IndisputableMonolith.Foundation.DAlembert.Unconditional
The Unconditional module rewrites the d'Alembert identity for the J-cost to show that the combiner P is computed directly from J with no extra assumptions. Researchers establishing the inevitability of the Recognition Composition Law in the forcing chain would cite it as the base layer for unconditional RCL statements. The module organizes its content as a sequence of lemmas that first compute P from J, then establish determination on ranges and non-negative values before stating the full unconditional identity.
claimThe d'Alembert identity for the cost function states that $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$, which directly computes the combiner $P(u,v)$ as the polynomial $2uv + 2u + 2v$ without range restrictions or sign assumptions on $P$.
background
This module sits inside the Foundation.DAlembert layer and imports the Cost module (which defines the J-cost function and its basic properties) together with Cost.FunctionalEquation (which supplies the T5 uniqueness lemmas). The local setting is the unconditional case of the Recognition Composition Law, where J satisfies the functional equation derived from the self-similar fixed point phi without any prior hypothesis on P. Key sibling declarations inside the module translate the identity into statements that J computes P, that P is determined on its range, and that the RCL holds with no non-negativity or surjectivity restrictions.
proof idea
The module first applies direct substitution from the upstream FunctionalEquation lemmas to obtain J_computes_P, then uses algebraic range analysis to reach P_determined_on_range and P_determined_nonneg. It next assembles these into rcl_unconditional and completes the forcing chain to P_uniqueness. All steps are algebraic reductions that rely on the T5 helpers imported from Cost.FunctionalEquation.
why it matters in Recognition Science
The module supplies the unconditional base statements that are imported by FullUnconditional (which proves both F and P are forced with no assumption on P), RightAffineFromFactorization (which closes Gap 4 via factorization), TriangulatedProof (which unifies the four gates), and Ultimate (which reduces the five assumptions to three primitives). It therefore occupies the precise slot between the T5 J-uniqueness result and the strongest forms of RCL inevitability in the forcing chain.
scope and limits
- Does not impose boundary conditions or zero-boundary hypotheses on P.
- Does not derive numerical values for constants such as alpha or G.
- Does not address conditional variants that assume non-negativity of P.
- Does not extend the identity to dimensions other than the forced D=3.