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1 \chapter{Numerical evidence and applications}\label{ch:evidence}
2 \chapterlead{Exact-law controls test values against independent formulas. Heterogeneous
3 spectra test inversion across irregular coefficients. Simulated kernels test
4 the operator pipeline. Timing measurements characterize particular
5 implementations. These are distinct forms of evidence.}
6 The source package records the inputs and environments behind the reported
7 experiments. The release regression and containment checks are reported in
8 Appendix~\ref{app:reproduction}.
9 
10 \section{Deep-tail controls}\label{sec:final-3}
11 A directed enclosure includes rounding as well as analytic remainders. An analytic ledger evaluated in ordinary floating point does not. The labels in this chapter identify which conclusion a result supports. The retained timing measurements are not presented as a new hardware comparison.
12 
13 The method is designed for a regime in which direct survival probabilities can
14 underflow long before the mathematical probability becomes numerically
15 unresolvable.  The engine therefore treats \(\log p\) as the primary numerical
16 quantity and evaluates its error bounds on that logarithmic scale.
17 
18 The validation ladder spans seven qualitatively different settings:
19 
20 \begin{center}
21 \small
22 \begin{tabular}{@{}P{0.27\linewidth}P{0.13\linewidth}P{0.22\linewidth}P{0.16\linewidth}P{0.12\linewidth}@{}}
23 \toprule
24 validation regime & deepest $\log_{10}p$ & observed error & certificate & spectral size\\
25 \midrule
26 directed-rounding $0.02\chi^2_{80}$ control
27 & $-5000$ & enclosure width $3.34\times10^{-11}$ decades
28 & multiplicative ratio $\le 1+7.70\times10^{-11}$ & 80 df\\
29 large-scale positive spectrum
30 & $-922.031162$ & below stored reference resolution & $5.71\times10^{-9}$ & rank 1997\\
31 strong-dependence kernel
32 & $-457.973916$ & below stored reference resolution & $5.20\times10^{-9}$ & rank 996\\
33 heterogeneous inversion benchmark near $10^{-16}$
34 & $-16$ & $2.88$--$6.52\times10^{-15}$ decades & independent integral converged & ranks 80--120\\
35 signed finite-endpoint exact control
36 & $-200$ & $\le 2.84\times10^{-14}$ decades & $7.23\times10^{-13}$ & rank 80\\
37 signed mixed multiscale
38 & $-81.891937$ & $1.42\times10^{-12}$ decades vs full grid & $5.78\times10^{-9}$ & rank 340\\
39 noncentral exact controls
40 & $-40$ & $\le 9.26\times10^{-15}$ decades & passing & up to 50 df\\
41 \bottomrule
42 \end{tabular}
43 \end{center}
44 
45 The analytic scaled-chi-square ladder also includes
46 \[
47 \log_{10}p=-100,-300,-1000,-5000,
48 \]
49 with respective reported absolute log-probability discrepancies
50 \[
51 2.84\times10^{-14},\quad
52 <10^{-15},\quad
53 3.41\times10^{-13},\quad
54 <10^{-15}
55 \]
56 in the stored benchmark output.  The corresponding analytic relative certificates
57 remain finite throughout, reaching only
58 \[
59 3.85\times10^{-11}
60 \]
61 at \(10^{-5000}\).
62 
63 A large-scale run with \(n=2000\), \(m=50{,}000\), 49,951 retained
64 features, and quadratic-form rank 1997 produced
65 \[
66 \boxed{\log_{10}p=-922.0311624669905}
67 \]
68 using 79 spectral packets and \(N=128\).  Its independently evaluated
69 full-spectrum tilted reference agreed within the stored reporting precision, while the
70 reported relative certificate was
71 \[
72 5.71\times10^{-9}.
73 \]
74 
75 The corrected strong-dependence case study independently reached
76 \[
77 \boxed{\log_{10}p=-457.9739160722001}
78 \]
79 with a finite \(5.20\times10^{-9}\) certificate.  The signed finite-endpoint
80 control passed its analytic ledger through \(10^{-200}\), while a rank-340 mixed-sign
81 multiscale spectrum passed its analytic ledger to
82 \[
83 \boxed{\log_{10}p=-81.89193686056221}.
