Foundation Dalembert Full Unconditional

Full unconditional inevitability

Full unconditional inevitability is the strongest form of a central Recognition Science result. It states that a single function F on positive real numbers, together with a composition rule P that combines two costs into one, are both uniquely forced by five conditions. The conditions are normalization, reciprocal symmetry, smoothness, calibration, and a multiplicative consistency law. No assumption is made about the form of P in advance. The theorem proves that F must be the reciprocal cost J(x) = (x + 1/x)/2 - 1, and P must be the polynomial 2uv + 2u + 2v on nonnegative inputs.

The result is established in the Lean 4 module FullUnconditional.lean. The key innovation is that P is not assumed to be a polynomial, nor is F assumed to equal J. Earlier versions required one of those assumptions. This version shows that the consistency equation F(xy) + F(x/y) = P(F(x), F(y)) alone, with the other four conditions, forces everything. The proof proceeds by showing P must be symmetric, then that the equation forces a function G(t) = F(exp(t)) to satisfy a differential equation. Uniqueness of the solution to that equation forces G(t) = cosh(t) - 1, hence F = J. Once F is known, P is computed directly.

The theorem is stated as full_unconditional_inevitability in the module. It combines two sub-results: the d'Alembert equation forces the hyperbolic cosine, and consistency forces the RCL form of P. Both are established with no additional assumptions on P. The result is a milestone in the forcing chain that derives the golden ratio, the eight-tick cycle, and three spatial dimensions from the single starting point of a forced recognition cost.

THEOREM full_unconditional_inevitability · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean

THEOREM full_unconditional_inevitability · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean

THEOREM P_symmetric_of_F_symmetric · log_consistency_of_mult_consistency · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean

THEOREM dAlembert_forces_cosh_is_theorem · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean

What this page does not claim

This answer does not claim that the full unconditional theorem is the only route to J in the library. This answer does not claim that the theorem derives any physical constants or dimensions directly. This answer does not claim that the module proves the full forcing chain beyond the cost and composition rule.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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