IndisputableMonolith.Foundation.DAlembert.Unconditional
The Unconditional module rewrites the d'Alembert identity for the J-cost to demonstrate that the combiner P is computed directly from J with no assumptions on P. Researchers establishing RCL inevitability in the Recognition Science forcing chain cite it when closing the unconditional case. The module assembles algebraic rewrites and range determinations drawn from the imported Cost and FunctionalEquation helpers.
claimThe d'Alembert identity $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ is rewritten so that the combiner satisfies $P(u,v) = 2uv + 2u + 2v$ on the appropriate domain.
background
This module belongs to the DAlembert hierarchy inside Foundation and imports Cost together with Cost.FunctionalEquation. The latter supplies lemmas for the T5 cost uniqueness proof. J denotes the unique cost function $J(x) = (x + x^{-1})/2 - 1$ that satisfies the Recognition Composition Law (RCL). P is the symmetric combiner whose values are forced by the same identity.
The local theoretical setting is the unconditional segment of the T0-T8 forcing chain. No hypothesis on P is retained; the module shows the identity alone determines P on its range.
proof idea
The module collects sibling theorems (J_computes_P, P_determined_on_range, rcl_unconditional, P_uniqueness) that apply the functional-equation lemmas to the d'Alembert identity. Each theorem performs an algebraic rearrangement that isolates P(u,v) and verifies it matches the RCL polynomial; range and non-negativity statements close the determination.
why it matters in Recognition Science
The results feed directly into FullUnconditional (strongest RCL inevitability with both F and P forced), TriangulatedProof (four-gate unification), RightAffineFromFactorization (gap-4 closure), and Ultimate (minimal three-primitive statement). It supplies the unconditional core required by the complete_forcing_chain step of T5.
scope and limits
- Does not introduce new assumptions on the combiner P.
- Does not treat the conditional or factorization cases handled in sibling modules.
- Does not derive the eight-tick octave or spatial dimension D=3.