# v0.19 liao2021 Floquet Spectral-Gap Reliability Mechanism Form ## Result (2026-06-03) **Verdict: `spectral_gap_mechanism_partial`.** The first-principles mechanism for the *reliability itself* is confirmed strongly; the per-cell bridge to AUC via the chosen aggregation is not. ```text reproduce-v0.18 gate: max_abs_diff 0.00e+00 -> PASS (re-measurement reproduces v0.18 vf/score/frame_spread exactly; spectrum data trustworthy) H1 (primary, label-blind): Spearman(re_gap, frame_spread) = -0.836, p = 1.0e-5 -> PASS (past the |0.45| bar); small spectral gap drives high frame-fragility falsifier guard: large-gap-high-spread fraction = 0.0052 (15/2880; GAP_CLEAR 0.0008368) -> clean (<= 5%) H2 (secondary): per-cell Spearman(median gap_reliability, AUC_cell) = 0.063, p = 0.40 -> NOT confirmed ``` **H1 is the headline and it lands hard: frame-fragility IS spectral near-degeneracy.** The argmax-of-`Re(lambda)` selection flips under reparameterization exactly where the top Floquet spectrum clusters, and that drives the four-frame `frame_spread` (rho -0.836, 2880 reproduce-gated orbits). The v0.18 phenomenological reliability map has a first-principles cause at the orbit level: the spectral gap that makes the selection well-posed. **Why H2 fails -- aggregation mismatch, not mechanism refutation.** v0.18 predicted per-cell AUC from the cell's frame-spread TAIL (`frame_p90`, rho 0.60). v0.19's H2 used the cell's MEDIAN `gap_reliability` -- a central measure that washes out the fragile minority that actually depresses AUC. The chain small-gap -> high-spread (H1, confirmed) -> AUC-degradation (v0.18, confirmed) is intact, but the specific per-cell MEDIAN-gap summary does not bridge it. A tail-based cell gap measure (cell min `re_gap`, or fraction of near-degenerate orbits) would likely bridge it -- but that is a new locked statistic, not a re-read, so v0.19 honestly stops at `partial`. **Cross-substrate export (chatv2 handoff):** supported at the FRAGILITY level -- an argmax-selected shadow's reparameterization-instability is its selection spectral gap, computable label-blind a-priori (Floquet Re-gap / residual-stream eigengap / control-basis gap). NOT yet supported as a full per-region transfer-strength predictor via this aggregation. **Independent verification.** A standalone hand-rolled rank/Pearson Spearman + falsifier recompute from `per_row_spectrum.csv` reproduced H1 rho -0.836220, GAP_CLEAR 0.000836786, falsifier 0.0052 (15/2880) BIT-FOR-BIT. Receipt + verifier: `results/isotrophy/k-facet-v19-spectral-gap-reliability/` (`manifest.json`, `gap_frame_spread_test.json`, `reproduce_v18_check.json`, `per_row_spectrum.csv`, `_independent_check.py`). **Bounded, unchanged:** Tier-2 / Li-Liao lineage / stable-support / within-cell / tail-resolved score / not theorem-facing. H1 is the load-bearing first-principles result; H2/H3 are downstream/exploratory. This explains the reliability mechanism (altitude 2, gap -> fragility); it does NOT explain why vf -> stability (altitude 1, deferred). **Provenance note:** the v0.19 chapter (runner + this form + npm aliases) was orphaned by the 2026-06-03 leak-remediation history rewrite (committed at the now-dangling `88241ec9`); recovered to the working tree from that commit 2026-06-03. The 2880-row measurement survived (gitignored results). --- Status: **OPERATOR LOCK 2026-06-03; VERDICT LANDED `spectral_gap_mechanism_partial` 2026-06-03.** Reviewed for self-consistency: the sign is confirmed (H1 negative on `re_gap` vs `frame_spread`; H2 positive on `gap_reliability = log10(re_gap)` vs `AUC_cell`); `re_gap` / `selected_group_re_width` / `selection_sv_gap` map onto `select_gamma_1`'s `Re >= max_real - 1e-6` group + cascade SVD; falsifier is procedure-pinned; the reproduce-v0.18 gate is binding. Runner implemented after lock; no v0.19 measurement has run and no v0.19 statistic computed at lock time. This is the first **first-principles mechanism** chapter of the isotrophy program: it does not collect more external-transfer evidence (the v0.11->v0.18 arc is closed/consolidated). It asks *why* the reliability map exists. ## Frame v0.18 registered that label-blind frame reliability (`-log10 frame_p90`) predicts per-cell transfer AUC (Spearman rho 0.5975, p 0.00523, reversal guard clean). That is a **phenomenological** reliability map: it says reliable cells transfer better, but not why reliability varies. v0.19 proposes and tests a first-principles cause. **The candidate mechanism is nearly forced by how the shadow is built.** `select_gamma_1` picks the Floquet eigenvector of largest `Re(lambda)`. A v0.13a/b frame perturbation re-integrates a *rotated copy* of the orbit; its monodromy is a similarity conjugate, so the **eigenvalue spectrum is identical** and only the numerical argmax can move. The argmax can move only when two distinct modes have near-equal `Re(lambda)` -- a small **Re-part spectral gap** (near-degeneracy). Therefore: ```text frame-fragility (argmax flips under reparameterization) <=> small Re-part spectral gap the selected-direction shadow is reliable exactly when the selection is well-posed, which is exactly when the top spectrum is gapped. ``` This is substrate-independent: it is the general law for any argmax / eigenvector / top-component-selected shadow (chatv2 residual-stream directions, Mesa control bases, NSE diagnostic projections). Confirming it would upgrade v0.18's reliability map into a **spectral-gap map** with a portable predictor. ## Primary Question > Does the Re-part Floquet spectral gap of the selected direction predict its > frame-fragility (and hence the per-cell transfer AUC)? Is the v0.18 reliability map a > spectral-gap map? ## Integrity Caveat ```text theory-derived hypothesis (NOT discovered on data): The spectral-gap mechanism is derived from the argmax-conditioning argument above, not fit to any prior sample. The primary statistic (gap -> frame_spread) is label-blind on both sides and was never measured in v0.11-v0.18. mechanism on the discovery rows is appropriate: A mechanism chapter EXPLAINS a found result on the rows where it was found. Unlike a confirmation chapter, it does not require a fresh outcome sample. The load-bearing primary (gap -> frame_spread) involves neither the stability label nor AUC, so it carries no discovery-sample circularity even on v0.18 rows. feature + grid frozen: score = median(vf_0, vf_37, vf_90, vf_211); frozen v0.7/v0.13a D5; v0.14 8x8 grid construction; seed 20260523. The ONLY new object is the per-orbit monodromy spectrum dump and the spectral-gap statistic derived from it. ``` Forbidden: ```text changing the score, frame set, D5 gates, grid, or seed using the stability label or AUC in the primary gap->frame_spread predictor reading the spectral gap off the selected direction's own vf (the gap is a spectrum property, computed before any velocity/position split) lowering the locked correlation bars after seeing the spectrum promoting to Tier-3, theorem-facing, or a controller claim reopening or revising any v0.11-v0.18 verdict (this chapter explains, it does not amend) ``` ## Sample -- Resolved Lock Default ```text LOCK DEFAULT: re-measure the exact v0.18 sample (2880 rows, the 18 8x8 cells). - The mechanism explains the v0.18 reliability map on the rows that produced it. - The re-measurement reproduces v0.18's vf / score / frame_spread BIT-FOR-BIT (deterministic same-orbit re-run) as a built-in validation, then ADDS the spectrum. - No fresh draw, so no holdout exhaustion. REJECTED ALTERNATIVE: fresh sixth-holdout draw. - A fresh 8x8 quintuple holdout is razor-thin: smallest eligible stable pool = 84 (only 4 above the 80 draw) -- abort-prone. A fresh draw would need 10x10 (23 cells, smallest pool 112, 3680 rows) for headroom. - Cleaner for the per-cell gap->AUC test, but the primary gap->frame_spread test gains nothing (it is already label-blind), and it costs ~30% more compute on an exhausting stable pool. ``` Rationale: the verdict-bearing primary is label-blind, so re-measuring the v0.18 rows is both clean and the most direct explanation of the map under study; the data limit makes a fresh 8x8 draw fragile. ## New Measurement (spectrum dump) Re-run the frozen four-frame ensemble; additionally persist, per