IndisputableMonolith.Foundation.DAlembert.Inevitability
This module defines the cost functional on positive reals as the starting object for DAlembert-style derivations of the Recognition Composition Law. Researchers tracing RCL inevitability from symmetry assumptions to the full unconditional theorem would cite it as the entry point. The module supplies definitions and basic properties that rest on convexity results imported from Cost.Convexity, with no proofs inside.
claimA cost functional $F:ℝ_+→ℝ$ on the positive reals that satisfies symmetry under inversion and normalization at the unit.
background
The module sits inside the Foundation.DAlembert namespace of Recognition Science and introduces the cost functional as the basic object whose properties force the Recognition Composition Law. It imports the Cost module and the Convexity submodule, which establishes that $Jcost(x)=½(x+x^{-1})-1$ is strictly convex on $ℝ_+$. This convexity supplies the uniqueness backbone referenced in the T5 J-uniqueness step of the forcing chain.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The definitions feed the Full Unconditional RCL Inevitability result, which shows both the cost functional and the combiner are forced with no assumption on the combiner. They also support the Right-Affine from Factorization closure and the polynomiality arguments in PolynomialityFromLogic and GeneralizedDAlembert by supplying the symmetric normalized object required for the factorization gate.
scope and limits
- Does not prove any theorem or contain sorry placeholders.
- Does not derive the explicit form of the Recognition Composition Law.
- Does not impose continuity or polynomial-degree assumptions.
- Does not extend the domain beyond positive reals.
used by (5)
depends on (2)
declarations in this module (13)
-
structure
CostFunctional -
def
IsSymmetric -
def
IsNormalized -
structure
PolynomialCombiner -
def
HasMultiplicativeConsistency -
theorem
F_div_swap_of_P_symmetric -
theorem
F_symmetric_of_P_symmetric -
theorem
symmetry_and_normalization_constrain_P -
theorem
P_symmetric_from_F_symmetric -
theorem
polynomial_form_forced -
theorem
bilinear_family_reduction -
theorem
bilinear_family_forced -
theorem
axiom_bundle_necessary