Encyclopedia Cost Cost Ndim Block Reduction Dinv

ARTICLE 3 claims 2 theorems 1 model

Cost Ndim Block Reduction Dinv

A simple diagonal matrix, the inverse of a metric built from hyperbolic cosines, turns out to be the key that lets a high-dimensional projector collapse exactly to a two-dimensional formula.

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The diagonal inverse

In the Recognition Science framework, the object called Dinv is the inverse of a diagonal matrix D. The matrix D has entries cosh(t_i) along its diagonal, where t is a vector of real numbers. Dinv is therefore also diagonal, with entries 1/cosh(t_i). This inverse is trivial to compute, but it is the foundation for a much larger result.

The framework's library proves a theorem about a projector built from this Dinv. A projector is a linear operator that selects a subspace. Here, the projector P_λ is built from Dinv and a vector α, with a parameter λ. The theorem states that if α is supported on only two coordinates, say i0 and i1, and if the background t has its i1-th coordinate equal to zero, then the (i0, i0) entry of this n-dimensional projector is exactly equal to a known two-dimensional closed form. This is not an approximation or an isomorphism; it is an equality of real numbers computed from the full n-dimensional definitions.

This reduction is what allows the framework to lift a key two-dimensional result to arbitrary dimension. The two-dimensional result, already proved, states that the projector is not parallel to the flat connection along a certain direction. The general-n theorem, PApply_not_parallel_gen, proves the same non-parallelism holds for any ambient dimension n, as long as the vector α is 2-sparse and the other coordinates of t are fixed. The derivative of the projector entry with respect to the i0-th coordinate is never zero, for any n, under these conditions.

In Recognition Science, this means a two-dimensional structural fact is not an artifact of low dimension. It survives embedding in a higher-dimensional space. The other n-2 coordinates can be present and fixed at arbitrary values; they do not wash out the effect. This is the algebraic content behind the phrase "the n-dimensional projector reduces to the 2D one on the 2-sparse slice."

What Dinv does not claim is broader. It does not claim that the full n-dimensional projector is block-diagonal in any global sense; the reduction holds only for the specific entry (i0, i0) and for the specific slice where t i1 = 0. It does not claim that the non-parallelism holds for arbitrary α, only for 2-sparse ones. And it does not claim that the framework has derived the physical origin of this structure; the bridge from recognition events to this specific metric remains an open question in the framework.

MODEL Dinv · IndisputableMonolith/Cost/Ndim/BlockReduction.lean
Dinv · IndisputableMonolith/Cost/Ndim/BlockReduction.lean:61 
/-- The inverse of the diagonal "undeformed" metric `D = diag(cosh(t 0), …, cosh(t
(n-1)))` on `ℝⁿ`. `D` itself is the Hessian of `∑ᵢ cosh(tᵢ)` (the `λ = 0`, un-coupled
part of the potential `Φ_λ`); `Dinv` is its (diagonal, hence trivially computable)
inverse. -/
noncomputable def Dinv {n : ℕ} (t : Vec n) : Fin n → Fin n → ℝ :=
  fun i j => if i = j then (Real.cosh (t i))⁻¹ else 0
THEOREM PApply_e_eq_P00Gen · IndisputableMonolith/Cost/Ndim/BlockReduction.lean
PApply_e_eq_P00Gen · IndisputableMonolith/Cost/Ndim/BlockReduction.lean:133 
/-- **The block-reduction identity.** For any ambient dimension `n`, any `α` supported
on two indices `i0 ≠ i1` with `α i0 ≠ 0`, any `λ ≠ 0`, and any background `t` with
`t i1 = 0`, the `(i0, i0)` entry of the abstract, `n`-dimensional projector `P_λ =
PApply lam (Dinv t) α` — applied to the indicator direction `e i0` — equals exactly the
2D closed form `P00Gen (α i0) (α i1) (t i0)` of `ScalarCertificates.lean`. This is the
algebraic content behind "the `n`-dimensional projector reduces to the 2D one on the
2-sparse slice": no approximation, no isomorphism-up-to-relabeling, an equality of
real numbers computed from the genuinely `n`-dimensional definitions. -/
theorem PApply_e_eq_P00Gen {n : ℕ} (t : Vec n) (α : Vec n) (lam : ℝ)
    (i0 i1 : Fin n) (hne : i0 ≠ i1) (h2 : TwoSparse α i0 i1)
    (ha : α i0 ≠ 0) (hlam : lam ≠ 0) (ht1 : t i1 = 0) :
    PApply lam (Dinv t) α (e i0) i0 = P00Gen (α i0) (α i1) (t i0) := by
  have hc : (0 : ℝ) < Real.cosh (t i0) := Real.cosh_pos _
  have hc' : Real.cosh (t i0) ≠ 0 := ne_of_gt hc
  have ha2 : (0 : ℝ) < α i0 ^ 2 := sq_pos_of_ne_zero ha
  have hAA : AApply lam (Dinv t) α (e i0) i0
      = lam * ((Real.cosh (t i0))⁻¹ * α i0) * α i0 := by
    show lam * sharp (Dinv t) α i0 * dot α (e i0) = _
    rw [sharp_Dinv_apply, dot_e]
  have hmu : mu lam (Dinv t) α
      = lam * ((Real.cosh (t i0))⁻¹ * α i0 ^ 2 + α i1 ^ 2) := by
    rw [mu_Dinv_twoSparse t α lam i0 i1 hne h2, ht1, Real.cosh_zero]
    norm_num
  have hmu_pos_part : (0 : ℝ) < (Real.cosh (t i0))⁻¹ * α i0 ^ 2 + α i1 ^ 2 := by
    have h1 : (0 : ℝ) < (Real.cosh (t i0))⁻¹ * α i0 ^ 2 := mul_pos (inv_pos.mpr hc) ha2
    nlinarith [sq_nonneg (α i1)]
  have hdenom_pos : (0 : ℝ) < α i0 ^ 2 + α i1 ^ 2 * Real.cosh (t i0) := by
    nlinarith [sq_nonneg (α i1), mul_nonneg (sq_nonneg (α i1)) (le_of_lt hc)]
  have hPapply : PApply lam (Dinv t) α (e i0) i0
      = (mu lam (Dinv t) α)⁻¹ * AApply lam (Dinv t) α (e i0) i0 := by
    show (mu lam (Dinv t) α)⁻¹ • AApply lam (Dinv t) α (e i0) i0 = _
    rw [smul_eq_mul]
  rw [hPapply, hAA, hmu]
  unfold P00Gen
  rw [eq_div_iff (ne_of_gt hdenom_pos)]
  field_simp
THEOREM PApply_not_parallel_gen · IndisputableMonolith/Cost/Ndim/BlockReduction.lean
PApply_not_parallel_gen · IndisputableMonolith/Cost/Ndim/BlockReduction.lean:170 
/-- **Theorem 1a, arbitrary ambient dimension `n`** (panel-greenlit general-`n`
extension). Embed a 2-sparse `α = (…, α i0, …, α i1, …, 0, …)` supported on indices
`i0 ≠ i1` inside an `n`-dimensional recognition space, and consider the slice `t` with
`t i1 = 0` (all other `n - 2` coordinates arbitrary and fixed). As the `i0`-th
coordinate varies, the `(i0, i0)` entry of the genuinely `n`-dimensional projector
`P_λ` obeys *exactly* the 2D scalar law `dP00Gen`, and — for `α i0 ≠ 0`, `α i1 ≠ 0` —
that derivative is never zero. Hence `P_λ` fails to be `D`-parallel along the `i0`
direction at every point of the slice, for every ambient dimension `n ≥ 2`, not just
`n = 2`. This is the direct general-`n` lift of `ScalarCertificates.dP00Gen_ne_zero`. -/
theorem PApply_not_parallel_gen {n : ℕ} (t : Vec n) (α : Vec n) (lam : ℝ)
    (i0 i1 : Fin n) (hne : i0 ≠ i1) (h2 : TwoSparse α i0 i1)
    (ha : α i0 ≠ 0) (hb : α i1 ≠ 0) (hlam : lam ≠ 0) (ht1 : t i1 = 0) (s : ℝ) :
    HasDerivAt (fun s' => PApply lam (Dinv (Function.update t i0 s')) α (e i0) i0)
        (dP00Gen (α i0) (α i1) s) s
      ∧ (s ≠ 0 → dP00Gen (α i0) (α i1) s ≠ 0) := by
  have hfun_eq : (fun s' => PApply lam (Dinv (Function.update t i0 s')) α (e i0) i0)
      = P00Gen (α i0) (α i1) := by
    funext s'
    have ht1' : Function.update t i0 s' i1 = 0 := by
      rw [Function.update_of_ne (Ne.symm hne)]
      exact ht1
    have hred := PApply_e_eq_P00Gen (Function.update t i0 s') α lam i0 i1 hne h2 ha hlam ht1'
    rwa [Function.update_self] at hred
  refine ⟨?_, fun hs => dP00Gen_ne_zero (α i0) (α i1) s ha hb hs⟩
  rw [hfun_eq]
  exact hasDerivAt_P00Gen (α i0) (α i1) s ha

What this page does not claim

The reduction holds only for the (i0, i0) entry on a specific slice, not for the entire n-dimensional projector as a block-diagonal operator. The non-parallelism result requires α to be supported on exactly two coordinates; it does not hold for arbitrary α. The framework has not yet derived the physical origin of this metric structure from recognition events.

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

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