IndisputableMonolith.Foundation.DAlembert.Ultimate
This module establishes the ultimate inevitability of the Recognition Composition Law by defining symmetry, normalization, and multiplicative consistency on the cost function F and chaining them to the unconditional RCL result. Researchers deriving T5 J-uniqueness in the forcing chain cite it. The structure defines the three properties then proves their essentiality and the final RCL inevitability without assumptions on P.
claimIsSymmetricComparison$(F) \coloneqq \forall x, F(x)=F(1/x)$; IsNormalizedCost and HasMultiplicativeConsistency are the companion properties; these three together imply the RCL holds unconditionally.
background
The module sits inside the DAlembert development of cost functions and imports Cost, Cost.FunctionalEquation (lemmas for T5 cost uniqueness), and DAlembert.Unconditional. The Unconditional module states: "if F is determined (by symmetry, normalization, calibration, smoothness), then P is COMPUTED from the functional equation, not assumed." Key definitions introduced are IsSymmetricComparison (the supplied doc-comment: "Symmetry: F(x) = F(1/x). This is the DEFINITION of symmetric comparison"), IsNormalizedCost, and HasMultiplicativeConsistency.
proof idea
The module opens with the three property definitions, then proves the essentiality lemmas (symmetry_is_essential, normalization_is_essential, consistency_defines_composition) followed by ultimate_inevitability and rcl_is_inevitable. Each essentiality step feeds the unconditional RCL theorem imported from DAlembert.Unconditional.
why it matters in Recognition Science
It supplies the final link showing RCL is forced once the three properties hold, directly supporting T5 J-uniqueness (J(x)=(x+x^{-1})/2-1) inside the UnifiedForcingChain. The module completes the "strongest possible form of RCL inevitability" described in the Unconditional upstream doc-comment and removes any residual hypothesis on P.
scope and limits
- Does not assume any particular form for the measure P.
- Does not include calibration or smoothness axioms.
- Does not derive the explicit J form; only the RCL step.
- Confined to the DAlembert cost setting inside Foundation.