Foundation Rs Forcing Chain Module 003

A shared template

In Recognition Science, a ledger, a discrete record of recognition events, assigns a cost to comparing two quantities. The framework's central result forces a specific cost function, J(x) = (x + 1/x)/2 - 1, from five plain conditions. Module 003 in the forcing chain defines a domain cost, a cost for comparing a measured quantity m with a reference e, as J(m/e). It then proves three general facts about that cost: it is zero when m equals e, it is never negative for positive inputs, and a threshold value, phi minus 3/2, is positive.

Those three facts are all the module proves, and its own documentation is explicit that they are generic. The threshold phi minus 3/2 is meant to mark a minimum depth for self-reference, and the number 45 appears in a research note as the product 9 times 5 at dimension 3. But the module defines the cost without any reference to what m and e actually are. It proves nothing about self-reference, depth, or dimension. The note records where the idea was meant to go, not a result that the machine-checked code establishes.

This body is shared verbatim with 2383 sibling modules. The content is stated once, universally quantified, in a central template called DomainCostTemplate. What would make module 003 a statement about its subject is a definition of m and e in that subject's own terms. Without that, the module is a placeholder: a correct but empty shell, awaiting a subject-specific definition to give it content.

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 nothing about self-reference, depth, or three-dimensional space. The threshold phi minus 3/2 is not established as a meaningful physical constant by 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/Foundation/RS_Forcing_Chain_Module_003.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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