Astrophysics Stellar Assembly J Nonneg

The nonnegative cost

In the Recognition Science framework, the ledger (a discrete record of recognition events) assigns a cost to each event. The declaration J_nonneg is a formal lemma stating that for any positive scale ratio x, the cost J(x) is never negative. The cost function itself is J(x) = (x + 1/x)/2 - 1, and the lemma establishes 0 ≤ J(x) whenever x > 0.

The proof is algebraic and exact. The lemma rewrites the cost as (x - 1)^2 / (2x). Since the numerator is a square and the denominator is positive for x > 0, the quotient is nonnegative. This is a theorem in the machine-checked library of formal theorems, meaning it is verified by a computer from the definitions, with no hidden assumptions beyond the usual logical axioms.

What J_nonneg does not claim is equally important. It does not assert that the cost is positive for all x; at x = 1, the cost is exactly zero. It does not say anything about which scale ratios actually occur in a stellar collapse. It only establishes a lower bound on the cost for any positive ratio. The lemma is a building block, not a physical prediction.

The consequence is that the framework's cost function is well-behaved: it never produces a negative cost, which would be meaningless in a ledger of events. This makes the cost usable as a basis for comparing different configurations, such as photon emission versus mass storage in a star, where the cost difference determines the mass-to-light ratio.

In the stellar assembly module, this lemma underpins the derivation of the mass-to-light ratio M/L. The module defines the cost difference Δδ between photon emission and mass storage, and shows that if Δδ equals n times a fundamental bit of cost, then M/L = φ^n, where φ is the golden ratio. The nonnegativity lemma ensures that the cost difference is a sensible quantity to minimize, though it does not by itself fix the value of n or the ratio.

/-- J is nonnegative for positive arguments -/
lemma J_nonneg {x : ℝ} (hx : 0 < x) : 0 ≤ J x := by
  unfold J
  have : Cost.Jcost x = (x - 1)^2 / (2 * x) := by
    unfold Cost.Jcost
    have hne : x ≠ 0 := ne_of_gt hx
    field_simp [hne]
    ring
  rw [this]
  exact div_nonneg (sq_nonneg _) (by linarith)

/-- J is nonnegative for positive arguments -/
lemma J_nonneg {x : ℝ} (hx : 0 < x) : 0 ≤ J x := by
  unfold J
  have : Cost.Jcost x = (x - 1)^2 / (2 * x) := by
    unfold Cost.Jcost
    have hne : x ≠ 0 := ne_of_gt hx
    field_simp [hne]
    ring
  rw [this]
  exact div_nonneg (sq_nonneg _) (by linarith)

/-- Recognition cost for scale ratio x: J(x) = ½(x + 1/x) - 1 -/
noncomputable def J (x : ℝ) : ℝ := Cost.Jcost x

/-- When Δδ = n · J_bit = n · log(φ), we get M/L = φ^n -/
theorem ml_is_phi_power (n : ℤ) (Δδ : ℝ) (h : Δδ = n * J_bit) :
    ml_from_cost_diff Δδ = φ ^ n := by
  simp only [ml_from_cost_diff, J_bit] at *
  rw [h]
  -- exp(n * log(φ)) = φ^n by definition of zpow for positive reals
  have hφ : 0 < φ := Constants.phi_pos
  rw [← Real.rpow_intCast φ n]
  rw [Real.rpow_def_of_pos hφ]
  ring

What this page does not claim

J_nonneg does not assert that the cost is positive for all x; at x = 1 it is zero. J_nonneg does not determine which scale ratios occur in a stellar collapse. The lemma does not by itself fix the value of the mass-to-light ratio.

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

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