linear_phase_justification

formal source

 204   cumulative information content grows linearly with tick number.
 205
 206The triangular sum T(n) = Σₖ₌₁ⁿ k is the natural cumulative for linear growth. -/
 207def linear_phase_justification : String :=
 208  "Linear phase accumulation follows from J-cost normalization J''(1) = 1. " ++
 209  "Each tick k adds phase ~ k due to cumulative ledger evolution. " ++
 210  "The triangular number T(n) is the natural sum for linear growth."
 211
 212/-! ## Why 5 Appears (Fibonacci Connection) -/
 213
 214/-- **WHY 5?**
 215
 216The factorization 45 = 9 × 5 includes 5 because:
 217
 2181. T(9) = 9 × 10 / 2 = 9 × 5 (the 10/2 becomes 5)
 2192. 10 = closure_number + 1 = (8 + 1) + 1 = 9 + 1
 2203. So T(9) = 9 × (9+1) / 2 = 9 × 5
 221
 222The Fibonacci connection (5 = Fib(5)) is a consequence of:
 223- 5 = 10/2 = (9+1)/2
 224- The "+1" comes from the triangular formula n(n+1)/2
 225
 226This explains why the Fibonacci derivation "works" — it's actually
 227computing T(9) in a different form. -/
 228def fibonacci_connection_explained : String :=
 229  "5 = (9+1)/2 from the triangular formula T(9) = 9 × 10 / 2. " ++
 230  "The Fibonacci interpretation is algebraically equivalent but " ++
 231  "the triangular number is the fundamental origin."
 232
 233/-! ## Complete Physical Derivation Chain -/
 234
 235/-- **THE COMPLETE DERIVATION**
 236
 2371. The 8-tick cycle comes from 2^D with D=3 (ledger coverage)