time_dilation_factor
360/37 quantifies the subjective time dilation factor arising from the least common multiple of the 8-tick octave and 45-fold phase structure in the Gap-45 barrier. Researchers modeling ledger versus experienced time in Recognition Science cite this bound when deriving the approximate 9.73-fold dilation from coprime periodic constraints. The proof is a one-line term-mode verification that splits the conjunction and normalizes the rational arithmetic to confirm the strict inequalities.
claimIn the Gap-45 setting the subjective time dilation factor satisfies $9 < 360/37 < 10$.
background
The Gap-45 Recognition Barrier module formalizes the mathematical core of the barrier induced by gcd(8,45)=1. No proper divisor of 360 simultaneously divides both periods, so any finite window shorter than 360 ticks encounters incompatible constraints from the 8-tick phase (defined via EightTick.phase as kπ/4) and the 45-fold structure. The fundamental time quantum is the tick τ₀=1 from Constants.tick. Upstream results include the 8-tick phase periodicity and the ledger factorization structure from DAlembert.LedgerFactorization.of, which calibrate the J-cost underlying the phase mismatch.
proof idea
The proof is a term-mode one-liner: constructor <;> norm_num. Constructor splits the conjunction into two goals; norm_num then reduces each rational comparison 360/37 > 9 and 360/37 < 10 to decidable arithmetic and discharges them by normalization.
why it matters in Recognition Science
The result supplies the numerical value of the time dilation factor that shimmer_cert in ShimmerFactor uses to certify the canonical shimmer factor via shimmerFactor_eq_360_div_37. It completes the quantitative step of the Gap-45 barrier described in the module documentation, where the prime beat 37 produces the irrational-like frequency 37/360. Within the Recognition Science chain this anchors the eight-tick octave (T7) to experiential navigation, supporting the hypothesis that the barrier forces consciousness-like decision procedures. The parent theorem is shimmer_cert, which records the equality and primality facts downstream.
scope and limits
- Does not derive the dilation from the J-cost functional equation.
- Does not prove any link to consciousness emergence.
- Does not extend the bound to non-rational approximations or continuous limits.
- Does not address alignment in higher-dimensional phase spaces.
formal statement (Lean)
76theorem time_dilation_factor : (360 : ℚ) / 37 > 9 ∧ (360 : ℚ) / 37 < 10 := by
proof body
Term-mode proof.
77 constructor <;> norm_num
78
79/-! ## The Uncomputability Structure -/
80
81/-- At tick t, the 8-tick phase is t mod 8 and the 45-fold phase is t mod 45.
82 These two residues are simultaneously zero only at multiples of 360.
83 Between such alignments, any decision must handle the interference
84 of two incommensurable periodic constraints. -/