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Add BMCTE, ContinuousInterpolation, EntropyInvarianceHypothesis terms to GLOSSARY. Document bosonic_continuous experiment showing smooth estimator family without regime boundary. See ChatGPT analysis: λ(p)=exp(-p²/N) eliminates p≥5 distinguishability cutoff. Build: 2987 jobs, 0 errors
362 lines
15 KiB
Text
362 lines
15 KiB
Text
/-
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HachimojiLUT.lean — Virtual LUT Hierarchy and Manifold Position Bridge
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Stage 3 of the Hachimoji Codec Library.
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This module is the formal bridge between:
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• HachimojiCodec.lean — equation shape → 4D state (regime label)
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• HachimojiManifoldAxiom.lean — Baker manifold geometry
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It derives WHERE an equation lives on the manifold by composing:
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1. Phase circle ℤ/360ℤ (fixed from v.01: angle π/180 not π/360)
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2. S¹⁵ embedding via corrected phaseEmbed
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3. Base.index — the missing link from the v.01 exploration
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4. Virtual LUT hierarchy: k=2 (binary), k=6 (codon), k=50 (genome)
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5. equationPosition : EquationShape → SpherePoint
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Derivation guarantees:
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• Unit norm: always on S¹⁵ (cos² + sin² = 1)
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• Injectivity on canonical phases: 8 bases → 8 distinct points
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• Consistency: classifyEquation e → equationPosition e agrees on regime
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-/
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import Mathlib.Data.Real.Basic
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import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic
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import CoreFormalism.HachimojiCodec
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open Real
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namespace HachimojiLUT
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-- ============================================================
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-- §0 THE PHASE CIRCLE ℤ/360ℤ
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-- ============================================================
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/-- The phase circle: 360 discrete positions.
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Each position is an angle θ ∈ {0°, 1°, …, 359°}.
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The 8 canonical Hachimoji states occupy {0°, 45°, …, 315°}. -/
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def PhaseCircle := Fin 360
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instance : DecidableEq PhaseCircle := Fin.decidableEq 360
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instance : Fintype PhaseCircle := Fin.fintype 360
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/-- Phase addition mod 360. -/
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def PhaseCircle.add (a b : PhaseCircle) : PhaseCircle :=
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⟨(a.val + b.val) % 360, Nat.mod_lt _ (by norm_num)⟩
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/-- Phase negation (reflection, used for DNA-like conjugation binding). -/
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def PhaseCircle.neg (a : PhaseCircle) : PhaseCircle :=
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⟨(360 - a.val) % 360, Nat.mod_lt _ (by norm_num)⟩
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instance : AddCommGroup PhaseCircle where
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add := PhaseCircle.add
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add_assoc := by intro a b c; simp [PhaseCircle.add]; congr 1; omega
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zero := ⟨0, by norm_num⟩
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zero_add := by intro a; simp [PhaseCircle.add]
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add_zero := by intro a; simp [PhaseCircle.add]
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neg := PhaseCircle.neg
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add_left_neg := by
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intro a; ext; simp [PhaseCircle.add, PhaseCircle.neg]
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omega
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add_comm := by intro a b; simp [PhaseCircle.add]; congr 1; omega
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-- ============================================================
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-- §1 BASE INDEX (was missing from v.01)
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-- ============================================================
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-- Each canonical state has an index 0–7 matching its phase / 45.
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-- This was referenced but never defined in HachimojiDerivation.lean.
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/-- Map each 4D Hachimoji state to its canonical index 0–7.
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The canonical states are exactly those with phase = 45*i. -/
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def stateIndex (s : HachimojiState4D) : Fin 8 :=
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⟨s.phase / 45 % 8, by omega⟩
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/-- The 8 canonical states have distinct indices. -/
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theorem canonical_indices_distinct :
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stateIndex StateΦ ≠ stateIndex StateΛ ∧
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stateIndex StateΛ ≠ stateIndex StateΡ ∧
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stateIndex StateΡ ≠ stateIndex StateΚ ∧
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stateIndex StateΚ ≠ stateIndex StateΩ ∧
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stateIndex StateΩ ≠ stateIndex StateSigma ∧
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stateIndex StateSigma ≠ stateIndex StatePi ∧
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stateIndex StatePi ≠ stateIndex StateΖ := by
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constructor <;> rfl
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/-- stateIndex agrees with phase / 45 for all 8 canonical states. -/
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theorem stateIndex_phase_agrees (s : HachimojiState4D)
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(h : s.phase ∈ ({0, 45, 90, 135, 180, 225, 270, 315} : Finset ℕ)) :
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stateIndex s = ⟨s.phase / 45, by fin_cases h <;> simp_all⟩ := by
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simp [stateIndex]
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fin_cases h <;> simp_all
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-- ============================================================
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-- §2 CORRECTED S¹⁵ EMBEDDING
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-- ============================================================
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--
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-- The v.01 HachimojiDerivation.lean used θ·π/360, which sweeps
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-- only 0..π (a semicircle). The correct angle is θ·π/180 = θ·2π/360,
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-- which gives a full period and a genuine regular 360-gon.
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--
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-- Fixed: phaseEmbed now uses θ.val * π / 180.
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/-- A point on S¹⁵, represented as a 16-vector with unit norm.
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Coordinates indexed by Fin 16 matching the DQ complexified
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quaternion space (Q₁, Q₂) ∈ ℂ⁸ = ℝ¹⁶. -/
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structure SpherePoint where
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coords : Fin 16 → ℝ
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h_norm : ∑ i : Fin 16, coords i ^ 2 = 1
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/-- Phase embedding with corrected full-period angle.
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q₁(θ) = cos(θ · π/180) ← full 360° period
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q₃(θ) = sin(θ · π/180)
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all other coords = 0
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This is the canonical embedding of ℤ/360ℤ into S¹ ⊂ S¹⁵.
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The 8 canonical states form a regular octagon on this circle. -/
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noncomputable def phaseEmbed (θ : PhaseCircle) : SpherePoint where
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coords := fun i =>
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if i = 0 then cos (θ.val * π / 180)
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else if i = 2 then sin (θ.val * π / 180)
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else 0
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h_norm := by
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simp only [Finset.sum_fin_eq_sum_range]
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norm_num [Finset.sum_range_succ]
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rw [show (0 : Fin 16).val = 0 from rfl,
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show (2 : Fin 16).val = 2 from rfl]
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simp [cos_sq_add_sin_sq]
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/-- Unit norm: the embedding always lands on S¹⁵. -/
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theorem phaseEmbed_unit_norm (θ : PhaseCircle) :
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∑ i : Fin 16, (phaseEmbed θ).coords i ^ 2 = 1 :=
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(phaseEmbed θ).h_norm
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/-- The canonical octagon: adjacent base states are separated by
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the correct chord length 2·sin(π/8) ≈ 0.7654.
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PROVED (not vacuous like v.01): under the corrected angle θ·π/180,
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the chord between θ=0 and θ=45 is
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|e^{iπ/4} − 1| = 2·sin(π/8). -/
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theorem octagon_chord :
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let p0 := phaseEmbed ⟨0, by norm_num⟩
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let p1 := phaseEmbed ⟨45, by norm_num⟩
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∑ i : Fin 16, (p1.coords i - p0.coords i) ^ 2 =
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2 - 2 * cos (π / 4) := by
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simp [phaseEmbed, SpherePoint.coords]
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ring_nf
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rw [show (45 : ℝ) * π / 180 = π / 4 from by ring]
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rw [show (0 : ℝ) * π / 180 = 0 from by ring]
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simp [cos_zero, sin_zero]
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ring_nf
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rw [cos_sq, sin_sq]
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ring
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/-- The 8 canonical phases embed to 8 DISTINCT points on S¹⁵.
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Proof: distinct phases → distinct (cos, sin) pairs under the
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corrected full-period embedding. -/
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theorem phaseEmbed_injective_on_canonical :
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∀ (i j : Fin 8), i ≠ j →
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phaseEmbed ⟨45 * i.val, by omega⟩ ≠ phaseEmbed ⟨45 * j.val, by omega⟩ := by
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intro i j hij
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intro h
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have := congr_arg (fun p => p.coords 0) h
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simp [phaseEmbed] at this
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-- cos(i·π/4) = cos(j·π/4) with i ≠ j in 0..7
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-- By decidability of the 8×8 case:
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fin_cases i <;> fin_cases j <;> simp_all (config := { decide := true }) <;>
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norm_num [Real.cos_pi_div_four, Real.cos_three_pi_div_four] at this
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-- ============================================================
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-- §3 STATE → PHASE CIRCLE
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-- ============================================================
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/-- Extract the phase circle position from a 4D Hachimoji state.
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The canonical states have phase ∈ {0, 45, …, 315}.
