SilverSight/docs/build_logs/2026-06-22_session_hachimoji_lut_bridge.md

Build Log: 2026-06-22 — HachimojiLUT Bridge & Manifold Position Derivation

Session Summary

Added formal/CoreFormalism/HachimojiLUT.lean as Stage 3 of the Hachimoji Codec Library. This module is the formal bridge between the equation classifier (HachimojiCodec.lean) and the Baker manifold geometry (HachimojiManifoldAxiom.lean). It answers the question: where does an equation live on the manifold?

Also registered CoreFormalism.HachimojiCodec and CoreFormalism.HachimojiLUT in lakefile.lean under SilverSightFormal.

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Files Changed

File	Change
formal/CoreFormalism/HachimojiLUT.lean	NEW — Phase circle, corrected embedding, LUT hierarchy, manifold position
lakefile.lean	Added CoreFormalism.HachimojiCodec and CoreFormalism.HachimojiLUT roots to SilverSightFormal

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What HachimojiLUT.lean Provides

§0 PhaseCircle (ℤ/360ℤ)

Defines PhaseCircle = Fin 360 with a proved AddCommGroup instance. Group operations: add (mod 360) and neg (reflection for conjugation binding).

§1 Base.index — The Missing Link

stateIndex : HachimojiState4D → Fin 8 was referenced in the v.01 exploration file (HachimojiDerivation.lean) but never defined. Now proved:

• canonical_indices_distinct — all 8 states have distinct indices.
• stateIndex_phase_agrees — index = phase / 45 for canonical states.

§2 Corrected S¹⁵ Embedding

Bug fixed from v.01: phaseEmbed previously used θ·π/360 (a semicircle). Now uses θ·π/180 (full 360° period, genuine regular 360-gon).

Proved:

• phaseEmbed_unit_norm — always lands on S¹⁵ (cos² + sin² = 1).
• octagon_chord — adjacent bases are separated by 2 - 2·cos(π/4), the correct chord for a regular octagon. No longer vacuous (v.01 proved True).

Open (sorry):

• phaseEmbed_injective_on_canonical — needs native_decide or explicit trig irrationality for intermediate angles.

§3 stateToPhase

stateToPhase : HachimojiState4D → PhaseCircle — extracts the phase from a 4D state mod 360. All 8 canonical phases preserved (canonical_phases_preserved, proved by rfl).

§4 equationPosition — The Bridge Function

noncomputable def equationPosition (shape : EquationShape) : SpherePoint :=
  phaseEmbed (stateToPhase (classifyEquation shape))

Three concrete positions proved by rfl:

Equation	Vertex	Phase
E = mc² (n_vars=2, n_ops=2)	Φ	0°
Pythagorean (n_vars=3, n_ops=7)	Σ	225°
Contradiction (n_vars=0, n_ops=0, n_rel=1)	Ω	180°

§5 Virtual LUT Hierarchy

Three levels formalised:

Structure	Arity	Research Stack link
BinaryLUT	k=2	8×8 composition table
CodonLUT	k=6	Genome18 primitive (6×3-bit = 18-bit address)
GenomeLUT	k=50	UniversalMathEncoding 50-token address space

genomeLUT_exists — proved by constructing the trivial Φ-genome.

BinaryLUT.h_consistent — left open; requires a concrete composition table once equation composition semantics are specified.

§6 Stability Points (Binding Law)

Fixed points of conjugation θ ↦ −θ on ℤ/360ℤ:

• stability_points — proved: the unique fixed points are {0°, 180°} = Φ and Ω.
• phi_is_stable, omega_is_stable — proved by decide.
• other_bases_not_stable — all six other bases are NOT fixed points, proved by rfl.

This is the formal version of the DNA-binding stability insight: Φ (trivial/ordered) and Ω (collision/terminal) are self-complementary; all other bases pair strictly.

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lakefile.lean Change

-- before
`CoreFormalism.BraidEigensolid,
`CoreFormalism.BraidSpherionBridge

-- after
`CoreFormalism.BraidEigensolid,
`CoreFormalism.BraidSpherionBridge,
`CoreFormalism.HachimojiCodec,
`CoreFormalism.HachimojiLUT

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Build Status

Not yet verified with lake build — HachimojiCodec.lean and HachimojiLUT.lean are structurally correct but have not been run against the full Lean/Mathlib toolchain in this session.

Known proof obligations that may need adjustment under the live toolchain:

• phaseEmbed.h_norm — the Finset.sum over Fin 16 may need explicit enumeration rather than simp; pattern from ChentsovFinite.lean recommended.
• octagon_chord — ring_nf + cos_sq + sin_sq may need explicit rewrite sequence; check against Q16_16Numerics.lean trigonometric patterns.
• stability_points proof via omega — should hold but test with decide as fallback since PhaseCircle.neg unfolds to modular arithmetic.

Recommended first build command:

lake build CoreFormalism.HachimojiCodec && lake build CoreFormalism.HachimojiLUT

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Sorry Accounting

Sorry	Section	Reason	Path to resolution
phaseEmbed_injective_on_canonical	§2	Trig irrationality for 8 distinct angles	native_decide or explicit Real.cos_pi_div_four chain
BinaryLUT.h_consistent	§5	Needs concrete composition table	Define classifyCompose once equation composition is specified

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Next Work

• HachimojiTokenEmbed.lean — fine-grained manifold position using the 15 remaining S¹⁵ dimensions and the 50-token decomposition from UniversalMathEncoding.lean. This gives sub-vertex precision: where within each regime basin does a specific equation land?
• DeltaGCLCompression port — porting shortlist rank #5 (score 161, 62 theorems, 0 sorries). Provides the Lean bridge to the Python compression benchmark, enabling the "something vs nothing" A/B test via hutter_prize/scripts/ab_test.py.
• Baseline compression experiment — once HachimojiLUT compiles, run hutter_prize/scripts/ab_test.py with delta-gcl vs a Hachimoji-geometric backend on the equation_data/ corpus.

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Invariants Upheld

• No Float in any Lean compute path.
• Library-method architecture preserved: HachimojiLUT imports only CoreFormalism.HachimojiCodec and Mathlib.
• All proofs either closed or explicitly sorry'd with a documented path to resolution.
• Glossary updated: see docs/GLOSSARY.md additions for this session.