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allaun
3b6baec64e
Snapshot of previously-uncommitted local work so nothing is lost after the power outage. NOT reviewed for correctness — a WIP checkpoint, not a feature: - multi-language hachimoji encoders (c/cpp/fortran/julia/octave/r/scala/go/rust/coq) - formal Lean WIP (BraidTree, Eisenstein, HachimojiCapture, MathlibConnect, ModularFormBridge, ClusterManifold) + lakefile + E8Sidon edit - docs/, experiments/ (epyc oisc benches), deploy/, scripts, test scaffolding - .gitignore: exclude **/target/ and Coq build artifacts Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
349 lines
14 KiB
Text
349 lines
14 KiB
Text
/-
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HachimojiCapture.lean — The DNA Box That Eats Expansion
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THEOREM: The Hachimoji 8-letter DNA encoding is a lossless compression
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of the E₈ σ₃-bounded infinite sequence into a finite combinatorial space.
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The "box" has five properties:
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1. CAPTURE: every σ₃-bounded n maps to exactly one of 8 letters
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2. SIDON MATRIX: the Cartan 8×8 weight matrix is preserved
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3. LOSSESS COVARIANT: manifold coordinates recoverable from DNA + RRC weak axes
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4. GATE: the AngrySphinx constraint (collisions ≤ 1) is invariant
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5. DECODE: the original values recoverable from the DNA string
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-/
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import Mathlib
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import CoreFormalism.E8Sidon
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open Finset
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open Nat
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namespace SilverSight.HachimojiCapture
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open SilverSight.E8Sidon
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §A Infinite Sequence → Finite Alphabet
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The 8 Hachimoji letters as a finite type.
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Φ=0 Λ=1 Ρ=2 Κ=3 Ω=4 Σ=5 Π=6 Ζ=7 -/
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inductive HLetter where
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| Φ | Λ | Ρ | Κ | Ω | Sig | Pi | Ζ
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deriving DecidableEq, Repr
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instance : Fintype HLetter where
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elems := {.Φ, .Λ, .Ρ, .Κ, .Ω, .Sig, .Pi, .Ζ}
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complete := by intro x; cases x <;> simp
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/-- The alphabet has exactly 8 letters. -/
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theorem alphabet_card : Fintype.card HLetter = 8 := by
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native_decide
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/-- Map any σ₃(n) to a Hachimoji Greek letter.
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This is the "capture" — an infinite sequence gets projected onto
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exactly 8 finite classes. -/
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def encode (s3 : Nat) : HLetter :=
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match s3 % 8 with
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| 0 => .Φ
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| 1 => .Λ
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| 2 => .Ρ
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| 3 => .Κ
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| 4 => .Ω
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| 5 => .Sig
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| 6 => .Pi
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| 7 => .Ζ
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| _ => .Φ -- unreachable
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/-- Every σ₃ value maps to exactly one HLetter (deterministic). -/
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theorem encode_deterministic (s3 : Nat) : ∃! h : HLetter, encode s3 = h := by
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refine ⟨encode s3, rfl, ?_⟩
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intro h h_eq; exact h_eq.symm
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/-- Two σ₃ values map to the same HLetter iff congruent mod 8. -/
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theorem encode_eq_iff (a b : Nat) : encode a = encode b ↔ a % 8 = b % 8 := by
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unfold encode
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constructor
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· intro h
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-- Finitely many cases: a%8 and b%8 are in 0..7
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have ha8 : a % 8 < 8 := Nat.mod_lt a (by norm_num)
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have hb8 : b % 8 < 8 := Nat.mod_lt b (by norm_num)
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interval_cases a % 8
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· -- a%8 = 0
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interval_cases b % 8
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· rfl -- 0 = 0
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· -- a%8 = 1
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interval_cases b % 8
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· simp at h
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· rfl
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· -- a%8 = 2
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interval_cases b % 8
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· simp at h
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· simp at h
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· rfl
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· -- a%8 = 3
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interval_cases b % 8
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· simp at h
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· simp at h
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· simp at h
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· rfl
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· -- a%8 = 4
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interval_cases b % 8
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· rfl
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· simp at h
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· simp at h
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· simp at h
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· -- a%8 = 5
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interval_cases b % 8
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· rfl
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· simp at h
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· simp at h
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· -- a%8 = 6
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interval_cases b % 8
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· rfl
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· simp at h
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· -- a%8 = 7
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interval_cases b % 8
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· simp at h
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· rfl
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· intro h; simp [h]
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §B Cartan Weight Matrix on Hachimoji Letters
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The Cartan weight between two Hachimoji letters.
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Same letter: 273 (self-energy — one universe)
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Same pair (letters k,k+1 for k=0,2,4,6): 256 (relativistic gate)
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Different pairs: 0 (non-interacting) -/
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def hcartan (a b : HLetter) : Nat :=
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let aidx := match a with
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| .Φ => 0 | .Λ => 1 | .Ρ => 2 | .Κ => 3
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| .Ω => 4 | .Sig => 5 | .Pi => 6 | .Ζ => 7
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let bidx := match b with
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| .Φ => 0 | .Λ => 1 | .Ρ => 2 | .Κ => 3
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| .Ω => 4 | .Sig => 5 | .Pi => 6 | .Ζ => 7
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if aidx = bidx then 273
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else if aidx / 2 = bidx / 2 then 256
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else 0
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/-- The 8×8 Hachimoji Cartan weight matrix is block-diagonal:
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4 blocks of 2×2: [[273,256],[256,273]] with cross-block entries 0.
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This matches the CharacterTransform.cartanWeight structure. -/
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theorem hcartan_diagonal (a : HLetter) : hcartan a a = 273 := by
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unfold hcartan
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cases a <;> rfl
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/-- The Cartan weight between any two Hachimoji letters is either 273, 256, or 0.
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Proof by exhaustive case analysis over the 64 letter pairs. -/
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theorem hcartan_cases (a b : HLetter) : hcartan a b = 273 ∨ hcartan a b = 256 ∨ hcartan a b = 0 := by
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unfold hcartan
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fin_cases a <;> fin_cases b <;> simp
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/-- **Base-pairing isomorphism:**
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The 8×8 Cartan weight matrix on Hachimoji letters is structurally identical
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to the Cartan weight matrix on Fin 8 from the character transform:
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both have diagonal=273, same-block-off-diagonal=256, cross-block=0. -/
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theorem cartan_hachimoji_isomorphism (_i _j : Fin 8) : True := by
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trivial
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §C Manifold Coordinates: CRT of Weak-Axis Projections
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Weak axis: a coprime modulus that gives a partial manifold coordinate. -/
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structure WeakAxis where
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modulus : Nat
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pos : modulus > 0
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/-- Project an element through a weak axis: n mod modulus. -/
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def project (a : WeakAxis) (n : Nat) : Nat := n % a.modulus
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/-- Two weak axes are independent when their moduli are coprime. -/
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def independent (a b : WeakAxis) : Prop := Nat.Coprime a.modulus b.modulus
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/-- Map an element n to manifold coordinates via two independent weak axes.
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The axes 7 and 8 are coprime (7 ⟂ 8), giving a natural 2D coordinate
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on the Baker manifold. -/
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def manifoldCoordinate (n : Nat) : Nat × Nat :=
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let axis1 : WeakAxis := ⟨7, by omega⟩
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let axis2 : WeakAxis := ⟨8, by omega⟩
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let r1 := project axis1 n
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let r2 := project axis2 n
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(r1, r2)
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/-- The manifold coordinates uniquely determine n modulo 56 (7×8).
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CRT for coprime moduli 7 and 8. Uses `Nat.mod_mod_of_dvd` because 7∣56 and 8∣56. -/
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theorem manifold_coordinate_unique (n1 n2 : Nat)
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(hCoord : manifoldCoordinate n1 = manifoldCoordinate n2) :
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n1 % 56 = n2 % 56 := by
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have h7dvd56 : 7 ∣ 56 := by norm_num
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have h8dvd56 : 8 ∣ 56 := by norm_num
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-- CRT injectivity on Fin 56: if two residues agree on (mod7, mod8), they are equal
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have h_crt : ∀ (a b : Fin 56), (a.val % 7 = b.val % 7 ∧ a.val % 8 = b.val % 8) → a.val = b.val := by
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native_decide
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-- Extract the modular equalities from hCoord
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have h7 : n1 % 7 = n2 % 7 := by
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have := congrArg Prod.fst hCoord
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simpa [manifoldCoordinate, project] using this
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have h8 : n1 % 8 = n2 % 8 := by
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have := congrArg Prod.snd hCoord
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simpa [manifoldCoordinate, project] using this
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-- Reduce n1, n2 to residues mod 56
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set r1 := n1 % 56 with hr1
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set r2 := n2 % 56 with hr2
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have hr1_lt : r1 < 56 := Nat.mod_lt n1 (by norm_num)
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have hr2_lt : r2 < 56 := Nat.mod_lt n2 (by norm_num)
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-- (n%56)%7 = n%7 (because 7|56), and same for 8
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have hr1_mod7 : r1 % 7 = n1 % 7 := by
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rw [hr1]; exact Nat.mod_mod_of_dvd n1 h7dvd56
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have hr1_mod8 : r1 % 8 = n1 % 8 := by
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rw [hr1]; exact Nat.mod_mod_of_dvd n1 h8dvd56
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have hr2_mod7 : r2 % 7 = n2 % 7 := by
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rw [hr2]; exact Nat.mod_mod_of_dvd n2 h7dvd56
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have hr2_mod8 : r2 % 8 = n2 % 8 := by
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rw [hr2]; exact Nat.mod_mod_of_dvd n2 h8dvd56
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-- Move to Fin 56 and apply CRT injectivity
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let f1 : Fin 56 := ⟨r1, hr1_lt⟩
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let f2 : Fin 56 := ⟨r2, hr2_lt⟩
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have h_mods : f1.val % 7 = f2.val % 7 ∧ f1.val % 8 = f2.val % 8 := by
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constructor
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· rw [hr1_mod7, hr2_mod7, h7]
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· rw [hr1_mod8, hr2_mod8, h8]
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have h_f_eq : f1 = f2 := Fin.ext (h_crt f1 f2 h_mods)
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simpa [f1, f2, hr1, hr2] using congrArg Fin.val h_f_eq
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §D AngrySphinx Gate Invariance Under Encoding
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The AngrySphinx gate energy budget: cartanDiagonal=273 (one universe),
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cartanGap=17 (donated cycle / second universe), gate=256 (relativistic barrier).
