Foundation Forcing Chain Completeness3

The completeness certificate

The file ForcingChainCompleteness3.lean is a certificate, not a derivation. It packages three proved facts about a cost function, a measure of how expensive it is for a system to recognize one value as another. The three facts are: the cost is zero when the two values are equal, the cost is never negative when both values are positive, and a certain threshold number is positive. The certificate bundles these three facts into a single structure, and then proves that this structure exists.

The cost function itself is defined as J(m/e), where J is a fixed mathematical function and m and e are real numbers. The three theorems are general facts about this definition. They hold for any positive inputs, with no further conditions. The file proves them cleanly, but it proves nothing about what m and e mean. The docstring inside the file says so plainly: it records where the idea was meant to go, not a result.

The name of the file suggests a much larger ambition. The research note attached to it sketches a chain of steps T0 through T8, from the uniqueness of J up to a cosmological constant. That chain is the subject of other files in the framework. This file does not contain that chain. It contains only the three general facts, and a certificate that bundles them.

What the certificate is good for is a foundation check. Before building a long chain of derivations, you want a compact object that asserts the basic properties your chain will rely on. This file provides that object. It is a small, solid base, not a tower.

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 derive any physical constants. This module does not prove the T0 through T8 forcing chain. The research note in the docstring is not a proved result.

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/ForcingChainCompleteness3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: