Cosmology Scale Invariance Selection Cert Rcl Equality

The cost of combining scales

Scale invariance is the idea that the laws of physics look the same at every size. A common naive hope is that the cost of a process is unchanged when you rescale it, so that the cost at size cx equals the cost at size x. The Recognition Science framework, which models reality as storing a discrete record of events with a forced cost, shows this hope fails in general. The framework's central cost function is J(x) = (x + 1/x)/2 - 1, a measure of how far a scale factor x is from unity, where the cost is zero.

What actually holds is a precise inequality. The theorem rcl_equality proves that for any two positive scale factors x and y, the cost of their product plus the cost of their ratio equals a specific combination of their individual costs: J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). This is the Recognition Composition Law in its equality form. Because costs are never negative, this equality immediately implies the useful bound J(cx) ≤ 2J(c)J(x) + 2J(c) + 2J(x), which says the cost of changing scale by a factor c is controlled by the cost of c itself and the cost of the original size. This is the "cost of scale change" principle: rescaling is never free, and its price is set by the scale factor alone.

The framework's library, a machine-checked collection of formal theorems, proves these statements with no unproved assumptions. The equality is derived directly from the definition of J using algebraic manipulation. The framework also proves two simple consequences: scaling by 1 costs nothing, and the cost is symmetric under inversion, so J(x) = J(1/x). These facts are collected into a certificate structure that packages the scale-invariance selection argument.

What the declaration does not claim is equally important. It does not establish that the cost function is scale-invariant in the naive sense; the equality shows the opposite, that costs do change under rescaling. It does not claim that the composition law itself forces the specific form of J; that uniqueness is a separate theorem requiring additional conditions. And it makes no statement about physical cosmology, only about the mathematical structure of the cost function. The declaration is a structural fact about a function, not a claim about the universe's history.

/-- 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

/-- 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)]

/-- Scale-invariance selection certificate. -/
def scaleInvarianceCert : ScaleInvarianceCert where
  rcl := rcl_equality
  scale_cost_bound := scale_change_cost
  free_at_unit := @no_scale_change_is_free
  log_symmetric := Jcost_symm

What this page does not claim

The declaration does not prove that the cost function is scale-invariant in the naive sense J(cx) = J(x). The declaration does not claim that the composition law alone forces the specific form of J. The declaration makes no statement about physical cosmology.

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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