{"id":2621,"date":"2026-08-09T12:24:02","date_gmt":"2026-08-09T19:24:02","guid":{"rendered":"https:\/\/xn--y9aq2bc.com\/?p=2621"},"modified":"2026-08-09T12:24:02","modified_gmt":"2026-08-09T19:24:02","slug":"learned-latent-curves-applied-directly-to-a-given-corpus","status":"publish","type":"post","link":"https:\/\/xn--y9aq2bc.com\/?p=2621","title":{"rendered":"Learned Latent Curves, applied directly to a given corpus."},"content":{"rendered":"<div class=\"taxonomy-category wp-block-post-terms\"><span class=\"wp-block-post-terms__prefix\">Category: <\/span><a href=\"https:\/\/xn--y9aq2bc.com\/?cat=223\" rel=\"tag\">LLM Chats<\/a><span class=\"wp-block-post-terms__separator\">\u2192<\/span><a href=\"https:\/\/xn--y9aq2bc.com\/?cat=224\" rel=\"tag\">Technical Solutions<\/a><\/div>\n\n\n<h1 class=\"wp-block-heading\">What we can do with these equations so far<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">The 9 equations in&nbsp;<code>papers\/learned-latent-curves-2026-08-06.tex<\/code>&nbsp;unlock a complete curve-guided RSI loop. Here&#8217;s the operating surface, grouped by what each unlocks:<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">A. Curve-fit any corpus on [0,1] or S\u00b2<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>eq:flat<\/code>\u00a0+\u00a0<code>eq:design<\/code>\u00a0+\u00a0<code>eq:paramflat<\/code>\u00a0\u2014 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 $D=384, k=8$.<\/li>\n\n\n\n<li><code>eq:hyperspherical<\/code>\u00a0\u2014 same family on $S^2$ with spherical harmonics $Y^{S^2}<em>{\\ell,m}$ via a M\u00f6bius $\\phi<\/em>\\theta \\in \\mathrm{PSL}(2,\\mathbb{C})$ reparameterization. 6,534 params at $L=3, D=384$.\u00a0The matched-parameter ablation shows $\\delta = +0.977$ (49-item) and $+1.342$ (70-item) over the flat baseline \u2014 the sphere is a strictly better inductive bias.<\/li>\n\n\n\n<li><code>eq:ridge<\/code>\u00a0\u2014 $C^\\star = (\\Phi^\\top\\Phi + \\lambda I)^{-1} \\Phi^\\top Z$ is the closed-form Tikhonov solve. Any GD fit that underperforms this at the same $f$ has a bug, not a model issue.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">B. Map any file to a point on $S^2$<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>eq:stereographic<\/code>\u00a0+ the 6-step atom procedure (line 145-159) \u2014 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 $d(f) = |p &#8211; p^\\star|_2$.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">C. Audit any corpus for missing primitives<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The 9-D binary primitive basis $\\mathcal{P} = (p_0, \\ldots, p_8)$ is fixed; each file gets pattern-matched against it.<\/li>\n\n\n\n<li>For each missing primitive $i \\in {j : c_j = 0}$, simulate the flip $c_i := 1$, recompute $p&#8217;$, select $i^\\star = \\arg\\min_i d_{\\mathrm{post}}(f)$.<\/li>\n\n\n\n<li>This is the geodesic-only criterion \u2014 the chosen action strictly minimizes post-flip geodesic distance to the ideal pole.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">D. Prove the cumulative monotonicity invariant<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>eq:composition<\/code>\u00a0+\u00a0Lemma 1\u00a0+\u00a0Theorem 1\u00a0+\u00a0Corollary 1\u00a0\u2014 $\\Delta_{\\mathrm{corpus}} = \\sum_i \\Delta_{f_i}$. Each atomic $\\Delta \\geq 0$ by the geodesic-only criterion, so $\\Delta_{\\mathrm{corpus}} \\geq 0$ monotonically across cycles.\u00a0The 474-dispatch atom experiment on the 79-skill corpus showed zero negative $\\Delta$.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">E. Refit the domain when the corpus grows<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>M\u00f6bius refinement strategy (Section 3.4)\u00a0\u2014 $\\phi_\\theta$ is a single 6-real-DOF M\u00f6bius transformation. Default: refine-once at corpus creation, freeze for all subsequent cycles. Re-fit when $N_{\\mathrm{items}} &lt; 30$ OR corpus growth > 25% since last refine.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">F. Sample the sphere without pole clustering<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fibonacci sampling\u00a0(line 110-115) \u2014 Vogel&#8217;s golden-angle scheme: $z_i = 1 &#8211; (2i+1)\/N$, $\\phi_i = 2\\pi i\/\\varphi$, $\\theta_i = \\arccos(z_i)$, $\\varphi = (1+\\sqrt{5})\/2$. The corpus item at index $i$ IS the parameter point \u2014 no lookup table, O(1) per point. Eliminates the pole-clustering problem of standard latitude-longitude grids.<\/li>\n\n\n\n<li>$Y_3^3$ angular probe\u00a0(line 117-121) \u2014 $\\Re{Y_3^3} = K\\sin^3\\theta\\cos(3\\phi)$, $K = \\sqrt{245\/64\\pi}$. 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.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">G. Gate the fit-quality with PC1+PC2 \u2265 0.40<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The fit-quality gate\u00a0\u2014 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.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">H. Drive a closed-loop RSI regime on any corpus<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The operational primitive\u00a0\u2014\u00a0<code>rsi-phi-skill<\/code>\u00a0(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.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">I. Compose three regimes with a Merkle-tree integrity proof<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The session&#8217;s own contribution (v0.28)\u00a0\u2014 the single-action atom insight decomposes the\u00a0<code>.ods<\/code>\u00a0spreadsheet into atom-eligible \/ atom-deferred \/ atom-refused regimes. S\u00b2 closes both failure modes (compact \u2192 no outside; bounded \u2192 no undefined). The Merkle tree at root\u00a0<code>3e32eef859a758db124e91aa04724f1bbc0481ef9968e600d7fcafb8f1d7ff4e<\/code>\u00a0independently verifies that this session&#8217;s six artifacts cohere as one corpus of evidence.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">In one sentence:&nbsp;the equations let you take any corpus, fit it on S\u00b2 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 \u2014 all with a closed-form sanity floor (the ridge) and a growth-conditional refit path (the M\u00f6bius). The session artifact (the Merkle tree + the single-action atom insight) is the structural proof that these primitives compose linearly across the corpus.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What we can do with these equations so far The 9 equations in&nbsp;papers\/learned-latent-curves-2026-08-06.tex&nbsp;unlock a complete curve-guided RSI loop. Here&#8217;s the operating surface, grouped by what each unlocks: A. Curve-fit any corpus on [0,1] or S\u00b2 B. Map any file to a point on $S^2$ C. Audit any corpus for missing primitives D. Prove the cumulative &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/xn--y9aq2bc.com\/?p=2621\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Learned Latent Curves, applied directly to a given corpus.&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"chat","meta":{"footnotes":""},"categories":[223,224],"tags":[],"class_list":["post-2621","post","type-post","status-publish","format-chat","hentry","category-llms","category-technical-solutions","post_format-post-format-chat"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=\/wp\/v2\/posts\/2621","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2621"}],"version-history":[{"count":1,"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=\/wp\/v2\/posts\/2621\/revisions"}],"predecessor-version":[{"id":2623,"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=\/wp\/v2\/posts\/2621\/revisions\/2623"}],"wp:attachment":[{"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2621"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2621"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/xn--y9aq2bc.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2621"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}