module module high

`IndisputableMonolith.Gap45.Derivation`

This module defines the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... to support the Gap45 derivation. Researchers addressing the 45-tick synchronization gap in Recognition Science would cite these definitions. The module supplies recursive definitions and small lemmas on Fibonacci numbers with no complex proofs.

claimThe Fibonacci sequence satisfies $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$.

background

The module sits inside the Gap45.Derivation component of Recognition Science and imports Mathlib. It introduces the Fibonacci sequence whose terms are used to construct cumulative phase counts that motivate the number 45 in the dimension-forcing argument. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21, ... and is developed through basic recursive clauses and elementary lemmas.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the Fibonacci definitions that feed directly into IndisputableMonolith.Gap45.PhysicalMotivation. That downstream module uses them to give a physically grounded derivation of the 45-tick synchronization, addressing the paper gap that the 45-tick argument remains physically unmotivated.

scope and limits

Does not derive the full 45-tick synchronization argument.
Does not connect Fibonacci numbers to J-cost, defectDist, or the phi-ladder.
Does not prove any Recognition Science forcing-chain theorems.
Does not address physical constants or mass formulas.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Gap45.PhysicalMotivation module_import

declarations in this module (38)

def fib L49
lemma fib_0 L54
lemma fib_1 L55
lemma fib_2 L56
lemma fib_3 L57
lemma fib_4 L58
lemma fib_5 L59
lemma fib_6 L60
theorem fibonacci_5_is_5 L63
theorem fibonacci_6_is_8 L66
theorem fib_coprime_4_5 L70
theorem five_eight_coprime L73
def eight_tick_period L78
def closure_factor L81
lemma closure_factor_eq L83
def fibonacci_factor L86
lemma fibonacci_factor_eq L88
theorem fibonacci_factor_is_fib L91
theorem fibonacci_factor_coprime_with_8 L94
def gap L100
theorem gap_eq_45 L103
theorem gap_factorization L106
theorem gap_forced_from_eight_tick_and_fibonacci L109
theorem gap_coprime_with_8 L116
theorem forty_five_eq_nine_times_five L121
theorem forty_five_factorization L124
def full_period L129
theorem full_period_eq_360 L132
theorem full_period_is_product L137
theorem cycles_of_eight L141
theorem cycles_of_gap L145
def power_of_two L151
theorem lcm_360_forces_D_eq_3 L154
theorem D_3_gives_8 L181
theorem D_3_forced_from_structure L184
def closure_interpretation L200
def fibonacci_interpretation L206
def derivation_summary L210