Cosmology Cosmogenesis Sim Trace Certificates Seed2

A seed of two

The number 2, when fed into the self-similar recurrence r ↦ 1 + 1/r, generates the sequence 2, 3/2, 5/3, 8/5, 13/8, and so on. These are the Fibonacci convergents, and they approach the golden ratio φ. In the Recognition Science framework, this recurrence is not merely a numerical curiosity; it is the engine of a cosmogenesis, a discrete record of eight recognition events that builds a universe from a seed.

The declaration trace_certificates_seed2 is a machine-checked certificate, a single object that bundles five proved facts about this seed-2 cosmogenesis. It certifies that the event list has exactly 16 entries (eight ticks, each posted with its reciprocal), that a conserved quantity called the flow product equals 1 at the starting agent, that the cadence is eight ticks, that the cycle closes over nine vertices, and that the first tick carries positive cost. Each fact is a theorem in the framework's machine-checked library of formal theorems, proved without unproved assumptions.

The conserved quantity deserves a plain picture. At each agent, the flow product multiplies the ratios of all events touching that agent. When an event is posted together with its reciprocal, the product gains a factor r · r⁻¹ = 1, so the flow product never changes. The theorem cosmogenesis_conserves proves this invariant holds exactly, for any positive seed, after the full eight-tick run. For seed 2, the certificate records the value at agent 0 as exactly 1.

The cost of each tick is also forced. The function qJ(x) = (x + 1/x)/2 − 1 measures the cost of a recognition event with ratio x. For seed 2, the first ratio is 2, and the theorem seed2_first_tick_cost_pos proves this cost is strictly positive. The certificate bundles this with the other four facts into a single TraceCertificates structure, so a reader can check the whole cosmogenesis in one glance.

What the certificate does not claim is just as important. It does not claim that seed 2 is special among seeds; the conservation law holds for any positive seed, and the recurrence converges to φ for any positive start. It does not claim that this computable rational simulation replaces the real-number dynamics proved elsewhere in the framework; it is a mirror over the rationals, chosen because rational arithmetic is exactly computable. And it does not claim that the eight-tick cycle is the only possible cosmogenesis; it certifies one specific, fully checked instance.

/-- **σ conservation, proved.** After the full cosmogenesis, the flow product is
exactly `1` at every agent, for any positive seed. No `decide`, no `sorry`. -/
theorem cosmogenesis_conserves (seed : ℚ) (hs : 0 < seed) (agent : ℕ) :
    flowProduct (cosmogenesis seed) agent = 1 :=
  flowProduct_foldl agent (cosmoEvent seed)
    (fun t => (recurSeq_pos seed hs t).ne') 8

/-- The cosmogenesis posts sixteen events (eight ticks, each paired). -/
theorem cosmogenesis_length (seed : ℚ) : (cosmogenesis seed).length = 16 := by
  have h : (cosmogenesis seed).length = 2 * 8 := foldl_addEvent_length (cosmoEvent seed) 8
  omega

/-- 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

/-- Kernel-checked trace certificates for the canonical seed-2 run. -/
theorem trace_certificates_seed2 : TraceCertificates where
  event_count := cosmogenesis_length 2
  sigma_at_zero := cosmogenesis_conserves 2 (by norm_num) 0
  cadence_eight_ticks := cosmogenesis_tick_count 2
  cyc_closed_nine_vertices := cyc_length
  first_tick_cost_pos := seed2_first_tick_cost_pos

What this page does not claim

The certificate does not claim seed 2 is the only seed that yields a valid cosmogenesis. It does not claim the rational simulation replaces the real-number dynamics proved elsewhere. It does not claim the eight-tick cycle is the only possible cosmogenesis structure.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: