rs_params_perturbative_proven
The declaration proves that the absolute value of the product of the RS-derived ILG exponent α and the lag coupling C_lag is strictly less than 0.1. Workers on the general-relativistic limit of the Recognition Science model cite this bound to confirm that the corrections remain perturbative. The argument invokes separate positivity and upper-bound lemmas on each factor, rewrites the absolute value, and applies a multiplication inequality followed by numerical comparison to reach 1/20 < 0.1.
claimLet $α = (1 - φ^{-1})/2$ and $C_{lag} = φ^{-5}$, where $φ$ is the golden ratio. Then $|α · C_{lag}| < 0.1$.
background
The module establishes bounds on the ILG parameters extracted from the Recognition Science spine. The exponent α arises from the geometry of the phi-ladder as (1 - 1/φ)/2. The lag constant C_lag equals φ^{-5}, matching the coherence energy scale E_coh = φ^{-5} eV. These parameters enter the general-relativistic limit through the GRLimitParameterFacts instance. Upstream results supply the positivity statement and the individual bounds α < 1/2 together with C_lag < 1/10. The module documentation notes that the product bound is required to justify treating the corrections as small.
proof idea
The proof begins by extracting positivity of both factors from the positivity lemma. It rewrites the absolute value via the nonnegativity rule applied to the product. Separate lemmas supply α < 1/2 and C_lag < 1/10. The calc block then invokes the multiplication inequality on these strict inequalities, reduces the product to 1/20 by numerical normalization, and concludes that 1/20 < 0.1 again by numerical normalization.
why it matters in Recognition Science
This result supplies the perturbative bound required by the GRLimitParameterFacts instance in the same module. It closes the claim that the RS-derived parameters remain small enough for the general-relativistic limit to be treated perturbatively, as stated in the module documentation. The bound relies on the eight-tick resonance that fixes C_lag = φ^{-5} and on the phi-derived expression for α. An open question noted in the proof comments is whether tighter estimates can reach a product below 0.02.
scope and limits
- Does not prove the product is smaller than 0.02.
- Does not derive the explicit numerical values of α or C_lag beyond the supplied bounds.
- Does not address higher-order corrections in the GR limit.
Lean usage
instance grLimitParameterFacts_proven : GRLimitParameterFacts where rs_params_perturbative := rs_params_perturbative_provenformal statement (Lean)
131theorem rs_params_perturbative_proven : |alpha_from_phi * cLag_from_phi| < 0.1 := by
proof body
Tactic-mode proof.
132 have hα_pos := rs_params_positive.1
133 have hC_pos := rs_params_positive.2
134 rw [abs_of_nonneg (mul_nonneg (le_of_lt hα_pos) (le_of_lt hC_pos))]
135 have hα_lt : alpha_from_phi < 1 / 2 := alpha_lt_half
136 have hC_lt : cLag_from_phi < 1 / 10 := cLag_lt_one_tenth
137 calc alpha_from_phi * cLag_from_phi
138 < (1 / 2) * (1 / 10) := by
139 apply mul_lt_mul'' hα_lt hC_lt (le_of_lt hα_pos) (le_of_lt hC_pos)
140 _ = 1 / 20 := by norm_num
141 _ < 0.1 := by norm_num
142
143/-- STATUS: Product < 0.02 needs tighter bounds.
144
145 PROGRESS: Proven product < 0.05 (since α < 1/2, C_lag < 1/10)
146 NEEDED: Either α < 1/5 OR C_lag < 1/11 to get product < 0.02
147
148 Current bounds:
149 - α = (1-1/φ)/2 where φ = (1+√5)/2
150 - Need to show α < 1/5 OR find tighter C_lag bound
151
152 Path forward:
153 - Prove φ < 1.62 ⟹ 1/φ > 0.617 ⟹ 1-1/φ < 0.383 ⟹ α < 0.192 < 1/5 ✓
154 - Requires proving √5 < 2.24 ⟹ φ < (1+2.24)/2 = 1.62
155 - This is doable with Mathlib's Real.sqrt inequalities
156-/