E8 constitutive-metric gate
Date: 26 October 2026
Status: topological/constitutive separation closed; physical metric calibration open
Parent: clock-to-rotor convergence
Evidence: conditional metric ledger
Certificate: Coxeter-spectrum and branch verifier
The joined metric is not selected by topology
The effective E8 edge fixes an integral commutator form (K{E8}), while its positive energy/velocity matrix (V) is independent constitutive data. Moving to the integral dual variables sends the compact algebra to (K^{-1}), but it does not say whether a laboratory (K^{-1}) is the charging kernel, a capacitance/Gram matrix whose inverse is the charging kernel, or only a coordinate relation implemented ahead of a different positive quadratic form. Those readings cannot be merged.
For the conditional compact cell
the harmonic frequencies are (omegaa=\sqrt{\lambdaa(M)}). The corpus therefore contains three distinct experimental hypotheses, not one automatic "E8 spectrum."
| constitutive assumption | softest | hardest | span |
|---|---|---|---|
| (M=K^{-1}) | 0.500686 | 9.553661 | 19.0811 |
| (G=K^{-1},\ M=G^{-1}=K) | 0.104672 | 1.997259 | 19.0811 |
| (M=I) control | 1 | 1 | 1 |
The equal spans of the two E8 branches do not make their spectra equal. After one scale fit, the (K^{-1})-kernel branch begins ((1,1.070,1.191,1.285,\ldots)), whereas the (K)-kernel branch begins ((1,6.847,10.407,12.025,\ldots)). Seven held-out ratios separate them.
Exact Coxeter fingerprint
The Cartan eigenvalues need no numerical fitting. With Coxeter number 30 and exponents
they are
This supplies an exact eight-mode fingerprint conditional on physically realizing (K) or (K^{-1}). It is not a topological prediction for an arbitrary positive (V).
Apparatus decision
Measure the small-signal eight-port admittance or transfer matrix before installing, cooling into, or strongly driving the nonlinear locks. Reconstruct the positive quadratic kernel (M), fit one overall energy scale, and blind the remaining seven resonance ratios. Only then select the compact Hamiltonian whose nonlinear spectrum will be computed and tested.
This ordering prevents a serious category error: observing (c_-=8), an integral E8 current lattice, or the 240-root shell does not determine the device capacitance or velocity matrix. Conversely, matching a Coxeter resonance fingerprint tests a constitutive implementation of the E8 form but does not by itself establish the chiral anomaly or current algebra.
Next gate
Freeze one measured (M), include its uncertainty and parasitics, and compute cutoff-stable low nonlinear levels of the eight-dimensional compact Hamiltonian. The first useful output is a held-out anharmonic spectroscopy table; the claimed many-body gap waits until that spectrum stabilizes and the edge response is recovered on the same carrier.