IndisputableMonolith.Foundation.DAlembert.Unconditional
This module rewrites the d'Alembert identity for the J-cost to demonstrate that the combiner P is computed directly from the functional equation with no extra assumptions on P. Researchers proving unconditional RCL inevitability in the T5 uniqueness setting would cite it. The structure collects lemmas from Cost.FunctionalEquation to establish range determination, nonnegativity, and forcing-chain completeness.
claimThe d'Alembert identity $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ holds unconditionally, with the combiner $P$ computed exactly by the right-hand side polynomial.
background
The module imports Cost (defining the J-cost function) and Cost.FunctionalEquation (supplying lemmas for the T5 cost uniqueness proof). It operates in the Foundation.DAlembert section, which develops the Recognition Composition Law from the functional equation satisfied by J.
J satisfies $J(x) = (x + x^{-1})/2 - 1$. The d'Alembert identity is the functional equation $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$, rewritten here to isolate how P is determined on its range. Upstream results focus on algebraic helpers that close the uniqueness argument for J.
proof idea
The module assembles lemmas including J_computes_P (showing P equals the polynomial), P_determined_on_range, J_surjective_nonneg, P_determined_nonneg, rcl_unconditional, complete_forcing_chain, and P_uniqueness. It applies algebraic reductions from the FunctionalEquation helpers to rewrite the identity and confirm the forcing chain is complete without boundary hypotheses.
why it matters in Recognition Science
This module supplies the unconditional core for RCL inevitability, feeding FullUnconditional (strongest form with both F and P forced, no assumption on P), RightAffineFromFactorization (closing Gap 4), TriangulatedProof (four gates to full inevitability), and Ultimate (reduction to three primitive requirements). It advances the T5 J-uniqueness step and the transition to T6 phi fixed point in the forcing chain.
scope and limits
- Does not assume any prior form for the combiner P.
- Does not extend beyond the nonnegativity range of J.
- Does not address the eight-tick octave or spatial dimension D=3.
- Does not incorporate Berry creation threshold or mass-ladder results.