta=0 $$ is absent, define： $$ \boxed{ \mathfrak H_d(q) = \sum_{ \alpha,\beta\in G_q \atop \beta\ne0 } \frac{ | \widehat{\mathfrak K}_{4,q}(\alpha,\beta) |^2 }{ |1-e(\beta)|^2 }. } $$ This is the mean-square norm of the periodic $d$-antiderivative。 If： $$ \mathfrak G_q(h,d+1)-\mathfrak G_q(h,d) = \mathfrak K_{4,q}(h,d), $$ with zero $d$-mean normalization, then Parseval gives： $$ \boxed{ \frac1{q^2} \sum_{h,d\bmod q} |\mathfrak G_q(h,d)|^2 = \mathfrak H_d(q). } $$ Reference values for primorial-like conductors grow slowly, but no uniform theorem is claimed here。 --- # 17. Finite-modulus coefficient factorization Let： $$ B_\beta(t) = \rho_q(t)\rho_q(t+\beta). $$ Then for： $$ \beta\ne0, $$ Theorem 3.1 can be rewritten： $$ \boxed{ \widehat{\mathfrak K}_{4,q}(\cdot,\beta) = B_{-\beta} \ast B_{\beta}. } $$ The convolution is on： $$ G_q. $$ This gives a concrete entry point for： - Young inequalities； - adding-fractions estimates； - conductor decompositions； - Sobolev bounds。 --- # 18. Basic coefficient norms One has： $$ \boxed{ \|\rho_q\|_2^2 = \sum_{ r\mid q,\ r>1 } \frac1{\phi(r)} = \frac q{\phi(q)}-1. } $$ Also： $$ \boxed{ \|\rho_q\|_4^4 = \sum_{ r\mid q,\ r>1 } \frac1{\phi(r)^3} \le \prod_p \left( 1+\frac1{(p-1)^3} \right) -1 <\infty. } $$ The $L^4$ norm is uniformly bounded in $q$。 This is consistent with bounded fourth-order local moments。 --- # 19. Limit-periodic structure of the full model Each raw singular-series factor for a fixed subset of： $$ \{0,h,d,h+d\} $$ is an Euler product of periodic local functions。 For a local factor with $k\le4$ forms： - outside collision lines： $$ L_p=1+O_k(p^{-2}); $$ - collision residues occupy only： $$ O_k(p) $$ of the： $$ p^2 $$ pairs； - on collisions： $$ L_p=1+O_k(p^{-1}). $$ Therefore： $$ \boxed{ \mathbb E_{h,d\bmod p} |L_p-1|^2 = O_k(p^{-3}). } $$ Since： $$ \sum_p p^{-3}<\infty, $$ finite-prime Euler products converge in Besicovitch： $$ B^2(\mathbb Z^2). $$ --- # 20. Pair-square term The pair singular series as a function of $h$ has finite moments of every fixed order；in particular finite-prime approximants are bounded in $B^4$。 Therefore： $$ \mu(h)^2 $$ exists in $B^2$。 Thus： ## Theorem 20.1 The full deterministic covariance： $$ \boxed{ \mathfrak K_4(h,d) = \mathfrak S_0(0,h,d,h+d) - [\mathfrak S(h)-1]^2 } $$ is a two-dimensional $B^2$ limit-periodic function。 --- # 21. Full zero mean Finite-prime approximants： $$ \mathfrak K_{4,q_y} $$ satisfy the stronger rowwise zero-mean theorem。 Passing to the $B^2$ limit gives no： $$ \beta=0 $$ Besicovitch Fourier component。 In particular： $$ \boxed{ \mathcal M( \mathfrak K_4 )=0, } $$ where $\mathcal M$ is the two-dimensional Besicovitch mean。 --- # 22. Weighted limit-periodic lemma Let： $$ F\in B^2(\mathbb Z^2) $$ be limit-periodic with mean zero。 Let： $$ W_N(h,d) $$ be supported in an： $$ O(N)\times O(N) $$ region and satisfy： $$ \|W_N\|_\infty \ll N^3. $$ Suppose： $$ \sum_{h,d}|W_N(h,d)|^2 \ll N^8. $$ Then periodic approximation plus Cauchy–Schwarz gives： ## Lemma 22.1 If every fixed periodic approximant has weighted discrepancy： $$ O_q(N^4), $$ then： $$ \boxed{ \sum_{h,d} W_N(h,d)F(h,d) = o(N^5). } $$ --- # 23. Proof mechanism of Lemma 22.1 Let： $$ F_q $$ be a fixed periodic approximation with： $$ \|F-F_q\|_{B^2}<\varepsilon. $$ Then： $$ \sum W_NF_q = O_q(N^4) = o(N^5). $$ For the error： $$ \begin{aligned} \left| \sum W_N(F-F_q) \right| &\le \left( \sum|W_N|^2 \right)^{1/2} \\ &\quad\times \left( \sum|F-F_q|^2 \right)^{1/2} \\ &\le O(N^4) \cdot O(\varepsilon N) \\ &= O(\varepsilon N^5), \end{aligned} $$ after taking the large-$N$ Besicovitch average。 