Cosmology Cosmogenesis Sim

A computable mirror of the first ticks

Cosmology cosmogenesis sim is a simulation module in the Recognition Science framework's machine-checked library of formal theorems. It gives a computable form to the framework's account of the earliest moments of a universe: a discrete record of events, called the ledger, in which each event pairs a source with a target and a ratio. The module runs this record over rational numbers, the fractions like 3/2 and 5/3, so that a computer can evaluate it exactly, with no rounding error.

The simulation's central result is a conservation law. The module defines a quantity called the flow product at an agent, the product of the ratios of all events touching that agent. Each event is posted together with its reciprocal, the inverse ratio, so the product multiplies by r times 1/r, which is 1. The theorem cosmogenesis_conserves proves that after the full eight-tick run, the flow product at every agent is exactly 1, for any positive starting seed. This is proved in the kernel, the core of the proof checker, not merely checked by running the code.

The module also makes the emergence of the golden ratio visible as an exact sequence of fractions. The recurrence r maps to 1 + 1/r, starting from a seed, produces the Fibonacci convergents 2, 3/2, 5/3, 8/5, 13/8, and so on. These fractions converge to phi, the golden ratio, about 1.618. So the framework's claim that phi is the unique self-similar scaling appears here as a concrete, computable sequence rather than an abstract theorem.

In Recognition Science, the full cosmogenesis runs over the real numbers in the PreBigBang and FirstTick modules. This sim module mirrors that dynamics over the rationals, making it a Lean object you can evaluate. The eight-tick cadence is fixed: the simulation posts 16 events, two per tick, one for each event and its reciprocal. The cost of each tick, measured by the framework's J function, is positive, and the module bundles these facts into a certificate structure, a single object that packages the key properties for seed 2.

What this establishes in plain language is that the framework's account of early cosmogenesis is not just a formal story. It is a concrete, checkable computation: a finite sequence of rational numbers, a proved conservation law, and a visible path to the golden ratio. The reader can see the first ticks happen, the ledger balance, and the self-similar pattern take shape, all without leaving the domain of exact arithmetic.

/-- **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

/-- 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

What this page does not claim

The module does not prove the real-number dynamics of cosmogenesis; it mirrors them over the rationals. The conservation law holds for the flow product, not for any other quantity such as total cost. The simulation does not derive the fine-structure constant or any other measured physical constant.

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: