# RH-W-17: Chamber-Aware Subdivision and Event Thin Layers **版本：** v0.1 **日期：** 2026-07-24 **狀態：** `CERTIFIED_CHAMBER_AWARE_EVENT_SUBDIVISION` **範圍聲明：** This document only investigates parametric chamber subdivision on a fixed finite-dimensional mixed-order Weil dictionary; it does not prove or disprove the Riemann Hypothesis. --- ## Abstract The parametric tubes in RH-W-14 through RH-W-16 all required the entire parameter box to be located within the same spline polynomial chamber; if any $\pm\log p^k$ sample potentially crossed a spline knot, the old pipeline would reject the entire box. This iteration establishes for the first time: $$ \boxed{ \text{Event surface localization} \rightarrow \text{Stable chamber / event thin layer subdivision} \rightarrow \text{Strict interval-by-interval matrices} \rightarrow \text{exact }LDL^T \rightarrow \text{Chamber adjacency graph} } $$ The demonstration event is $$ \boxed{4d=\log 2}, $$ which causes the furthest lag $n=2$ sample to simultaneously cross the central spline knot of correlation degrees $3,5,7$. Throughout the entire process, the active prime powers remain as $$ \{2,3,4\}, $$ What changes is the identity of the polynomial piece, not the entry or exit of arithmetic samples. The system subdivides the master interval into three closed intervals: $$ C_0=[0.17328669,0.17328679], $$ $$ C_1=[0.17328679,0.17328680], $$ $$ C_2=[0.17328680,0.17328690]. $$ where $C_1$ strictly contains $\log2/4$. The purely rational verifier proves: $$ \boxed{ \lambda_{\min}(M(d),G(d))>10^{-8} \quad \forall d\in C_0\cup C_1\cup C_2. } $$ Thus, for the first time, a complete closed interval cover is obtained without deleting the event surface. --- ## 1. Fixed Dictionary Fixed scale and relative translation: $$ h=\frac{1797}{10000}, \qquad \sigma=0. $$ Two families of bases are used: - basis degree $m=1$, five translations; - basis degree $m=3$, five translations. The total dimension is ten, and the correlation kernel degrees are $$ 1\times1\to3, \qquad 1\times3\to5, \qquad 3\times3\to7. $$ The center positions are $$ t_j=(j-2)d, \qquad j=0,1,2,3,4. $$ The center difference of the furthest pair of bases is $$ c=-4d. $$ --- ## 2. Event Equation For the sample $x=-\log2$, the equation of the central knot is $$ x=c, $$ therefore $$ -\log2=-4d, $$ that is, $$ \boxed{d_*=\frac{\log2}{4}}. $$ Strict rational interval computation gives $$ d_* \in [0.173286795139986327354308\ldots, 0.173286795139986327354309\ldots]. $$ It is completely contained within the event thin layer $$ C_1=[0.17328679,0.17328680]. $$ --- ## 3. Simultaneous Multiple Regularity Event The event does not act on just a single block. For the furthest lag, the following oriented entries simultaneously hit the central knot: - $m=1$ self block, correlation degree $3$; - $m=1/3$ cross block, correlation degree $5$; - $m=3$ self block, correlation degree $7$. Accounting for orientation and matrix symmetry positions, there are a total of eight oriented event witnesses. Therefore, this is a: $$ \boxed{ \text{SIMULTANEOUS CENTRAL SPLINE KNOT CROSSING} } $$ rather than an accidental switching of a single matrix entry. --- ## 4. Why Event Thin Layers Can Be Directly Verified A degree-$r$ cardinal B-spline belongs to $$ C^{r-1}. $$ The minimum correlation degree of this dictionary is $r=3$, so all correlation kernels are at least $$ C^2. $$ Therefore, even if the interval crosses a knot, the matrix entries still possess globally bounded second derivatives. The linear interpolation remainder $$ |F(d)-L(d)| \le \frac12 \sup|F''| \rho_d^2 $$ still holds. This is the core of the current iteration: > Fixing the polynomial piece is a convenient condition for generating concise formulas, but it is not a necessary condition for cross-knot positivity certificates; as long as the second-order regularity of the kernel is sufficient, the switching surface can be sealed using an event thin layer. --- ## 5. Three-Interval Subdivision ### 5.1 Left Chamber $C_0$ $$ C_0=[0.17328669,0.17328679]. $$ Strictly