Foundation Multi Axis Robustness Multi Axis Robustness

The dimension route

In the Recognition Science framework, the physical world is described by maintaining a discrete record of recognition events, and the cost of that record is forced by a proved theorem. A separate route to three spatial dimensions starts with a recognized object of dimension p, and asks what dimension the surrounding substrate must have. The framework's machine-checked library of formal theorems proves that the substrate dimension is given by the formula 2p + 1. For p = 1, this gives 3.

The central declaration, multi_axis_robustness, bundles two proved facts. First, if the recognized-object dimension p is at least 1 and is not 1, then the substrate dimension 2p + 1 is not 3. Second, when p is exactly 1, the substrate dimension is exactly 3. Together these say that along this axis, called axis P, the value p = 1 is the only one that produces three dimensions. The proof is a direct computation from the definition, and it is checked by the machine.

The declaration also records a compatibility result: the codimension route, once p = 1, agrees with the framework's existing forced dimension result. This is a consistency statement between two routes to the same conclusion, not a new independent derivation.

What the declaration does not claim is as important as what it proves. The source that introduced this route also names three other axes of stability: the coefficient ring, the tracked invariant, and acyclicity. Those are described in that source as stability claims under perturbation, but they are not formalized here. Earlier stub declarations that simply asserted these as true by definition were removed, because a statement that cannot fail is not a real theorem. They remain open targets, to be reopened only when a richer mathematical substrate can support a claim that could actually be false.

/-- The codimension formula for a recognized object of dimension `p`. -/
def CodimensionDimension (p : ℕ) : ℕ := 2 * p + 1

/-- The `p = 1` codimension case is `D = 3`. -/
theorem p_one_gives_D3 :
    CodimensionDimension 1 = 3 := by
  rfl

/-- If `p ≥ 1` and `p ≠ 1`, the codimension dimension `2p+1` is not `3`. -/
theorem axis_P_moves_D (p : ℕ) (_hp : 1 ≤ p) (hne : p ≠ 1) :
    CodimensionDimension p ≠ 3 := by
  unfold CodimensionDimension
  omega

/-- Compatibility with the existing dimension forcing result: once `p = 1`,
the codimension route agrees with the existing forced dimension. -/
theorem p_one_route_agrees_with_dimension_forced :
    ∃! D : DimensionForcing.Dimension,
      D = CodimensionDimension 1 ∧ DimensionForcing.RSCompatibleDimension D := by
  refine ⟨3, ?_, ?_⟩
  · constructor
    · rfl
    · exact DimensionForcing.D3_compatible
  · intro D hD
    exact hD.1.trans (by rfl)

What this page does not claim

The declaration does not claim that axes C, I, and A are proved; they are named stability obligations of the source. It does not claim that the formula 2p + 1 applies when p is less than 1. It does not claim that the compatibility result is a new independent derivation of three dimensions.

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/MultiAxisRobustness.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: