IndisputableMonolith.Papers.DIF.CausalClosure
The module supplies the algebraic core for Gap 3, establishing a ballistic causality bound on mode refresh frequency inside the DIF causal closure analysis. Researchers deriving frequency limits from the Recognition Composition Law and J-uniqueness would cite these results. The module organizes a collection of definitions and lemmas around the phi-ladder scaling and eight-tick octave without a single central proof.
claimThe ballistic causality bound asserts that mode refresh frequency $f$ satisfies $f < J^{-1}(1)$ where $J(x) = (x + x^{-1})/2 - 1$, enforced by the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ under the phi fixed-point scaling.
background
Recognition Science derives all physics from the single functional equation whose solutions are governed by the J-cost function $J(x) = (x + x^{-1})/2 - 1$, also written as cosh(log x) - 1. The module introduces the ballistic_bound as the constraint that prevents superluminal mode refresh in the causal closure setting, together with the scale_free_causal_closure that removes explicit scale dependence. It operates inside the T5-T8 forcing chain, using the self-similar fixed point phi and the eight-tick octave period $2^3$ to fix spatial dimension D = 3.
proof idea
This is a definition module whose argument is carried by a sequence of sibling declarations. It first states the kernel_correction_positive to guarantee positivity, then derives the solar_ratio_bound and scale_free_causal_closure by direct algebraic reduction of the Recognition Composition Law, and finally packages the results into CausalClosureForced. No external lemmas beyond Mathlib are required; each step is an algebraic identity on the J-function.
why it matters in Recognition Science
The module fills Gap 3 of the DIF paper by supplying the algebraic core of the ballistic causality bound. It feeds the parent results on causal closure that close the forcing chain to D = 3 and the alpha inverse band (137.030, 137.039). The construction directly supports the mass formula on the phi-ladder and the Berry creation threshold at phi^{-1}.
scope and limits
- Does not compute explicit numerical values for the bound in SI units.
- Does not extend the bound to non-standard J-cost functions.
- Does not include empirical comparison with measured mode frequencies.
- Does not address higher-dimensional or curved-space extensions.