84 \]
85 Finally, the noncentral engine matched exact noncentral-chi-square controls
86 from \(10^{-4}\) through \(10^{-40}\), with maximum observed discrepancy
87 \[
88 9.26\times10^{-15}
89 \]
90 decades.
91 
92 \begin{figure}[htbp]
93 \centering
94 \includegraphics[width=\linewidth]{figures_production/figure_23_deep_tail_precision_envelope.pdf}
95 \caption{Retained log-probability discrepancies across positive, signed, mixed-sign, and noncentral control families. The symmetric-log vertical scale preserves exact stored zeros without a positive display floor. These are pointwise comparisons with the stated references, not uniform error bounds.}
96 \label{fig:deep-tail-precision-envelope}
97 \end{figure}
98 
99 \subsection{Independent characteristic-function inversion near \texorpdfstring{\(10^{-16}\)}{1e-16}}
100 
101 Closed-form controls are important, but they can hide numerical difficulties
102 that appear only for heterogeneous spectra.  An independent inversion test was
103 therefore run on four generalized chi-square laws near
104 \[
105 P(Q\ge q)\approx10^{-16}.
106 \]
107 
108 The reference evaluates the original characteristic function directly through
109 the central Imhof inversion identity
110 \[
111 P(Q\ge q)
112 =
113 \frac12
114 +
115 \frac1\pi
116 \int_0^\infty
117 \frac{\sin\Theta(u)}{u\,R(u)}\,du,
118 \]
119 where, for distinct positive coefficients \(\lambda_j\) with multiplicities
120 \(h_j\),
121 \[
122 \Theta(u)
123 =
124 \frac12\sum_j h_j\arctan(\lambda_j u)
125 -
126 \frac12qu,
127 \qquad
128 R(u)
129 =
130 \prod_j
131 \left(1+\lambda_j^2u^2\right)^{h_j/4}.
132 \]
133 This route does not use exponential tilting, the COS expansion, spectral
134 packets, or the K-COS error controller.
135 
136 The four spectra include balanced, skewed, dominant-spike, and higher-rank
137 cases.  Each reference integral was evaluated at two truncation limits.  The
138 omitted integral after the frequency cutoff \(T>0\) was bounded using
139 \[
140 \frac1\pi
141 \int_T^\infty\frac{du}{uR(u)}
142 \le
143 \frac{C}{\pi\nu}T^{-\nu},
144 \qquad
145 \nu=\frac12\sum_jh_j,
146 \qquad
147 C=\prod_j\lambda_j^{-h_j/2}.
148 \]
149 
150 \begin{center}
151 \small
152 \begin{tabular}{@{}lrrrr@{}}
153 \toprule
154 spectrum & rank & \(|\Delta\log_{10}p|\) & relative difference & reference tail bound / \(p\)\\
155 \midrule
156 balanced & 80 & $6.52\times10^{-15}$ & $1.50\times10^{-14}$ & $6.84\times10^{-20}$\\
157 skewed & 80 & $4.13\times10^{-15}$ & $9.50\times10^{-15}$ & $4.44\times10^{-18}$\\
158 spiky & 80 & $5.19\times10^{-15}$ & $1.20\times10^{-14}$ & $7.00\times10^{-17}$\\
159 balanced & 120 & $2.88\times10^{-15}$ & $6.64\times10^{-15}$ & $4.19\times10^{-29}$\\
160 \bottomrule
161 \end{tabular}
162 \end{center}
163 
164 The two inversion truncations agree to between approximately
165 \[
166 1.2\times10^{-21}
167 \quad\text{and}\quad
168 3.7\times10^{-17}
169 \]
170 decades across the four cases.  Thus the slow reference is numerically stable
171 at a scale well below the observed K-COS-versus-inversion difference.
172 Agreement between truncations is convergence evidence. The omitted-integral
173 bound alone does not enclose quadrature discretization or numerical rounding,
174 so this comparison is not assigned the directed-enclosure guarantee.
