Isotrophy · K_facet · v0.3h
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.
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.
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.
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 theorem of universal isotropy.
K_facet at m₃ = 1 is one slice. The wider isotrophy roadmap (descent from Z₃-choreographies into m₁ = m₂ piano-trios) remains the live, longer-running program.
Inspection Trail