Cost Ndim Block Reduction Sharp Dinv Apply

The sharp operation

The sharp operation is a standard move in differential geometry: it uses a metric to convert a covector (a row of numbers) into a vector (a column of numbers). In Recognition Science, the framework's ledger, a discrete record of events, works with a special diagonal metric D whose entries are hyperbolic cosines of the coordinates. The declaration sharp_Dinv_apply states a simple fact about the inverse of that metric: when you apply the sharp operation using the inverse, the i-th component of the result is just the i-th component of the original covector, scaled by the reciprocal of the hyperbolic cosine of the i-th coordinate. This holds for any covector, in any dimension, with no further conditions.

The fact is almost trivial to prove because the metric is diagonal: each coordinate direction is independent, so the sharp operation simply divides each component by its own metric entry. The statement's importance comes from what it enables. In the framework's cost, the forced price of recognition, a projector operator is built from this inverse metric and a covector that is supported on only two coordinates. The sharp operation's simple diagonal form is what allows the full n-dimensional sum defining the projector to collapse, term by term, into the exact two-dimensional closed form. That collapse is the algebraic core of the block-reduction theorem, which shows that a genuinely n-dimensional object behaves identically to its two-dimensional counterpart on a carefully chosen slice.

What the declaration does not claim is just as precise. It does not assert anything about the projector itself, about parallelism, or about any derivative. It is a lemma about the sharp operation alone, a single step in a longer chain. It also does not require the covector to be sparse: the identity holds for every covector, sparse or not. The two-sparse condition appears only later, in the theorems that use this lemma to collapse sums. Finally, the declaration says nothing about the physical meaning of the coordinates or the metric; it is a purely algebraic statement about real-valued functions on a finite index set.

/-- `sharp (Dinv t) α` picks out the `i`-th component of `α` scaled by `(cosh(t i))⁻¹`,
for every `i` — a direct consequence of `Dinv t` being diagonal. This holds for *any*
`α`, not just 2-sparse ones; it is the general-`n` fact underlying the whole reduction. -/
theorem sharp_Dinv_apply {n : ℕ} (t : Vec n) (α : Vec n) (i : Fin n) :
    sharp (Dinv t) α i = (Real.cosh (t i))⁻¹ * α i := by
  unfold sharp Dinv
  rw [Finset.sum_eq_single i]
  · simp
  · intro b _ hb
    have : ¬ (i = b) := fun h => hb h.symm
    simp [this]
  · intro h
    exact absurd (Finset.mem_univ i) h

/-- `sharp (Dinv t) α` picks out the `i`-th component of `α` scaled by `(cosh(t i))⁻¹`,
for every `i` — a direct consequence of `Dinv t` being diagonal. This holds for *any*
`α`, not just 2-sparse ones; it is the general-`n` fact underlying the whole reduction. -/
theorem sharp_Dinv_apply {n : ℕ} (t : Vec n) (α : Vec n) (i : Fin n) :
    sharp (Dinv t) α i = (Real.cosh (t i))⁻¹ * α i := by
  unfold sharp Dinv
  rw [Finset.sum_eq_single i]
  · simp
  · intro b _ hb
    have : ¬ (i = b) := fun h => hb h.symm
    simp [this]
  · intro h
    exact absurd (Finset.mem_univ i) h

What this page does not claim

The declaration does not establish any property of the projector P_λ itself, such as non-parallelism or a derivative condition. The declaration does not require the covector α to be supported on only two coordinates. The declaration does not assign physical meaning to the coordinates or the metric; it is a purely algebraic statement.

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/BlockReduction.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: