Isotrophy · K_facet · v0.3-v0.20
The audit chain catches itself working.
Twenty structural-zero receipts and one named quarantine. A pre-registered three-stage audit chain ran 21 strict G.2 single-curve choreographies through the K_facet test. For 20 rows the standard D₃ sector is structurally absent — the prediction is zero by construction, not by tolerance or post-hoc pruning. The 21st row (O_617) was quarantined, not failed: the deep dive locates the defect in a bridge direction outside the valid D3 representation, not in the audit chain itself.
That v0.3h result became the first chapter of a
pre-registered program that now runs through v0.20. The
later chain sharpened the Floquet velocity-fraction finding:
it stratifies stability inside fixed mass-ratio strata
(AUC_cond = 0.6783, exact p = 2.046e-7)
but fails as a mass-marginal held-out predictor
(AUC = 0.4125). The honest shape is conditional,
not universal. The external-transfer arc then split the
instrument from the signal: the coarse v0.11 zone form did not
transfer cleanly, but the continuous tail-resolved score did
(AUC_cond = 0.647, p = 1e-5) on a
fresh Tier-2 liao2021 stable-support sample. A second fresh
triple-holdout sample then replicated both the pooled transfer
and the seven-cell heterogeneity ordering
(rho = 1.0, exact p_rho = 1/5040). A
third fresh sample on a wider 8×8 grid then explained the
heterogeneity: label-blind frame reliability predicts per-cell
transfer strength (rho = 0.60, p = 5e-3),
and no frame-stable cell defies it. v0.19 then explains that
frame-fragility at the orbit level: small Floquet spectral gap
predicts high frame spread (rho = -0.836,
p = 1e-5), while the direct median-gap cell summary
misses the AUC bridge. v0.20 then recovers that bridge by reading
the gap's low tail instead of its median (rho = 0.882,
p = 1e-5) — a confirmatory re-analysis on the
same cells, so the median was the wrong aggregation, not a missing
mechanism.
The Load-Bearing Statement
20/21 structural zeros plus one quarantined O_617 defective-D₃ bridge.
The phrasing matters. This is not a closed 21/21 theorem-facing result, and any promo or external write-up that collapses the verdict to 21/21 silently retires the discipline that makes the result interesting.
Program Chapter Arc
From structural zeros to a conditional stability signal.
v0.3h is the bead-maze pillar. The full isotrophy program continued through v0.20, asking whether progressively richer projections of the Li & Liao piano-trio catalog carry stability information. The chain now has a sharper boundary: velocity-fraction ranks stability inside fixed mass-ratio bins, fails as a mass-marginal score, and transfers externally only after the saturated coarse zone is replaced by a tail-resolved continuous score. Its transfer reliability is now traced to a spectral-gap mechanism at the orbit level, and the direct gap→AUC bridge is recovered once the gap is read at its low tail rather than its median (v0.20, confirmatory, same cells).
v0.3 Domain-of-applicability
Gamma_i is D3-equivariant; the strict G.2 input set gives 20 structural zeros plus the named O_617 quarantine. Supplementary-B is Z2-or-smaller and outside this mechanism's domain.
v0.4 Structural-negative
Natural Z2-shadow candidates fail on the piano-trio catalog; orbit-level Z2 cleanliness does not carry stability.
v0.5 Projection-limit
The branch hash on (m_3, z_0) stratifies
in-sample at p about 1e-7, then fails the held-out
mass-bin predictor.
v0.6 Conditional-independence
The continuous conserved-quantity shadow
(E, |L|) passes in-sample but collapses
under the alignment-breaking within-branch audit.
v0.7 Qualified positive
The Floquet velocity-fraction direction shadow stratifies S/U on the 250-row analyzable subset: chi^2 = 16.43, p about 9.3e-4, alignment 0.698.
v0.8 Structural-negative
Unsigned direction-purity fails; the v0.7 signal is real, but it lives in signed velocity fraction rather than distance from mixed.
v0.9 Meta-finding
The U-shape mechanism fails under physical cutpoints, while a monotone-increasing pattern remains: positional 11% S, mixed 34% S, velocity-heavy 44% S.
v0.10 Trend + boundary
Jonckheere-Terpstra registers the monotone trend in-sample (p = 7.304e-3), while the mass-marginal held-out predictor fails cleanly (AUC = 0.4125).
v0.11 Conditional rank
Within fixed m3 strata, the frozen vf zone order ranks stable rows above unstable rows: AUC_cond = 0.6783, exact p = 2.046e-7. Conditional signal, not a global predictor.
v0.12 Supplementary-A blocked
Taking the frozen rule to supplementary-A, an unbiased uniform probe read attrition 0.3433 (Wilson95 [0.2919, 0.3987], above the 0.20 gate). The frozen D5 measurement is intractable on ~1/3 of the target — a feasibility limit, not a falsification. v0.13 opens a signal-blind search for a tractable external target.
v0.13 Tractable target found
The search locks liao2021 as a leakage-clean Tier-2 target. Frame-zone stability and the D5 feasibility probe clear: the transfer question can be asked on a fresh external stable-support sample.
v0.14-v0.15 Coverage wall, then saturation
v0.14 is undecidable because the target's outcome x feature geometry lacks enough support. v0.15 defeats coverage but lands directional-weak: the coarse three-zone feature is almost saturated in liao2021.
v0.16 Tail-resolved transfer pass
On 1,120 fresh doubly-held-out rows, the continuous four-frame velocity-fraction score clears the external floor: AUC_cond = 0.647, p = 1e-5, zero attrition. The pass is Tier-2, stable-support, pooled, and heterogeneous by mass cell.
v0.17 Heterogeneity replicated
A fresh triple-holdout 80/80 draw repeats the pooled transfer (AUC_cond = 0.646875, p = 1e-5) and the seven-cell AUC rank order exactly (rho = 1.0, exact p_rho = 1/5040). Post-hoc anatomy reads the two sub-0.5 cells as a frame-fragile region and a near-null region, not a competing mechanism.
v0.18 Reliability map confirmed
A fresh quadruple-holdout 8×8 grid (18 cells, 2,880 rows) replicates the transfer a third time (AUC_cond = 0.620, p = 1e-5) and confirms the heterogeneity's cause: label-blind frame reliability predicts per-cell AUC (rho = 0.60, p = 5e-3), with no frame-stable cell defying it — an instrument-reliability map, not a competing law.
v0.19 Spectral gap explains fragility
Re-measuring the exact v0.18 rows, the reproduce gate is bit-for-bit clean and the first-principles H1 lands hard: small Floquet Re-gap predicts high frame spread (rho = -0.836, p = 1e-5). The locked verdict remains partial because median gap does not bridge directly to per-cell AUC — the effect lives in the fragile tail.
v0.20 Tail-gap bridge confirmed
A confirmatory re-analysis on the same 18 cells swaps the median gap for a pre-registered q10 low-tail summary: the direct gap→AUC bridge returns at full strength (rho = 0.882, p = 1e-5), leave-one-cell-out 0.86–0.92. The median washed the mechanism out of view; it did not break the bridge.
The Stability Signal
Floquet direction composition stratifies stability after conditioning and tail resolution.
The v0.7a' result opened the substantive positive in the chain. It uses the velocity fraction of the largest-real-part Floquet eigenvector as a geometric direction in phase space, after center-of-mass reduction and mass-weighted normalization. It does not use eigenvalue magnitude, spectral radius, unstable-pair count, or unit-circle status as a feature.
The caveats carry the result. This is not catalog-wide: 23 of 273 rows attrited at the locked variational precision, mostly in the long-period high-mass regime. It is not a mass-marginal held-out predictor: v0.10b tested that directly and failed at AUC = 0.4125. It is not the U-shape mechanism: v0.9 showed that the physical-zone pattern is monotone-increasing rather than a true U. And the external confirmation is bounded: supplementary-A was blocked by D5 attrition, the coarse v0.11 zone form stayed saturated and sub-threshold on liao2021, and the v0.16/v0.17 pass holds only for the continuous tail-resolved score on Tier-2 stable-support samples.
The durable public claim is narrower and better: this is the first non-branch-aligned low-dimensional signal in the isotrophy chain, now localized and externally probed. Floquet velocity-fraction stratifies stability within mass-ratio strata, not as a single global score; read continuously in the tail, it transfers and replicates on the liao2021 target as a pooled but mass-cell-heterogeneous rank signal.
The Audit Chain
Receipt-first, closure-relative, three stages.
The audit was deliberately constructed so that constants and outcome categories were registered in code and spec before interpretation. No row-specific knobs were introduced. Each row either became a structural-zero receipt, a named refinement case, or an excluded defective case.
Stage 1 Sentinel / Γ runner
Built Mi, typed D₃ operators, reduced ω, and the closed-form ∂ε Mi; gated D₃ relations, leakage, and finite-difference consistency.
Stage 2 Adaptive-floor reprocessor
Reused saved .npy artifacts (no
integration). Chose the smallest pre-registered floor
satisfying three guards: D₃ leakage ≤ 1e-3, gap ratio
≤ 1e-3, and first-rejected singular value ≥ 1e-3.
Stage 3 Bridge audit
Examined any row not resolved by the adaptive-floor rule. Tested bridge vectors for neutral overlap, Jordan behavior, and defective D₃ signatures.
The Named Quarantine
Why O_617 is held back, and where the defect actually lives.
At the bridge-admitted kernel, O_617's representation reads
T(2) + S(6) + E(1), which is not a valid real
standard D₃ block. The Jordan-chain test amplifies rather
than drops (90.04) and the order-3 relation
fails at about four percent
(‖σ₃³ − I‖∞ = 3.96e-2).
The companion deep dive corrects the first attribution:
O_617 is a clean opposite-strict catalog row with admission
residual 1.01e-8. The earlier
1.62e-1 value is the canonical residual and is
diagnostic-only for this opposite-strict row. The defect
therefore lives in the bridge representation, not in the
orbit's Γi. The audit chain itself is intact.
The signed isotypic check locates that bridge near the
sign sector, not the trivial sector:
<v,F_beta v> = -0.9999997, with
F_beta^2 v - v = 0. The odd
E(1) readout is projector contamination from
that near-S bridge under imperfect sigma3 closure, not a
valid standard block.
O_617 is quarantined, not counted for or against the prediction. Naming the quarantine is the point: a discipline that absorbs out-of-scope cases into a clean number is not the discipline this audit was built around.
Claim Boundary
Not a closed 21/21 theorem-facing result.
The audit chain is intact. The theorem-facing result is not closed. These are different statements.
Not a proof of Sundog.
Structural zeros at m₃ = 1 for the strict G.2 catalog are a specific, defensible, narrow result. They do not extend to a universal theorem of choreography isotropy.
Not "we caught the edge case."
The O_617 quarantine is explained with a specific representation-level diagnosis at the bridge boundary, not marketed as proof of care. The deep dive runs six no-integration probes to triangulate the defect.
Not a catalog-wide v0.7 claim.
The v0.7a' positive holds on 250 analyzable rows, not all 273 supplementary-B rows. The attrited rows remain visible on the receipt.
Not a mass-marginal stability predictor.
The v0.10b predictor chapter tested a global leave-one-m3-bin-out score and failed at AUC = 0.4125. v0.11 is a conditional rank pass inside fixed mass-ratio strata, not a rescue of that held-out null.
Not a universal external transfer claim.
v0.16/v0.17/v0.18/v0.19/v0.20 confirm, replicate, and explain the tail-resolved continuous score on Tier-2 liao2021 stable-support samples. They do not confirm the coarse zone rule, the full catalog, a Tier-3 independent target, or a uniform per-cell law; v0.20's direct gap→AUC tail bridge is a confirmatory re-analysis on the same cells, not a fresh-evidence upgrade.
Not a retired program.
The isotrophy program reopened through v0.20 and earned a replicated, mechanism-explained bounded external confirmation. The next honest upgrade is Tier-3 data, a paper-side consolidation, or a separate cross-substrate reliability test.
Inspection Trail