alpha_from_self_similarity

formal source

 191
 192/-- From self-similarity and the fixed-point equation φ² = φ + 1,
 193    we can derive constraints on α. This is a placeholder for the full derivation. -/
 194theorem alpha_from_self_similarity (hSS : SelfSimilarKernel)
 195    (h_constraint : hSS.alpha = (1 - 1 / phi) / 2) :
 196    hSS.alpha = alphaLock := by
 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