Cosmology Cosmogenesis Sim Flow Contribution Pair

The double-entry rule

In the Recognition Science framework's cosmogenesis simulation, the basic unit of activity is a recognition event, a discrete record of one agent (a numbered point) registering a ratio against another. The framework models the ledger, a discrete record of such events, as a list of these records. Each event has a source, a target, and a positive rational ratio. The simulation's recognition ledger, the framework's term for this record of events, is built by a rule of double-entry posting: every event is added together with its reciprocal, the same two agents with the ratio inverted.

The declaration flowContribution_pair proves a conservation law about this double-entry rule. For any single event and any agent, the product of the event's contribution to that agent and its reciprocal's contribution is exactly one. The contribution of an event to an agent is the event's ratio if the agent is one of its two endpoints, and one otherwise. Since the reciprocal event has the inverted ratio, the product of the two contributions is r times r⁻¹, which is 1. This is the kernel-checked reason why the whole simulation conserves a quantity called the flow product, the product of the ratios of all events touching an agent, which the framework proves stays exactly 1 at every agent after the full 8-tick cosmogenesis, for any positive seed.

The theorem is a formal statement in the framework's machine-checked library of formal theorems, meaning its proof is verified by a computer kernel rather than checked only at runtime. It is a statement about the simulation's arithmetic, not about physical cosmology. The framework's library proves this conservation law for the rational-number simulation, and the same law is stated for the real-number dynamics in the framework's earlier modules. The simulation also runs the self-similar recurrence r ↦ 1 + 1/r over the rationals, producing the exact Fibonacci convergents 2, 3/2, 5/3, 8/5, 13/8, which converge to the golden ratio φ, making the emergence of that constant visible as an exact rational sequence.

What the declaration does not claim is broader than what it proves. It does not claim that the double-entry rule is the only possible accounting convention, nor that the conservation law holds for events with a zero ratio, which the theorem explicitly excludes. It does not claim that the flow product being 1 has any empirical content about the actual universe; it is a statement about the framework's own model. The theorem is a precise, narrow result: a local algebraic identity that makes the simulation's global conservation law go through.

/-- 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

/-- **Conservation step.** Posting a paired event leaves the flow product
unchanged at every agent (the `r · r⁻¹ = 1` cancellation). -/
theorem flowProduct_addEvent (L : List QEvent) (e : QEvent) (he : e.ratio ≠ 0)
    (agent : ℕ) : flowProduct (addEvent L e) agent = flowProduct L agent := by
  unfold flowProduct addEvent
  simp only [List.map_cons, List.prod_cons]
  rw [← mul_assoc, flowContribution_pair e he agent, one_mul]

/-- 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 double-entry rule is the only possible accounting convention for a recognition ledger. The conservation law holds for events with a zero ratio, which the theorem explicitly excludes. The flow product being 1 has any empirical content about the actual universe.

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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