Gravity Ilgspatial Kernel C Competing Violates Budget

The budget test

The Information-Limited Gravity (ILG) framework modifies the standard Newton-Poisson relation in Fourier space with a kernel term: `w_ker(k) = 1 + C · (k_0 / k)^α`, where `α ≈ 0.191` is already derived. The question is what value the amplitude `C` takes. The framework's answer is `C = φ⁻² ≈ 0.382`, where `φ` is the golden ratio. A competing value, `C = φ⁻³ᐟ² ≈ 0.486`, has also been proposed. The declaration `C_competing_violates_budget` is the formal, machine-checked statement that the competing value fails a specific structural test.

The test is a budget identity. In the framework's ledger, crossing one golden-ratio rung carries a cost penalty `J(φ) = φ - 3/2 ≈ 0.118`. The kernel amplitude `C` is interpreted as a cost-saving from finite-latency closure. The framework proves the identity `J(φ) + C = 1/2`: the penalty plus the saving exactly fills the half-rung interval. The declaration `C_competing_violates_budget` proves that the competing value breaks this identity, giving `J(φ) + φ⁻³ᐟ² > 1/2`. It is a theorem, not a hypothesis, and it is verified with zero unproven assumptions in the framework's machine-checked library of formal theorems.

The theorem does not, by itself, prove that `C = φ⁻²` is the correct physical value. That requires a separate, empirical argument. The framework's prediction is that a fit to galaxy rotation curves will find `C ≈ 0.382`. The machine-checked result only establishes the algebraic fact about the budget. The competing value's failure of this identity is one reason it is rejected, but the ultimate arbiter is the data. The framework states a falsifier: any future SPARC-class fit that finds the kernel amplitude inconsistent with `φ⁻² = 0.382` at better than 5% (a >3σ deviation) would falsify the prediction.

What this means in practice is a sharpened choice. The framework's internal logic forces the amplitude to sit in a narrow band, 0.380 < C < 0.390, and the competing value sits far outside it. The theorem `C_competing_violates_budget` is the formal gate that excludes one candidate on structural grounds, leaving the empirical fit as the final discriminator between the surviving value and any future alternative.

/-- 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

/-- **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

/-- **THEOREM.** Numerical band: `0.380 < C < 0.390` from
    `1.61 < φ < 1.62` via the `2 - φ` closed form. -/
theorem C_kernel_band :
    (0.380 : ℝ) < C_kernel ∧ C_kernel < (0.390 : ℝ) := by
  rw [C_kernel_eq_two_minus_phi]
  have h_lo : 1.61 < phi := phi_gt_onePointSixOne
  have h_hi : phi < 1.62 := phi_lt_onePointSixTwo
  refine ⟨?_, ?_⟩ <;> linarith

What this page does not claim

The theorem does not prove that the competing value is physically wrong, only that it fails the budget identity. The theorem does not establish the empirical value of the kernel amplitude from data. The theorem does not derive the kernel exponent α, which is taken as already derived elsewhere.

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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