Cosmology Cosmogenesis Sim Cosmogenesis Conserves

The conservation theorem

In mathematics, a conservation law is a quantity that does not change as a system evolves. The declaration cosmogenesis_conserves is a formal theorem proving such a law for a specific computational model of cosmogenesis. The model is a finite list of events, each event recording a source, a target, and a positive rational ratio. The quantity conserved, called the flow product at an agent, multiplies together the ratios of all events touching that agent. The theorem states that after the full eight-tick creation process, this flow product equals exactly 1 at every agent, for any positive starting seed.

The proof works by double-entry bookkeeping. Each event is posted together with its reciprocal, an event that swaps source and target and inverts the ratio. For any agent, the contribution of an event and its reciprocal multiply to 1, because a ratio times its inverse is 1. Since every event is posted in such a pair, the total product at each agent remains unchanged from its initial empty value, which is 1. The theorem is proved in the kernel of the machine-checked library of formal theorems, with no runtime checks and no unproved assumptions.

This conservation law is not a statement about the physical universe. It is a property of a computational simulation, a Lean object that can be executed to produce explicit rational numbers. The simulation mirrors a broader framework's dynamics, but the theorem itself only concerns this discrete, computable model. It does not claim that the flow product is conserved in any actual physical process, nor that the eight-tick sequence is the unique way to model cosmogenesis. The theorem's significance is that it gives a kernel-checked, executable witness to a conservation principle within a specific formal system.

The conservation law also connects to a familiar sequence. The simulation's self-similar recurrence, r maps to 1 + 1/r, generates the Fibonacci convergents 2, 3/2, 5/3, 8/5, 13/8, and so on, which converge to the golden ratio. This means the emergence of the golden ratio is visible as an exact rational sequence within the simulation. However, the conservation theorem itself does not depend on this convergence; it holds for any positive seed, regardless of whether the sequence approaches the golden ratio.

/-- **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

/-- An event and its reciprocal contribute a factor of exactly `1` at every
agent: either both touch it (factor `r · r⁻¹ = 1`) or neither does (factor `1`). -/
theorem flowContribution_pair (e : QEvent) (he : e.ratio ≠ 0) (agent : ℕ) :
    flowContribution agent e * flowContribution agent (qreciprocal e) = 1 := by
  simp only [flowContribution, qreciprocal]
  by_cases h : e.source = agent ∨ e.target = agent
  · rw [if_pos h, if_pos h.symm]
    exact mul_inv_cancel₀ he
  · rw [if_neg h, if_neg (mt Or.symm h)]
    ring

/-- **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 self-similar recognition recurrence `r ↦ 1 + 1/r` over `ℚ`.
Exact Fibonacci convergents to `φ`. -/
def recurSeq (seed : ℚ) : ℕ → ℚ
  | 0 => seed
  | (n + 1) => 1 + (recurSeq seed n)⁻¹

What this page does not claim

The theorem does not assert that the flow product is conserved in any actual physical process. The theorem does not claim that the eight-tick sequence is the unique way to model cosmogenesis. The conservation law does not depend on the sequence converging to the golden 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/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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