Masses Baseline Derivation Nontriviality From Cost

A zero-cost transition leaves no record

In the Recognition Science framework, a ledger (a discrete record of events) assigns a cost to each possible transition between states. The cost function J(x) = (1/2)(x + x⁻¹) - 1 measures how expensive it is to change a quantity by a factor x. The theorem nontriviality_from_cost states a simple consequence: if a transition costs strictly more than zero, then its ratio x cannot equal 1. In other words, a zero-cost transition is the identity transition, and the identity transition leaves no record in the ledger.

This is a formal theorem in the machine-checked library of formal theorems. Its proof is short: J(1) = 0 by direct calculation, so if J(x) > 0, then x ≠ 1. The contrapositive also holds: if x = 1, then J(x) = 0. The theorem assumes x > 0, which is natural since ratios of positive quantities are positive. The result is not a claim about any particular physical system; it is a structural fact about the cost function itself, derived from the definition of J.

The theorem matters because it gives the framework a way to distinguish real events from non-events. If a transition costs nothing, it is indistinguishable from no transition at all. This is what the docstring calls “non-triviality”: the ledger only records changes that cost something. Without this property, the ledger would be cluttered with zero-cost transitions that carry no information. The theorem is a baseline rung in a chain that derives particle mass baselines from the geometry of a 3-cube, but the nontriviality result itself stands alone as a constraint on any cost-based accounting of change.

What the theorem does not claim is equally important. It does not say that all real transitions have positive cost, only that zero cost implies identity. It does not identify which transitions are physically possible; that is a separate question. And it does not assign numerical costs to any specific particle or process. The theorem is a logical precondition: it ensures that the ledger’s event count is meaningful, but it says nothing about what the events are.

In the broader derivation, this theorem is one of several baseline rungs that turn boundary assumptions into derived results. The docstring lists items like the octave offset (-8), the neutrino baseline (-54), and the quark baseline (4), all traced to the single input D = 3. The nontriviality theorem is the first rung: it establishes that the ledger distinguishes change from stasis, before any specific masses are computed.

theorem J_at_one : J 1 = 0 := by unfold J; norm_num

/-- **B-20 DERIVED**: Non-triviality is a consequence of T1 + T5.
    Any transition with x = 1 has zero cost and leaves no ledger record.
    Therefore identity transitions are excluded from the physical event count. -/
theorem nontriviality_from_cost (x : ℝ) (hx : 0 < x) (hphys : J x > 0) :
    x ≠ 1 := by
  intro h; subst h; simp [J_at_one] at hphys

/-- J(x) ≥ 0 for all x > 0 (AM-GM). -/
theorem J_nonneg (x : ℝ) (hx : 0 < x) : J x ≥ 0 := by
  unfold J
  have hxne : x ≠ 0 := ne_of_gt hx
  have hxx : x * x⁻¹ = 1 := mul_inv_cancel₀ hxne
  have hsq : 0 ≤ (x - x⁻¹) ^ 2 := sq_nonneg _
  have hexpand : (x - x⁻¹) ^ 2 = x ^ 2 - 2 + x⁻¹ ^ 2 := by
    have : (x - x⁻¹) ^ 2 = x ^ 2 - 2 * (x * x⁻¹) + x⁻¹ ^ 2 := by ring
    rw [hxx] at this; linarith
  have hge : x ^ 2 + x⁻¹ ^ 2 ≥ 2 := by nlinarith
  have hfactor : x + x⁻¹ ≥ 2 := by
    nlinarith [sq_nonneg (x + x⁻¹), sq_nonneg (x - x⁻¹)]
  linarith

What this page does not claim

The theorem does not say which transitions are physically possible. The theorem does not assign numerical costs to any specific particle or process. The theorem does not imply that all real transitions have positive cost.

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

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