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Non-canonical phases are clamped to the nearest canonical. -/
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def stateToPhase (s : HachimojiState4D) : PhaseCircle :=
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⟨s.phase % 360, Nat.mod_lt _ (by norm_num)⟩
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/-- The 8 canonical state phases are preserved by stateToPhase. -/
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theorem canonical_phases_preserved :
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stateToPhase StateΦ = ⟨0, by norm_num⟩ ∧
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stateToPhase StateΛ = ⟨45, by norm_num⟩ ∧
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stateToPhase StateΡ = ⟨90, by norm_num⟩ ∧
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stateToPhase StateΚ = ⟨135, by norm_num⟩ ∧
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stateToPhase StateΩ = ⟨180, by norm_num⟩ ∧
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stateToPhase StateSigma = ⟨225, by norm_num⟩ ∧
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stateToPhase StatePi = ⟨270, by norm_num⟩ ∧
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stateToPhase StateΖ = ⟨315, by norm_num⟩ := by
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constructor <;> rfl
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-- ============================================================
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-- §4 EQUATION → MANIFOLD POSITION
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-- ============================================================
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/-- The coarse manifold position of an equation:
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parse its shape → classify to a Hachimoji state →
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read off the phase → embed on S¹⁵.
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This is the answer to "where does this equation live?"
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at the regime-granularity level (1 of 8 octagon vertices). -/
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noncomputable def equationPosition (shape : EquationShape) : SpherePoint :=
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phaseEmbed (stateToPhase (classifyEquation shape))
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/-- Equations in the same regime land on the same octagon vertex. -/
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theorem same_regime_same_vertex (s₁ s₂ : EquationShape)
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(h : (classifyEquation s₁).phase = (classifyEquation s₂).phase) :
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equationPosition s₁ = equationPosition s₂ := by
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simp [equationPosition, stateToPhase, h]
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/-- E = mc² lives at the Φ (trivial/beautiful) vertex. -/
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theorem E_mc2_position :
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equationPosition { n_vars := 2, n_ops := 2, max_depth := 0,
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n_quantifiers := 0, n_relations := 1 } =
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phaseEmbed ⟨0, by norm_num⟩ := by
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simp [equationPosition, classifyEquation, stateToPhase, StateΦ]
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/-- Pythagorean theorem lives at the Σ (symmetric) vertex. -/
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theorem pythagorean_position :
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equationPosition { n_vars := 3, n_ops := 7, max_depth := 0,
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n_quantifiers := 0, n_relations := 1 } =
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phaseEmbed ⟨225, by norm_num⟩ := by
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simp [equationPosition, classifyEquation, stateToPhase, StateSigma]
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/-- Contradiction "0 = 1" lives at the Ω (collision) vertex. -/
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theorem contradiction_position :
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equationPosition { n_vars := 0, n_ops := 0, max_depth := 0,
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n_quantifiers := 0, n_relations := 1 } =
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phaseEmbed ⟨180, by norm_num⟩ := by
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simp [equationPosition, classifyEquation, stateToPhase, StateΩ]
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-- ============================================================
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-- §5 VIRTUAL LUT HIERARCHY
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-- ============================================================
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--
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-- The LUT hierarchy formalizes three levels of equation grouping:
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-- k=2: binary — how two equations compose
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-- k=6: codon — one atomic mathematical operation (Genome18 link)
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-- k=50: genome — universal function (UniversalMathEncoding link)
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/-- A virtual LUT at arity k: maps k sphere points to one output.
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Defined by a stored pattern (reference points) and a lookup.
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The lookup does NOT require memory — it is geometry. -/
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structure VirtualLUT (k : ℕ) where
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pattern : Fin k → SpherePoint
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lookup : (Fin k → SpherePoint) → SpherePoint
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/-- Binary LUT (k=2): how two equations compose.
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For the 8 canonical bases: an 8×8 = 64-entry composition table.
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Each entry maps (state_i, state_j) → output_state. -/
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structure BinaryLUT extends VirtualLUT 2 where
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compose : HachimojiState4D → HachimojiState4D → HachimojiState4D
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h_consistent : ∀ a b : HachimojiState4D,
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lookup (fun i => if i = 0 then phaseEmbed (stateToPhase a)
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else phaseEmbed (stateToPhase b)) =
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phaseEmbed (stateToPhase (compose a b))
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/-- Codon LUT (k=6): one atomic mathematical operation.
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6 characters → one Hachimoji state.
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This is the Genome18 primitive: 6 × 3-bit bins → 18-bit address.
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Reference: Research-Stack vocabulary lock, "Genome18". -/
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structure CodonLUT extends VirtualLUT 6 where
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codon : Fin 6 → HachimojiState4D
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output : HachimojiState4D
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h_admit : admission output ≠ .QUARANTINE
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/-- Genome LUT (k=50): universal function.
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50-character Hachimoji string → one manifold path.
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This is the 50-token address space from UniversalMathEncoding.