orbit, at the **identity frame** monodromy: ```text floquet_eigs_real, floquet_eigs_imag full 18-multiplier spectrum, not top-k max_abs_lambda spectral radius -> elliptic vs hyperbolic type selected_re, selected_im the select_gamma_1 eigenvalue re_gap Re(selected) - Re(nearest competing DISTINCT mode), using the existing select_gamma_1 degeneracy tolerance to exclude the conjugate partner / the selected near-degenerate group degenerate_eigenvalue_count (the v0.7/v14 count, as a gap cross-check) eigenspace_dim_used (cross-check) selected_group_re_width max-min Re(lambda) inside the selected group selection_sv_gap report-only: cascade-step singular gap inside the selected group, when eigenspace_dim_used > 1 frame_spread reproduced from the four frames (must match v0.18) ``` The spectrum is frame-invariant, so it is computed once at the identity frame; the four frames are still run to reproduce `frame_spread`. ## Spectral-Gap Definition (locked) ```text re_gap = Re(lambda_selected) - max{ Re(lambda_j) : mode_j not in the selected near-degenerate group, at the select_gamma_1 degeneracy tolerance } gap_reliability = log10(max(re_gap, 1e-9)) (large gap -> larger value -> reliable) gap_unreliability = -gap_reliability (small gap -> larger value -> fragile) ``` The conjugate partner and any modes within the selection's degeneracy tolerance are excluded from the competitor, because they are handled by `select_gamma_1` as one selected group. This makes the primary test specifically about the Re-part gap to the next competing group. `degenerate_eigenvalue_count`, `selected_group_re_width`, and `selection_sv_gap` are reported as cross-checks so any within-group cascade fragility is visible instead of being silently folded into H1. ## Hypotheses (locked directions + bars) ```text H1 (PRIMARY, verdict-bearing, label-blind): per-orbit, a SMALL Re-part gap drives HIGH frame_spread. statistic: Spearman rho(re_gap, frame_spread) over all success orbits. bar: rho <= -0.45 AND one-sided permutation p <= 0.01 (100000 label-free shuffles of the gap/spread pairing, seed 20260523; p tail = shuffled rho <= observed rho). H2 (SECONDARY, downstream): the v0.18 reliability map is a spectral-gap map. statistic: per-cell Spearman(cell median gap_reliability, AUC_cell) over the cells. bar (report + soft): rho >= 0.45 AND one-sided p <= 0.05 (100000 cell-level shuffles, seed 20260523; p tail = shuffled rho >= observed rho; mirrors v0.18). On v0.18 rows this is partly mediated by H1 + the known frame_spread->AUC link; it is confirmatory, not independent, and is labeled as such. H3 (EXPLORATORY, report-only -- altitude 1, the deeper why): spectral TYPE and selected-eigenvector geometry vs vf and stability: - is unstable ~ a separated hyperbolic mode (max_abs_lambda > 1+eps), position-heavy selected vector (low vf)? - is stable ~ clustered elliptic modes (all |lambda| ~ 1), velocity-heavy (high vf)? Reported as a descriptive panel (per-orbit type, gap, vf, label). It cannot set the verdict; it scopes a possible dynamical "why vf->stability" for a later chapter. ``` ## Reproduce-v0.18 Validation (gate) Before any H1/H2 interpretation, assert the re-measurement reproduces v0.18 on the shared columns: ```text re-measured vf_0/37/90/211, score, frame_spread == v0.18 per_row_sample values, max abs diff <= 1e-9 over all 2880 orbits. ``` If this fails, verdict is `blocked_by_nonreproduction` -- the spectrum dump perturbed the measurement and nothing is interpreted. ## Falsification Guard The mechanism must be falsifiable: ```text GAP_CLEAR = 66.7th percentile of re_gap over successful v0.18 re-measure rows (procedure-pinned, label-blind) SPREAD_HIGH = 0.10 large_gap_high_spread_orbit = re_gap >= GAP_CLEAR AND frame_spread >= SPREAD_HIGH ``` A non-trivial population of large-gap-yet-fragile orbits would mean the argmax flips for reasons other than near-degeneracy, falsifying H1's mechanism even if the pooled rho is negative. Report the count and fraction; pre-register that `> 5%` of success orbits in this box demotes H1 to `mechanism_partial`. ## Verdict Tree ```text blocked_by_receipt v0.18 receipts / sample frame / spectrum dump unavailable or inconsistent blocked_by_nonreproduction re-measured frame_spread/vf/score do not match v0.18 within 1e-9 spectral_gap_explains_reliability H1 passes (rho <= -0.45, p <= 0.01) AND large-gap-high-spread fraction <= 5% AND H2 confirmatory (rho >= 0.45, p <= 0.05) spectral_gap_mechanism_partial H1 passes OR is borderline, but H2 weak OR large-gap-high-spread fraction > 5% spectral_gap_mechanism_not_supported H1 fails (rho > -0.45 OR p > 0.01) ``` ## Required Outputs ```text results/isotrophy/k-facet-v19-spectral-gap-reliability/ manifest.json per_row_spectrum.csv per-orbit: eigs, re_gap, gap_reliability, max_abs_lambda, vf, frame_spread, degenerate_count, eigenspace_dim, selected_group_re_width, selection_sv_gap, stability, mass_cell reproduce_v18_check.json bit-for-bit validation vs v0.18 gap_frame_spread_test.json H1 Spearman + permutation null + falsification-guard counts per_cell_gap_reliability.csv H2 per-cell gap_reliability vs AUC_cell spectral_type_panel.csv H3 elliptic/hyperbolic + vf + label (report-only) operator_commands.md ``` `manifest.json` must include verdict, the H1 rho + p + N, `GAP_CLEAR`, `SPREAD_HIGH`, the large-gap-high-spread fraction, the H2 per-cell rho + p, the reproduce-v18 max abs diff, gitCommit, target_sha256, sample provenance (`v0.18_remeasure`), and the claim boundary. ## Claim Boundary If `spectral_gap_explains_reliability`: ```text The v0.18 frame-reliability map is, first-principles, a Floquet Re-part spectral-gap map: the selected-direction shadow is fragile exactly where the top spectrum is near-degenerate, and that fragility is what depresses per-cell transfer AUC. This is a mechanism for the RELIABILITY, computed label-blind from the monodromy spectrum. ``` Bounded, unchanged: Tier-2 / Li-Liao lineage / stable-support / within-cell / tail-resolved score / not coarse-zone / not full-catalog / not Tier-3 / not theorem-facing. H1 is the load-bearing first-principles result; H2 is confirmatory on the discovery rows; H3 is an exploratory scope for a later "why vf->stability" dynamical chapter, not a claim. This chapter does NOT explain *why velocity-fraction tracks stability* (altitude 1) -- it explains *why the transfer's reliability varies* (altitude 2). The two are distinct. ## Cross-Substrate Export If H1 holds, the exportable principle for CROSS_SUBSTRATE_NOTES / chatv2 is concrete: ```text an argmax / top-component / eigenvector-selected shadow transfers in proportion to the spectral gap that makes its selection well-posed; compute that gap and you have a label-blind, a-priori reliability predictor -- in three-body (Floquet Re-gap), in an LLM residual stream (top-direction eigengap), in Mesa (control-basis gap), in NSE (diagnostic projection gap). ``` This is the chatv2-facing payoff: a portable reliability predictor, not a re-run. ## Resolved Decisions Before Lock ```text 1. Sample: re-measure the exact v0.18 rows; fresh 10x10 is rejected for this mechanism chapter because H1 is label-blind and the fresh pool is needlessly exhausting. 2. Spectrum persistence: write the full 18 multipliers; top-k is rejected because full spectrum is cheap and needed for honest H3 elliptic/hyperbolic typing. 3. H1 bar: keep rho <= -0.45 and p <= 0.01; stricter -0.5 is rejected because this is the first first-principles mechanism test and -0.45 already mirrors the v0.18 reliability magnitude. 4. Falsification box: GAP_CLEAR = success-row 66.7th percentile of re_gap; SPREAD_HIGH = 0.10; large-gap-high-spread fraction > 5% demotes to partial. 5. H3 stays strictly report-only; altitude-1 "why vf->stability" deserves its own chapter. ``` ## Lock-In Statement Committed as a draft before any v0.19 runner exists. Any change to the spectral-gap definition, the hypotheses, the bars, the falsification guard, the reproduce-v0.18 gate, the verdict tree, or the claim boundary after execution is a re-registration. The runner reuses the frozen v0.16/v0.18 ensemble + D5 + grid symbols unchanged and adds only the identity-frame spectrum dump, the `re_gap` statistic, and the gap/frame_spread/AUC tests.