module module high

`IndisputableMonolith.Gap45.Derivation`

Gap45.Derivation defines the Fibonacci sequence starting 1, 1, 2, 3, 5, 8 and verifies initial terms plus coprimality. Researchers closing the physical motivation gap for 45-tick synchronization in Recognition Science cite it as the arithmetic substrate. The module relies on recursive definitions and direct checks for small indices.

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

background

The Gap45.Derivation module sits inside the Recognition Science treatment of the 45-tick synchronization gap. It introduces the Fibonacci sequence exactly as stated in its documentation: 1, 1, 2, 3, 5, 8, 13, 21, .... Definitions cover the recursive function, base cases, explicit evaluations at small indices, and coprimality of selected pairs.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the Fibonacci sequence to the PhysicalMotivation module. That module in turn supplies the physically grounded derivation of the number 45, addressing the paper gap that the 45-tick synchronization argument remains physically unmotivated. The sequence supplies the cumulative-phase substrate for the dimension-forcing step.

scope and limits

Does not derive the integer 45 from the sequence.
Does not connect Fibonacci terms to phi, alpha, or the T0-T8 chain.
Does not supply physical interpretations or phase accumulation rules.

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