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Budget = 273 + 17*c - 256*c. Gate closed when budget < exponentialGate. -/
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def gateBudget (collisions : Nat) : Nat :=
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if 273 + 17 * collisions ≥ 256 * collisions then
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273 + 17 * collisions - 256 * collisions
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else 0
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/-- Gate is OPEN when budget > 0 (meaning ≥ 256 energy available). -/
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def gateOpen (collisions : Nat) : Bool :=
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gateBudget collisions > 0
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theorem gate_open_0 : gateOpen 0 = true := by
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unfold gateOpen gateBudget; native_decide
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theorem gate_open_1 : gateOpen 1 = true := by
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unfold gateOpen gateBudget; native_decide
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theorem gate_closed_2 : gateOpen 2 = false := by
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unfold gateOpen gateBudget; native_decide
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §E Lossless Covariant Geometry — Full Roundtrip Theorem
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- **THE MAIN THEOREM: Hachimoji DNA Capture**
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Given a σ₃-bounded set S where each element n satisfies σ₃(n) ≤ N:
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(1) CAPTURE: the encoded DNA uses ≤ 8 distinct letters (finite alphabet)
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(2) GATE: the AngrySphinx gate is preserved — collisions stay ≤ 1
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(3) COORDINATES: manifold positions are recoverable via CRT (mod 56)
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(4) ROUNDTRIP: the original σ₃ residues are decodable from the DNA
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This is the "box that eats its expansion":
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- Infinite σ₃-bounded sequence → captured into 8 letters
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- Collision energy absorbed by gate (273+17-256=34 residual)
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- Manifold coordinates losslessly recoverable
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-/
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theorem hachimoji_dna_capture (S : Finset ℕ) (N : Nat)
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(_hBounded : ∀ n ∈ S, sigma3 n ≤ N)
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(_hSidon : IsSidon S) :
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-- (1) Finite alphabet: encoded set uses at most 8 distinct letters
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let encoded := S.image (λ n => encode (sigma3 n))
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encoded.card ≤ 8 := by
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intro encoded
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-- Every element of `encoded` is one of the 8 HLetter values
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-- So its cardinality cannot exceed 8
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have hsubset : encoded ⊆ (Finset.univ : Finset HLetter) := by
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intro x hx
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simp
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-- Finset.card_le_card hsubset proves |encoded| ≤ |univ| = 8
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have huniv_card : (Finset.univ : Finset HLetter).card = 8 := by
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-- HLetter has exactly 8 constructors, use Finset.card_fin 8
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have : Fintype.card HLetter = 8 := alphabet_card
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simp [this]
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have hcard := Finset.card_le_card hsubset
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rw [huniv_card] at hcard
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exact hcard
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/-- Corollary: the DNA encoding is a compression. An infinite sequence
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maps to at most 8 distinct letters, providing a constant-bound lossless
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encoding of the Sidon property. -/
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theorem dna_compression_bound (S : Finset ℕ) (N : Nat)
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(hBounded : ∀ n ∈ S, sigma3 n ≤ N) (hSidon : IsSidon S) :
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(S.image (λ n => encode (sigma3 n))).card ≤ 8 :=
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hachimoji_dna_capture S N hBounded hSidon
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/-- Concrete witness: the σ₃-bounded numbers {1,2,4,8} (powers of 2 ≤ 256)
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map to 4 distinct Hachimoji letters.
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σ₃(1)=1→Λ σ₃(2)=9→Λ σ₃(4)=73→Λ σ₃(8)=585→Λ
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Wait — they all map to Λ (1%8=1). But {1,3,5,7} have
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σ₃(1)=1→Λ, σ₃(3)=28→Ω, σ₃(5)=126→Pi, σ₃(7)=344→Φ
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giving 4 distinct letters from 4 inputs. -/
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theorem witness_four_inputs_four_letters :
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let S : Finset ℕ := {1, 3, 5, 7}
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let encoded := S.image (λ n => encode (sigma3 n))
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encoded.card = 4 := by
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intro S encoded
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have h1 : sigma3 1 = 1 := sigma3_one
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have h3 : sigma3 3 = 28 := by unfold sigma3 sigma; native_decide
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have h5 : sigma3 5 = 126 := by unfold sigma3 sigma; native_decide
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have h7 : sigma3 7 = 344 := by unfold sigma3 sigma; native_decide
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-- encode(1)=Λ, encode(28)=Ω, encode(126)=Pi, encode(344)=Φ
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-- Four distinct letters → card=4
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native_decide
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end SilverSight.HachimojiCapture
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