Cosmology Bitkernel Shape Forcing

The forced kernel shape

In cosmology, the dark energy equation of state is often written as a simple power law: the ratio of pressure to energy density, w(z), evolves with redshift z. A common parameterization, called CPL, uses two constants, w0 and wa, to describe this evolution as a straight line in a particular plane. The question is whether this shape is a convenient fit to data or a consequence of deeper physics.

In Recognition Science, the framework models the universe's expansion as a discrete ledger of recognition events, and the cost of each recognition is forced by a proved theorem. The new module, BITKernelShapeForcing, derives the shape of this kernel from two premises that mirror the machinery already used to force the dark energy density. The first premise says that the aging charge carried across multiple rungs of cosmic scale is the product of the sub-attenuations. The second says that one rung attenuates by a self-similar factor, which forces that factor to be the inverse of the golden ratio, φ⁻¹. The substrate is self-similar with ratio φ, so rung n sits at 1 + z = φⁿ. Iterating these premises gives attenuation φ⁻ⁿ at rung n, which equals 1/(1+z) on that lattice.

The continuum interpolation is pinned inside the scale-free class: every scale-free kernel is a power law (1+z)^(-s). The per-rung value φ⁻¹ forces the exponent s = 1 exactly, excluding the spatial-volume competitor s = 3 and the spacetime competitor s = 4. The canonical kernel K(z) = 1/(1+z) is therefore no longer a modeling choice but the unique single-channel scale-free φ-dilution law. This is a proved theorem in the machine-checked library of formal theorems.

With the shape forced, the RS equation of state becomes a one-parameter family w(z) = -1 + δw0/(1+z), which is exactly CPL with w0 = -1 + δw0 and wa = -δw0, hence w0 + wa = -1. The prediction is a line segment in the (w0, wa) plane: wa = -(1 + w0) with -1 < w0 < -0.88. The module proves that w(z) never drops below -1, so any confirmed phantom crossing at any redshift would falsify the mechanism. It also proves a ceiling on the deviation, and the sum rule w0 + wa = -1. The today-amplitude δw0 remains open, but the shape no longer depends on it.

This result changes what the framework can claim: the shape of dark energy's evolution is derived, not assumed, and it makes a sharp, dated prediction for upcoming surveys. The module also retires an earlier hypothesis that the BIT correction could explain the gap between the bare RS value of ΩΛ and the Planck measurement. The honest position is that the bare value stands on its own, and the BIT correction is a prediction about w(z), not a patch for ΩΛ.

/-- **THE RUNG DILUTION LAW IS FORCED: `occ n = φ⁻ⁿ`.** Derived by
induction from the two premises alone. -/
theorem occ_forced : ∀ n : ℕ, L.occ n = (1 / Constants.phi) ^ n := by
  intro n
  induction n with
  | zero => simpa using L.occ_zero
  | succ k ih =>
      have h := L.composes k 1
      rw [h, ih, L.occ_one_forced]
      ring

/-- 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 pinned kernel equals the canonical `DeltaWKernel` shape pointwise on
the physical redshift domain: `(1+z)^(−1) = f_canonical(z)` for `z ≥ 0`. -/
theorem powerKernel_one_eq_canonical (z : ℝ) (hz : 0 ≤ z) :
    powerKernel 1 z = f_canonical z := by
  unfold powerKernel f_canonical
  rw [if_neg (not_lt.mpr hz), 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

/-- The CPL sum rule `w₀ + wₐ = −1`: exact ΛCDM recovery in the early
universe (no accumulated cosmic Z). -/
theorem cpl_sum_rule (dw0 : ℝ) : (-1 + dw0) + (-dw0) = -1 := by 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

What this page does not claim

This answer does not claim that the dark energy mechanism itself is proved; it is tagged as a hypothesis with named falsifiers. This answer does not claim that the today-amplitude δw0 is derived; it remains open. This answer does not claim that the framework explains the Planck-RS ΩΛ gap; that hypothesis is retired.

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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