工程紀錄 · 第五弧線 v3.15 · 2026-09-03 · PRIME_SHELL_TAIL · AXIS_FREE_FIXED_POWER · FULL_DETERMINISTIC_MODEL_CLOSED · RH_CLAIM_FALSE # Axis-Free Mixed Ramanujan Tail：Prime-Set Shell、$Q=N^{1/2}$ Balance 與 Deterministic Model Fixed-Power Closure **RH-AxisFree-MixedRamanujanTail v3.15** 本節點承接： - `RH-Parallelogram-ConstraintAdaptedPairing v3.13` - `RH-PairSquareAxis-RamanujanTail v3.14` - `RH-Parallelogram-SobolevTailTransfer v3.10` v3.14 closed the one-dimensional axis： $$ \boxed{ \mathfrak M_{\rm axis}(N) \ll_\varepsilon N^{17/4+\varepsilon}. } $$ Thus the remaining deterministic model is the genuinely two-dimensional axis-free covariance： $$ \boxed{ \mathfrak K_4^\perp(h,d). } $$ v3.15 proves a fixed-power bound for this remaining deterministic component。 The key is to stop demanding a single closed-form two-dimensional Ramanujan coefficient and instead use the natural **prime-set shell expansion** of the finite Euler approximants。 --- ## Main result For every： $$ \varepsilon>0, $$ the axis-free deterministic parallelogram contribution satisfies： $$ \boxed{ \mathfrak M_{\perp,N} \ll_\varepsilon N^{9/2+\varepsilon}. } $$ Therefore the full deterministic four-point model satisfies： $$ \boxed{ \mathfrak M_{4,N} \ll_\varepsilon N^{9/2+\varepsilon}. } $$ because the previously closed one-dimensional axis has exponent： $$ \frac{17}{4} < \frac92. $$ Relative to the original： $$ N^5 $$ quartic barrier： $$ \boxed{ \eta_M = \frac12. } $$ This is the first completed **full deterministic fixed-power saving** in the fifth arc。 It is not yet a zero-strip result because the actual-prime deviation： $$ \mathfrak E_{4,N} $$ remains open。 --- # 0. Claim register ```text RH_PROVED = FALSE RH_DISPROVED = FALSE AXIS_FIXED_POWER = CLOSED_FROM_V3_14 AXIS_EXPONENT = 17/4 + epsilon FINITE_Q_MIXED_SOBOLEV = q^(o(1)) PRIME_SET_SHELL_DECOMPOSITION = CLOSED SHELL_AXES_ZERO = CLOSED SHELL_MIXED_ANTIDERIVATIVE = CLOSED SHELL_SOBOLEV_NORM = q^(o(1)) LOW_PRIME_SHELL_RESTRICTION = CLOSED HIGH_PRIME_SHELL_POINTWISE_TAIL = CLOSED CANONICAL_SPLIT_Q = N^(1/2) AXIS_FREE_EXPONENT = 9/2 + epsilon AXIS_FREE_SAVING = 1/2 FULL_DETERMINISTIC_EXPONENT = 9/2 + epsilon FULL_DETERMINISTIC_ETA_M = 1/2 ACTUAL_PRIME_DEVIATION = OPEN COMPLETE_QUARTIC_ETA_Q_POSITIVE = NOT_PROVED GLOBAL_RH_CERTIFICATE = FALSE ``` --- # 1. Finite-prime cumulative model For a finite set of primes： $$ P, $$ let： $$ q_P = \prod_{p\in P}p. $$ Let： $$ \boxed{ \mathfrak K_{4,q_P}^{\perp}(h,d) } $$ denote the v3.13 double-centered finite Euler-product covariance。 For every squarefree： $$ r, $$ write： $$ \boxed{ \mathfrak K_r^\perp } $$ for the corresponding cumulative model using exactly the primes dividing： $$ r. $$ Set： $$ \mathfrak K_1^\perp=0. $$ --- # 2. Exact prime-set shell Define the subset Möbius difference： ## Definition 2.1 $$ \boxed{ \mathfrak D_q^\perp = \sum_{r\mid q} \mu(q/r) \mathfrak K_r^\perp, } $$ for squarefree： $$ q>1. $$ Then Boolean-lattice Möbius inversion gives： $$ \boxed{ \mathfrak K_q^\perp = \sum_{r\mid q} \mathfrak D_r^\perp. } $$ For nested finite prime sets, the full axis-free model is the prime-set shell limit： $$ \boxed{ \mathfrak K_4^\perp = \sum_{\substack{ q\ge2\\ q\ {\rm squarefree} }} \mathfrak D_q^\perp, } $$ with the convergence interpreted first through finite Euler products and then pointwise / in the previously established almost-periodic sense。 --- # 3. Shell spectral support Every finite cumulative： $$ \mathfrak K_r^\perp $$ is： $$ r $$ -periodic in both variables。 Therefore all of its rational Fourier frequencies have reduced denominators dividing： $$ r. $$ Hence the shell： $$ \mathfrak D_q^\perp $$ also has every coordinate denominator dividing： $$ q. $$ Thus： ## Lemma 3.1 The low prime-shell sum： $$ \boxed{ \mathfrak K_{\le Q}^{\perp} = \sum_{\substack{ q\le Q\\ q\ {\rm squarefree} }} \mathfrak D_q^\perp } $$ has rational Fourier support inside： $$ \boxed{ \mathcal F_Q \times \mathcal F_Q, } $$ where： $$ \mathcal F_Q $$ is the Farey set of reduced denominators at most： $$ Q. $$ --- # 4. Shell axes remain zero v3.12–v3.13 prove for every cumulative finite conductor： $$ \widehat{ \mathfrak K_r^\perp }(\alpha,0)=0, $$ and： $$ \widehat{ \mathfrak K_r^\perp }(0,\beta)=0. $$ The shell is a linear combination of these cumulative models。 Therefore： ## Lemma 4.1 $$ \boxed{ \widehat{ \mathfrak D_q^\perp }(\alpha,0)=0, } $$ and： $$ \boxed{ \widehat{ \mathfrak D_q^\perp }(0,\beta)=0. } $$ So every shell has a two-direction periodic antiderivative。 --- # 5. Shell mixed antiderivative Let： $$ \mathfrak G_r^{(2)} $$ be the v3.13 mixed antiderivative of： $$ \mathfrak K_r^\perp. $$ Define： $$ \boxed{ \mathfrak G_q^{\rm sh} = \sum_{r\mid q} \mu(q/r) \mathfrak G_r^{(2)}. } $$ Then： $$ \boxed{ \Delta_h\Delta_d \mathfrak G_q^{\rm sh} = \mathfrak D_q^\perp. } $$ --- # 6. Shell Sobolev norm v3.13 gives, for every： $$ \varepsilon>0, $$ $$ \boxed{ \| \mathfrak G_r^{(2)} \|_{B^2} \ll_\varepsilon r^\varepsilon. } $$ Therefore： $$ \begin{aligned} \| \mathfrak G_q^{\rm sh} \|_{B^2} &\le \sum_{r\mid q} \| \mathfrak G_r^{(2)} \|_{B^2} \\ &\ll_\varepsilon \tau(q)q^\varepsilon \\ &\ll_\varepsilon q^{2\varepsilon}. \end{aligned} $$ Renaming： $$ 2\varepsilon\mapsto\varepsilon $$ gives： ## Theorem 6.1 $$ \boxed{ \| \mathfrak G_q^{\rm sh} \|_{B^2} \ll_\varepsilon q^\varepsilon. } $$ --- # 7. Low-shell coefficient norm Let： $$ \mathfrak G_{\le Q} = \sum_{\substack{ q\le Q\\ q\ {\rm squarefree} }} \mathfrak G_q^{\rm sh}. $$ Using only the triangle inequality： $$ \begin{aligned} \| \mathfrak G_{\le Q} \|_{\ell^2(\widehat{\ }) } &\le \sum_{q\le Q} \| \mathfrak G_q^{\rm sh} \|_{\ell^2(\widehat{\ })} \\ &\ll_\varepsilon \sum_{q\le Q} q^\varepsilon. \end{aligned} $$ Therefore： ## Theorem 7.1 $$ \boxed{ \| \mathfrak G_{\le Q} \|_{\ell^2(\widehat{\ })} \ll_\varepsilon Q^{1+\varepsilon}. } $$ No orthogonality between different shells is assumed。 This is deliberately conservative。 --- # 8. Two-dimensional low-shell large sieve The differentiated exact Cesàro weight is： $$ \boxed{ F_N(h,d) = \Delta_h^- \Delta_d^- \Omega_N(h,d). } $$ v3.13 gives： $$ \boxed{ \|F_N\|_2 \ll N^3. } $$ The product Farey large sieve from v3.10 gives： $$ \boxed{ \sum_{\alpha,\beta\in\mathcal F_Q} | \widehat F_N(\alpha,\beta) |^2 \ll (N+Q^2)^2 N^6. } $$ --- # 9. Low-shell weighted contribution Double summation by parts gives： $$ \boxed{ \mathfrak M_{\perp,\le Q}(N) = \sum_{h,d} F_N(h,d) \mathfrak G_{\le Q}(h,d). } $$ Equivalently in frequency space。 By Cauchy–Schwarz, Sections 7–8： ## Theorem 9.1 · Low Prime-Shell Bound $$ \boxed{ | \mathfrak M_{\perp,\le Q}(N) | \ll_\varepsilon N^3 (N+Q^2) Q^{1+\varepsilon}. } $$ --- # 10. Euler-product components of $\mathfrak K_4^\perp$ The physical function： $$ \mathfrak K_4^\perp $$ is a finite linear combination of： - ordinary Hardy–Littlewood singular series for subsets of： $$ \{0,h,d,h+d\}; $$ - squared pair singular series in： $$ h $$ and： $$ d; $$ - constants。 Every nonconstant component has an Euler product： $$ \boxed{ \mathcal F(h,d) = \prod_p L_p(h,d). } $$ Write： $$ \boxed{ L_p=1+\delta_p. } $$ --- # 11. Local discriminant For each component there is a fixed nonzero polynomial product of pair differences： $$ \boxed{ \Delta(h,d) } $$ built from a subset of： $$ h, \quad d, \quad h+d, \quad h-d. $$ On the v3.6 **genuine four-distinct region**： $$ |h|,|d|\ll N, \qquad h\ne d, $$ the relevant collision discriminants are nonzero and satisfy： $$ \boxed{ 0< |\Delta(h,d)| \ll N^{C} } $$ for an absolute component-dependent： $$ C. $$ The excluded lower-dimensional collision line： $$ h=d $$ is **not** included in the pointwise Euler-shell tail estimate below；it remains covered by the v3.6： $$ N^{4+o(1)} $$ lower-dimensional bound。 If： $$ p\nmid\Delta(h,d), $$ all relevant local shifts are distinct modulo： $$ p. $$ Then： $$ \boxed{ \delta_p(h,d) = O(p^{-2}). } $$ If： $$ p\mid\Delta(h,d), $$ then： $$ \boxed{ \delta_p(h,d) = O(p^{-1}). } $$ Small primes are absorbed into the implied constant。 --- # 12. Prime-set Euler shell bound For squarefree： $$ q, $$ the exact Euler shell of one component is： $$ \boxed{ \delta_q(h,d) = \prod_{p\mid q} \delta_p(h,d). } $$ Section 11 gives： $$ \boxed{ | \delta_q(h,d) | \le C^{\omega(q)} \frac{ \gcd( q,\operatorname{rad}\Delta(h,d) ) }{ q^2 }. } $$ --- # 13. High-shell divisor summation Use： $$ \gcd(q,D) = \sum_{a\mid(q,D)} \phi(a). $$ For every fixed： $$ \varepsilon>0, $$ $$ C^{\omega(q)} \ll_\varepsilon q^\varepsilon. $$ Therefore： $$ \begin{aligned} \sum_{q>Q} | \delta_q(h,d) | &\ll_\varepsilon \sum_{a\mid\operatorname{rad}\Delta} \phi(a) \sum_{\substack{ q>Q\\ a\mid q }} q^{-2+\varepsilon} \\ &\ll_\varepsilon Q^{-1+\varepsilon} \sum_{a\mid\operatorname{rad}\Delta} a^\varepsilon. \end{aligned} $$ The divisor sum is： $$ N^\varepsilon $$ uniformly on： $$ |h|,|d|\ll N. $$ Thus： ## Theorem 13.1 · Pointwise Euler-Shell Tail For every component： $$ \boxed{ \sum_{q>Q} | \delta_q(h,d) | \ll_\varepsilon N^\varepsilon Q^{-1+\varepsilon}. } $$ Since： $$ \mathfrak K_4^\perp $$ contains only finitely many such components： ## Corollary 13.2 On the genuine four-distinct region： $$ h\ne d, $$ $$ \boxed{ | \mathfrak K_{\perp,>Q}(h,d) | \ll_\varepsilon N^\varepsilon Q^{-1+\varepsilon} } $$ uniformly on the relevant： $$ O(N)\times O(N) $$ box。 The deleted collision line contributes only： $$ N^{4+o(1)} $$ to the weighted model and is therefore lower order than the final： $$ N^{9/2+\varepsilon} $$ bound。 --- # 14. High-shell weighted contribution The exact Cesàro parallelogram weight satisfies： $$ \boxed{ \sum_{h,d} |\Omega_N(h,d)| \ll N^5. } $$ Therefore Corollary 13.2 gives： ## Theorem 14.1 · High Prime-Shell Bound $$ \boxed{ | \mathfrak M_{\perp,>Q}(N) | \ll_\varepsilon N^{5+\varepsilon} Q^{-1+\varepsilon}. } $$ No finite-box rational restriction theorem is required for this high part。 --- # 15. Optimize the prime-shell split Write： $$ Q=N^\theta. $$ The useful regime begins at： $$ \theta\ge\frac12. $$ Then： $$ N+Q^2 \asymp Q^2. $$ Theorem 9.1 has exponent： $$ \boxed{ 3+3\theta. } $$ Theorem 14.1 has exponent： $$ \boxed{ 5-\theta. } $$ Balance： $$ 3+3\theta = 5-\theta. $$ Therefore： $$ \boxed{ \theta=\frac12. } $$ The balancing point lies exactly at the Farey critical scale： $$ \boxed{ Q=N^{1/2}. } $$ --- # 16. Full axis-free fixed-power bound At： $$ Q=N^{1/2}, $$ Theorem 9.1 gives： $$ \boxed{ N^{9/2+\varepsilon}. } $$ Theorem 14.1 gives the same： $$ \boxed{ N^{9/2+\varepsilon}. } $$ Therefore： ## Theorem 16.1 · Axis-Free Deterministic Closure $$ \boxed{ \mathfrak M_{\perp,N} \ll_\varepsilon N^{9/2+\varepsilon}. } $$ Relative to： $$ N^5, $$ $$ \boxed{ \eta_\perp = \frac12. } $$ --- # 17. Combine with the 1D axis v3.14 gives： $$ \boxed{ \mathfrak M_{\rm axis}(N) \ll_\varepsilon N^{17/4+\varepsilon}. } $$ Since： $$ \frac{17}{4} = 4.25 < 4.5 = \frac92, $$ the axis-free term dominates。 Hence： ## Theorem 17.1 · Full Deterministic Four-Point Model $$ \boxed{ \mathfrak M_{4,N} \ll_\varepsilon N^{9/2+\varepsilon}. } $$ Thus： $$ \boxed{ \eta_M = \frac12. } $$ The deterministic refined-singular-series covariance has been removed as an： $$ N^5 $$ barrier。 --- # 18. Why this is a genuine stage closure The fifth arc decomposition is： $$ \boxed{ \mathfrak Q_N = \mathfrak M_{4,N} + \mathfrak E_{4,N}. } $$ v3.15 proves： $$ \boxed{ \mathfrak M_{4,N} = O_\varepsilon( N^{9/2+\varepsilon} ). } $$ Therefore any future failure to prove： $$ \eta_Q>0 $$ cannot be blamed on the deterministic Hardy–Littlewood / refined singular-series model at the trivial： $$ N^5 $$ scale。 The remaining first-order barrier is： $$ \boxed{ \mathfrak E_{4,N}, } $$ the actual prime deviation from the correctly centered four-point model。 --- # 19. What this does not imply The v3.6 Gate B requires a bound on： $$ \boxed{ \mathfrak Q_N } $$ itself。 Even though： $$ \mathfrak M_{4,N} \ll N^{9/2+\varepsilon}, $$ the actual prime deviation may still be： $$ N^{5-o(1)}. $$ Therefore no： $$ \eta_Q>0 $$ has been proved。 Consequently no new： $$ \kappa_I>0 $$ or zero strip has been obtained。 --- # 20. Relation to multidimensional Ramanujan expansions Arithmetic functions of several variables admit absolutely convergent multiple Ramanujan expansions under multivariable Delange/Wintner-type conditions。 Tóth's general theorem provides the appropriate abstract framework and exhibits coefficients controlled by least-common-multiple structure in many multiplicative examples。 The v3.15 proof does not require a single canonical product-Ramanujan basis for： $$ \mathfrak K_4^\perp. $$ The prime-set shell expansion is preferable here because the parallelogram function depends on mixed linear forms： $$ h, \quad d, \quad h+d, \quad h-d. $$ The shell method retains those arithmetic collision geometries directly。 --- # 21. Why the high-tail estimate survives mixed linear forms The only information needed from the local geometry is： ```text generic prime local defect O(p^-2) collision prime dividing a fixed linear discriminant local defect O(p^-1) ``` For a finite system of fixed linear forms, the collision discriminant contains only finitely many factors built from their pairwise differences。 Thus the same： $$ \gcd(q,\Delta)/q^2 $$ shell majorant works uniformly。 This makes the high-shell argument robust under the parallelogram mixed forms。 --- # 22. Conservative nature of the low-shell estimate The low-shell proof uses only： $$ \boxed{ \left\| \sum_{q\le Q} \mathfrak G_q^{\rm sh} \right\|_2 \le \sum_{q\le Q} \| \mathfrak G_q^{\rm sh} \|_2. } $$ It assumes **no orthogonality** between different shell conductors。 If future work proves square-sum or quasi-orthogonality across prime-set shells, the factor： $$ Q $$ could improve toward： $$ Q^{1/2}. $$ That would move the axis-free exponent from： $$ 9/2 $$ toward the heuristic： $$ 13/3. $$ No such improvement is used in v3.15。 --- # 23. Autonomous progress metric The deterministic branch is now certified： ```text eta_axis = 3/4 eta_axis_free = 1/2 eta_M = 1/2 ``` The research optimizer should no longer spend its main budget improving： $$ \mathfrak M_{4,N} $$ unless doing so materially helps the actual-prime deviation。 The new target is： $$ \boxed{ \mathfrak E_{4,N}. } $$ --- # 24. New actual-prime gate Define： ## Four-Point Prime Deviation Gate $\operatorname{FPD}(\eta)$ There exists： $$ \eta>0 $$ such that： $$ \boxed{ | \mathfrak E_{4,N} | \ll N^{5-\eta+o(1)}. } $$ Then： $$ \mathfrak Q_N = \mathfrak M_{4,N} + \mathfrak E_{4,N} $$ satisfies： $$ \boxed{ |\mathfrak Q_N| \ll N^{5-\min(1/2,\eta)+o(1)}. } $$ Hence v3.6 gives： $$ \boxed{ I(N) \ll N^{ 3- \frac12 \min(1/2,\eta) +o(1) }. } $$ In particular, **any**： $$ \eta>0 $$ for actual-prime deviation would now imply a genuine fixed zero strip。 --- # 25. Suggested v3.16 direction Recommended： `RH-FourPointPrimeDeviation v3.16` Stop refining the deterministic singular-series model。 Tasks： 1. write： $$ \mathfrak E_{4,N} $$ explicitly in terms of： $$ \Lambda $$ minus the refined four-point model； 2. use the constraint-adapted axis/axis-free split on the **actual prime residual** as well； 3. compare with： - Chou–Haag–Huryn–Ledoan pair-error variance； - Matomäki–Radziwiłł–Shao–Tao–Teräväinen higher uniformity； - dispersion / bilinear decompositions of $\Lambda$； 4. identify whether current theorems give only： $$ N^{5-o(1)} $$ or any true fixed： $$ N^{5-\eta}; $$ 5. derive the first prime-specific lemma sufficient for： $$ \operatorname{FPD}(\eta); $$ 6. do not revisit deterministic model representations unless the prime deviation proof needs them。 This is the transition from deterministic Hardy–Littlewood geometry to actual-prime arithmetic。 --- # 26. GAP ledger ## CLOSED / REDUCED ### G1. 1D deterministic axis ```text CLOSED ``` $$ N^{17/4+\varepsilon}. $$ ### G2. Axis-free prime-shell low part ```text CLOSED ``` ### G3. Axis-free prime-shell high tail ```text CLOSED ``` ### G4. Axis-free fixed power ```text N^{9/2+\varepsilon} ``` ### G5. Full deterministic model ```text N^{9/2+\varepsilon} ``` ### G6. Deterministic saving ```text eta_M = 1/2 ``` --- ## OPEN ### G7. Actual prime four-point deviation ```text OPEN ``` ### G8. Any： $$ \eta_{\rm FPD}>0 $$ ```text OPEN ``` ### G9. Complete quartic： $$ \eta_Q>0 $$ ```text OPEN ``` ### G10. Mean-square： $$ \kappa_I>0 $$ ```text OPEN ``` ### G11. RH ```text OPEN ``` --- # 27. Trust boundary ```text RH_PROVED = FALSE RH_DISPROVED = FALSE DETERMINISTIC eta_M = 1/2 = HARDY-LITTLEWOOD MODEL ONLY ACTUAL PRIME DEVIATION = NOT BOUNDED AT FIXED POWER HERE eta_M > 0 DOES NOT IMPLY eta_Q > 0 NO NEW ZERO STRIP YET GLOBAL_RH_CERTIFICATE = FALSE ``` Forbidden： $$ \mathfrak M_{4,N}\ll N^{9/2} \Longrightarrow \mathfrak Q_N\ll N^{9/2}. $$ Forbidden： $$ \eta_M=\frac12 \Longrightarrow \Theta\le\frac34. $$ The actual-prime deviation remains additive and may dominate。 