alpha_from_self_similarity
The theorem establishes that any self-similar ILG kernel whose exponent is fixed to (1 - 1/phi)/2 equals the canonical locked value alphaLock. ILG cosmologists cite it to anchor the perturbation exponent from the phi fixed point. The proof is a one-line simplification that equates the hypothesis directly to the definition of alphaLock.
claimLet $K$ be a self-similar kernel with exponent $alpha$. If $alpha = (1 - phi^{-1})/2$, then $alpha$ equals the locked constant $alpha_{lock}$.
background
The ILG kernel takes the form $w(k,a) = 1 + C (a/(k tau_0))^alpha$, where $tau_0$ is the fundamental recognition tick. SelfSimilarKernel is the structure that encodes the scaling property: the kernel evaluated at scale factor $phi a$ equals the original kernel scaled by $phi^alpha$. The upstream constant alphaLock is defined exactly as $(1 - 1/phi)/2$, the value forced by the phi fixed-point equation $phi^2 = phi + 1$.
proof idea
The proof is a one-line term-mode simplification that substitutes the constraint hypothesis into the definition of alphaLock and verifies syntactic identity.
why it matters in Recognition Science
This result discharges the placeholder derivation noted in the SelfSimilarKernel structure, confirming the exponent matches the locked value used throughout the kernel module. It supports downstream kernel properties such as positivity and monotonicity in scale factor. In the Recognition framework it instantiates the T6 self-similar fixed point for the ILG exponent, consistent with the eight-tick octave.
scope and limits
- Does not derive the exponent value from self-similarity without the golden-ratio constraint hypothesis.
- Does not prove the self-similarity scaling property of the kernel.
- Does not supply numerical bounds or observational tests for alpha.
formal statement (Lean)
194theorem alpha_from_self_similarity (hSS : SelfSimilarKernel)
195 (h_constraint : hSS.alpha = (1 - 1 / phi) / 2) :
196 hSS.alpha = alphaLock := by
proof body
Term-mode proof.
197 simp [h_constraint, alphaLock]
198
199/-! ## Causality bounds (Beltracchi 2026 resolution)
200
201Two pathologies of the literal Riemann–Liouville / Fourier-only ILG formulation
202were identified by P. Beltracchi (April 2026 internal note):
203
2041. **Cumulative-time growth.** Reading the time-domain RL form
205 `ρ_eff(t) = ρ(t) + C τ₀⁻ᵅ Iᵅ[ρ(t)]` literally for an isolated mass `M`,
206 the gravitational acceleration grows as `t^α` without bound.
207
2082. **Infrared divergence.** The Fourier-space kernel
209 `w(k,a) = 1 + C (a / (k τ₀))^α` diverges as `k → 0`, making the
210 homogeneous-background limit incoherent and naive perturbation theory
211 ill-defined.
212
213Both pathologies are forced by reading the kernel as (i) a cumulative-time
214convolution and (ii) a free-running k-space multiplier. The resolution from
215the recognition-operator forcing chain is structural:
216
217- The ledger lag is **per recognition tick**, not per cumulative cosmic time.
218 At rest in equilibrium, the ledger is at `J(1) = 0` and there is no
219 "memory backlog" to integrate. The working kernel is a **dynamical-time**
220 kernel `w(T_dyn)`, not a cumulative-time RL convolution.
221- The kernel acts on **gradient flow** of the ledger, not on the ledger value.
222 A homogeneous distribution sits at `J = 0` with no flow, so the background
223 mode is unaffected (`kernel_background = 1`).
224- The IR is bounded by the **recognition horizon** `R_H = c/H`. Below
225 `k_min = a H / c`, no causal ledger update can occur, and the kernel
226 saturates at `kernel(k_min)`.
227
228This section formalizes the IR-bounded perturbation kernel, proves the
229boundedness theorem, and exposes the perturbation/background split that
230makes ILG self-consistent for cosmological perturbation theory.
231
232The original `kernel` definition above is preserved unchanged for backward
233compatibility; the new `kernel_perturbation` and `kernel_background`
234distinguish the two physical regimes.
235-/
236
237/-- The IR-bounded ILG perturbation kernel:
238
239 `w_pert(k_min, k, a) = 1 + C · (a / (max(k_min, k) · τ₀))^α`
240
241For `k ≥ k_min` this collapses to the original `kernel P k a`. For `k < k_min`
242the wavenumber saturates at `k_min`, capping the enhancement at
243`kernel(k_min)`. The IR cutoff `k_min` is physically the inverse recognition
244horizon `a H / c`. -/