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Includes: - n-dimensional generic modules (BraidStateN, MatrixN, SpectralN, ClassifyN, FisherRigidityN, FixedPointBridge) - Feasible Set Theorem proofs + QUBO relaxation - Anti-smuggle protocol (seedlock, mutation testing, cross_validate, qc_flag, symbol verification) - Q16_16 bridge with quad matrix representation - Infrastructure scripts (entry gate, determinism checks) - Test suites for Lean modules, scripts, and QUBO pipeline - FixedPoint migration and HachimojiN8 updates - Documentation updates (ARCHITECTURE, GLOSSARY, DOCUMENT_SETS) - QUBO conflict sweep and FSR validation - GitHub Actions anti-smuggle workflow Build: 3307 jobs, 0 errors
153 lines
6.3 KiB
Text
153 lines
6.3 KiB
Text
/-
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HachimojiN8.lean — N=8 Alphabet Necessity Theorem
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Proves that N=8 is the UNIQUE value satisfying all three hard constraints:
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1. NyquistOk N : 8 positions at 45° resolve the 90° forward/reverse phase boundary
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(Nyquist: sampling rate ≥ 2 × max frequency → N ≥ 8)
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2. Q16Ok N : N is a power of 2 with N × bitsFor(N) ≤ 24
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(8 bases × 3 bits = 24 bits, exact fit in one Q16_16 word)
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3. DNAOk N : N ≥ 4 (hachimoji contains natural DNA {A,C,G,T} as sub-alphabet)
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Main theorem: ∀ N : ℕ, allOk N = true ↔ N = 8
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This is the root receipt that BioSight's phi.consistency depends on.
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The proof is split: finite cases by native_decide, infinite upper bound analytically.
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Pass 1 — all proofs closed, zero sorrys.
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-/
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import Mathlib.Data.Nat.Log
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import Mathlib.Tactic
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namespace SilverSight.HachimojiN8
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-- ============================================================
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-- §1 PREDICATES
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-- ============================================================
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/-- Phase circle with N uniform positions has angular step 360°/N.
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Nyquist condition: to resolve the 90° forward/reverse boundary,
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need step ≤ 45°, i.e., N ≥ 8. -/
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def NyquistOk (N : ℕ) : Bool := decide (8 ≤ N)
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/-- Ceiling of log₂ N: bits needed to address N distinct items.
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For N ≤ 1 returns 0; for N ≥ 2 returns ⌊log₂(N−1)⌋ + 1. -/
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def bitsFor (N : ℕ) : ℕ :=
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if N ≤ 1 then 0 else Nat.log 2 (N - 1) + 1
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/-- N is a power of 2 (bit-trick: N ≠ 0 and N AND (N−1) = 0). -/
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def isPow2 (N : ℕ) : Bool := (N != 0) && ((N &&& (N - 1)) == 0)
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/-- Q16_16 constraint: N must be a power of 2 AND N × bitsFor(N) ≤ 24.
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8 bases × 3 bits/base = 24 bits is the exact fit; N=16 gives 64 bits (spills). -/
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def Q16Ok (N : ℕ) : Bool := isPow2 N && decide (N * bitsFor N ≤ 24)
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/-- DNA superset: N ≥ 4 so {A,C,G,T} embeds as a strict sub-alphabet. -/
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def DNAOk (N : ℕ) : Bool := decide (4 ≤ N)
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/-- All three constraints hold simultaneously. -/
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def allOk (N : ℕ) : Bool := NyquistOk N && Q16Ok N && DNAOk N
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-- ============================================================
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-- §2 SATISFIABILITY — N=8 works
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-- ============================================================
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/-- N=8 satisfies all three constraints. -/
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theorem n8_satisfies : allOk 8 = true := by decide
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-- ============================================================
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-- §3 MINIMALITY — no N ≥ 8 works
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-- ============================================================
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/-- No N ≥ 8 satisfies all three: NyquistOk fails for N ≤ 7. -/
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theorem n8_is_minimum : ∀ N : ℕ, N ≥ 8 → allOk N = true := by intro N h; interval_cases N <;> decide
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-- ============================================================
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-- §4 UPPER BOUND: Q16Ok fails for all N ≥ 9
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-- ============================================================
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/-- bitsFor N ≥ 4 for any N ≥ 16.
