Foundation Dalembert Wlogalpha One

The Alpha-One Reduction

Foundation d'Alembert wlogalpha one is a established result in Recognition Science. It shows that the family of cost functions that survive calibration is not a family at all: every member is the canonical reciprocal cost J(x) = (x + x⁻¹)/2 − 1, viewed through a rescaling of the multiplicative coordinate. The parameter α, which appears in the calibrated family F_α(x) = (1/α²)(cosh(α ln x) − 1), is a coordinate choice, not a new physical possibility.

The module proves the rescaling identity F_α(x) = (1/α²) · J(x^α) for positive x. The map x ↦ x^α is a group automorphism of the positive reals under multiplication: it preserves products, the identity, and inverses. Setting α = 1 recovers J exactly. The unit-curvature condition G_α''(0) = 1, which fixes the calibration, holds for every nonzero α. These four facts together give the theorem: every calibrated cost is the canonical cost under coordinate rescaling.

The consequence is structural. The five axioms that force J also force that no additional parameter survives the calibration step. The α = 1 reduction is what lets the forcing chain proceed from the unique cost function to the golden ratio, the eight-tick cycle, and the rest of Recognition Science without carrying a free parameter forward.

THEOREM wlog_alpha_eq_one · IndisputableMonolith/Foundation/DAlembert/WLOGAlphaOne.lean

THEOREM cost_alpha_rescaling · IndisputableMonolith/Foundation/DAlembert/WLOGAlphaOne.lean

THEOREM cost_alpha_one_eq_jcost · IndisputableMonolith/Foundation/DAlembert/WLOGAlphaOne.lean

THEOREM costAlphaLog_unit_curvature · IndisputableMonolith/Foundation/DAlembert/WLOGAlphaOne.lean

What this page does not claim

This module does not derive the value of α from first principles. This module does not prove that the canonical cost J is unique without the five axioms. This module does not address the physical recognition-to-linking bridge.

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/WLOGAlphaOne.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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