# RH-W-13: Cross-Regularity Continuation and Gauge Parameters **版本：** v0.1 **日期：** 2026-07-23 **狀態：** Finite-dimensional rigorous certificate completed; does not constitute an RH proof or counterexample --- ## Abstract RH-W-12 discovered cross-regularity cancellation modes in the degree-$1$ and degree-$3$ B-spline dictionaries. The original plan for this round was to continue along $(h,d,\alpha)$, where $\alpha$ controls the relative weight of the two channels. Rigorous analysis first points out: if $\alpha$ merely multiplies the entire $m=3$ channel by a non-zero constant, it is simply an invertible change of basis and does not alter the generalized spectrum. Therefore, $\alpha$ is a gauge/preconditioning parameter, not a true mathematical continuation direction. This round replaces $\alpha$ with a relative translation $\sigma$ between the two channels, and establishes an explorer with consistent parameters. The explorer previously generated false negative generalized eigenvalues because it only quantized $M$ without synchronously quantizing the Gram matrix $G$. After correcting the error, we selected $$ h=\frac{1797}{10000},\qquad d=\frac{893}{5000},\qquad \sigma=0, $$ and constructed a ten-dimensional real Weil interval matrix using a jump-resolved Archimedean tail bound. Finally, the exact verifier proves $$ \boxed{10^{-8}<\lambda_{\min}(M,G)<5\times10^{-8}}. $$ This is a strictly positive, near-zero finite-dimensional spectral band; it is neither an RH proof nor an RH counterexample. --- ## 1. Mixed Dictionary The two channels are $$ v_j^{(1)}(x)=h^{-1/2}\beta_1\!\left(\frac{x-t_j}{h}\right), $$ $$ v_j^{(3)}(x)=h^{-1/2}\beta_3\!\left(\frac{x-t_j}{h}\right), $$ where $$ t_j=\left(j-2\right)d,\qquad j=0,\dots,4. $$ The correlation kernels are respectively $$ \beta_1*\beta_1=\beta_3, $$ $$ \beta_1*\beta_3=\beta_5, $$ $$ \beta_3*\beta_3=\beta_7. $$ Thus, the full matrices are $$ M= \begin{pmatrix} M^{11}&M^{13}\\ M^{31}&M^{33} \end{pmatrix}, \qquad G= \begin{pmatrix} G^{11}&G^{13}\\ G^{31}&G^{33} \end{pmatrix}. $$ --- ## 2. $\alpha$ is Not a Continuation Direction Let $$ T_\alpha=\operatorname{diag}(I_5,\alpha I_5),\qquad \alpha\ne0. $$ If we merely multiply the entire $m=3$ channel by $\alpha$, then $$ M_\alpha=T_\alpha^TMT_\alpha, $$ $$ G_\alpha=T_\alpha^TGT_\alpha. $$ For any $y\ne0$, let $c=T_\alpha y$, then $$ \frac{y^TM_\alpha y}{y^TG_\alpha y} = \frac{c^TMc}{c^TGc}. $$ Meanwhile, $$ \det(M_\alpha-\lambda G_\alpha) = \det(T_\alpha)^2\det(M-\lambda G). $$ So the set of generalized eigenvalues remains completely unchanged: $$ \boxed{\operatorname{spec}(M_\alpha,G_\alpha)=\operatorname{spec}(M,G)}. $$ Therefore, $\alpha$ can improve or worsen the coordinate condition number, but it cannot generate a new mathematical subspace. --- ## 3. The True Continuation Parameter $\sigma$ Instead, let the two channels be relatively translated: $$ t_j^{(1)}=t_j-\frac\sigma2, $$ $$ t_j^{(3)}=t_j+\frac\sigma2. $$ At this point, the self-blocks remain unchanged, but the centers of the cross-blocks become $$ (t_i-t_j)-\sigma, $$ Thus, $\sigma$ truly changes the subspace spanned by the dictionary. By reflection symmetry, the generalized spectrum is an even function of $\sigma$: $$ \lambda_k(\sigma)=\lambda_k(-\sigma). $$ Exploratory scanning shows that near the $(h,d)$ selected in this round, $\sigma=0$ is a local minimum within the sampled range; this is merely an exploratory conclusion, not a global minimum theorem. --- ## 4. The Consistent Parameter Principle The generalized eigenvalue problem is $$ Mc=\lambda Gc. $$ If $M$ uses parameters $\theta_q$, while $G$ uses another extremely close set of parameters $\theta$, the computation is not actually for the same dictionary: $$ M(\theta_q)c=\lambda G(\theta)c. $$ When the Gram matrix is close to linear dependence, this mismatch will be significantly amplified. Error case in this round: - Original parameters: $(0.17919521,0.17823161,0.01146941)$; - Cache key for $M$ quantized to: $(0.179195,0.178232,0.011469)$; - $G$ still used the unquantized values; - Erroneous output: $-3.3206\times10^{-7}$; - After consistent quantization: $+8.7636\times10^{-8}$; - 80-bit independent check: $+8.7621\times10^{-8}$. Therefore, the fixed rule is: > Any generalized spectrum computation must first generate a single canonical parameter object, and then simultaneously construct $M$ and $G$ from this same object. --- ## 5. Selected Rational Chamber The final rigorous candidate is $$ h=\frac{1797}{10000}=0.1797, $$ $$ d=\frac{893}{5000}=0.1786, $$ $$ \frac dh=\frac{1786}{1797}\approx0.9938786867, $$ $$ \sigma=0. $$ The maximum correlation support radius is $$ 4d+4h=\frac{3583}{2500}=1.4332<\log5. $$ So the complete non-zero von Mangoldt indices are exactly $$ \boxed{2,3,4}. $$ where $4=2^2$ must be treated as an independent prime-power layer. --- ## 6. Exact Spectral Bounds The maximum row error of the interval matrix is $$ \epsilon \approx2.9107439530551\times10^{-12}. $$ The purely rational $LDL^T$ verifier proves $$ C-10^{-8}G-\epsilon I\succ0. $$ So $$ \lambda_{\min}(M,G)>10^{-8}. $$ Additionally, there is an integer witness $$ \begin{aligned} c={}&(68190193,137154794,187700175,137154794,68190193,\\ &-3577963013,-7569824004,-10000000000,-7569824004,-3577963013)^T, \end{aligned} $$ satisfying $$ \frac{c^TMc}{c^TGc} \in [3.9958420713,3.9959698087]\times10^{-8}. $$ Therefore, $$ \boxed{10^{-8}<\lambda_{\min}(M,G)<5\times10^{-8}}. $$ --- ## 7. Channel Cancellation and Arithmetic Cancellation For the above witness, the normalized channel decomposition is approximately $$ Q_{11}\approx2.55526\times10^{-7}, $$ $$ 2Q_{13}\approx-5.12593\times10^{-7}, $$ $$ Q_{33}\approx2.97025\times10^{-7}. $$ The sum of the three leaves only about $$ 3.9959\times10^{-8}. $$ The prime-power contributions are approximately $$ P_2\approx-0.1161577733, $$ $$ P_3\approx-0.00115199678, $$ $$ P_4\approx-6.87721\times10^{-10}. $$ Therefore, the prime-free background must use approximately $$ +0.1173098107 $$ to cancel the complete arithmetic block. The near-zero value simultaneously contains: 1. Cross-block cancellation between the $m=1$ and $m=3$ channels; 2. Arithmetic cancellation between the prime-free background and the prime-power blocks. --- ## 8. Isolated Channels The exact verifier additionally proves $$ \lambda_{\min}^{(m=1)}>4\times10^{-4}, $$ $$ \lambda_{\min}^{(m=3)}>10^{-7}. $$ So the mixed gap is strictly lower than any proven isolated lower scale. In particular, $$ \lambda_{\min}^{\mathrm{mixed}}<5\times10^{-8}<10^{-7}<\lambda_{\min}^{(m=3)}. $$ This supports that the "cross-regularity low spectral band" is not a direct replica of a single-channel low mode. --- ## 9. Conclusions and Limitations Completed in this round: - Proved that the $\alpha$ channel scale is a gauge parameter; - Introduced the true continuation direction $\sigma$; - Discovered and eliminated a false negative Weil candidate; - Established a jump-resolved Archimedean tail bound; - Rigorously proved that the ten-dimensional mixed gap lies in $(10^{-8},5\times10^{-8})$. However, we still only have $$ \boxed{\text{Finite-dimensional positivity and near-zero spectral band}\centernot\Longrightarrow RH}. $$ No true negative Weil witness was found. --- ## Reference Background - E. Bombieri, *Problems of the Millennium: The Riemann Hypothesis*. - M. Suzuki, *Weil's quadratic form via the screw function*, arXiv:2606.09096, 2026. - A. Groskin, *A finite Guinand–Weil dictionary and archimedean tail order for the truncated Weil quadratic form*, arXiv:2607.02828, 2026.