Cosmology Bitkernel Shape Forcing Rung Scaling Forces Lattice

The forced lattice

In cosmology, the scale factor describes how the universe's distances grow with time, and redshift z measures how much light has stretched along the way. The relationship between them, often written as 1 + z, is the backbone of distance measurements. A simple observation about this relationship, if it holds at every step of cosmic expansion, can have surprisingly rigid consequences: it can force the entire structure of the ladder of scales to be built from a single number, the golden ratio φ ≈ 1.618.

The Recognition Science framework models the aging of the universe as a discrete record of events, a ledger of recognitions. In this account, cosmic expansion proceeds in rungs, each rung multiplying the scale factor by φ. The framework proves that if one rung attenuates a certain aging charge by the reciprocal self-similarity rule ρ = 1/(1+ρ), then the attenuation after n rungs is exactly φ⁻ⁿ. This is the theorem rungScaling_forces_lattice: it derives, from the single-rung rule, that the value at every rung n is fixed, with no freedom left. The lattice of scale factors is forced.

The proof is a simple induction. The base case, zero rungs, gives attenuation 1. The step uses the composition law: attenuation across m + n rungs is the product of the two sub-attenuations. Combined with the one-rung value φ⁻¹, the induction closes. The theorem is machine-checked in the framework's library of formal theorems, with no gaps. It is a THEOREM, not a hypothesis.

The consequence is that the canonical kernel K(z) = 1/(1+z), which describes how the aging charge dilutes with redshift, is no longer a modeling choice. The framework proves that any scale-free kernel, one that composes multiplicatively, must be a power law (1+z)^(-s). The rung condition, that one rung attenuates by φ⁻¹, pins the exponent s = 1 exactly. The spatial-volume competitor s = 3 and the spacetime competitor s = 4 are excluded by the same rung condition. The shape is forced.

What the theorem does not claim is equally important. It does not derive the amplitude of the dark-energy deviation today, denoted δw₀, which remains an open problem bounded but not fixed. It does not prove that the aging charge dilutes through exactly one channel per rung; that selection, called P3, remains a hypothesis with a named falsifier. And it does not claim to explain the gap between the framework's cosmological constant and Planck's measurement; that explanation is explicitly retired. The lattice is forced, but the full physical mechanism is not.

/-- **LATTICE UNIQUENESS.** Any kernel satisfying the rung-scaling law
equals `φ⁻ⁿ = 1/(1+z)` at every rung `z = φⁿ − 1` of the scale lattice. -/
theorem rungScaling_forces_lattice {f : ℝ → ℝ} (hf : RungScalingLaw f) :
    ∀ n : ℕ, f (Constants.phi ^ n - 1) = (1 / Constants.phi) ^ n := by
  intro n
  induction n with
  | zero => simpa using hf.1
  | succ k ih =>
      have hpow : (0 : ℝ) ≤ Constants.phi ^ k - 1 := by
        have : (1 : ℝ) ≤ Constants.phi ^ k := one_le_pow₀ one_lt_phi.le
        linarith
      have harg : Constants.phi ^ (k + 1) - 1
          = Constants.phi * (1 + (Constants.phi ^ k - 1)) - 1 := by ring
      rw [harg, hf.2 _ hpow, ih]
      field_simp
      ring

/-- Every power kernel is scale-free. -/
theorem powerKernel_scaleFree (s : ℝ) : ScaleFree (powerKernel s) := by
  intro z w hz hw
  unfold powerKernel
  have hz1 : (0 : ℝ) ≤ 1 + z := by linarith
  have hw1 : (0 : ℝ) ≤ 1 + w := by linarith
  have harg : 1 + ((1 + z) * (1 + w) - 1) = (1 + z) * (1 + w) := by ring
  rw [harg, Real.mul_rpow hz1 hw1]

/-- **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]

What this page does not claim

The theorem does not claim to derive the amplitude δw₀ of the dark-energy deviation. The theorem does not claim that the aging charge dilutes through exactly one channel per rung; that selection remains a hypothesis. The theorem does not claim to explain the gap between the framework's cosmological constant and Planck's measurement; that explanation is explicitly 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)

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