What we can do with these equations so far

The 9 equations in papers/learned-latent-curves-2026-08-06.tex unlock a complete curve-guided RSI loop. Here’s the operating surface, grouped by what each unlocks:

A. Curve-fit any corpus on [0,1] or S²

• eq:flat + eq:design + eq:paramflat — fit a flat Fourier curve $z_j(t) = a_{j,0} + \sum_m (a_{j,m}\sin 2\pi f_m t + b_{j,m}\cos 2\pi f_m t)$ on $[0,1]$ with $k$ shared frequencies and $D$ output dims. 6,528 params at $D=384, k=8$.
• eq:hyperspherical — same family on $S^2$ with spherical harmonics $Y^{S^2}{\ell,m}$ via a Möbius $\phi\theta \in \mathrm{PSL}(2,\mathbb{C})$ reparameterization. 6,534 params at $L=3, D=384$. The matched-parameter ablation shows $\delta = +0.977$ (49-item) and $+1.342$ (70-item) over the flat baseline — the sphere is a strictly better inductive bias.
• eq:ridge — $C^\star = (\Phi^\top\Phi + \lambda I)^{-1} \Phi^\top Z$ is the closed-form Tikhonov solve. Any GD fit that underperforms this at the same $f$ has a bug, not a model issue.

B. Map any file to a point on $S^2$

• eq:stereographic + the 6-step atom procedure (line 145-159) — given a corpus item, compute its 9-D binary coverage vector, weighted aggregate (thresholded at 0.5), PCA top-2 via SVD, stereographic lift, chordal distance to the ideal pole. The result: one point $p = \sigma(\bar u, \bar v) \in S^2$ per file, with a defined geodesic gap $d(f) = |p – p^\star|_2$.

C. Audit any corpus for missing primitives

• The 9-D binary primitive basis $\mathcal{P} = (p_0, \ldots, p_8)$ is fixed; each file gets pattern-matched against it.
• For each missing primitive $i \in {j : c_j = 0}$, simulate the flip $c_i := 1$, recompute $p’$, select $i^\star = \arg\min_i d_{\mathrm{post}}(f)$.
• This is the geodesic-only criterion — the chosen action strictly minimizes post-flip geodesic distance to the ideal pole.

D. Prove the cumulative monotonicity invariant

• eq:composition + Lemma 1 + Theorem 1 + Corollary 1 — $\Delta_{\mathrm{corpus}} = \sum_i \Delta_{f_i}$. Each atomic $\Delta \geq 0$ by the geodesic-only criterion, so $\Delta_{\mathrm{corpus}} \geq 0$ monotonically across cycles. The 474-dispatch atom experiment on the 79-skill corpus showed zero negative $\Delta$.

E. Refit the domain when the corpus grows

• Möbius refinement strategy (Section 3.4) — $\phi_\theta$ is a single 6-real-DOF Möbius transformation. Default: refine-once at corpus creation, freeze for all subsequent cycles. Re-fit when $N_{\mathrm{items}} < 30$ OR corpus growth > 25% since last refine.

F. Sample the sphere without pole clustering

• Fibonacci sampling (line 110-115) — Vogel’s golden-angle scheme: $z_i = 1 – (2i+1)/N$, $\phi_i = 2\pi i/\varphi$, $\theta_i = \arccos(z_i)$, $\varphi = (1+\sqrt{5})/2$. The corpus item at index $i$ IS the parameter point — no lookup table, O(1) per point. Eliminates the pole-clustering problem of standard latitude-longitude grids.
• $Y_3^3$ angular probe (line 117-121) — $\Re{Y_3^3} = K\sin^3\theta\cos(3\phi)$, $K = \sqrt{245/64\pi}$. Per-item basis $b_i = (\sin^3\theta_i\cos(m_1\phi_i), \ldots)$ with $m_k = 3k$ for $k=1..128$ (384 = $2^7 \cdot 3$). Both orderings $(\ell=128, m=256)$ and $(\ell=256, m=128)$ tested per cycle, pick higher PC1+PC2.

G. Gate the fit-quality with PC1+PC2 ≥ 0.40

• The fit-quality gate — for any corpus fit, the PC1+PC2 explained-variance ratio must clear 0.40 or the curve is rejected as uninformative. This is the structural test that the corpus has signal worth modeling.

H. Drive a closed-loop RSI regime on any corpus

• The operational primitive — rsi-phi-skill (added 2026-08-07 in PR #200) operationalizes all of the above as one agent skill. It accepts a corpus, fits the sphere, runs the atom on each file, applies the geodesic-only action, accumulates $\Delta_{\mathrm{corpus}}$, and refits $\phi_\theta$ on corpus growth. The 79-skill corpus dataset (Section 4) is the canonical example.

I. Compose three regimes with a Merkle-tree integrity proof

• The session’s own contribution (v0.28) — the single-action atom insight decomposes the .ods spreadsheet into atom-eligible / atom-deferred / atom-refused regimes. S² closes both failure modes (compact → no outside; bounded → no undefined). The Merkle tree at root 3e32eef859a758db124e91aa04724f1bbc0481ef9968e600d7fcafb8f1d7ff4e independently verifies that this session’s six artifacts cohere as one corpus of evidence.

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In one sentence: the equations let you take any corpus, fit it on S² with a strictly better inductive bias than flat $[0,1]^2$, audit every file for missing 9-D primitives, prove each individual action strictly improves geodesic distance, and accumulate non-negative deltas across cycles — all with a closed-form sanity floor (the ridge) and a growth-conditional refit path (the Möbius). The session artifact (the Merkle tree + the single-action atom insight) is the structural proof that these primitives compose linearly across the corpus.