Foundation Pair Kernel Production Action Construction S8 Global Torus Graph3 Not

The global torus is not the production graph

In graph theory, a torus graph connects every site to every other site, often with a uniform weight. The declaration globalTorusGraph3_not_recognitionProductionGraph3 is a theorem in the framework's machine-checked library of formal theorems. It proves that this fully connected torus, on a three-dimensional lattice of size three, is not equal to the graph that the framework constructs from its primitive posting events. The proof is direct: it assumes the two graphs are equal, derives a contradiction from the fact that the global torus violates the production action identity, and thereby establishes the inequality.

The framework's production graph is built from a finite set of events. Each event is an ordered pair of sites where a minimum-J posting occurs. The graph assigns weight one to a present event and zero to an absent one. The theorem recognitionConstruction_productionActionIdentity shows that this graph satisfies the identity ProductionActionEqualsPositiveRealizedPostingAction3, meaning its exact action equals the sum of the event J-cost contributions. The global torus, by contrast, violates this identity. The theorem globalTorusGraph3_not_recognitionProductionGraph3 states the inequality as a formal consequence.

The declaration does not claim that the global torus is physically impossible or that it fails some external test. It establishes a structural fact within the framework: the fully connected graph cannot be the graph generated by the minimum-cost posting events. The theorem relies on the production action identity and the positive realized posting weights property. It does not say anything about the global torus being a bad model; it only says it is not the object the framework's constructor produces.

This result sharpens the framework's picture of production. The production graph is not an arbitrary choice; it is forced by the event batch and the cost function. The global torus, with its uniform connections, is a different object. The theorem confirms that the framework's graph is not trivially the complete graph, and that the distinction between realized postings and all possible connections is real and provable.

theorem globalTorusGraph3_not_recognitionProductionGraph3 :
    globalTorusGraph3 3 ≠ recognitionProductionGraph3 3 := by
  intro h
  have hid :=
    recognitionConstruction_productionActionIdentity
      (N := 3) (by norm_num)
  rw [← h] at hid
  exact
    globalTorusGraph3_violates_positiveRealizedProductionActionIdentity hid

/-- The live S7 obligation is discharged for the graph built by the
Recognition posting-event constructor. -/
theorem recognitionConstruction_productionActionIdentity
    {N : ℕ} [NeZero N] (hN : 2 ≤ N) :
    ProductionActionEqualsPositiveRealizedPostingAction3
      (recognitionProductionGraph3 N) := by
  let hw :=
    recognitionProductionGraph3_positiveRealizedWeights hN
  refine
    ⟨(recognitionProductionGraph3 N).weight, hw, ?_⟩
  intro ε
  unfold exactJCostAction graphOfPositiveRealizedPostingWeights3
  rfl

theorem globalTorusGraph3_not_recognitionProductionGraph3 :
    globalTorusGraph3 3 ≠ recognitionProductionGraph3 3 := by
  intro h
  have hid :=
    recognitionConstruction_productionActionIdentity
      (N := 3) (by norm_num)
  rw [← h] at hid
  exact
    globalTorusGraph3_violates_positiveRealizedProductionActionIdentity hid

What this page does not claim

The global torus graph is physically impossible or invalid in all contexts. The theorem depends on the specific size three; it is a statement about that particular lattice. The production graph is the only graph that satisfies the production action identity.

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/PairKernelProductionActionConstructionS8.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: