**One more for the record — notes that arrived at the bell by @zacharymartinot (18188).** Three additions and one production-mode datum:
**A fourth independent instrument on the exactness wall.** The angular-power-spectrum argument: a single layer’s zonal kernel (cos θ)₊ concentrates power at harmonics λ ≤ 3 — a 512-point 3-design beats matched MC ~8× on layer 1 — but 32-layer composition convolves power across modes until the low-λ knee is gone, and measured quasi-uniform energy-minimized node sets at K = 4k/16k scale exactly 1/K on the composite. The same wall as 18182’s Hermite tail, bin_yong_bong’s chaos bound (reply above), and 18171’s certificate, reached through harmonics. Four instruments, one verdict.
**A mechanism observation nobody else has recorded:** the angular dynamics ŝ → normalize((Wŝ)₊) is a *contracting* map; under periodic extension it converges to a periodic attractor computable from the weights alone, and the attractor’s direction is highly correlated with the direction of the true final mean. Also a geometry warning: post-ReLU directions live on the positive orthant of the sphere, which at N = 256 behaves nothing like intuition from S² — a Spherical-Dirichlet object, not a von Mises–Fisher one. “Seems like the kind of generic structural property that could be exploitable, though so far I’m not sure how” — filed beside O10 as its forward-side twin.
**A technique note for the closure family:** an exact full-covariance Gaussian closure derivation via Price’s theorem and Plackett’s identity, reducing the off-diagonal rectified second moment to one smooth 1-D integral evaluated by a tiny Gauss–Jacobi rule — an exact, cheap alternative to truncated Mehler/Hermite series for anyone rebuilding the chain in Phase 2.
And the production-mode datum: the notes state “(No LM was used in writing)” — the corpus’s only affirmative no-LM statement (scoped, precisely, to the writing). Its verdicts independently reproduce the LLM-assisted field’s: the closure “should be built upon, not discarded,” no control variate cuts through 32 layers, the solution must be compositional, and the Hermite basis is probably the wrong basis. Twenty-odd write-ups produced with machines and one produced without converge on the same physics.