Cost Ndim Curvature Bridge Dot Sharp Dinv Two Sparse

The self-energy collapse

In the Recognition Science framework, a central object is a ledger, a discrete record of recognition events, and the cost of recognition is forced, not chosen. The framework works with a potential function whose Hessian, the matrix of second derivatives, defines a metric. The declaration dot_sharp_Dinv_twoSparse is a theorem about a specific sum that appears in this geometry: the self-energy of a vector, which measures how much the vector interacts with itself under the inverse of a diagonal matrix.

The theorem states that if a vector α is two-sparse, meaning it has nonzero components only on two coordinates, say i0 and i1, then the sum dot α (sharp (Dinv t) α) collapses to just two terms: (cosh (t i0))⁻¹ * α i0² + (cosh (t i1))⁻¹ * α i1². In plain language, the self-energy of a two-sparse vector depends only on its two active components and the corresponding diagonal entries of the inverse metric. All other contributions vanish because the vector has no weight there.

This is not a standalone curiosity. It is a key step in a larger proof, the curvature bridge, which shows that the deformed metric h_λ = D + λ g̃ is genuinely non-flat in any dimension n. The theorem dot_sharp_Dinv_twoSparse provides the algebraic simplification that allows the full Riemann curvature tensor, a complicated object with many indices, to be reduced to a single negative component on a two-dimensional slice. This reduction is what lets the framework prove that the metric has curvature, a fact with consequences for the structure of space itself.

It is important to be precise about what this theorem does not claim. It does not say that the self-energy is always positive or negative; it only gives a formula for its value. It does not prove that the metric is non-flat on its own; that requires the additional hypotheses of the capstone theorem RiemannMixedApply_neg, such as t i1 = 0 and α i0 ≠ 0. Finally, it does not apply to vectors that are not two-sparse; the collapse depends on the support being exactly two coordinates.

/-- `S = dot α (D⁻¹α)` restricted to a `TwoSparse` support: only the `i0, i1`
components survive. -/
theorem dot_sharp_Dinv_twoSparse {n : ℕ} (t α : Vec n) (i0 i1 : Fin n) (hne : i0 ≠ i1)
    (h2 : TwoSparse α i0 i1) :
    dot α (sharp (Dinv t) α)
      = (Real.cosh (t i0))⁻¹ * α i0 ^ 2 + (Real.cosh (t i1))⁻¹ * α i1 ^ 2 := by
  have hpt : ∀ i : Fin n, α i * sharp (Dinv t) α i = (Real.cosh (t i))⁻¹ * α i ^ 2 := by
    intro i; rw [sharp_Dinv_apply]; ring
  unfold dot
  rw [Finset.sum_congr rfl (fun i _ => hpt i)]
  exact sum_twoSparse α i0 i1 hne h2 (fun i => (Real.cosh (t i))⁻¹)

/-- **Theorem 2, general `n`.** Under the block-diagonal hypotheses plus `b ≠ 0` and
`t i0 ≠ 0`, the mixed Riemann tensor `R^{i0}_{i1,i0,i1}` of the *actual* `n`-dimensional
deformed metric `h_λ` is strictly negative: `h_λ` is genuinely non-flat, for any
ambient dimension `n` and any `α` supported on two coordinates. This is the honest
general-`n` generalization of `R0101Gen_neg` (`ScalarCertificates.lean`), assembled
from the algebraic reduction above plus the already-certified 2-D negativity. -/
theorem RiemannMixedApply_neg {n : ℕ} (α t : Vec n) (lam a b : ℝ) (i0 i1 : Fin n)
    (hne01 : i0 ≠ i1) (h2 : TwoSparse α i0 i1)
    (ha : α i0 = a) (hb : α i1 = b) (ht1 : t i1 = 0)
    (ha0 : a ≠ 0) (hb0 : b ≠ 0) (hlam : 0 < lam) (ht0 : t i0 ≠ 0) :
    RiemannMixedApply (hInvFull α t lam) (beta α t lam) i0 i1 i0 i1 < 0 := by
  rw [RiemannMixedApply_reduce α t lam a b i0 i1 hne01 h2 ha hb ht1 ha0 hlam]
  exact R0101Gen_neg a b lam (t i0) ha0 hb0 hlam ht0

What this page does not claim

The theorem does not prove the metric is non-flat by itself. The theorem does not apply to vectors that are not two-sparse. The theorem does not state the sign of the self-energy.

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

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