Foundation Multi Axis Robustness Axis P Selects D

The dimension selector

In geometry, the dimension of a space is often a starting assumption, not a result. This framework treats dimension as something that must be derived from a more basic structure: a discrete record of recognition events, called the ledger. The question is what space the ledger must live in, and the answer turns out to be a piece of arithmetic.

The rule is stark. If a recognized object has dimension p, then the space around it has dimension 2p + 1. This is the codimension formula, a term for how many extra dimensions the surrounding space has beyond the object itself. The theorem axis_P_selects_D states this formula exactly: for any p at least 1, the substrate dimension is 2p + 1. The proof is a direct computation, a reflexivity check in the machine-checked library of formal theorems.

The consequence lands when p equals 1. A one-dimensional recognized object forces a three-dimensional substrate, since 2 times 1 plus 1 equals 3. No other value of p gives 3. If p is 2, the substrate is 5-dimensional; if p is 3, it is 7-dimensional. The theorem axis_P_moves_D makes this exclusion precise: whenever p is at least 1 and not equal to 1, the resulting dimension is not 3. The bundled theorem multi_axis_robustness packages both facts: only p = 1 yields three dimensions, and p = 1 does yield three dimensions.

This is the Axis P content of a broader robustness claim. The framework's paper names three other axes: the coefficient ring, the tracked invariant, and acyclicity. Those are stability obligations, not proved theorems here. Their former formal statements were trivial placeholders, each equivalent to the proposition True, and they have been removed from the library rather than kept as empty claims. The framework does not assert them until a real substrate model can support a statement that can actually fail.

What this means for a reader is a precise division of labor. The dimension-selection theorem is fully proved, and it agrees with the framework's earlier forced-dimension result once p is set to 1. The other axes remain open targets for future formalization. The arithmetic is settled; the surrounding stability story is not.

/-- Axis P is dimension-selecting: for recognized-object dimension `p`, the
codimension formula gives substrate dimension `2p + 1`. -/
theorem axis_P_selects_D (p : ℕ) (_hp : 1 ≤ p) :
    CodimensionDimension p = 2 * p + 1 := rfl

/-- 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

/-- 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⟩

What this page does not claim

The other axes C, I, and A are not proved here; they are named stability obligations only. The framework does not claim that a recognized object of dimension p must exist for any p; the formula is conditional on such an object. No claim is made that this arithmetic rule applies outside the framework's ledger model.

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)

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Derived articles

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