Cosmology Scale Invariance Selection Cert Scale Invariance Cert

Scale invariance and its cost

Scale invariance is the idea that a law of physics looks the same whether you measure it in meters or kilometers, seconds or hours. In many theories, this symmetry is exact. In the Recognition Science framework, the central object is a cost, a number assigned to each possible recognition event, and the framework's cost function J(x) is not scale invariant in that naive sense. The declaration ScaleInvarianceCert, a machine-checked collection of formal theorems, establishes exactly what scale invariance does and does not mean here.

The core result is a bound on the cost of changing scale. If you multiply a scale x by a factor c, the cost of the new scale J(cx) is not equal to J(x). Instead, it is controlled by the costs of the two pieces: J(cx) ≤ 2·J(c)·J(x) + 2·J(c) + 2·J(x). This is the cost of scale change principle. The cost of combining two scales is capped by the individual costs, a direct consequence of the framework's composition law. For positive x and y, that law states J(xy) + J(x/y) = 2·J(x)·J(y) + 2·J(x) + 2·J(y).

The certificate also records two simpler facts. If c = 1, meaning no scale change at all, the cost is unchanged: J(1·x) = J(x). And the cost function is symmetric under inversion in log-space: J(x) = J(1/x). These four properties together form the certificate: the composition law, the scale-change bound, the free unit, and the log-space symmetry. The library proves all four with no unproved assumptions and no axioms beyond the standard logical ones.

What the certificate does not claim is that scale invariance is selected. The argument that scale invariance is the unique cost-minimizing choice is not formalized here. The certificate only establishes the structural facts about the cost function's behavior under scaling. It also does not claim that J(cx) = J(x) holds in general; the bound is an inequality, not an equality. The certificate is a precise statement about the cost of scale change, not a proof that scale invariance is the only possible law.

/-- 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

/-- 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

/-- 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

What this page does not claim

The certificate does not prove that scale invariance is selected by cost-minimisation. It does not claim that J(cx) = J(x) holds for all c and x. It does not establish any physical consequence of the scale-change bound.

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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