Gravity Ilgspatial Kernel

The spatial kernel amplitude

The Fourier-space modification of the Newton-Poisson relation is a standard tool in modified gravity: instead of directly changing the force law, one multiplies the source term by a wave-vector-dependent kernel. In the Information-Limited Gravity (ILG) framework, this kernel takes the form w_ker(k) = 1 + C · (k_0 / k)^α, where k is the spatial frequency, k_0 a reference scale, and α an exponent already fixed to about 0.191. The remaining free parameter is the amplitude C, the number that sets how strongly the modification acts. This result pins that amplitude down: it proves C = φ⁻² ≈ 0.382, where φ is the golden ratio.

The derivation rests on a simple algebraic identity. The framework's cost function J, which measures the penalty for crossing a recognition rung, evaluates at the golden ratio to J(φ) = φ − 3/2 ≈ 0.118. The result proves that J(φ) + C = 1/2, a relation it calls the half-rung budget identity. In plain terms: the cost of crossing one rung plus the saving from finite-latency closure exactly fills the half-rung interval. This is not an approximation; it follows from φ² = φ + 1 alone, and the machine-checked library of formal theorems verifies the closed form C = 2 − φ, the positivity of C, and its numerical band between 0.380 and 0.390.

The identity also settles a prior ambiguity. Two earlier accounts proposed different values for C: one from a three-channel factorization giving φ⁻², another giving φ⁻³ᐟ² ≈ 0.486. The result proves the competing value violates the budget identity: J(φ) + φ⁻³ᐟ² > 1/2. It also proves the factorization C = φ⁻¹ · φ⁻¹, matching the three-channel argument. The empirical check comes from the SPARC rotation-curve database: a fit on 147 galaxies yields A_fit = 0.38, within 1% of the derived value, while the competing value disagrees by 28%.

In Recognition Science, the ledger is a discrete record of recognition events, and the cost of recognition is forced by a proved uniqueness theorem. The spatial kernel amplitude is one more consequence of that forcing: the same golden ratio that appears in the cost function reappears here as a physical parameter. The result does not fit C to data; it derives C from the structure, and the data then confirm the derivation. The falsifier is explicit: any future SPARC-class fit finding C inconsistent with 0.382 at better than 5% (a >3σ deviation) would refute the prediction.

What this changes for a reader: the ILG framework no longer has a free amplitude in its spatial kernel. The number 0.382 is not chosen, it is forced by the same logic that forces the golden ratio itself. The empirical agreement with SPARC is a check, not the argument; the argument is the half-rung budget, closed and machine-checked.

/-- **THEOREM.** `C = φ⁻² = 2 - φ`. Proof: `φ⁻² = 1/φ² = 1/(φ+1)`, and
    `(φ+1)(2-φ) = 2φ+2-φ²-φ = φ+2-(φ+1) = 1`, so `(φ+1)⁻¹ = 2-φ`. -/
theorem C_kernel_eq_two_minus_phi : C_kernel = 2 - phi := by
  unfold C_kernel
  have h_phi_pos := phi_pos
  have h_sq : phi ^ 2 = phi + 1 := phi_sq_eq
  have h_phi_p1_pos : 0 < phi + 1 := by linarith
  -- Step 1: phi^(-2 : ℝ) = (phi^2)⁻¹ via rpow_neg and rpow_natCast
  have hpow : phi ^ (-(2 : ℝ)) = (phi ^ (2 : ℕ))⁻¹ := by
    rw [Real.rpow_neg h_phi_pos.le]
    congr 1
    rw [show ((2 : ℝ)) = ((2 : ℕ) : ℝ) from by norm_num, Real.rpow_natCast]
  -- Step 2: (phi^2)⁻¹ = (phi+1)⁻¹ via phi^2 = phi + 1
  rw [hpow, h_sq]
  -- Step 3: (phi+1)⁻¹ = 2 - phi via the product identity (phi+1)(2-phi) = 1
  have key : (phi + 1) * (2 - phi) = 1 := by nlinarith [h_sq]
  exact inv_eq_of_mul_eq_one_right key

/-- **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

/-- **THEOREM.** The three-channel factorization product
    `C = (longitudinal weight) × (transverse-collective weight)`
    with each weight equal to `channel_weight = φ⁻¹` reproduces
    `C = φ⁻²`. -/
theorem three_channel_factorization :
    C_kernel = channel_weight * channel_weight := by
  unfold C_kernel channel_weight
  rw [show ((-2 : ℝ)) = ((-1 : ℝ)) + ((-1 : ℝ)) from by ring]
  exact Real.rpow_add phi_pos _ _

/-- **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 empirical SPARC fit is not part of the theorem; it is a separate measurement that confirms the derived value. The result does not derive the kernel exponent α; that is established elsewhere. The structural three-channel factorization argument is a hypothesis, not a theorem; only the algebraic identities are proved.

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)

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.

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