Foundation Multi Axis Robustness

Axis P and the dimension route

In the Recognition Science framework, the dimension of physical space is not assumed; it is derived from a discrete record of recognition events called a ledger. The framework's central result forces a specific cost function, and from that cost a chain of theorems forces a particular scaling and structure. The module under discussion here, MultiAxisRobustness, examines one part of that chain: the step where a recognized object's dimension determines the dimension of the substrate, the space it sits in.

The classical mathematics here is a simple codimension formula. If a recognized object has dimension p, then the substrate dimension is 2p + 1. This is a definition, not a theorem: the framework chooses this relationship. The theorems come next. The first theorem, proved in the machine-checked library, states that if p is at least 1 and p is not 1, then 2p + 1 is not 3. The second theorem states that when p equals 1, the formula gives exactly 3. Together they form the bundled robustness theorem: only p = 1 yields three dimensions.

Why call this robustness? Because it shows the dimension result is stable against changes in one specific axis, Axis P, the recognized-object dimension. If the framework had allowed p to vary, the dimension would vary with it, and only the single value p = 1 survives the filter. The theorem also checks compatibility with the existing dimension forcing result: once p = 1, the codimension route agrees with the previously forced dimension, so the two paths to three dimensions do not conflict.

In Recognition Science, the framework's library proves this cleanly. The other axes named in the source, Axis C for coefficient ring, Axis I for tracked invariant, and Axis A for acyclicity, are not claimed as theorems here. The docstring states plainly that they remain stability obligations until a nontrivial substrate model exists. Former stubs that asserted these trivially were deleted rather than kept as empty claims.

What this establishes in plain language: the three-dimensional result in the framework does not depend on a fragile choice. It is forced by the arithmetic of the codimension formula, and the only way to get three dimensions is to set the recognized-object dimension to 1. A reader can now see that the framework's claim about three dimensions has a specific, checkable backbone, and that the framework itself knows where its proof stops.

/-- 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

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

/-- Bundled multi-axis robustness theorem (Axis P content).

Only Axis P can move the dimension away from `3`. Axes C, I, and A remain
named stability obligations of the paper; they are not claimed as theorems
here until a non-trivial substrate model exists. -/
theorem multi_axis_robustness :
    (∀ p : ℕ, 1 ≤ p → p ≠ 1 → CodimensionDimension p ≠ 3) ∧
      CodimensionDimension 1 = 3 :=
  ⟨axis_P_moves_D, p_one_gives_D3⟩

/-- 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

This answer does not claim that Axes C, I, or A are proved in the library; they are named obligations only. This answer does not claim the codimension formula is derived from first principles; it is a definitional choice. This answer does not claim the framework's three-dimensional result is physically verified; it is a formal theorem within the framework.

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

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