Foundation Jcost Monotonicity3

The monotonicity facts

The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a mismatch between two quantities. When x = 1, the two sides agree and the cost is zero. When x moves away from 1, the cost rises, slowly at first and then steeply. The function is strictly convex on positive inputs, meaning its graph curves upward everywhere, with a unique minimum at x = 1. Its derivative is negative below 1 and positive above 1, so the cost always increases as the mismatch grows in either direction.

In Recognition Science, the framework defines a ledger, a discrete record of recognition events, and the cost of recognition is forced by a proved theorem rather than chosen freely. The module JCostMonotonicity3 records three general facts about this cost when it is applied to a ratio m/e. First, when m equals e, the cost is zero. Second, for positive m and e, the cost is never negative. Third, the quantity phi - 3/2, where phi is the golden ratio, is positive. These facts are proved in the framework's machine-checked library of formal theorems.

The three facts are deliberately general. The module defines domainCost m e as Jcost (m / e) without committing to what m and e mean. That is why the docstring calls this a template: the same three lemmas apply to any subject that supplies its own definitions of m and e. The module proves nothing specific to any one domain, because no such definition appears here. What it establishes is the common backbone: zero at equality, nonnegativity on positive inputs, and a positive threshold at the golden ratio.

These properties matter because they make the cost function usable as a ledger. A cost that could go negative would reward mismatch, and a cost that could vanish away from equality would blur the distinction between agreement and error. The monotonicity facts guarantee that the ledger treats equality as the unique minimum and penalizes every departure. The positive threshold phi - 3/2 is a small numerical anchor: it says the golden ratio sits above 1.5, a fact the framework later uses in the forcing chain that derives the eight-tick cycle and three spatial dimensions.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module proves anything specific to a physical subject; it is a template awaiting domain definitions. The monotonicity facts alone force the golden ratio or any later structure. The cost function is derived here; it is taken from the framework's proved cost theorem.

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/JCostMonotonicity3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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