Foundation Cpt Theorem3 From Jcost

Three cost facts

In physics, the CPT theorem states that nature's laws stay the same when you flip charge, mirror space, and run time backwards. Recognition Science models each of those reversals as a transformation on a recognition event, a discrete record of a comparison. The framework's forced cost function, J(x) = (x + 1/x)/2 - 1, yields three general facts when applied to a ratio of two quantities.

The first fact is a zero at equality. If the two quantities m and e are the same, their ratio is 1, and the cost J(1) is exactly 0. The second is nonnegativity: for any two positive quantities, the cost is never negative; it is always zero or above. The third fact is a positive threshold: the golden ratio phi minus 1.5 is greater than zero, which the framework reads as a canonical separation between scales. These three facts hold for any positive real inputs, with no further assumptions.

What the framework does not do is claim these facts apply to any specific physical system. The definition of the cost here is generic: it is J applied to the ratio m/e, where m and e are just real numbers. The framework's own docstring states this plainly. To make it a theorem about, say, particle masses or charges, one would need to define m and e in those terms. The three facts are a template, proved once and shared across many modules, not a derivation of CPT symmetry itself.

The value of the framework is that it isolates what the cost function guarantees on its own. Before any physical interpretation, the cost has these three structural properties. That is the foundation on which later, subject-specific claims would have to build.

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]

What this page does not claim

This framework does not prove CPT invariance for any physical system. It does not define m and e in terms of charge, parity, or time. It does not derive the golden ratio; it only proves a positivity statement about it.

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/CPT_Theorem3FromJCost.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: