Cosmology Scale Invariance Selection Cert Log Space Symmetry

The symmetry in log-space

Scale invariance is the idea that a physical law looks the same at every size. A circle is scale invariant: shrink it or enlarge it, and it is still a circle. In the Recognition Science framework, the question is whether the cost of recognition, the price reality pays to register a change, can itself be scale invariant. The framework's cost function is J(x) = ½(x + x⁻¹) − 1, and the theorem log_space_symmetry states that J(x) = J(x⁻¹) for any positive x. In plain terms, scaling a value up by a factor and scaling it down by the same factor carry identical costs.

This symmetry is not the naive scale invariance J(cx) = J(x) that a first glance might expect. That stronger property fails for this cost function. What holds instead is a bounded relationship: the cost of a scaled value J(cx) is controlled by the cost of the scale change J(c) and the original cost J(x), through the inequality J(cx) ≤ 2J(c)J(x) + 2J(c) + 2J(x). This is the "cost of scale change" principle, and it follows from the Recognition Composition Law, the equality J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The composition law says the cost of combining two values is governed by their individual costs.

The log-space symmetry itself is a direct consequence of the cost function's definition, and the framework's machine-checked library records it as a proved theorem with no axioms and no unfinished proofs. The symmetry is one of four properties bundled into the framework's ScaleInvarianceCert structure, a certificate that the cost function satisfies the structural facts the pre-Big-Bang paper's scale-invariance argument relies on. The other properties are the composition law, the scale-change cost bound, and the fact that scaling by 1, no change at all, is free.

What the theorem does not claim is that physical laws are scale invariant. The theorem is about a cost function, not about the laws of physics. It does not claim that J(cx) = J(x) holds, and it does not claim that the cost of any scale change is zero. It establishes a symmetry under inversion and a bound on scaling costs, nothing more. The broader cosmological argument that scale invariance is selected by cost minimisation is a separate claim, built on these structural facts, not established by them.

/-- Scale invariance in log-space: J is symmetric under inversion. -/
theorem log_space_symmetry {x : ℝ} (hx : 0 < x) :
    Jcost x = Jcost x⁻¹ := Jcost_symm hx

/-- Scale-change cost: J(cx) is controlled by J(x) and J(c). -/
theorem scale_change_cost {c x : ℝ} (hc : 0 < c) (hx : 0 < x) :
    Jcost (c * x) ≤ 2 * Jcost c * Jcost x + 2 * Jcost c + 2 * Jcost x := by
  have h := rcl_equality hc hx
  -- J(cx) + J(c/x) = 2J(c)J(x) + 2J(c) + 2J(x)
  -- J(cx) ≤ 2J(c)J(x) + 2J(c) + 2J(x) since J(c/x) ≥ 0
  linarith [Jcost_nonneg (div_pos hc hx)]

/-- The Recognition Composition Law (RCL) in inequality form:
    J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y).
    The cost of combining x and y is controlled by their individual costs. -/
theorem rcl_equality {x y : ℝ} (hx : 0 < x) (hy : 0 < y) :
    Jcost (x * y) + Jcost (x / y) = 2 * Jcost x * Jcost y + 2 * Jcost x + 2 * Jcost y := by
  rw [Jcost_eq_sq hx.ne', Jcost_eq_sq hy.ne',
      Jcost_eq_sq (mul_pos hx hy).ne',
      Jcost_eq_sq (div_pos hx hy).ne']
  field_simp [hx.ne', hy.ne']
  ring

What this page does not claim

The theorem does not claim that physical laws are scale invariant. The theorem does not claim that J(cx) = J(x) holds for arbitrary scale factors. The theorem does not claim that the cost of any scale change is zero.

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/ScaleInvarianceSelectionCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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