Foundation Rs Forcing Chain Module 004

A template for cost

The golden ratio φ, roughly 1.618, is the unique positive number that satisfies r² = r + 1. It appears in pentagon geometry, in the Fibonacci sequence, and in many natural proportions. In the Recognition Science framework, φ also marks a special value of a cost function, the number that measures the price of a recognition event.

A recognition event, in this framework, is a discrete record of something being matched or compared. The cost of that event is forced by a proved theorem, not chosen freely. The framework's central result derives the cost function J(x) = (x + 1/x)/2 - 1 from five plain conditions. At x = 1, the cost is zero: matching something to itself costs nothing. At x = φ, the cost equals φ - 3/2, a positive number approximately 0.118.

Module 004 in the forcing chain is a small, honest piece of that structure. Its machine-checked library proves three general facts about the cost function: it vanishes when the two inputs are equal, it never goes negative for positive inputs, and the threshold value φ - 3/2 is positive. These are all true for any positive numbers m and e, because the module defines its cost simply as J(m/e).

The module proves nothing specific to any subject. To make it a theorem about, say, particles or geometry, a later definition must say what m and e mean in that subject's own terms. The module is a template, a reusable scaffold, not a finished result. Its value is that it isolates what is universally true about the cost function before any particular interpretation is added.

In Recognition Science, this is how the forcing chain is meant to be read: each link proves a general fact, and the chain only becomes a theory of the physical world when later modules supply the missing definitions. Module 004 establishes the floor, the basic properties any cost must have, and leaves the subject-specific meaning open for the next module to supply.

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

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_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 does not prove that the golden ratio is the unique self-similar scaling. This module does not derive any physical constant or particle mass. This module does not establish that any particular subject exists to which the cost function applies.

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/RS_Forcing_Chain_Module_004.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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