--- id: triangle-regression-v12-foundation type: log title: Triangle Regression — v12's Geometric Foundation (Scale-Indifferent) date_published: 2026-04-30 date_updated: 2026-05-12 project: cipher_v12 status: confirmed log_subtype: framework_foundation tags: [triangle-regression, v12, scale-indifferent, snap-as-regression, geometric-foundation] author: Jonathan Shelton data_supporting: - cipher-corrections-hurt-accuracy - ft-snap-tunneling-mechanism see_also: - cipher-corrections-hurt-accuracy - ft-snap-tunneling-mechanism - cipher-version-progression-audit --- ## Author notes Cipher v12's leap over v11 (91/107 with 0 misses vs v11's 71/107 corrected) came from replacing v11's *framerate calculation* with v12's **triangle regression** — a purely geometric foundation that is *scale-indifferent*. ### What changed: framerate → triangle regression **v11 architecture:** the cipher computed framerate at each dimensional position, then used framerate ratios to derive properties. Framerate has a *scale* (it's a frequency); predictions implicitly carried that scale, and small scale-mismatches across elements caused some of v11's misses. **v12 architecture:** the cipher uses the regression of a triangle (the geometric object whose vertices represent f, |t, and r in the framework's three-axiom architecture) onto its constituent geometric primitives. The regression doesn't carry an absolute scale — it operates on the *ratios* of the triangle's geometric features. Scale-indifferent: doubling all measurements halves all features proportionally, leaving the regression result unchanged. ### How this fixed v11's misses v11's 36 misses (107 − 71) clustered in two regions: 1. High-Z elements where framerate scaling broke down (the cipher's framerate calculation became unstable as Z increased). 2. Elements near the snap topology where the framerate calculation discontinuously switched between f-state and |t-state. v12's triangle regression: 1. Has no scale instability at high Z because it operates on ratios, not absolute frequencies. 2. Treats the snap as a *regression onset* (the point where the triangle's geometric regression begins to favor the |t-vertex over the f-vertex) — a continuous geometric phenomenon rather than a discontinuous calculation switch. ### Snap = regression onset The [snap-tunneling mechanism](/research/notes/ft-snap-tunneling-mechanism.html) identified the snap as a tunneling event between f-state and |t-state. v12's triangle regression gives the *geometric* picture: at the snap topology, the triangle's regression begins favoring the |t-vertex; below the snap, the f-vertex dominates. The regression onset *is* the tunneling probability transition. This reformulation didn't change any predictions but made the mechanism *geometric* rather than *frequency-dependent*. That was the load-bearing piece for v12's improvement. ### Scale-indifference and cross-scale unification The framework's cross-scale unification claim (same mechanism at atomic, mesoscale, macroscale, cosmic) is now grounded in v12's scale-indifference. Triangle regression operates the same way at any scale — only the absolute geometric features change with scale; the regression result depends on ratios only. This means: - The cipher predicts atomic-scale crystal structure. - The same triangle regression predicts mesoscale void resonance. - The same triangle regression predicts (in principle) cosmic-scale galactic geometry. The "in principle" qualifier matters: the cosmic-scale application requires cycle-2 / cycle-3 / cycle-4 frameworks (Tribonacci / Pentanacci / Octanacci recurrences) that the cipher hasn't fully worked out yet. But the *core* mechanism (triangle regression on {f, |t, r} vertices) is scale-indifferent and applies at any cycle. ### Why corrections hurt accuracy The [corrections-hurt-accuracy finding](/research/notes/cipher-corrections-hurt-accuracy.html) makes sense under v12's triangle-regression foundation: the underlying geometry is *exact* (it's the regression of a triangle onto its vertices — there's nothing approximate about it). Correction layers (spherical-limit blending, eigenvalue-based correction, shell-completion logic) all imposed *approximate* fixes on top of an exact mechanism. Approximate-on-exact = worse than exact alone. v12 stripped the corrections, kept the exact mechanism, and outperformed v11. ### Confirmation v12's 91/107 with 0 misses on the same 107-element bench that v11 scored 71/107 corrected is the empirical confirmation. The trajectory of v9 → v10 → v11 → v12 (each adding/refining corrections, v12 stripping them) gives the discipline test: the framework's mechanism is *exact* when the geometric foundation is right. ## Summary Cipher **v12's leap over v11** (91/107 with 0 misses vs v11's 71/107 corrected) came from replacing v11's framerate calculation with v12's **triangle regression** — a purely geometric foundation that is *scale-indifferent*. **Architecture change:** - **v11:** computed framerate at each dimensional position; used framerate ratios to derive properties. Framerate has a scale (frequency); predictions carried that scale and broke down at high Z and near snap topology. - **v12:** regression of a triangle (vertices = f, |t, r) onto its geometric primitives. Operates on *ratios*, not absolute scale. Doubling all measurements halves all features proportionally, leaving regression unchanged. **How this fixed v11's misses:** 1. No scale instability at high Z (ratios, not absolute frequencies). 2. The snap is a *regression onset* (continuous geometric phenomenon), not a discontinuous calculation switch between f-state and |t-state. **Snap as regression onset:** the [snap-tunneling mechanism](/research/notes/ft-snap-tunneling-mechanism.html) identifies snap as a tunneling event. v12's triangle regression gives the geometric picture: at the snap topology, regression begins favoring the |t-vertex; below the snap, f-vertex dominates. The regression onset *is* the tunneling probability transition. **Scale-indifference enables cross-scale unification.** Triangle regression operates the same way at any scale. The framework's cross-scale claim (same mechanism at atomic, mesoscale, macroscale, cosmic) is now grounded in v12's scale-indifferent foundation. **Why corrections hurt accuracy:** under v12's exact-geometry foundation, correction layers impose *approximate* fixes on an *exact* mechanism. Approximate-on-exact = worse than exact alone. v12 stripped corrections, kept exact mechanism, beat v11 cleanly. **Status: confirmed.** v12's 91/107 with 0 misses on the same bench v11 scored 71/107 is empirical confirmation. The geometric foundation is exact.