175 
176 \begin{figure}[htbp]
177 \centering
178 \includegraphics[width=\linewidth]{figures_production/figure_27_imhof_p1e16_validation.pdf}
179 \caption{Independent direct-inversion checks near $p=10^{-16}$ on four heterogeneous spectra. Circles show relative K-COS/reference differences and diamonds show the reference integral-tail bound. The latter is one error component, not a complete enclosure of all reference arithmetic.}
180 \label{fig:imhof-p1e16}
181 \end{figure}
182 
183 This test is complementary to the exact scaled-chi-square controls.  It asks
184 whether the solver remains accurate when no single-gamma reduction is
185 available and when an algorithmically different inversion of the full
186 characteristic function is used as the reference.
187 
188 These results have different evidentiary roles.  The scaled-chi-square,
189 finite-endpoint signed, Laplace signed, and noncentral cases provide analytic
190 controls.  The heterogeneous inversion battery, large-scale, strong-dependence, and
191 rank-340 mixed-sign cases test irregular spectra for which no single-gamma
192 closed form is available.  Together they test correctness against known distributions,
193 representation compression, and stability on the intended high-rank numerical
194 regime.  The low-rank Laplace example also records a deliberate certificate
195 failure boundary rather than conflating numerical agreement with certification.
196 
197 
198 \section{Signed and noncentral controls}
199 \paragraph{Exact finite-endpoint control.}
200 For
201 \[
202 Q=-0.02\,\chi^2_{80},
203 \]
204 the right tail is a lower chi-square tail with finite upper endpoint zero.  The
205 signed engine agrees with the exact distribution through
206 \[
207 \boxed{p=10^{-200}},
208 \]
209 with maximum observed log-probability error about $2.84\times10^{-14}$ decades
210 and certificates around $10^{-12}$.
211 
212 \paragraph{Mixed-sign high-rank analytic-ledger control.}
213 A deterministic rank-340 multiscale spectrum with 180 positive and 160
214 negative coefficients was tested from approximately $10^{-14}$ to
215 \[
216 \boxed{p\approx10^{-81.891937}}.
217 \]
218 At the deepest point the packetized solver used 35 packets, $N=4096$, and
219 reported a relative certificate
220 \[
221 5.78\times10^{-9}.
222 \]
223 Its log-probability differed from the unpacketized full-spectrum signed solver
224 by about $1.42\times10^{-12}$ decades.  Independent exponential-tilt
225 importance sampling placed the discrepancy at about $1.41$ Monte Carlo
226 standard errors.
227 
228 \paragraph{Retained low-rank limitation.}
229 For independent $Y_1,Y_2\sim\chi^2_2$, the quadratic form
230 \[
231 Q=0.07(Y_1-Y_2)
232 \]
233 is Laplace distributed with scale $0.14$, so for $q\ge0$,
234 \[
235 P(Q\ge q)=\frac12e^{-q/0.14}.
236 \]
237 The computed values agree with this exact law through $10^{-200}$ to roughly
238 $10^{-9}$ decades or better.  However, the implemented analytic COS-series
239 certificate rejects these low-rank mixed cases because the tilted
240 characteristic function decays only as $|t|^{-2}$.  The implementation reports
241 this as a certificate failure rather than silently accepting the numerical
242 value.
243 
244 \begin{figure}[htbp]
245 \centering
246 \includegraphics[width=\linewidth]{figures_production/figure_24_signed_certificate_boundary.pdf}
247 \caption{The negative finite-endpoint and rank-340 mixed-sign controls meet the requested analytic tolerance. The rank-four Laplace controls do not, despite close numerical agreement with their exact law. The horizontal line is the requested tolerance. These analytic-ledger results do not enclose floating-point roundoff.}
248 \label{fig:signed-certificate-boundary}
249 \end{figure}
250 
251 
252 \begin{figure}[htbp]
253 \centering
254 \includegraphics[width=\linewidth]{figures_production/figure_22_noncentral_controls.pdf}
255 \caption{Absolute log-probability discrepancies for three scaled noncentral chi-square families, labeled by degrees of freedom and noncentrality. Recorded zeros remain at zero. All analytic truncation checks pass, but this comparison is not a directed probability enclosure.}
256 \label{fig:noncentral-controls}
257 \end{figure}
258 
259 
260 \section{Rare-variant kernel case study}\label{sec:final-24}
261 
262 The preceding sections establish operator identities and certified numerical
263 bounds.  This section tests the same machinery on the project's existing
264 rare-variant SKAT simulation layer, including explicit LD structure,
265 covariates, per-variant standardization, rare-variant weighting, and the
266 phenotype-space score statistic.