First： $$ N\to\infty, $$ then： $$ \varepsilon\to0. $$ --- # 24. Full deterministic subpower closure Apply Lemma 22.1 with： $$ F=\mathfrak K_4, $$ and： $$ W_N=\Omega_N. $$ The diagonal： $$ h=d $$ and other collision lines contribute only： $$ N^{4+o(1)} $$ by the v3.6 lower-dimensional count。 Therefore： ## Theorem 24.1 · Full Model Subpower Closure $$ \boxed{ \mathfrak M_{4,N} = o(N^5). } $$ Thus the full deterministic singular-series model has no persistent $N^5$ main term。 --- # 25. What Theorem 24.1 does not prove It does **not** imply any fixed： $$ \eta_M>0. $$ The bound is compatible with： $$ \frac{N^5}{\log N}, $$ or： $$ N^{5-o(1)}. $$ Therefore it does not yet close the v3.6 Gate B。 --- # 26. High-conductor rational tail norm The exact surviving spectrum motivates： $$ \boxed{ \mathfrak C_\beta(N) = \sum_{ \substack{ (\alpha,\beta)\in\operatorname{Spec}(\mathfrak K_4) \\ \beta\ne0 } } | \widehat{\mathfrak K}_4(\alpha,\beta) | M_N(\beta), } $$ interpreted through finite conductor truncation / a justified summation procedure。 Using： $$ |\widehat\Omega_N| \ll N^4M_N(\beta), $$ a sufficient deterministic gate is： ## $\operatorname{PRT}(\eta)$ · Parallelogram Rational Tail $$ \boxed{ \mathfrak C_\beta(N) \ll N^{1-\eta+o(1)} } $$ for some： $$ \eta>0. $$ Then： $$ \boxed{ \mathfrak M_{4,N} \ll N^{5-\eta+o(1)}. } $$ This is deliberately stronger than necessary but exposes the remaining conductor problem in one spectral direction。 --- # 27. Two-dimensional stronger gate Using the sharper Theorem 13.1, define： $$ \boxed{ \mathfrak C_{\alpha,\beta}(N) = \sum_{ \beta\ne0 } | \widehat{\mathfrak K}_4(\alpha,\beta) | M_N(\alpha) M_N(\beta). } $$ If： $$ \boxed{ \mathfrak C_{\alpha,\beta}(N) \ll N^{2-\eta+o(1)}, } $$ then again： $$ \boxed{ \mathfrak M_{4,N} \ll N^{5-\eta+o(1)}. } $$ --- # 28. Why adding-fractions enters exactly here The refined singular-series expansion sums reduced fractions： $$ \theta_i=\frac{a_i}{q_i}, $$ with： $$ \theta_1+\theta_2+\theta_3+\theta_4=0. $$ The surviving frequency： $$ \beta=\theta_3+\theta_4 $$ is nonzero。 The dangerous spectral region is： $$ \|\beta\|\lesssim N^{-1}, $$ where： $$ M_N(\beta)\asymp N. $$ Thus the hard counting problem is precisely： > how many weighted rational tuples satisfy the integer-sum condition while producing a nonzero but very small two-fraction sum $\theta_3+\theta_4$？ This is an adding-fractions problem。 Bloom–Kuperberg's work supplies strong counting machinery for rational linear equations and denominator separation, but no theorem is asserted here that directly closes $\operatorname{PRT}(\eta)$ for this even four-fraction covariance。 --- # 29. Spectral pairing interpretation The rational spectrum separates the Wick-like channels cleanly。 ### First pairing $$ (0,h)(d,d+h) $$ is exactly： $$ \beta=0. $$ It is completely removed by the pair-square subtraction。 ### Second pairing $$ (0,d)(h,d+h) $$ appears on： $$ \alpha=0, \qquad \beta\ne0. $$ ### Third pairing $$ (0,d+h)(h,d) $$ occupies genuinely two-dimensional rational frequencies。 Thus v3.8 gives a coefficient-level explanation of the v3.7 local cancellation theorem。 --- # 30. Finite-conductor Sobolev diagnostics For： $$ q=6,30,210,2310, $$ the reference package computes： $$ \mathfrak H_d(q). $$ The values grow slowly in these samples。 This suggests that a quantitative $d$-antiderivative estimate may exist。 But no asymptotic bound such as： $$ \mathfrak H_d(q)\ll(\log q)^A $$ or： $$ \mathfrak H_d(q)\ll q^\varepsilon $$ is promoted to theorem in this node。 This is a candidate v3.9 direction。 --- # 31. Stronger future Sobolev route If one can prove that the full limit-periodic covariance possesses a $d$-antiderivative： $$ \boxed{ \Delta_d\mathfrak G_4 = \mathfrak K_4 } $$ with quantitatively controlled Besicovitch $B^2$ norm, then summation by parts can exploit： $$ \Delta_d\Omega_N=O(N^2) $$ instead of： $$ \Omega_N=O(N^3). $$ This has the potential to gain an additional power of $N$。 