E8 clock-to-rotor convergence

Date: 26 October 2026

Status: local compact-mode convergence closed; joined eight-channel metric open

Parent: E8 bounded-degree clock hardware

Evidence: convergence ledger

Certificate: clock/pendulum spectral verifier

The scaling correction

The finite clock certificate normalized both Weyl penalties by (dq=1-\cos(2\pi/q)). That keeps the elementary clock violation gap fixed, but at fixed symbols it sends the effective rotor potential to (u{\rm eff}\sim q^2). The finite-(q) result remains exact; it simply is not the same fixed-coupling continuum limit as (q\to\infty).

For a physical rotor with fixed (U=u=1), use

Only the kinetic clock penalty carries the (d_q^{-1}) scaling. For every fixed integer (n), its eigenvalue tends to (n^2/2), while the phase potential retains fixed strength. The target is the quantum pendulum

Spectral convergence

At (q=129), the relative gap error is (1.106\times10^{-4}), and the gap remains positive at every rung. In the ideal product strip, adding width only adds disjoint convergent locking modes, so widths 2 through 34 retain sixteen unpaired boundary modes and the same local first gap.

What this closes

The finite clock is no longer separated from the rotor by an unspecified analogy. One effective compact locking mode converges spectrally to the fixed-coupling pendulum, and the precise energy renormalization is known. This also identifies which part of the earlier uniform clock gap cannot be carried unchanged into a fixed-physics continuum statement.

Remaining join

The E8 dual basis has the dense integral charging/commutator form (K^{-1}). The physical eight-channel cell is therefore not eight independent pendula. The next certificate must insert that coupled metric, compute its compact and harmonic spectra, and test perturbations between channels. Only after that can edge dispersion, (c_-=8) and the 240-current detector be transported onto the rotor carrier.