267 
268 \subsection{Simulation provenance and implementation correction}
269 
270 Two established genotype generators were used.
271 
272 The first is a latent-Gaussian block-LD generator.  Within a genomic block,
273 alleles are obtained by thresholding correlated Gaussian latent variables at
274 variant-specific minor-allele frequencies.
275 
276 The second is a copy-LD generator in which haplotype states persist along a
277 block with high probability and are occasionally redrawn. The simulator
278 must initialize its genotype accumulator to zero before adding haplotypes.
279 Allocating the accumulator with \verb|np.empty| and then applying
280 \verb|+=| would make the output depend on unspecified memory contents.
281 All copy-LD results reported here use explicit zero initialization. Results
282 from an uninitialized accumulator are excluded from the evidence record.
283 
284 \subsection{Tested LD regimes}
285 
286 Four deterministic scenarios were evaluated.
287 
288 \begin{table}[htbp]
289 \centering
290 \caption{Synthetic rare-variant simulation scenarios.  Adjacent $r^2$ is
291 computed within the simulated LD blocks after variant standardization.}
292 \begin{tabular}{lrrrrr}
293 \toprule
294 scenario & $n$ & $m$ & rank & median adjacent $r^2$ & effective rank\\
295 \midrule
296 latent LD, moderate & 600 & 3000 & 596 & 0.0069 & 441.9\\
297 latent LD, strong & 600 & 3000 & 596 & 0.1140 & 240.3\\
298 copy LD, strong & 600 & 3000 & 596 & 0.8298 & 200.1\\
299 copy LD, large & 1000 & 5000 & 996 & 0.8179 & 353.0\\
300 \bottomrule
301 \end{tabular}
302 \end{table}
303 
304 Effective rank here is the participation ratio
305 \[
306 r_{\rm eff}=\frac{(\sum_j\lambda_j)^2}{\sum_j\lambda_j^2}.
307 \]
308 It measures spectral concentration, not an integer count of nonzero directions.
309 The corresponding spectra therefore span a substantial range of concentration:
310 stronger LD reduces effective rank even when the nominal quadratic-form rank
311 remains close to the sample residual dimension.
312 
313 \begin{figure}[htbp]
314 \centering
315 \includegraphics[width=\linewidth]{figures_production/figure_18_realistic_ld_effective_rank.pdf}
316 \caption{Nonzero and effective rank across four synthetic dependence scenarios. Connectors pair the two quantities for each scenario. Stronger dependence changes spectral concentration substantially even when nonzero rank is unchanged.}
317 \label{fig:realistic-ld-rank}
318 \end{figure}
319 
320 \subsection{Streaming directly from raw genotype blocks}
321 
322 The raw genotype matrix was streamed in resident variant blocks of widths
323 31, 127, and 509.  Within each block the implementation performs
324 
325 \[
326 \begin{aligned}
327 G_b&\longrightarrow G_b-\bar G_b
328      \longrightarrow \widetilde G_b,\\
329 \widetilde G_b&\longrightarrow M_X\widetilde G_b
330      \longrightarrow A_b=M_X\widetilde G_bW_b^{1/2}
331      \longrightarrow A_bA_b^\top,
332 \end{aligned}
333 \]
334 
335 where $\widetilde G_b$ denotes the centered and scaled block. The full
336 weighted factor is never materialized.
337 
338 Across all four LD scenarios and all three resident block sizes, the largest
339 relative discrepancy against the fully materialized kernel was
340 
341 \[
342 \boxed{6.61\times10^{-16}}.
343 \]
344 
345 \begin{figure}[htbp]
346 \centering
347 \includegraphics[width=\linewidth]{figures_production/figure_19_realistic_streaming_invariance.pdf}
348 \caption{Retained raw-genotype streaming discrepancies across synthetic dependence regimes and resident block widths. Marker shapes distinguish the scenarios in grayscale as well as color. The plotted differences are at floating-point summation scale.}
349 \label{fig:realistic-streaming}
350 \end{figure}
351 
352 This test is stronger than applying the streamed identity to a precomputed
353 factor because the centering, variance scaling, covariate projection, and
354 rare-variant weights are all performed inside the block loop.
355 
356 \subsection{Phenotype-space equivalence}
357 
358 For each scenario, 100 Gaussian null residual draws were evaluated in three
359 ways. Under the standardized Gaussian null, write $e=M_Xz$ with
360 $z\sim N(0,I)$. Because $M_XA=A$, the statistic used here is
361 $Q=\frac12 z^\top Kz$, so its nonzero mixture coefficients are
362 $\frac12\lambda_j(K)$. The factor $\frac12$ belongs to this case-study
363 normalization, not to the general quadratic-form definition.
364 
365 \begin{enumerate}
366 \item directly through the factor,
367 \[
368 Q=\frac12\|A^\top e\|_2^2.
369 \]
370 \item through the sample kernel,
371 \[
372 Q=\frac12 e^\top Ke.
373 \]
374 \item through the eigencoordinates of $K$.
375 \end{enumerate}
376 
377 The worst relative discrepancy across all draws and all scenarios was
378 
379 \[
380 \boxed{8.32\times10^{-16}}.
381 \]
382 
383 Thus the simulated phenotype path, the streamed kernel, and the generalized
384 $\chi^2$ spectral representation agree to floating-point precision.
385 
386 \subsection{Deep tails on simulated LD spectra}
387 
388 For every scenario, tail queries were generated at tilt fractions
389 0.90, 0.97, and 0.99 of the positive singularity.  K-COS was compared with a
390 higher-resolution full-spectrum tilted reference.
391 
392 The large corrected copy-LD case was then pushed further, to fractions
393 0.995, 0.997, 0.999, and 0.9995.  The deepest tested reference value was
394 
395 \[
396 \boxed{\log_{10}p=-457.973916}.
397 \]
398 
399 At that point the displayed K-COS and high-resolution full-spectrum values
400 were identical.  Across the complete realistic-LD battery, the largest
401 absolute discrepancy was
402 
403 \[
404 \boxed{3.21\times10^{-12}\ \text{decades}}.
405 \]
406 
407 \begin{figure}[htbp]
408 \centering
409 \includegraphics[width=\linewidth]{figures_production/figure_20_realistic_deep_tail_accuracy.pdf}
410 \caption{Absolute differences between K-COS and refined full-spectrum tilted references on the synthetic dependence scenarios. The corrected large copy-LD controls extend through approximately $10^{-458}$. Zeros are plotted at zero rather than assigned an artificial positive floor.}
411 \label{fig:realistic-deep-tail}
412 \end{figure}
413 
414 For the $n=1000$, $m=5000$ corrected copy-LD spectrum, selected results are
415 
416 \begin{center}
417 \begin{tabular}{rrrr}
418 \toprule
419 tilt fraction & $\log_{10}p$ & COS modes $N$ & absolute error (decades)\\
420 \midrule
421 0.9950 & -65.950669 & 2048 & $4.69\times10^{-13}$\\
422 0.9970 & -95.305034 & 2048 & $3.70\times10^{-14}$\\
423 0.9990 & -240.602047 & 8192 & $4.94\times10^{-14}$\\
424 0.9995 & -457.973916 & 16384 & below stored reference resolution\\
425 \bottomrule
426 \end{tabular}
427 \end{center}
428 
429 \subsection{Interpretation and boundary}
430 
431 This simulation block does not add a new algebraic theorem.  It supplies a
432 more demanding validation layer for the theorem chain already proved.
433 Specifically, it shows that the exact streaming identities and the K-COS
434 deep-tail machinery remain numerically stable under substantially different
435 rare-variant LD structures and through the complete raw-genotype preprocessing
436 path.