The missing point is a quantitative conductor-uniform Sobolev estimate, not the algebraic existence of nonzero frequencies。 --- # 32. Current deterministic status After v3.8： ```text constant spectral mode CLOSED entire beta=0 axis CLOSED full deterministic N^5 main term EXCLUDED: o(N^5) fixed power eta_M > 0 OPEN dangerous modes beta != 0 and beta very close to an integer analytic interface adding fractions / conductor Sobolev ``` This is a substantially narrower high-conductor GAP than v3.7。 --- # 33. Suggested v3.9 direction Recommended： `RH-Parallelogram-SobolevConductor v3.9` Tasks： 1. use the exact finite-$q$ coefficient： $$ \widehat K_q(\cdot,\beta) = B_{-\beta}\ast B_\beta; $$ 2. bound： $$ \mathfrak H_d(q) = \sum_{\beta\ne0} \frac{ \|\widehat K_q(\cdot,\beta)\|_2^2 }{ |1-e(\beta)|^2 }; $$ 3. exploit： $$ \|\rho_q\|_2^2=q/\phi(q)-1, $$ and uniform： $$ \|\rho_q\|_4; $$ 4. classify $\beta$ by reduced denominator； 5. use adding-fractions / Farey-spacing estimates to control near-zero $\beta$； 6. determine whether： $$ \mathfrak H_d(q) \ll_\varepsilon q^\varepsilon $$ or stronger； 7. convert such a bound into a weighted $\Omega_N$ power saving； 8. only then return to actual prime deviation。 This is the canonical deterministic continuation。 --- # 34. GAP ledger ## CLOSED / REDUCED ### G1. Rational parallelogram phase ```text CLOSED ``` $$ h\alpha+d\beta. $$ ### G2. Finite four-point coefficient ```text CLOSED ``` ### G3. Entire $\beta=0$ spectral axis ```text ZERO EXACTLY ``` ### G4. Rowwise $d$ mean ```text ZERO EXACTLY ``` ### G5. $\Omega_N$ Fourier formula ```text CLOSED ``` ### G6. $\Omega_N$ multiplier bound ```text CLOSED ``` ### G7. Full deterministic model ```text o(N^5) ``` ### G8. Nonzero constant $N^5$ model term ```text EXCLUDED ``` --- ## OPEN ### G9. Quantitative nonzero-$\beta$ conductor tail ```text OPEN ``` ### G10. Full deterministic fixed $\eta_M>0$ ```text OPEN ``` ### G11. Actual prime four-point deviation ```text OPEN ``` ### G12. Complete quartic $\eta_Q>0$ ```text OPEN ``` ### G13. RH ```text OPEN ``` --- # 35. Trust boundary ```text RH_PROVED = FALSE RH_DISPROVED = FALSE FINITE-Q AXIS CANCELLATION = EXACT FULL B2 LIMIT-PERIODIC CLOSURE = DETERMINISTIC MODEL RESULT o(N^5) != N^(5-eta) FINITE-Q SOBOLEV NUMERICS = DIAGNOSTIC ONLY NO FULL ETA_M > 0 PROVED NO PRIME ETA_Q > 0 PROVED GLOBAL_RH_CERTIFICATE = FALSE ``` Forbidden： $$ \beta=0\text{ axis absent} \Longrightarrow \eta_M>0. $$ Forbidden： $$ o(N^5) \Longrightarrow \text{fixed zero-strip progress}. $$ Forbidden： $$ \mathfrak H_d(q)\text{ looks bounded numerically} \Longrightarrow \sup_q\mathfrak H_d(q)<\infty. $$ --- # 36. One-line status > v3.8 specializes the refined singular-series rational-frequency expansion to the parallelogram and strengthens the finite-modulus centering theorem from one vanished zero mode to an entire vanished spectral axis. Writing the four rational phases under the constraint $\theta_1+\theta_2+\theta_3+\theta_4=0$ gives the two-dimensional phase $h\alpha+d\beta$, with coefficient $C_q(\alpha,\beta)=\sum_t\rho_q(t)\rho_q(\alpha-t)\rho_q(\beta-t)\rho_q(t-\alpha-\beta)$. The pair-square subtraction has coefficient $P_q(\alpha)$ only at $\beta=0$, and $C_q(\alpha,0)=P_q(\alpha)$ identically. Hence $\widehat{\mathfrak K}_{4,q}(\alpha,0)=0$ for every $\alpha$, equivalently every fixed row has zero mean in the $d$ direction. The surviving $\alpha=0,\beta\neq0$ axis is exactly the second Wick-like pairing channel. The endpoint-Cesàro parallelogram weight has exact transform $\widehat\Omega_N=\sum_{n