IndisputableMonolith.Verification.FibSubstCert

   1import Mathlib
   2import IndisputableMonolith.Verification.Necessity.FibSubst
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace FibSubstCert
   7
   8open IndisputableMonolith.Verification.Necessity.FibSubst
   9
  10/-!
  11# Fibonacci Substitution Certificate
  12
  13This certificate packages the proof that Fibonacci recurrence arises from a
  14canonical 2-letter substitution system:
  15- σ(0) = 01
  16- σ(1) = 0
  17
  18Starting from the seed word [0], iteration yields words whose symbol counts
  19follow the Fibonacci sequence exactly:
  20- `countFalse (iter n) = fib (n+1)`
  21- `countTrue (iter n) = fib n`
  22
  23## Why this matters for the certificate chain
  24
  25The Fibonacci sequence is central to Recognition Science's derivation of φ:
  26- The ratio F_{n+1}/F_n converges to φ as n → ∞
  27- This substitution system provides a concrete discrete process that generates Fibonacci numbers
  28- The proof shows this is not imposed by fitting, but emerges from the simplest non-trivial substitution rule
  29
  30This certificate verifies the core mathematical structure connecting discrete
  31dynamics (substitution systems) to the golden ratio's emergence.
  32
  33## Proven results
  34
  351. **Substitution recurrence**: counts_sub_word shows how symbol counts transform under substitution
  362. **Fibonacci identity**: counts_iter_fib proves the Fibonacci recursion emerges from iteration
  37-/
  38
  39structure FibSubstCert where
  40  deriving Repr
  41
  42/-- Verification predicate: Fibonacci counts emerge from substitution iteration.
  43
  44This certifies:
  451. The substitution rules are standard (σ(0) = 01, σ(1) = 0)
  462. Starting from [0], the counts satisfy the Fibonacci recurrence
  473. At step n, countFalse = fib(n+1) and countTrue = fib(n)
  48-/
  49@[simp] def FibSubstCert.verified (_c : FibSubstCert) : Prop :=
  50  -- Core substitution rules are defined (structural fact)
  51  (fibSub false = [false, true]) ∧
  52  (fibSub true = [false]) ∧
  53  -- Fibonacci identity: iteration yields Fibonacci counts
  54  (∀ n : ℕ, (countFalse (iter n), countTrue (iter n)) = (fib (n+1), fib n)) ∧
  55  -- Base case: iter 0 = [false] with counts (1, 0) = (fib 1, fib 0)
  56  (countFalse (iter 0), countTrue (iter 0)) = (1, 0) ∧
  57  -- Recurrence: counts_iter_succ shows how counts evolve
  58  (∀ n : ℕ, countFalse (iter (n+1)) = countFalse (iter n) + countTrue (iter n))
  59
  60/-- Top-level theorem: the certificate verifies. -/
  61@[simp] theorem FibSubstCert.verified_any (c : FibSubstCert) :
  62    FibSubstCert.verified c := by
  63  constructor
  64  · -- fibSub false = [false, true]
  65    rfl
  66  constructor
  67  · -- fibSub true = [false]
  68    rfl
  69  constructor
  70  · -- Fibonacci identity
  71    exact counts_iter_fib
  72  constructor
  73  · -- Base case
  74    simp [iter, fib]
  75  · -- Recurrence
  76    intro n
  77    exact (counts_iter_succ n).1
  78
  79end FibSubstCert
  80end Verification
  81end IndisputableMonolith
  82