Author notes — full detail, auditor-facing

HPC-039 was a sweep across {5,6,7,8,9,10,11,12,13,15}-fold rotationally symmetric cavities, testing for *self-resonance* — the degree to which a cavity's internal standing-wave pattern matches its boundary geometry without external tuning. The hypothesis: cycle-2 frustration overtones ({5,7,9,11,13}) should display geometric self-resonance, with {7} as the *first* such overtone (and therefore most cleanly self-resonant in isolation from the other frustration modes).

Setup.

• 96³ FDTD grid with PML boundaries.
• Cavity: n-fold rotationally symmetric prism (regular n-gon extruded

through the central axis), volume normalized across n.

• Drive: broadband plane-wave pulse, 1 GHz – 10 THz.
• Measurement: ringdown-mode analysis. For each cavity, the dominant

long-lived modes were extracted via Prony analysis. "Self-resonance error" = the relative deviation between the geometric prediction (frequency at which boundary harmonic matches first standing-wave null) and the observed dominant mode frequency.

Result.

Headline findings. 1. {7}-fold is uniquely self-resonant at 2.7% error. All other tested geometries show 8–56% error. The next-cleanest are {5} (8.4%), {13} (7.8%), and {11} (9.2%) — all cycle-2 frustration overtones. The cycle-2 frustration set {5,7,9,11,13} clusters at the cleaner end of the distribution; non-frustration values ({6,8,10,12,15}) cluster at the worse end. 2. {7} also shows the strongest void contrast at 11.5× — the ratio of EM intensity in the cavity void vs the boundary walls when driven at the self-resonance frequency. This makes {7}-fold cavities the most efficient at *concentrating* energy in the void.

Why this matters.

This is the empirical foundation for the √7 structural factor in the magic-numbers derivation. Before HPC-039, the framework predicted that {7} should be the first cycle-2 frustration overtone but had no independent FDTD confirmation. HPC-039 provided that confirmation: {7} *is* uniquely self-resonant, 2.7% error vs the 8–56% spread of comparison geometries.

The biological-resonance implication (separate project, deferred): rotating biological machines — ATP synthase (10-fold), GroEL (7-fold), proteasome (7-fold) — preferentially use {7}-fold symmetry for the *rotor* component, while {10} is the *stator*. The {7}-fold rotational self-resonance gives a geometric reason: low frictional loss to rotational vibrational modes when the rotor is {7}-fold. This was an incidental finding from HPC-039 that aligned with structural biology observations.

Reproducibility. Full FDTD driver attached. The Prony analysis script is included separately. Sweep takes ~12 hours on Hetzner. Result table should match to within ~1% across grid resolutions; the {7}-fold unique-cleanness is robust under 64³, 96³, and 128³ runs (tested).

What this is and is not.

• IS: empirical confirmation that {7}-fold geometry is uniquely

self-resonant among tested n-fold cavities.

• IS NOT: a claim that {7} is the *only* self-resonant geometry —

the cycle-2 frustration set as a whole ({5,7,9,11,13}) clusters in the cleaner half of the distribution.

• IS: the FDTD foundation for the √7 structural factor in the

magic-numbers derivation.

• IS NOT: a claim about biological rotation directly — the biology

connection is an incidental observation, not a tested hypothesis.