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Spectral filters: verified controls and limits
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# SPECTRAL-001 findings, protocol version 2 Executed 2026-10-05. All **21 CUDA heat cases** and **12 CPU phase/loss controls** passed; trusted artifact verification passed. The numerical results support the bounded [protocol](SPECTRAL_FILTER_SPEC.md). They establish no RH result, compression advantage, linguistic capability or interval certificate. ## Results and interpretation Maximum input-normalized heat L2 error: **1.03163768e-07**. The acceptance ceiling remains analytical truncation error plus 3e-5. All eleven retained output byte hashes (first call plus ten warm repetitions) agreed within each case. The coefficient-series tail is a mathematical bound; floating coefficient evaluation, GPU recurrence and spectral-bound evaluation are not rigorously enclosed. Maximum operator roundtrip error among identity, endpoint-matched linear phase and arithmetic phase controls: **6.20924323e-15**. The 256-term zeta gate's maximum discrepancy at the four declared 50-digit probes was **0.000236123785**, below the ideal truncation bound 1/1024. This is a numerical crosscheck, not an interval certificate. Every lossy projection commuted with its Laplacian while losing a unit direction. This is the intended negative control: commutation alone does not establish recoverability. Identity and linear phase also passed, so the experiment does not show an advantage specific to zeta. Full complex coordinates remain required. No coefficients, graph, purpose, projection or acceptance limits were optimized against these results. There was no training or held-out semantic evaluation. ## All retained heat cases GPU times are validated-call CUDA-event milliseconds, reported as median [min,max] over ten warm repetitions. Validation synchronizations and dispatch gaps can be included; this is not pure active-kernel time. Each case's first call shares the worker with prior cases and is not a fresh-process benchmark. Small controls compare with dense spectral heat; the 10,000-edge cases compare with CPU float64 polynomial recurrence. The latter do not independently certify large-graph heat truncation error. Timings are descriptive, not speedup claims. | Case | Normalized L2 error | Tail bound | Warm GPU ms median [min,max] | First call ms | |---|---:|---:|---:|---:| | path-31-t0.5-k16 | 5.4126e-08 | 1.9892e-30 | 17.779 [17.116,18.647] | 15018.212 | | path-31-t0.5-k32 | 5.4126e-08 | 2.8853e-67 | 35.067 [33.751,51.620] | 34.787 | | path-31-t2-k16 | 3.6179e-08 | 2.0843e-20 | 19.764 [17.186,28.985] | 45.477 | | path-31-t2-k32 | 3.6179e-08 | 1.2855e-47 | 34.718 [34.086,35.890] | 34.276 | | path-31-t8-k16 | 1.0316e-07 | 8.2913e-10 | 17.783 [17.195,18.808] | 17.746 | | path-31-t8-k32 | 1.0316e-07 | 2.1021e-27 | 34.723 [34.436,35.401] | 34.731 | | path-10000-t2-k32 | 3.4905e-08 | 1.2855e-47 | 36.277 [35.558,52.849] | 49.288 | | cycle-31-t0.5-k16 | 5.321e-08 | 1.9892e-30 | 13.057 [12.906,13.181] | 13.231 | | cycle-31-t0.5-k32 | 5.321e-08 | 2.8853e-67 | 24.750 [24.153,35.524] | 55.310 | | cycle-31-t2-k16 | 3.5736e-08 | 2.0843e-20 | 12.955 [12.834,13.283] | 13.288 | | cycle-31-t2-k32 | 3.5736e-08 | 1.2855e-47 | 25.221 [24.982,28.891] | 25.902 | | cycle-31-t8-k16 | 9.7959e-08 | 8.2913e-10 | 13.009 [12.712,13.594] | 13.609 | | cycle-31-t8-k32 | 9.7959e-08 | 2.1021e-27 | 25.190 [24.856,30.423] | 62.603 | | cycle-10000-t2-k32 | 3.49e-08 | 1.2855e-47 | 26.207 [25.890,27.206] | 63.142 | | star-31-t0.5-k16 | 5.5159e-08 | 1.9892e-30 | 20.531 [20.271,22.626] | 22.169 | | star-31-t0.5-k32 | 5.5159e-08 | 2.8853e-67 | 41.746 [40.198,66.746] | 41.336 | | star-31-t2-k16 | 4.3106e-08 | 2.0843e-20 | 20.600 [20.288,21.286] | 20.338 | | star-31-t2-k32 | 4.3106e-08 | 1.2855e-47 | 39.596 [39.352,42.457] | 40.442 | | star-31-t8-k16 | 8.4672e-08 | 8.2913e-10 | 20.119 [19.629,49.121] | 32.490 | | star-31-t8-k32 | 8.4672e-08 | 2.1021e-27 | 40.036 [39.789,41.144] | 39.875 | | star-10000-t2-k32 | 8.3469e-08 | 1.2855e-47 | 55.354 [50.337,73.288] | 115.049 | Batch wall time: 28.343 seconds. Peak measured Torch allocation: 2.73 MiB; this excludes driver and other-process memory. Preparation, operator construction/H2D, D2H/hashing, CPU reference times and supervised telemetry are retained separately in the run. Degree 32 did not materially improve the observed errors over degree 16 in the small controls, although its theoretical truncation bound was much smaller. That is consistent with numerical/reference precision dominating this error range. It does not establish the same behavior for other times, maps or graphs. The similar small-graph and 10,000-edge times also motivate a separate study of dispatch, validation and kernel costs before any throughput optimization claim. ## Correction history Version 1 run `spectral-cbcf5dbbc012` passed all case numerics but failed its artifact verifier. The verifier required exact equality of a derived norm metric; replay differed by roughly 4.1e-22 due to CPU floating reduction behavior, while all actual errors remained below the scientific threshold. Its original source, receipt and failed verification remain unchanged. It is not labelled verified. Version 2 froze bounded metric-consistency tolerance before re-execution. Both saved and recomputed errors must independently satisfy the unchanged acceptance ceiling. Tests reject threshold crossings, negative/nonfinite metrics and forged values. New source and the new run below establish the corrected delivery. ## Source and local receipts | Identity | Value | |---|---| | Run | `spectral-95755ed3c71b` | | Implementation commit | `cde74b32c1b48a4b93aaed378f8d7b39e99972ce` | | Source SHA-256 | `8687642e5beefe5f5b6dc4efb040a72f0e37044d0bbfba50749d50c28500fb8d` | | Receipt SHA-256 | `7eefc9c42f151ee9e4823ae236a1a943c2a00756c89923bd7c0d968e350ccae4` | | Protocol SHA-256 | `15bb7a94b7a9bb3c00f96c998ae0759249e35f6e78df6f74d82c7efb930e1b3a` | Raw source snapshots, arrays, per-case outputs and telemetry: `G:/RenigmaLLM_Artifacts/spectral-runs/spectral-95755ed3c71b/`. Validation, both verification attempts, audit, figure and delivery receipt: `G:/RenigmaLLM_Artifacts/sheaf-stack-audit/`. Only original implementation, synthetic tests, research summaries and findings are included in source publication. Uploaded scripts were not copied or executed. Final full CPU suite: **1,488 passed, 26 skipped, 27 subtests** in 147.70 seconds. The skips cover absent SciPy/Ripser and CUDA tests in the CPU environment. Focused GPU-environment suite: **106 passed** in 4.21 seconds, including the new numerical and actual CUDA tests. Math checks passed. Supervised CLI execution and saved artifact replay passed. File hashes are in `delivery-receipt.json`. Draft [PR #12](https://github.com/ZuluYokohama/renigma-operational-semantics/pull/12) is stacked on #11. CodeRabbit was requested, but its [response](https://github.com/ZuluYokohama/renigma-operational-semantics/pull/12#issuecomment-6004089704) reports the developer-seat limit and summary-only service. No completed line review or human merge is claimed. Repository visibility remains unchanged. Independent scientific replication remains pending. This does not deploy the website or enable product diffusion, remote inference, NPU or WebGPU execution. ## Next decision Define one concrete storage/retrieval task and freeze its observation decoder, lossy operation, complete byte budget and held-out inputs. Compare arithmetic phase with identity, Fourier/elementary phase and matched random-frequency controls under identical budgets. Include graph/order/normalization metadata and both complex components. A new experiment must show less storage or lower cost at equal retained observable accuracy before claiming useful distillation. The [research review](SPECTRAL_RESEARCH.md) records the underlying primary sources and alternative Cayley-transform connection.