Foundation Pair Kernel Coherence Event Constructor S15 Cycle Distributed Energy

The eight-event energy rule

A coherence event is a unit of recognition in the framework: one discrete record of a posting, the smallest step the ledger keeps. The framework's ledger, a discrete record of events, assigns each such event a coherence energy, written E_coh. The declaration cycleDistributedEnergy_eight_events_eq_coherence proves a simple accounting fact: if you spread that energy evenly across eight ticks, each tick carries E_coh divided by 8, and eight of those shares sum exactly back to E_coh. The proof is a direct algebraic identity, 8 times (E_coh / 8) equals E_coh, checked by the machine.

The statement matters because it pins down one layer of how the framework accounts for energy across its eight-tick recognition cycle. The theorem itself does not say that any real posting actually has this energy, nor that the energy is physically delivered in eight equal installments. It is a consistency condition: if a cycle distributes the coherence energy uniformly, the arithmetic closes. The framework keeps this as a separate, target-blind predicate, meaning the claim is about the accounting identity, not about the physical realization of energy flow.

What the declaration does not claim is equally precise. It does not assert that each individual event carries the full coherence energy E_coh; in fact, a companion theorem, cycleDistributedEnergy_not_per_event, shows the opposite: the distributed energy is not equal to E_coh per event, since E_coh divided by 8 is strictly less than E_coh. It also does not claim that the distributed energy is the correct or actual energy of any posting; that would require a separate theorem tying the distribution to the framework's forcing chain. The theorem only establishes the arithmetic relation between the distributed shares and the total.

In plain terms, the declaration is a ledger-balance guarantee. It ensures that if the framework chooses to model an eight-tick cycle as carrying one coherence event's energy spread evenly, the accounts add up. It is a necessary piece of the framework's internal consistency, not a physical law about how energy behaves. The distinction matters because the framework's claims are only as strong as their proofs, and this proof is deliberately narrow: it closes the arithmetic, and leaves the physics to other declarations.

theorem cycleDistributedEnergy_eight_events_eq_coherence
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N) :
    8 * cycleDistributedPostingEnergy3 event =
      Constants.E_coh := by
  unfold cycleDistributedPostingEnergy3
  ring

theorem cycleDistributedEnergy_not_per_event :
    ¬ RealizedPostingEnergyEqualsCoherence3
      (@cycleDistributedPostingEnergy3 3 _) := by
  intro henergy
  obtain ⟨event, hevent⟩ :=
    realizedPrimitivePostingPair3_exists
  have h := henergy event hevent
  unfold cycleDistributedPostingEnergy3 at h
  have hpos := Constants.E_coh_pos
  nlinarith

What this page does not claim

The theorem does not claim that any real posting carries the distributed energy. The theorem does not claim that the distributed energy is the actual energy of a coherence event. The theorem does not claim that the eight-tick cycle is physically realized.

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/Foundation/PairKernelCoherenceEventConstructorS15.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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