Gravity Corrected Taylor Higher Cardinality Homogeneous Quadratic Is Even

The evenness lemma

A quadratic function is one where scaling the input by any factor scales the output by the square of that factor. The declaration homogeneous_quadratic_is_even proves that any such function on a real vector space is even: flipping the sign of the input leaves the output unchanged, so Q(−ξ) = Q(ξ). The proof is immediate: apply the scaling property with the factor −1, and the square of −1 is 1. This is a theorem in the machine-checked library of formal theorems, with no unproven obligations and no framework-internal assumptions.

The lemma is not an isolated curiosity. It is a necessary algebraic condition for the corrected local-Taylor gate, a finite identity that the framework uses to connect a discrete recognition ledger to continuous gravity. The gate concerns a stencil on a periodic torus, and the flat configuration must be a critical point of the Regge action, which is even in the displacement. The evenness lemma guarantees that any exactly quadratically homogeneous functional satisfies that requirement at every cardinality, without needing a finite certificate or a case check.

What the lemma does not claim is just as important. It does not establish the all-cardinality gate itself, which remains open; it only supplies a necessary condition. It does not prove that the Regge action is even, only that any quadratic homogeneous functional is. And it says nothing about the reverse direction, that evenness forces quadratic homogeneity, which is false. The lemma is a single clean step in a larger parameterized reduction, not the whole argument.

/-- Any exactly quadratically homogeneous functional on a real vector space
is *even*: `Q(-ξ) = Q(ξ)`. This is a necessary algebraic condition for the
correspondence at any cardinality, since the Regge action is even in the
displacement (the flat configuration is a critical point). The proof uses
only the homogeneity hypothesis with `a = -1` and `norm_num` — no
`native_decide`, no finite certificate. -/
theorem homogeneous_quadratic_is_even
    {V : Type*} [AddCommGroup V] [Module ℝ V]
    (Q : V → ℝ)
    (hQ : ∀ (a : ℝ) (ξ : V), Q (a • ξ) = a ^ (2 : ℕ) * Q ξ)
    (ξ : V) :
    Q (-ξ) = Q ξ := by
  have h : (-1 : ℝ) • ξ = -ξ := by
    rw [neg_smul, one_smul]
  rw [← h, hQ]
  norm_num

/-- Any exactly quadratically homogeneous functional on a real vector space
is *even*: `Q(-ξ) = Q(ξ)`. This is a necessary algebraic condition for the
correspondence at any cardinality, since the Regge action is even in the
displacement (the flat configuration is a critical point). The proof uses
only the homogeneity hypothesis with `a = -1` and `norm_num` — no
`native_decide`, no finite certificate. -/
theorem homogeneous_quadratic_is_even
    {V : Type*} [AddCommGroup V] [Module ℝ V]
    (Q : V → ℝ)
    (hQ : ∀ (a : ℝ) (ξ : V), Q (a • ξ) = a ^ (2 : ℕ) * Q ξ)
    (ξ : V) :
    Q (-ξ) = Q ξ := by
  have h : (-1 : ℝ) • ξ = -ξ := by
    rw [neg_smul, one_smul]
  rw [← h, hQ]
  norm_num

/-- Any exactly quadratically homogeneous functional on a real vector space
is *even*: `Q(-ξ) = Q(ξ)`. This is a necessary algebraic condition for the
correspondence at any cardinality, since the Regge action is even in the
displacement (the flat configuration is a critical point). The proof uses
only the homogeneity hypothesis with `a = -1` and `norm_num` — no
`native_decide`, no finite certificate. -/
theorem homogeneous_quadratic_is_even
    {V : Type*} [AddCommGroup V] [Module ℝ V]
    (Q : V → ℝ)
    (hQ : ∀ (a : ℝ) (ξ : V), Q (a • ξ) = a ^ (2 : ℕ) * Q ξ)
    (ξ : V) :
    Q (-ξ) = Q ξ := by
  have h : (-1 : ℝ) • ξ = -ξ := by
    rw [neg_smul, one_smul]
  rw [← h, hQ]
  norm_num

What this page does not claim

The all-cardinality corrected gate is proved; it remains open. The Regge action itself is proved even; only the general quadratic lemma is established. Evenness forces quadratic homogeneity; the implication runs only one way.

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/Gravity/CorrectedTaylorHigherCardinality.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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