Gravity Ilgspatial Kernel Ilg Spatial Kernel One Statement

The kernel amplitude

In classical physics, gravity obeys the inverse-square law: the force between two masses weakens with the square of the distance between them. Some modified-gravity proposals add a small correction to this law, often expressed in Fourier space, where the source-potential relation is multiplied by a factor that depends on the spatial frequency k. The Information-Limited Gravity (ILG) framework writes this correction as w_ker(k) = 1 + C · (k₀/k)^α, where the exponent α ≈ 0.191 is already derived from the golden ratio.

The declaration ilg_spatial_kernel_one_statement establishes the value of the amplitude C in this correction. It proves, in a single machine-checked theorem, that C equals the golden ratio raised to the power -2, which is approximately 0.382. This is the closed form C = 2 - φ, where φ is the golden ratio (1.618...). The theorem also verifies that C is positive, that it lies in the numerical band between 0.380 and 0.390, and that it satisfies a specific algebraic identity called the half-rung budget.

The half-rung budget identity is the structural core. It states that the cost penalty for crossing one golden-ratio rung, written J(φ), plus the amplitude C, equals exactly one half. This identity follows from the defining property of the golden ratio, φ² = φ + 1, and it singles out C = φ⁻² as the structurally forced value. The theorem also shows that a competing value, C = φ^(-3/2) ≈ 0.486, violates this budget identity, making it inconsistent with the framework's internal logic.

In Recognition Science, the framework models reality as maintaining a discrete ledger of recognition events, where each event carries a forced cost. The amplitude C here is interpreted as a cost-saving available from finite-latency closure, complementing the crossing penalty J(φ). The machine-checked library of formal theorems proves the algebraic identities and the numerical band. The three-channel factorization, which writes C as the product of two equal channel weights, is also proved as a theorem.

What the declaration does not claim is that this value is the final word on modified gravity. The fit to the SPARC galaxy rotation-curve data, which gives A_fit = 0.38, is an empirical check, not a theorem. The theoretical prediction that the amplitude should be exactly φ⁻² is a hypothesis, and it carries a named falsifier: any future SPARC-class fit that determines the amplitude empirically and finds it inconsistent with 0.382 at better than 5% (a >3σ deviation) would falsify the prediction.

/-- **ILG SPATIAL-KERNEL AMPLITUDE: ONE-STATEMENT THEOREM.**

The spatial-kernel amplitude `C = φ⁻²` in
`w_ker(k) = 1 + C · (k_0/k)^α` is structurally forced by the half-rung
budget identity `J(φ) + C = 1/2`, has the closed form `C = 2 - φ`,
admits the three-channel factorization `C = (1/φ)·(1/φ)`, lies in
the numerical band `(0.380, 0.385)` from `φ ∈ (1.61, 1.62)`, and
matches the SPARC empirical fit `A_fit = 0.38` to better than 1%.
The competing value `C' = φ⁻³ᐟ² ≈ 0.486` violates the budget
identity and is excluded. -/
theorem ilg_spatial_kernel_one_statement :
    -- (1) Closed form
    C_kernel = 2 - phi ∧
    -- (2) Positivity
    0 < C_kernel ∧
    -- (3) Half-rung budget identity
    Jphi_penalty + C_kernel = 1 / 2 ∧
    -- (4) Numerical band
    (0.380 : ℝ) < C_kernel ∧ C_kernel < (0.390 : ℝ) ∧
    -- (5) Three-channel factorization
    C_kernel = channel_weight * channel_weight ∧
    -- (6) Competing value violates budget
    Jphi_penalty + C_kernel_competing > 1 / 2 :=
  ⟨C_kernel_eq_two_minus_phi, C_kernel_pos, half_rung_budget,
   C_kernel_band.1, C_kernel_band.2, three_channel_factorization,
   C_competing_violates_budget⟩

/-- **THE HALF-RUNG BUDGET IDENTITY.** `J(φ) + C = 1/2`, the structural
    forcing of `C = φ⁻²` as the unique spatial-kernel amplitude
    consistent with the first-rung cost penalty. -/
theorem half_rung_budget : Jphi_penalty + C_kernel = 1 / 2 := by
  rw [Jphi_penalty_eq_phi_minus_three_halves, C_kernel_eq_two_minus_phi]
  ring

/-- The competing amplitude `C'` PLUS `J(φ)` exceeds the half-rung
    budget, violating the structural identity `J(φ) + C = 1/2`. -/
theorem C_competing_violates_budget :
    Jphi_penalty + C_kernel_competing > 1 / 2 := by
  have h1 : Jphi_penalty + C_kernel = 1 / 2 := half_rung_budget
  have h2 : C_kernel < C_kernel_competing := C_competing_gt_C_kernel
  linarith

/-- **ILG SPATIAL-KERNEL AMPLITUDE: ONE-STATEMENT THEOREM.**

The spatial-kernel amplitude `C = φ⁻²` in
`w_ker(k) = 1 + C · (k_0/k)^α` is structurally forced by the half-rung
budget identity `J(φ) + C = 1/2`, has the closed form `C = 2 - φ`,
admits the three-channel factorization `C = (1/φ)·(1/φ)`, lies in
the numerical band `(0.380, 0.385)` from `φ ∈ (1.61, 1.62)`, and
matches the SPARC empirical fit `A_fit = 0.38` to better than 1%.
The competing value `C' = φ⁻³ᐟ² ≈ 0.486` violates the budget
identity and is excluded. -/
theorem ilg_spatial_kernel_one_statement :
    -- (1) Closed form
    C_kernel = 2 - phi ∧
    -- (2) Positivity
    0 < C_kernel ∧
    -- (3) Half-rung budget identity
    Jphi_penalty + C_kernel = 1 / 2 ∧
    -- (4) Numerical band
    (0.380 : ℝ) < C_kernel ∧ C_kernel < (0.390 : ℝ) ∧
    -- (5) Three-channel factorization
    C_kernel = channel_weight * channel_weight ∧
    -- (6) Competing value violates budget
    Jphi_penalty + C_kernel_competing > 1 / 2 :=
  ⟨C_kernel_eq_two_minus_phi, C_kernel_pos, half_rung_budget,
   C_kernel_band.1, C_kernel_band.2, three_channel_factorization,
   C_competing_violates_budget⟩

What this page does not claim

The declaration does not prove that the ILG modification to gravity is physically correct. The declaration does not derive the kernel exponent α from the same half-rung budget identity. The empirical agreement with SPARC is not part of the theorem; it is a separate measured check.

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

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