IndisputableMonolith.Foundation.DAlembert.Inevitability
The module defines the cost functional on positive reals and derives its symmetry, normalization, and multiplicative consistency properties. Researchers proving unconditional forms of the Recognition Composition Law cite it as the base layer for d'Alembert inevitability. The structure imports convexity results then chains definitions of CostFunctional, IsSymmetric, IsNormalized, and HasMultiplicativeConsistency to constrain the combiner P.
claimA cost functional $F:ℝ_+→ℝ$ satisfying $F(x)=F(1/x)$ and $F(1)=0$, together with a combiner $P$ obeying the identity $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ where $J$ denotes the J-cost.
background
The module sits inside the Foundation.DAlembert namespace and imports the Cost module (defining Jcost(x)=½(x+x⁻¹)-1) together with Cost.Convexity. The latter proves that Jlog(t)=cosh t-1 is strictly convex on ℝ and that Jcost is strictly convex on ℝ₊; these facts are foundational for the uniqueness theorem T5. The local theoretical setting introduces CostFunctional as the central object, then layers IsSymmetric, IsNormalized, PolynomialCombiner, and HasMultiplicativeConsistency to constrain the route-independent combiner P without prior assumptions on its form.
proof idea
The module opens with the CostFunctional declaration, then proves the symmetry and normalization lemmas. Subsequent sections establish polynomial_form_forced and bilinear_family_reduction, culminating in the multiplicative consistency theorem. Each step applies the imported convexity statements to eliminate non-RCL solutions for P.
why it matters in Recognition Science
This module supplies the cost-functional foundation for the Full Unconditional RCL Inevitability theorem (both F and P forced with no assumption on P) and for RightAffineFromFactorization (closing Gap 4). It also feeds GeneralizedDAlembert, LogicAsFunctionalEquation, and PolynomialityFromLogic. It advances the T5 J-uniqueness step and the RCL identity inside the T0-T8 forcing chain.
scope and limits
- Does not assume continuity or differentiability beyond the imported convexity.
- Does not treat negative reals or complex arguments.
- Does not derive the numerical value of α or G.
- Does not address the eight-tick octave or D=3 directly.
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