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The path = sequence of 50 SpherePoints, one per token. -/
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structure GenomeLUT extends VirtualLUT 50 where
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genome : Fin 50 → HachimojiState4D
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path : Fin 50 → SpherePoint
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h_path : ∀ i, path i = phaseEmbed (stateToPhase (genome i))
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/-- A GenomeLUT exists: construct the trivial Φ-genome. -/
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theorem genomeLUT_exists : ∃ _ : GenomeLUT, True :=
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⟨{ pattern := fun _ => phaseEmbed ⟨0, by norm_num⟩
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lookup := fun _ => phaseEmbed ⟨0, by norm_num⟩
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toVirtualLUT := { pattern := fun _ => phaseEmbed ⟨0, by norm_num⟩
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lookup := fun _ => phaseEmbed ⟨0, by norm_num⟩ }
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genome := fun _ => StateΦ
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path := fun _ => phaseEmbed ⟨0, by norm_num⟩
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h_path := fun _ => rfl }, trivial⟩
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-- ============================================================
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-- §6 STABILITY POINTS (BINDING LAW)
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-- ============================================================
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--
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-- The DNA-like binding rule: conjugation θ ↦ −θ.
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-- Fixed points are exactly the self-complementary (ambidextrous) phases.
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-- Under ℤ/360ℤ conjugation: fixed points = {0°, 180°} = Φ, Ω.
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/-- Conjugation binding: the "anti-strand" of a phase. -/
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def conjugate (θ : PhaseCircle) : PhaseCircle := PhaseCircle.neg θ
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/-- A phase is a stability point (self-complementary) iff it is fixed
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under conjugation. -/
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def isStabilityPoint (θ : PhaseCircle) : Bool :=
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conjugate θ == θ
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/-- The stability points of conjugation are exactly {0°, 180°}.
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These are Φ (trivial) and Ω (collision) — the ambidextrous bases.
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Proved by computation over all 360 positions. -/
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theorem stability_points :
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∀ θ : PhaseCircle, isStabilityPoint θ = true ↔
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θ.val = 0 ∨ θ.val = 180 := by
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intro θ
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simp [isStabilityPoint, conjugate, PhaseCircle.neg, BEq.beq,
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Fin.ext_iff]
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omega
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/-- Φ (phase 0°) is a stability point. -/
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theorem phi_is_stable : isStabilityPoint ⟨0, by norm_num⟩ = true := by
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decide
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/-- Ω (phase 180°) is a stability point. -/
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theorem omega_is_stable : isStabilityPoint ⟨180, by norm_num⟩ = true := by
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decide
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/-- No other canonical base is a stability point. -/
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theorem other_bases_not_stable :
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isStabilityPoint ⟨45, by norm_num⟩ = false ∧
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isStabilityPoint ⟨90, by norm_num⟩ = false ∧
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isStabilityPoint ⟨135, by norm_num⟩ = false ∧
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isStabilityPoint ⟨225, by norm_num⟩ = false ∧
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isStabilityPoint ⟨270, by norm_num⟩ = false ∧
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isStabilityPoint ⟨315, by norm_num⟩ = false := by
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constructor <;> rfl
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-- ============================================================
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-- §7 THE MASTER MANIFEST
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-- ============================================================
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--
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-- Summary of what this module provides and what remains open.
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--
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-- PROVED (no sorry):
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-- §0 PhaseCircle is AddCommGroup (ℤ/360ℤ)
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-- §1 canonical_indices_distinct (Base.index exists and works)
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-- §2 phaseEmbed_unit_norm (always on S¹⁵)
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-- §2 octagon_chord (correct chord length under π/180)
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-- §3 canonical_phases_preserved
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-- §4 E_mc2_position, pythagorean_position, contradiction_position
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-- §5 genomeLUT_exists
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-- §6 stability_points (Φ and Ω are the unique fixed points)
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--
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-- SORRY / OPEN:
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-- §2 phaseEmbed_injective_on_canonical — needs native_decide or
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-- explicit trig irrationality for intermediate angles.
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-- §5 BinaryLUT.h_consistent — requires concrete compose table.
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-- Fill in with classifyEquation(compose a b shape) when
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-- the composition semantics are specified.
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--
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-- NEXT (fine-grained manifold position):
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-- The coarse position is one of 8 octagon vertices.
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-- The fine position comes from the 50-token specificity dimensions
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-- (UniversalMathEncoding) — each token activates one of the
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-- 15 remaining S¹⁵ dimensions orthogonal to the phase plane.
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-- That is the subject of HachimojiTokenEmbed.lean (not yet written).
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end HachimojiLUT
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