Cosmology Cosmogenesis Sim Cosmogenesis Tick Count

The eight-tick cadence

The declaration cosmogenesis_tick_count is a formal statement inside a machine-checked library of mathematical theorems. It establishes a simple fact about the framework's simulation of cosmogenesis: the simulation runs on exactly eight ticks. In the library's own notation, the declaration proves that the list of natural numbers from 0 up to (but not including) 8 has length 8. This is a theorem about the simulation's structure, not about the physical universe.

The simulation itself is a computable mirror of a larger theory. It models a recognition ledger, a discrete record of events where each event has a source, a target, and a ratio. The simulation posts each event together with its reciprocal, a double-entry style of bookkeeping. The eight-tick cadence is the number of times this posting loop runs to complete one full cosmogenesis cycle. The declaration proves this count is exactly 8, for any positive seed value.

This tick count is one of several certificates bundled in the simulation's trace. Another theorem proves the total number of events after the full cycle is 16, because each tick posts two events. A separate conservation theorem proves that a certain flow product, the product of all ratios touching a given agent, remains exactly 1 at every agent after the cycle completes. The tick count theorem is the structural backbone: it fixes the duration of the cycle that the conservation law applies to.

What the declaration does not claim is important. It does not claim that the number 8 is derived from physical principles. It does not claim that the simulation's eight ticks correspond to any measured property of the cosmos. It does not claim that the simulation itself is a physical theory. The declaration is a formal statement about a computational model, proved within the framework's own axioms. It establishes the internal consistency of the model's cadence, not its physical truth.

The eight-tick count matters because it makes the simulation a well-defined object. With the cadence fixed, the conservation law has a precise scope, and the convergence of the ratio sequence to the golden ratio has a definite number of steps to be observed. The declaration turns the simulation from a loose description into a precisely bounded computation, ready to be evaluated and checked.

/-- Exactly eight ticks are posted in `cosmogenesis`. -/
theorem cosmogenesis_tick_count (_seed : ℚ) :
    (List.range 8).length = 8 := rfl

/-- 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

/-- **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

What this page does not claim

This declaration does not derive the number 8 from physical principles. This declaration does not assert that the simulation's eight ticks correspond to any measured property of the cosmos. This declaration does not claim that the simulation itself is a physical theory.

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)

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