shimmerFactor_error_bound
The theorem bounds the shimmer factor, obtained as lcm(8,45) divided by the difference 45-8, to within 0.001 of the decimal 9.73. Workers on subjective-time scales or Gap-45 arithmetic in Recognition Science cite the bound when treating the observed ratio as a fixed numerical input. The proof rewrites the factor to its exact rational form 360/37 then invokes abs_lt and norm_num to confirm the inequality holds.
claim$|360/37 - 9.73| < 0.001$
background
The module defines gapLCM as lcm(8,45) = 360 and gapDiff as 45-8 = 37, with the shimmer factor their ratio 360/37. These arise from the eight-tick body clock and the 45-tick consciousness window whose realignment period is the least common multiple; the difference 37 is prime and coprime to 360. The local setting is pure Gap-45 arithmetic: every identity reduces to native_decide on natural-number facts or norm_num on real arithmetic, with no empirical calibration or hidden physical input. Upstream results supply the structural facts (coprime gap, primality of 37) from RecognitionBarrier; the present bound merely observes the numerical consequence once those facts are in place.
proof idea
The term proof first rewrites shimmerFactor via the already-proved equality shimmerFactor_eq_360_div_37, converting the expression to the concrete rational 360/37. It then applies abs_lt to split the absolute-value inequality into two one-sided bounds and finishes with norm_num on both sides.
why it matters in Recognition Science
The declaration supplies the concrete numerical error control required whenever the meta-claim (shimmer factor equals 360/37 with no extra physics) is invoked downstream. It closes the Gap-45 arithmetic package by confirming that the displayed precision 9.73 is accurate to three decimal places, consistent with the eight-tick octave and the coprimality already established in the barrier module. No parent theorem is listed among the used-by edges; the result therefore functions as a terminal numerical witness rather than an intermediate lemma.
scope and limits
- Does not derive the factor from any dynamical equation or forcing chain beyond the two Gap-45 primitives.
- Does not assert that 9.73 carries independent physical meaning outside the arithmetic ratio.
- Does not extend the bound to higher precision or to other gap pairs.
formal statement (Lean)
131theorem shimmerFactor_error_bound :
132 |shimmerFactor - (9.73 : ℝ)| < (1 / 1000 : ℝ) := by
proof body
Term-mode proof.
133 rw [shimmerFactor_eq_360_div_37]
134 rw [abs_lt]
135 refine ⟨?_, ?_⟩ <;> norm_num
136
137/-! ## The meta-claim as a single packaged theorem
138
139The full content of the user's statement — that the Shimmer
140subjective-time factor `~9.73` is just `lcm(8,45) / (45 - 8)` with no
141extra physics — is packaged below. Nothing in the proof uses anything
142beyond:
143
144* `native_decide` on ℕ-valued identities (lcm, subtraction);
145* `norm_num` on ℝ-valued inequalities;
146* the already-proved Gap-45 primitives (coprimality, `37` prime).
147
148The proof is deliberately short; its brevity is the content of the
149claim. -/
150
151/-- **Master statement** of the meta-claim:
152
153> The Shimmer subjective-time factor `~9.73` (i.e., `360/37`) is a
154> numerical observation from the Gap-45 arithmetic, not a named theorem.
155
156Concretely, the factor equals `lcm(8,45) / (45 - 8)`, reduces to
157`360 / 37`, is bounded by `9.72 < · < 9.73`, is irreducible because the
158denominator `37` is prime, and sits inside the Gap-45 barrier whose
159coprimality is `gcd(8, 45) = 1`. No extra input is used. -/