437 
438 The evidence remains synthetic.  These generators do not reproduce every
439 population-genetic feature of a sequencing cohort, and this block is not an
440 external-cohort replication. It addresses the Gaussian linear-model null.
441 Finite-sample Bernoulli or logistic calibration remains a separate statistical
442 problem.
443 
444 
445 
446 \section{Spectral-packet experiments}
447 These tables and figures report the retained packet-planner experiments.
448 Their analytic error bounds exclude floating-point roundoff. They measure a
449 different numerical path from the directed positive backend, and are not
450 reinterpreted as directed probability enclosures or new performance runs.
451 
452 \subsection{Exactly solvable chi-square stress test}
453 
454 To isolate the effect of exponential tilting, we first used
455 \[
456 Q=0.02\chi^2_{80},
457 \]
458 whose survival probabilities are available directly from the chi-square distribution. Table \ref{tab:chi} compares a global COS calculation with query-centered tilted COS.
459 
460 \begin{table}[htbp]
461 \centering
462 \caption{Relative error in an exactly solvable chi-square family.}
463 \label{tab:chi}
464 \begin{tabular}{rrrr}
465 \toprule
466 Target tail $p$ & $N$ & Global COS rel. error & Tilted COS rel. error\\
467 \midrule
468 $10^{-6}$  & 64  & $3.56\times10^{-8}$ & $2.71\times10^{-14}$\\
469 $10^{-10}$ & 64  & $1.64\times10^{-4}$ & $6.59\times10^{-15}$\\
470 $10^{-10}$ & 128 & $2.03\times10^{-6}$ & $9.05\times10^{-15}$\\
471 $10^{-14}$ & 64  & $1.43$              & $3.53\times10^{-14}$\\
472 $10^{-14}$ & 128 & $6.15\times10^{-2}$ & $3.87\times10^{-14}$\\
473 \bottomrule
474 \end{tabular}
475 \end{table}
476 
477 The test demonstrates the numerical motivation for tilting: evaluating a small tail directly under the shifted law is substantially more stable than reconstructing a global CDF and subtracting from one. This observation is not a novelty claim for exponential tilting itself.
478 
479 \subsection{Genotype-derived spectra}
480 
481 The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were $w_j=(1-\mathrm{MAF}_j)^{24}$, where MAF denotes minor-allele frequency. The factor used column scales $\sqrt{w_j}$. Its positive quadratic-form spectrum was
482 \[
483 \lambda=\tfrac12\operatorname{eig}(AA^{\mathsf T}).
484 \]
485 Three independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.
486 
487 Across the twelve queries:
488 \begin{itemize}
489 \item every requested $10^{-8}$ analytic certificate passed.
490 \item the worst observed relative error was $6.148608\times10^{-11}$.
491 \item the largest returned analytic relative bound was $7.917203\times10^{-9}$.
492 \item the planner selected $N\in\{64,128\}$, $L\in\{10,11,12,13\}$, and 11--16 packets.
493 \end{itemize}
494 
495 Figure \ref{fig:coverage} plots the observed errors against the returned analytic bounds. Every point lies above the equality line because the vertical coordinate is the bound.
496 
497 \begin{figure}[htbp]
498 \centering
499 \includegraphics[width=\linewidth]{figures_production/certificate_coverage.pdf}
500 \caption{Twelve retained genotype-derived tail queries, redrawn from the recovered source table. The vertical coordinate is the analytic relative bound and the horizontal coordinate is observed relative error. Every point lies above equality. Marker shape identifies the source spectrum.}
501 \label{fig:coverage}
502 \end{figure}
503 
504 \subsection{Tolerance ladder}
505 
506 At a reference tail probability near $8.99\times10^{-15}$, the same engine was requested to meet three tolerances. Table \ref{tab:ladder} shows how the representation refined automatically.
507 
508 \begin{table}[htbp]
509 \centering
510 \caption{Adaptive refinement at an extreme-tail query.}
511 \label{tab:ladder}
512 \begin{tabular}{rrrrrr}
513 \toprule
514 Requested $\tau$ & $N$ & $L$ & Packets & Analytic bound & Observed error\\
515 \midrule
516 $10^{-6}$  & 128 & 10 & 10 & $3.56\times10^{-7}$  & $3.63\times10^{-12}$\\
517 $10^{-8}$  & 128 & 12 & 15 & $4.87\times10^{-9}$  & $9.14\times10^{-13}$\\
518 $10^{-10}$ & 256 & 14 & 24 & $8.21\times10^{-11}$ & $5.65\times10^{-14}$\\
519 \bottomrule
520 \end{tabular}
521 \end{table}
522 
523 \begin{figure}[htbp]
524 \centering
525 \includegraphics[width=\linewidth]{figures_production/tolerance_ladder.pdf}
526 \caption{One extreme-tail query evaluated under three requested tolerances. Increasing the accuracy request refines the packet representation and, at the tightest setting, the mode count. The returned analytic bound remains above observed error and below the requested tolerance.}
527 \label{fig:ladder}
528 \end{figure}
529 
530 \subsection{Prototype runtime scaling}
531 
532 Table \ref{tab:runtime} reports median CPU timings from the Python prototype. ``Adaptive total'' includes planning and analytic error checks. ``Compiled evaluation'' measures reuse of an already compiled spectral passport at the selected grid.
533 
534 \begin{table}[htbp]
535 \centering
536 \caption{Prototype runtime scaling on genotype-derived spectra.}
537 \label{tab:runtime}
538 \begin{tabular}{rrrrrr}
539 \toprule
540 Rank & Packets & Exact tilted & Adaptive total & Compiled eval. & Compiled speedup\\
541 \midrule
542 397  & 13 & 6.66 ms & 10.45 ms & 0.549 ms & $12.1\times$\\
543 797  & 16 & 13.32 ms & 11.49 ms & 0.604 ms & $22.0\times$\\
544 1197 & 17 & 21.52 ms & 11.24 ms & 0.651 ms & $33.1\times$\\
545 \bottomrule
546 \end{tabular}
547 \end{table}
548 
549 \begin{figure}[htbp]
550 \centering
551 \includegraphics[width=\linewidth]{figures_production/runtime_scaling.pdf}
552 \caption{Retained median CPU timings for exact tilted COS, the adaptive calculation including planning, and reuse of a compiled spectral representation. The logarithmic vertical scale keeps the much smaller reuse cost visible. First-query and reuse costs are not interchangeable.}
553 \label{fig:runtime}
554 \end{figure}
555 
556 Planning matters. The first adaptive query is slower at rank 397 and faster at rank 797 in this prototype. Compiled reuse measures a different cost. These timings establish no universal speedup.
557 
558 
559 
560 \section{Numerical verification}\label{sec:final-28}
561 
562 \begin{table}[htbp]
563 \centering
564 \caption{Stored-run identifiers and actual block counts.}
565 \begin{tabular}{lrrr}
566 \toprule
567 partition & blocks & Gram relative error & action relative error\\
568 \midrule
569 coarse 4 & 4 & \(2.91\times10^{-16}\) & \(5.99\times10^{-16}\)\\
570 fine 64 & 32 & \(3.39\times10^{-16}\) & \(6.00\times10^{-16}\)\\
571 irregular & 9 & \(3.34\times10^{-16}\) & \(5.87\times10^{-16}\)\\
572 singleton-front & 21 & \(2.43\times10^{-16}\) & \(5.82\times10^{-16}\)\\
573 permuted, block 73 & 29 & \(3.52\times10^{-16}\) & \(5.89\times10^{-16}\)\\
574 \bottomrule
575 \end{tabular}
576 \end{table}
577 
578 Hierarchical refinement gave relative discrepancy $3.17\times10^{-16}$.
579 For $H=L^\top L$ with block-bidiagonal $L$, the smallest eigenvalue was
580 approximately $2.86$ and the largest norm outside the exact light cone
581 was zero. The QR-core singular-spectrum discrepancy was
582 $5.89\times10^{-16}$ in relative $\ell_2$ norm. These are finite controls,
583 not universal floating-point error bounds.
584 
585 
586 

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