Cosmology Cosmogenesis Sim Seed2 First Tick Cost Pos

The first tick's cost

The declaration seed2_first_tick_cost_pos is a theorem in a machine-checked library of formal theorems. It concerns a specific, computable model of the first event in a simulated cosmogenesis. The claim itself is narrow: for a particular starting value, the cost of the first recognition event is strictly greater than zero. A recognition event, in this framework, is a basic unit of interaction recorded in a ledger, a discrete record of events. The cost of such an event is given by a specific function, J(x) = (x + 1/x)/2 - 1, where x is the ratio between the two agents involved.

The theorem proves that for the seed value 2, the cost of the first event is positive. This is not a deep cosmological statement; it is a precise, checkable fact about a rational number. The proof relies on two simpler facts. First, the seed value 2 is positive, and the recurrence relation that generates subsequent ratios, r → 1 + 1/r, keeps all ratios positive. Second, the seed value 2 is not equal to 1, and the cost function J(x) is zero only when x = 1. For any positive x other than 1, J(x) is positive, because it can be rewritten as (x-1)² / (2x), a square divided by a positive number.

In the broader context of the simulation, this theorem is one of five certificates that together assert the simulation runs correctly. The others verify the event count, the conservation of a flow product, the eight-tick cadence, and the closure of the cycle. The positive cost of the first tick is a necessary condition for the simulation to be meaningful: it ensures that the very beginning of the process is not a free, zero-cost step. This matters because the framework's central result, the forcing of the cost function J, derives the structure of reality from the assumption that recognition is never free.

What the theorem does not claim is equally important. It does not claim that the seed value 2 is the only possible starting point, nor does it claim that the cost of the first event is large or has any particular magnitude beyond being positive. It does not, by itself, establish the entire forcing chain or the emergence of the golden ratio. Those are separate theorems. This declaration is a single, verified step in a much larger construction, a guarantee that one specific number in a simulation is on the right side of zero.

/-- First-tick cost is positive for the canonical seed-2 cosmogenesis run. -/
theorem seed2_first_tick_cost_pos : 0 < qJ (recurSeq 2 0) :=
  qJ_pos (recurSeq_pos 2 (by norm_num) 0) seed2_distinction

/-- Positive ratio distinct from unity incurs positive `qJ` cost. -/
theorem qJ_pos {x : ℚ} (hx : 0 < x) (hne : x ≠ 1) : 0 < qJ x := by
  have h : qJ x = (x - 1) ^ 2 / (2 * x) := by
    unfold qJ
    field_simp [hx.ne']
    ring
  have h01 : x - 1 ≠ 0 := sub_ne_zero.mpr hne
  have hsq : 0 < (x - 1) ^ 2 := by
    rw [pow_two]
    exact mul_self_pos.mpr h01
  rw [h]
  exact div_pos hsq (by linarith)

/-- Default seed-2 run posts a genuine distinction at tick 0. -/
theorem seed2_distinction : (recurSeq 2 0) ≠ 1 := by native_decide

What this page does not claim

The theorem does not claim that seed 2 is the only valid starting value for the simulation. It does not claim that the first tick's cost is large or has any specific magnitude beyond being positive. It does not, by itself, establish the emergence of the golden ratio or the three-dimensionality of space.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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