IndisputableMonolith.Skeleton.Cost
Skeleton packaging for the Recognition cost functional and its algebraic forcing gates. It surfaces the T5 uniqueness helpers for the unique J-cost together with the d'Alembert factorization gate that forces the combiner to be affine in its second argument. Anyone tracing how J enters the skeleton layer lands here. The module is organizational: imports and re-export surface, not a new theorem.
claimSkeleton surface for the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) and the factorization-plus-associativity gate that forces the Recognition combiner to respond affinely in its second argument, as used by the T5 and B2 forcing steps.
background
Recognition Science forces a unique nonnegative cost on positive reals from a single functional equation (the Recognition Composition Law). The T5 landmark identifies that cost as $J(x)=(x+x^{-1})/2-1$, also written $\cosh(\log x)-1$. The imported FunctionalEquation module supplies the lemma pack used in that uniqueness argument.
Separately, the B2 closure program needs an algebraic gate: factorization together with three-way compatibility implies the combiner is affine in its second argument. Once affinity is in hand, the remaining forcing is pure algebra. That gate lives in FactorizationForcing and is the hard analytic step quoted in the paper's B2 writeup.
Skeleton.Cost sits in the Skeleton domain and wires these two cost-side foundations into one import surface for downstream skeleton developments.
proof idea
This is a skeleton module, not a theorem module. Its content is the import graph: Cost.FunctionalEquation (T5 cost-uniqueness helpers) and Foundation.DAlembert.FactorizationForcing (factorization and associativity gate for B2). No local proof obligations are discharged here; argument structure lives in the imported modules.
why it matters in Recognition Science
Gives the skeleton layer a single entry point to the cost functional that T5 uniquely forces and to the affine-combiner gate that B2 needs after factorization. Without this surface, skeleton developments would reach into Cost and Foundation.DAlembert directly. Downstream used-by edges are empty at present, so the module is infrastructure rather than a cited lemma. It touches the T5 J-uniqueness landmark and the B2 algebraic core of the forcing chain, keeping the Recognition Composition Law and the passage to an affine combiner visible at the skeleton boundary.
scope and limits
- Does not prove T5 J-uniqueness; that lives in the FunctionalEquation import chain.
- Does not prove the affine-combiner forcing; that is in FactorizationForcing.
- Does not introduce a new cost functional or alter the RCL.
- Does not record downstream skeleton theorems; used-by is currently empty.
- Does not address mass ladder, alpha band, or dimensional forcing (T6–T8).