Cosmology Scale Invariance Selection Cert Scale Change Cost

The cost of changing scale

In mathematics and physics, a symmetry often means that something stays the same under a transformation. Scale invariance would mean that the laws of nature look identical whether you measure them in meters or kilometers. The Recognition Science framework, which studies the cost of recognition events, proves a precise statement about this idea: the cost function J(x) is not scale-invariant in the naive sense, but the ratio J(cx)/J(x) is bounded by J(c), the cost of the scale change itself. This is the theorem scale_change_cost.

The theorem states that for any positive numbers c and x, the cost of the scaled value J(cx) is at most 2·J(c)·J(x) + 2·J(c) + 2·J(x). Here J is the unique cost function forced by the framework's axioms, J(x) = ½(x + x⁻¹) − 1. The bound says that scaling by a factor c costs no more than a combination of the cost of c and the cost of x. When c = 1, meaning no scale change at all, the theorem gives J(x) = J(x), which is the free_at_unit result: doing nothing costs nothing.

The proof is a direct consequence of the Recognition Composition Law, which states that J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). Since J(x/y) is always nonnegative, dropping it from the left side gives the inequality. The framework's machine-checked library of formal theorems verifies this with no unproved assumptions. The result is part of a larger argument in the pre-Big-Bang work that scale invariance of physical laws is selected by cost-minimisation, though the theorem itself only establishes the bound, not the selection claim.

The importance of this bound is that it makes precise what scale invariance cannot mean in this framework. A naive scale invariance, where J(cx) = J(x) for all c, would make the cost of changing scale zero. The theorem shows that is false in general. Instead, the cost of changing scale is controlled by the cost of the scale factor itself. This gives a quantitative sense in which the framework's cost function resists arbitrary rescaling, while still allowing the composition law to hold.

What the theorem does not claim is that scale invariance is actually selected in cosmology. The theorem only establishes the structural bound. The selection argument, that among all possible cost functions only the scale-invariant form combined with the compositional constraint forces the unique solution, is a separate claim from that work. The theorem also does not say that J(cx) is always greater than J(x), only that it is bounded above. It leaves open the possibility that scaling down could be cheaper than scaling up, and it does not address what happens when c or x is not positive.

/-- 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)]

/-- If c = 1 (no scale change), cost is zero. -/
theorem no_scale_change_is_free {x : ℝ} (hx : 0 < x) :
    Jcost (1 * x) = Jcost x := by simp

/-- 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 prove that scale invariance is actually selected in cosmology. The theorem does not state that J(cx) is always greater than J(x), only that it is bounded above. The theorem does not address the behavior of the cost function for non-positive values of c or x.

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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