Cost Fixed Point

Cost fixed point

A cost fixed point in Recognition Science is a value of the recognition cost that reproduces itself when the cost function is applied. The cost function J(x) = (x + 1/x)/2 - 1 measures the mismatch between two recognitions. A fixed point is a number x where J(x) = x, meaning the cost of recognizing that cost is the cost itself.

The canonical lemma phi_is_cost_fixed_point establishes that the golden ratio φ is the positive solution of x = 1 + 1/x. This equation is equivalent to the fixed point condition for the cost function. The golden ratio is therefore the unique self-similar scaling in the framework: it is the only value at which the cost of recognition equals the recognition itself.

This fixed point is not incidental. The golden ratio emerges as the unique self-similar scaling forced by the cost function's axioms, and from it the framework derives the eight-tick recognition cycle, the number 2^3, and three spatial dimensions. The fixed point is the hinge on which the forcing chain turns.

THEOREM phi_is_cost_fixed_point · IndisputableMonolith/Cost/FixedPoint.lean

What this page does not claim

This answer does not claim that the golden ratio is derived from the cost function in this module, only that the fixed point equation holds. This answer does not claim that the cost fixed point is the only fixed point of J, only that φ satisfies the stated equation. This answer does not claim that the forcing chain from the fixed point to spatial dimensions is established in this module.

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

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