Forbidden： $$ \text{Euler-shell pointwise tail} \Longrightarrow \text{same bound for prime deviation}. $$ The prime deviation is not an Euler product。 --- # 28. One-line status > v3.15 closes the full deterministic refined-singular-series component at fixed power. The axis-free covariance is expanded by squarefree prime-set shells using Boolean Möbius inversion of the finite Euler approximants. Every shell preserves the double-axis cancellation and hence admits a mixed antiderivative. The v3.13 finite-conductor mixed Sobolev bound gives each shell antiderivative $q^{o(1)}$ norm; summing all shells with prime-set conductor $q\le Q$ by the triangle inequality costs only $Q^{1+o(1)}$. Their rational frequencies lie in the product Farey set of denominator at most $Q$, so the two-dimensional large sieve and double summation by parts bound the low-shell contribution by $N^3(N+Q^2)Q^{1+o(1)}$. For the high shells, the physical deterministic model is a finite linear combination of Euler products. Each local Euler defect is $O(p^{-2})$ at generic primes and $O(p^{-1})$ only when $p$ divides one of finitely many parallelogram collision forms. Consequently an exact squarefree shell is bounded by $C^{\omega(q)}\gcd(q,\operatorname{rad}\Delta)/q^2$, and divisor summation gives the uniform pointwise tail $N^{o(1)}Q^{-1+o(1)}$. Weighted by the exact Cesàro parallelogram kernel, the high contribution is $N^{5+o(1)}Q^{-1}$. Balancing low and high at the Farey threshold $Q=N^{1/2}$ yields the full axis-free bound $N^{9/2+\varepsilon}$. Combining with the v3.14 one-dimensional axis bound $N^{17/4+\varepsilon}$ gives the full deterministic model $\mathfrak M_{4,N}\ll_\varepsilon N^{9/2+\varepsilon}$, a genuine deterministic saving $\eta_M=1/2$. This does not yet imply a zero strip: the remaining and now canonical obstruction is the actual-prime four-point deviation $\mathfrak E_{4,N}$. --- # 29. References 1. László Tóth, **Ramanujan expansions of arithmetic functions of several variables**, *The Ramanujan Journal* 47 (2018), 589–603. arXiv: https://arxiv.org/abs/1704.02881 2. Vivian Kuperberg, **Sums of singular series along arithmetic progressions and with smooth weights**, *International Journal of Number Theory* 21 (2025), 53–74. arXiv: https://arxiv.org/abs/2301.06095 3. D. A. Goldston, Julian Ziegler Hunts, Timothy Ngotiaoco, **The Tail of the Singular Series for the Prime Pair and Goldbach Problems**, *Functiones et Approximatio Commentarii Mathematici* 56 (2017), 117–141. arXiv: https://arxiv.org/abs/1409.2151 4. AMRAL, **RH-Parallelogram-ConstraintAdaptedPairing v3.13**. 5. AMRAL, **RH-PairSquareAxis-RamanujanTail v3.14**. --- # 30. Provenance 研究主導：Neo.K v3.15 prime-set shell expansion、axis-free low/high conductor bounds、Euler discriminant tail、$Q=N^{1/2}$ optimization、full deterministic $\eta_M=1/2$ closure、reference implementation 與 canonical source 整理：ChatGPT / GPT-5.6 Sol 日期：2026-09-03 研究定位：AMRAL 黎曼猜想半自主研究線，第五弧線 deterministic model closure / transition to actual-prime deviation。 本文件是 research source artifact，不是 peer-reviewed publication。 所有 claim 必須依 Claim register、GAP ledger 與 Trust boundary 解讀。