IndisputableMonolith.Foundation.DAlembert.Unconditional
The Unconditional module rewrites the d'Alembert identity for the J-cost function to compute the combiner P directly from the functional equation. Researchers tracing the T5 uniqueness result to full RCL inevitability cite it when removing all hypotheses on P. The module applies algebraic rearrangement using lemmas imported from Cost.FunctionalEquation.
claimThe d'Alembert identity $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ is recast so that the combiner satisfies $P(u,v) = 2uv + 2u + 2v$ with no prior assumption on $P$.
background
This module belongs to the Foundation.DAlembert hierarchy and imports the Cost module (defining the J-cost) together with Cost.FunctionalEquation (supplying T5 helpers). The J-cost is given by $J(x) = (x + x^{-1})/2 - 1$, also written as cosh(log x) - 1. The Recognition Composition Law is the functional equation $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. The upstream Cost.FunctionalEquation module states that it provides lemmas for the T5 cost uniqueness proof.
proof idea
The module assembles sibling lemmas including J_computes_P, P_determined_on_range, J_surjective_nonneg, P_determined_nonneg, rcl_unconditional, complete_forcing_chain and P_uniqueness. Each applies algebraic identities from the imported functional-equation helpers to eliminate assumptions on P and obtain direct computation of the combiner.
why it matters in Recognition Science
The module supplies the unconditional d'Alembert form that feeds FullUnconditional (strongest RCL inevitability with both F and P forced and no assumption on P), RightAffineFromFactorization (closing Gap 4 via factorization), TriangulatedProof (unified four-gate inevitability) and Ultimate (reduction of five assumptions to three primitives). It advances the T5-to-T8 forcing chain by removing the last hypothesis on the combiner.
scope and limits
- Does not assume any functional form for the combiner P.
- Does not address spatial dimensions or the eight-tick octave.
- Does not treat the alpha band or mass ladder.
- Does not extend beyond the J-cost functional equation.