Cosmology Bitkernel Shape Forcing Occ Eq Inv One Plus Z

The forced kernel

In cosmology, the factor 1/(1+z), where z is redshift, is the standard way to describe how the wavelength of light stretches with the expansion of the universe. The Recognition Science declaration occ_eq_inv_one_plus_z proves, within its framework, that this factor is not a free choice but the unique outcome of two premises about how a quantity called the aging charge attenuates across cosmic scale. The theorem states that after n rungs of scale, where each rung is a factor of the golden ratio φ in the scale factor, the occupancy L.occ n equals 1/(1+z) with z = φ^n − 1. This is a THEOREM: it is proved in the framework's machine-checked library of formal theorems, with no unproved axioms beyond the standard three.

The two premises are simple. First, attenuation across m+n rungs is the product of the attenuations across m and n rungs, a multiplicative composition law. Second, one rung attenuates by the reciprocal self-similarity fixed point ρ = 1/(1+ρ), which forces ρ = φ⁻¹. Iterating these gives attenuation φ⁻ⁿ at rung n, which equals 1/(1+z) on the rung lattice. The continuum interpolation is then pinned: every scale-free kernel is a power law (1+z)^(−s), and the per-rung value forces the exponent s = 1 exactly. The spatial-volume competitor s = 3 and the spacetime competitor s = 4 are excluded by the same rung condition.

This result matters because it turns a modeling choice into a forced prediction. The dark-energy equation of state becomes w(z) = −1 + δw₀/(1+z), which is exactly the CPL parametrization with w₀ = −1 + δw₀ and wₐ = −δw₀, hence w₀ + wₐ = −1. The framework predicts a line segment in the (w₀, wₐ) plane: wₐ = −(1+w₀) with −1 < w₀ < −0.88. The amplitude δw₀ is bounded in (0, J(φ)] with J(φ) < 0.12, but its today-value is OPEN, not derived.

What the theorem does not claim is equally precise. It does not claim that the BIT mechanism itself is physical; that is a HYPOTHESIS with named falsifiers. The sharpest falsifier is any confirmed phantom crossing w(z) < −1 at any redshift, which would falsify the mechanism outright. A DESI Y3+ or Roman or Euclid CPL posterior excluding the predicted line segment at high confidence would also falsify the forced kernel. The theorem also does not claim to explain the Ω_Λ gap: the forced kernel keeps δw(z) ≥ 0, so dark energy was denser in the past, and the correction moves the effective Ω_Λ down, away from Planck, for every admissible amplitude. That direction is structurally fixed, and the gap-explanation hypothesis is retired.

/-- The rung-`n` attenuation in redshift form: at rung `n` the scale factor
has contracted by `φⁿ`, i.e. `1 + z = φⁿ`, and the attenuation equals
`1/(1+z)` exactly. -/
theorem occ_eq_inv_one_plus_z (n : ℕ) :
    L.occ n = 1 / (1 + (Constants.phi ^ n - 1)) := by
  rw [L.occ_forced n]
  have harg : 1 + (Constants.phi ^ n - 1) = Constants.phi ^ n := by ring
  rw [harg, div_pow, one_pow]

/-- **EXPONENT PINNED: the power kernel satisfies the φ-rung condition iff
`s = 1`.** With the rung attenuation forced to `φ⁻¹` (§1), the unique
scale-free kernel is `K(z) = (1+z)^(−1) = 1/(1+z)`. Spatial-volume (`s=3`)
and spacetime (`s=4`) dilution are excluded. -/
theorem powerKernel_rung_condition_iff (s : ℝ) :
    RungCondition (powerKernel s) ↔ s = 1 := by
  unfold RungCondition powerKernel
  have harg : 1 + (Constants.phi - 1) = Constants.phi := by ring
  rw [harg]
  constructor
  · intro h
    -- take logs: −s·log φ = −log φ, and log φ > 0
    have hlogpos : 0 < Real.log Constants.phi := Real.log_pos one_lt_phi
    have hlhs : Real.log (Constants.phi ^ (-s)) = -s * Real.log Constants.phi :=
      Real.log_rpow phi_pos (-s)
    have hrhs : Real.log (1 / Constants.phi) = -Real.log Constants.phi := by
      rw [one_div, Real.log_inv]
    have hkey : -s * Real.log Constants.phi = -Real.log Constants.phi := by
      rw [← hlhs, ← hrhs, h]
    have := mul_right_cancel₀ (ne_of_gt hlogpos) (by linarith : -s * Real.log Constants.phi = -1 * Real.log Constants.phi)
    linarith
  · intro h
    subst h
    rw [Real.rpow_neg_one, one_div]

/-- **The forced kernel is exactly CPL** with `w₀ = −1 + δw₀`, `wₐ = −δw₀`. -/
theorem w_RS_is_CPL (dw0 z : ℝ) (hz : -1 < z) :
    w_RS dw0 z = w_CPL (-1 + dw0) (-dw0) z := by
  have h1z : (0 : ℝ) < 1 + z := by linarith
  unfold w_RS w_CPL
  field_simp
  ring

/-- **F1 (sign falsifier): no phantom crossing.** Under the forced kernel
with non-negative amplitude, `w(z) ≥ −1` at every physical redshift. A
confirmed `w < −1` measurement falsifies the BIT mechanism. -/
theorem no_phantom (dw0 z : ℝ) (h0 : 0 ≤ dw0) (hz : -1 < z) :
    -1 ≤ w_RS dw0 z := by
  have h1z : (0 : ℝ) < 1 + z := by linarith
  have : 0 ≤ dw0 / (1 + z) := div_nonneg h0 h1z.le
  unfold w_RS
  linarith

/-- **RETIREMENT CERTIFICATE.** The "BIT explains the Planck-RS Ω_Λ gap"
hypothesis is structurally dead: under the forced kernel the correction at
any admissible amplitude moves `Ω_Λ_eff` below the bare RS value (certified
quadrature, arc 11), i.e. away from Planck. Bundles the arc-11 findings
with the forced-sign theorem. -/
theorem omega_gap_explanation_retired :
    -- (i) at maximum amplitude the corrected value is below the bare RS value
    omega_lambda_corrected_max_amplitude < CosmologicalConstantDerivation.Omega_Lambda_RS ∧
    -- (ii) and outside Planck 1σ, in the wrong direction
    planck_sigma < |omega_lambda_corrected_max_amplitude - planck_central| ∧
    -- (iii) while the forced kernel pins the deviation sign: w(z) ≥ −1 always
    (∀ dw0 z : ℝ, 0 ≤ dw0 → -1 < z → -1 ≤ w_RS dw0 z) :=
  ⟨max_amplitude_below_bare_interval,
   max_amplitude_outside_one_sigma,
   fun dw0 z h0 hz => no_phantom dw0 z h0 hz⟩

What this page does not claim

The BIT cosmic-aging mechanism itself is not proved; it is a hypothesis with named falsifiers. The today-amplitude δw₀ is not derived; only its bounds (0, J(φ)] are proved. The theorem does not explain the Ω_Λ gap; it retires that hypothesis. The framework does not claim to derive the fine-structure constant α; its expression is an identification, not a derived coupling.

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/BITKernelShapeForcing.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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