Cosmology Dark Energy Phi Dilution Derivation Occ One Eq Inv Phi

The forced dilution step

The golden ratio φ, roughly 1.618, appears throughout geometry and number theory. Its reciprocal, φ⁻¹ ≈ 0.618, solves a simple self-referential equation: a positive number ρ equals 1/(1+ρ) exactly when ρ = φ⁻¹. This is a classical fact, checkable by substituting the quadratic solution into the equation.

In Recognition Science, this classical fact becomes a structural constraint. The framework models reality as maintaining a discrete record of events, called a ledger, where each event has a recognition cost. The theorem occ_one_eq_inv_phi states: any positive occupancy value for a single dimension that satisfies the self-similarity equation ρ = 1/(1+ρ) must equal φ⁻¹. The proof routes through the same φ-forcing theorem that fixes the framework's fundamental scale, so the dilution factor is not chosen; it is forced by the equation alone.

The declaration does more than identify a number. It is the first step in a derivation. A dimension-uniform dilution law is a structure with two premises: occupancy across independent dimensions multiplies, and a single dimension attenuates by the self-similarity fixed point. From these, the theorem occ_eq_pow derives by induction that occupancy after n dimensions is (φ⁻¹)ⁿ. The exponent is then fixed to 4 by the framework's forced spacetime dimension, giving the dark-energy occupancy θ = φ⁻⁴.

What the declaration does not claim is equally precise. It does not assert that dark energy exists or that its measured value matches φ⁻⁴; those are empirical questions outside this theorem. It does not claim the dilution law itself is the only possible structure, only that any structure satisfying the two premises must take this form. The theorem is a conditional statement: if the premises hold, the conclusion follows. The framework's library of machine-checked formal theorems proves this implication with no gaps, but the premises remain definitional choices within the framework.

/-- **The self-similar attenuation ratio is forced to `φ⁻¹`.** Any positive `ρ` solving the
reciprocal self-similarity equation `ρ = 1/(1+ρ)` equals `1/φ`. The proof routes through the
*scale-forcing* theorem: `1/ρ` solves `x = recipShift x` with `x > 1`, hence `1/ρ = φ`. -/
theorem self_similar_attenuation_forced {ρ : ℝ} (hpos : 0 < ρ)
    (hfp : ρ = 1 / (1 + ρ)) : ρ = 1 / Constants.phi := by
  have hsum_ne : (1 + ρ) ≠ 0 := by positivity
  have key : ρ * (1 + ρ) = 1 := by rwa [eq_div_iff hsum_ne] at hfp
  have hρ_lt_one : ρ < 1 := by nlinarith [key, hpos]
  have hx_gt_one : (1 : ℝ) < 1 / ρ := by
    have h := one_div_lt_one_div_of_lt hpos hρ_lt_one
    simpa using h
  have hfix : ReciprocalGenerator.recipShift (1 / ρ) = 1 / ρ := by
    unfold ReciprocalGenerator.recipShift ReciprocalGenerator.recip
    rw [one_div ρ, inv_inv]
    rw [eq_comm, inv_eq_one_div, div_eq_iff hpos.ne']
    linear_combination -key
  have hxphi : 1 / ρ = Constants.phi :=
    (ReciprocalGenerator.recipShift_fixed_iff hx_gt_one).mp hfix
  calc ρ = 1 / (1 / ρ) := (one_div_one_div ρ).symm
    _ = 1 / Constants.phi := by rw [hxphi]

/-- **The dilution law, derived.** `n` independent dimensions dilute by `φ⁻ⁿ`. -/
theorem occ_eq_pow : ∀ n : ℕ, L.occ n = (1 / Constants.phi) ^ n := by
  intro n
  induction n with
  | zero => rw [pow_zero]; exact L.occ_zero
  | succ k ih =>
      have hc := L.composes k 1
      rw [hc, ih, L.occ_one_eq_inv_phi, pow_succ]

/-- The forced exponent equals `4` (1 temporal octave + 3 spatial from Alexander duality). -/
theorem dilutionExponent_eq_four : dilutionExponent = 4 :=
  SpacetimeEmergence.spacetime_dim_eq_four

/-- **Headline: `θ = φ⁻⁴` is forced.** Any dimension-uniform dilution law yields exactly the
four-dimensional φ-dilution `θ = φ⁻⁴` at the forced spacetime dimension. -/
theorem derivedTheta_eq_phiFour (L : DimensionUniformDilution) :
    derivedTheta L = DarkEnergyThetaPhiFour.thetaPhiFour := by
  unfold derivedTheta
  rw [L.occ_eq_pow, dilutionExponent_eq_four]
  unfold DarkEnergyThetaPhiFour.thetaPhiFour
  rw [div_pow, one_pow]

What this page does not claim

The theorem does not assert that dark energy exists or that its measured density equals φ⁻⁴. The theorem does not claim the dilution law is the only possible structure, only that any structure satisfying the two premises takes this form. The theorem does not derive the value of the fine-structure constant or any other coupling constant.

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/Cosmology/DarkEnergyPhiDilutionDerivation.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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