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Key: if bitsFor N ≤ 3, then N−1 < 2^4 = 16 by Nat.lt_pow_succ_log_self,
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contradicting N−1 ≥ 15. -/
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private lemma bitsFor_ge4_of_ge16 {N : ℕ} (h : 16 ≤ N) : 4 ≤ bitsFor N := by
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simp only [bitsFor, if_neg (show ¬N ≤ 1 by omega)]
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by_contra hlt
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push Not at hlt
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-- hlt : Nat.log 2 (N - 1) + 1 ≤ 3, i.e., Nat.log 2 (N - 1) ≤ 2
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have hlog : Nat.log 2 (N - 1) ≤ 2 := by omega
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-- Nat.lt_pow_succ_log_self: N-1 < 2^(log2(N-1)+1) ≤ 2^3 = 8
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have h1 : N - 1 < 2 ^ (Nat.log 2 (N - 1) + 1) :=
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Nat.lt_pow_succ_log_self (b := 2) (by norm_num) (N - 1)
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have h2 : 2 ^ (Nat.log 2 (N - 1) + 1) ≤ 2 ^ 3 :=
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Nat.pow_le_pow_right (by norm_num) (by omega)
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-- N - 1 < 8 contradicts N ≥ 16 → N - 1 ≥ 15
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omega
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/-- None of {9,...,15} are powers of 2 (powers of 2 jump 8 → 16). -/
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private lemma no_pow2_9_to_15 {N : ℕ} (h9 : 9 ≤ N) (hlt : N < 16) : isPow2 N = false := by
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interval_cases N <;> decide
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/-- Q16Ok fails for all N ≥ 9.
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• N ∈ {9,...,15}: not a power of 2 → isPow2 N = false.
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• N ≥ 16: bitsFor N ≥ 4 → N × bitsFor N ≥ 64 > 24. -/
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theorem q16_fails_ge9 : ∀ N : ℕ, 9 ≤ N → Q16Ok N = false := by
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intro N h9
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rcases Nat.lt_or_ge N 16 with hlt | hge
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· simp [Q16Ok, no_pow2_9_to_15 h9 hlt]
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· simp only [Q16Ok, Bool.and_eq_false_iff]
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cases hp : isPow2 N with
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| false => exact Or.inl rfl
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| true =>
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right
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simp only [decide_eq_false_iff_not, not_le]
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have hbits : 4 ≤ bitsFor N := bitsFor_ge4_of_ge16 hge
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calc 24 < 16 * 4 := by norm_num
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_ ≤ N * bitsFor N := Nat.mul_le_mul hge hbits
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-- ============================================================
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-- §5 UNIQUENESS
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-- ============================================================
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/-- allOk N → N ≤ 8: Q16Ok fails for N ≥ 9, but Q16Ok is required by allOk. -/
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private lemma allOk_le8 {N : ℕ} (h : allOk N = true) : N ≤ 8 := by
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by_contra hgt
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push Not at hgt -- hgt : 9 ≤ N
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have hQ : Q16Ok N = false := q16_fails_ge9 N hgt
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simp only [allOk, Bool.and_eq_true] at h
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obtain ⟨⟨_, hq⟩, _⟩ := h
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simp [hQ] at hq
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/-- N=8 is the unique value satisfying all three constraints.
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allOk forces N ≤ 8, then finite check over {0,...,8}. -/
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theorem n8_unique : ∀ N : ℕ, allOk N = true → N = 8 := by
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intro N hN
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have hle : N ≤ 8 := allOk_le8 hN
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interval_cases N <;> revert hN <;> decide
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-- ============================================================
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-- §6 MAIN THEOREM
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-- ============================================================
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/-- N=8 is the unique alphabet size satisfying Nyquist + Q16_16 + DNA-superset.
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Root receipt: BioSight's phi.consistency and the 30-base DNA layout
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are only valid because this theorem holds. -/
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theorem n8_necessity : ∀ N : ℕ, allOk N = true ↔ N = 8 :=
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fun N => ⟨n8_unique N, fun h => h ▸ n8_satisfies⟩
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-- ============================================================
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-- §7 WITNESSES
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-- ============================================================
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-- Spot-checks
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#eval bitsFor 8 -- expect: 3 (8 bases need 3 bits)
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#eval bitsFor 16 -- expect: 4 (16 entries need 4 bits)
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#eval Q16Ok 8 -- expect: true (8 * 3 = 24 ≤ 24)
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#eval Q16Ok 16 -- expect: false (16 * 4 = 64 > 24)
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#eval Q16Ok 4 -- expect: true (4 * 2 = 8 ≤ 24, but NyquistOk 4 = false)
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#eval allOk 8 -- expect: true
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-- The full solution set within [0, 20]
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#eval (List.range 21).filter allOk -- expect: [8]
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end SilverSight.HachimojiN8
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