Gravity Ledger Superposition Cost Gradient Functoriality Inhabited

The linear extension theorem

In linear algebra, a common construction takes a function defined on a set of basis vectors and extends it to a linear map on the whole vector space. The extension is unique: any two linear maps that agree on the basis must agree everywhere. The Recognition Science declaration costGradientFunctoriality_inhabited establishes that this construction is available in a specific setting, where the basis vectors represent definite configurations of a discrete ledger, a record of events that updates in steps.

The setting uses two index types, ι for density configurations and κ for gravity configurations. A classical map g sends each basis element of the density space to a vector in the gravity space. The theorem states that g extends uniquely to a ℂ-linear operator between the free complex vector spaces generated by the basis sets. The declaration costGradientFunctoriality_inhabited proves that the structure CostGradientFunctoriality, which packages this extension with its basis agreement and uniqueness properties, is nonempty. In plain terms, the linear extension always exists and is forced to be the one that agrees with the classical map on basis elements.

The physical interpretation, tagged MODEL, identifies this linear extension with the cost-gradient response in an extended ledger update that includes a gravitational channel. The claim is that this response must be the unique linear extension, not a nonlinear classical readout. The mathematical theorem itself is unconditional: it does not depend on any physical assumptions about gravity or recognition. It is a fact about free modules and linear maps.

What the declaration does not claim is equally precise. It does not assert that the physical interpretation is correct, nor that the cost-gradient response in nature is linear. It only establishes the mathematical fact of unique linear extension. The theorem also does not specify what the classical map g is; it works for any such map. The existence of the extension is guaranteed, but its physical relevance is a separate modeling choice.

theorem costGradientFunctoriality_inhabited (ι κ : Type) [DecidableEq ι] :
    Nonempty (CostGradientFunctoriality ι κ) :=
  ⟨costGradientFunctoriality ι κ⟩

/-- **T2 (uniqueness).** Any two `ℂ`-linear maps that agree on the
finsupp basis agree everywhere. This is the uniqueness half of the
universal property. -/
theorem costGradient_linear_unique {ι κ : Type*} [DecidableEq ι]
    (f₁ f₂ : (ι →₀ ℂ) →ₗ[ℂ] (κ →₀ ℂ))
    (h : ∀ α : ι, f₁ (Finsupp.single α (1 : ℂ)) = f₂ (Finsupp.single α (1 : ℂ))) :
    f₁ = f₂ := by
  apply Finsupp.lhom_ext'
  intro α
  apply LinearMap.ext
  intro c
  have h1 : Finsupp.single α c = c • Finsupp.single α (1 : ℂ) := by
    ext β
    by_cases hβ : β = α <;> simp [hβ]
  have hα := h α
  have step :
      f₁ (Finsupp.single α c) = f₂ (Finsupp.single α c) := by
    rw [h1, f₁.map_smul, f₂.map_smul, hα]
  show (f₁.comp (Finsupp.lsingle (R := ℂ) α)) c
        = (f₂.comp (Finsupp.lsingle (R := ℂ) α)) c
  simp [Finsupp.lsingle, step]

/-- **T2 (universal property of free linear extension).** Given any
classical map `g : ι → (κ →₀ ℂ)` from definite densities to definite
gravity configurations (or `ℂ`-combinations thereof), there exists a
canonical `ℂ`-linear map `ĝ : (ι →₀ ℂ) →ₗ[ℂ] (κ →₀ ℂ)` extending `g` in
the sense that `ĝ |α⟩ = g α` for every basis element `α : ι`. -/
def costGradientLinearExtension {ι κ : Type*} [DecidableEq ι]
    (g : ι → (κ →₀ ℂ)) : (ι →₀ ℂ) →ₗ[ℂ] (κ →₀ ℂ) :=
  Finsupp.lift (κ →₀ ℂ) ℂ ι g

/-- The canonical inhabitant of `CostGradientFunctoriality`. -/
def costGradientFunctoriality (ι κ : Type) [DecidableEq ι] :
    CostGradientFunctoriality ι κ where
  extend := costGradientLinearExtension
  basis_agreement := costGradient_linear_basis
  unique_on_basis := costGradient_linear_unique

What this page does not claim

The physical interpretation of the linear extension as the cost-gradient response is not asserted as a theorem. The theorem does not specify the classical map g or its physical meaning. The declaration does not prove that the cost-gradient response in nature is linear.

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/LedgerSuperposition.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

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