Cost Ndim Curvature Bridge Riemann Mixed Apply Neg

Non-flatness in any dimension

In differential geometry, a space is flat when it can be stretched or bent into a Euclidean shape without changing distances measured within it. The Riemann tensor is the standard mathematical tool that detects curvature: if every component of this tensor is zero, the space is flat; if any component is negative, the space is genuinely curved at that point. The declaration RiemannMixedApply_neg in the Recognition Science library proves that a particular family of spaces, built from a specific formula, is always non-flat in this sense.

The spaces in question are defined by a potential function that mixes ordinary coordinates with a special coupling term. The library constructs the metric (the rule for measuring distances) from this potential, computes its inverse, and then applies a standard formula from Hirohiko Shima's theory of Hessian metrics to obtain the Riemann tensor. The key result is that under a handful of plain conditions, one specific component of this tensor, written in symbols as R^{i0}_{i1,i0,i1}, is strictly less than zero. This component being negative means the space curves in a way that cannot be flattened away.

The proof works for any ambient dimension n. Earlier work had shown the result only for a two-dimensional slice; this declaration extends it to arbitrary dimension by showing that the full n-dimensional construction, when restricted to a pair of coordinates, collapses algebraically to the already-proven two-dimensional formula. The conditions are: the coupling vector has exactly two nonzero entries (called a "two-sparse" support), one of the two coordinates is set to zero, the coupling strength is positive, and the other coordinate is not zero. Under these, the negativity is forced, not observed.

In Recognition Science, this result is a step toward showing that the framework's core cost function generates spaces with genuine geometric structure rather than trivial flat ones. The library's broader project is to derive physical constants from a single cost function; this theorem contributes by showing that the geometry arising from that cost is not degenerate. For a reader outside the framework, the takeaway is simpler: a machine-checked proof that a certain family of spaces is curved, in any number of dimensions, under explicit and checkable conditions.

/-- **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

/-- **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

/-- Shima's formula for the doubly-lowered Riemann tensor of a Hessian metric:
`R_{ijkl} = (1/4) Σ_{p,q} h^{pq}(β_{jkp}β_{ilq} - β_{ikp}β_{jlq})`. -/
def RiemannLowerApply {n : ℕ} (ginv : Fin n → Fin n → ℝ) (b : Fin n → Fin n → Fin n → ℝ)
    (i j k l : Fin n) : ℝ :=
  (1 / 4) * ∑ p : Fin n, ∑ q : Fin n, ginv p q * (b j k p * b i l q - b i k p * b j l q)

What this page does not claim

The declaration does not claim the entire Riemann tensor is nonzero, only the single mixed component R^{i0}_{i1,i0,i1}. It does not claim the space is non-flat for all possible coupling vectors, only for those with two nonzero entries. It does not claim any physical interpretation of the curvature; that remains a separate question.

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)

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.

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