satisfies $$ 4d<\log2. $$ The $n=2$ sample is located on a fixed side of the central knot, and all event entries use the left polynomial piece. ### 5.2 Event Thin Layer $C_1$ $$ C_1=[0.17328679,0.17328680]. $$ Contains $$ 4d=\log2. $$ This interval does not claim a fixed piece, but instead directly covers the entire switching layer using a convex combination of the left and right endpoint matrices plus a second-order remainder. ### 5.3 Right Chamber $C_2$ $$ C_2=[0.17328680,0.17328690]. $$ Strictly satisfies $$ 4d>\log2. $$ The $n=2$ sample is located on the other side of the central knot, using the right polynomial piece. --- ## 6. Arithmetic Chamber Remains Unchanged The maximum correlation support radius of the master interval is $$ R_{\max}=4(0.17328690)+4(0.1797)=1.4119476. $$ and $$ R_{\max}<\log5. $$ so the active von Mangoldt indices throughout the entire interval remain constantly as $$ \boxed{2,3,4}. $$ The activation graphs of the four endpoint matrices are completely identical. Therefore: $$ \boxed{ \text{activation graph remains unchanged} \quad\text{but}\quad \text{polynomial piece changes}. } $$ This proves that the "prime-power chamber" and the "spline-piece chamber" are two distinct layers of discrete structures. --- ## 7. Strict Matrix Certificate Each cell uses: $$ M(d) = (1-t)M(d_-)+tM(d_+)+R_M(d), $$ $$ G(d) = (1-t)G(d_-)+tG(d_+)+R_G(d). $$ Adopting the second-order Weil bounds corrected in RH-W-15: $$ L_3^{(2)}=2494, \qquad L_5^{(2)}=3110, \qquad L_7^{(2)}=3697. $$ The maximum combined row remainder for the left and right stable chambers is $$ 2.5526250009290203\times10^{-10}, $$ while the event thin layer only has $$ 2.5526250009290200\times10^{-12}. $$ For each endpoint, the purely rational verifier checks $$ C-10^{-8}G-(\epsilon_{\rm point}+\epsilon_{\rm cell})I\succ0. $$ All three closed cells pass. Therefore: $$ \boxed{ M(d)-10^{-8}G(d)\succ0 \quad \forall d\in[0.17328669,0.17328690]. } $$ --- ## 8. Numerical Observations The following is for exploration only and does not enter the certificate: | $d$ | midpoint $\lambda_0$ | |---:|---:| | 0.17328669 | $2.1880721932\times10^{-7}$ | | 0.17328679 | $2.1880755243\times10^{-7}$ | | 0.17328680 | $2.1880758603\times10^{-7}$ | | 0.17328690 | $2.1880791895\times10^{-7}$ | The lowest mode smoothly crosses the event surface without exhibiting any sharp corners, negative values, or mode swaps. This is consistent with the regularity of B-splines: the central knot switches the polynomial formula, but the function and its first two derivatives remain continuous. --- ## 9. Chamber Adjacency Graph This iteration generates the first machine-readable adjacency graph: $$ \boxed{ C_0^{\rm left} \longleftrightarrow C_1^{\rm event} \longleftrightarrow C_2^{\rm right} } $$ Its data is saved in: - `chamber_adjacency_graph.json` - `event_surface_catalog.csv` - `chamber_subdivision_certificate.json` This is the minimal prototype for the automatic subdivision of multi-parameter boxes in the future. --- ## 10. Computational Contract The jump-resolved exponential power recurrence uses the first $$ K=1500 $$ terms, instead of the previously fixed $K=5000$. The remainder is still strictly enveloped by a positive geometric tail; thus, what is reduced is the cost, not the strictness. The widest saved matrix interval is approximately $$ 1.936\times10^{-12}, $$ and the 80-digit `mpmath` evaluations for the degree-$3/5/7$ event-sensitive entries at the four endpoints all fall within the strict intervals. --- ## 11. Conclusion RH-W-17 closes the previous restriction of "whole-box rejection across event surfaces": $$ \boxed{ \text{Fixed chamber} + \text{Event thin layer} + \text{Adjacency graph} } $$ These can now be composed into a complete closed parameter domain certificate. It has not found a Weil negative direction, nor has it advanced to a proof of RH; its research value lies in allowing parametric continuation to legally cross a spline chamber boundary for the first time, rather than stopping before the boundary.