IndisputableMonolith.Foundation.InevitabilityEquivalence
InevitableEquivalence module states the four concrete conditions under which the inevitability theorem holds in Recognition Science: J-cost uniqueness from T5, uniqueness of the positive golden-ratio root phi, defect vanishing only at unity, and finite costs everywhere. Foundation researchers cite it when checking that abstract inevitability reduces to these verifiable properties. The module assembles results from the imported Cost, LawOfExistence, and d'Alembert gate modules into a single equivalence statement.
claimInevitableEquivalence holds precisely when $J$ is the unique solution of the Recognition Composition Law, $\phi$ is the unique positive root of $\phi=1+\phi^{-1}$, $\mathrm{defect}(x)=0\iff x=1$, and $\mathrm{cost}(x)<\infty$ for all $x$.
background
The module sits in the foundation layer where inevitability is relocated from MP postulates to cost axioms. It imports the Law of Existence stating $x$ exists iff defect$(x)=0$, the Cost module supplying the J-cost and defectDist structures, and the two d'Alembert modules that encode the four gates. Upstream, FourthGate formalizes the normalized closure $G''(t)=G(t)+1$; TriangulatedProof combines the interaction, entanglement, curvature, and d'Alembert gates into one inevitability theorem; InevitabilityStructure identifies the choke points where degrees of freedom must be eliminated.
proof idea
This is a definition and equivalence module. It lists the four conditions in its doc-comment and links them, via the five module imports, to the sibling declarations concrete_inevitability, inevitability_holds, and NoAlternatives. No internal proofs appear; the argument structure is supplied entirely by the imported gate and law modules.
why it matters in Recognition Science
The module supplies the minimal concrete assumptions that close the inevitability argument and feed the unified theorem in TriangulatedProof. It directly instantiates the choke-point analysis of InevitabilityStructure by naming the T5 J-uniqueness, T6 phi uniqueness, Law-of-Existence defect property, and finite-cost requirement. It thereby removes free parameters and connects the forcing chain T5-T8 to the final inevitability statement.
scope and limits
- Does not derive J-uniqueness from axioms below T5.
- Does not compute numerical values inside the alpha band.
- Does not extend the argument beyond the eight-tick octave or D=3.
- Does not supply counter-models when any listed condition fails.
depends on (5)
declarations in this module (15)
-
structure
ConcreteInevitability -
theorem
phi_unique_pos -
def
concrete_inevitability -
theorem
inevitability_holds -
def
NoAlternatives -
def
NoFreeParameters -
def
SingleCalibration -
theorem
loglift_contDiff_of_cost_contDiff -
theorem
concrete_implies_no_alternatives -
theorem
inevitability_chain -
theorem
noFreeParameters -
structure
ScaffoldStatus -
def
current_scaffold_status -
def
scaffolds_remaining -
theorem
summary