Encyclopedia Foundation Foundation Pair Kernel Event Action Ancestry S14 Realized Primitive Posting Pair

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Event Action Ancestry S14 Realized Primitive Posting Pair

A single formal theorem pins down the smallest possible accounting step in a discrete ledger and proves it carries exactly one unit of cost.

Verify this page

The committed posting

The theorem realizedPrimitivePostingPair_has_legalAtomicTick_and_cost concerns a ledger, a discrete record of events. It states that every realized primitive posting pair, a basic two-sided entry in the ledger, has an underlying posting step that is legal, atomic, and cost-minimal. The step's ledger cost is exactly 1, its exact cost under the framework's cost function is Jlog(1), and the sum of its elementary postings over the torus is zero, which is the double-entry conservation condition.

In plain terms, the theorem proves that each primitive posting can be decomposed into a smallest possible unit step. This step is a LegalAtomicTick, a single indivisible ledger operation. The cost of that operation is forced to be 1, not chosen. The conservation condition means the ledger's records balance: what is added on one side is removed on the other.

The theorem is proved in the machine-checked library of formal theorems. It uses no hydrogen, electromagnetic, SI, or protein input, and it introduces no new axioms. The proof is complete, with no gaps.

What the theorem does not claim is equally important. It does not attach any physical duration or energy to the posting step. The step is purely a ledger operation. The theorem also does not mention the reduced Planck constant hbar. A separate theorem in the same module shows that if one assigns a coherence energy and a fundamental duration to a posting, the energy-time action equals hbar, but that assignment is a separate step, not part of this theorem.

The theorem's significance is that it isolates the smallest target-blind missing arrow in the framework. It shows that a primitive posting has a well-defined, minimal cost before any physical interpretation is added. This is the foundation on which the framework's physical constants and scales are later built, but the theorem itself stays within the pure ledger.

THEOREM realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:59 
/-- A realized primitive spatial event is an actual legal atomic ledger tick,
has the exact ledger cost `Jlog 1`, and is double-entry conserved. This is the
strongest committed ancestry before physical energy and duration are added. -/
theorem realizedPrimitivePostingPair_has_legalAtomicTick_and_cost
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N)
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    ∃ A B : LedgerPostingAdjacency.LedgerState 3,
      PostingStep A B ∧
        LegalAtomicTick A B ∧
        JMinimalPostingStep A B ∧
        ledgerL1Cost A B = 1 ∧
        ledgerJlogCost A B = Cost.Jlog 1 ∧
        (∑ k : Fin (TorusCard3 N),
          divF (elementaryPosting event.1 event.2) k) = 0 := by
  obtain ⟨A, B, hpost, hmin, hdiv⟩ :=
    realizedPrimitivePostingPair_has_ledger_event event hevent
  have hlegal : LegalAtomicTick A B :=
    postingStep_implies_legalAtomicTick hpost
  refine ⟨A, B, hpost, hlegal, hmin, hlegal.2, ?_, hdiv⟩
  exact ledgerJlogCost_eq_Jlog1_of_postingStep hpost
THEOREM realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:59 
/-- A realized primitive spatial event is an actual legal atomic ledger tick,
has the exact ledger cost `Jlog 1`, and is double-entry conserved. This is the
strongest committed ancestry before physical energy and duration are added. -/
theorem realizedPrimitivePostingPair_has_legalAtomicTick_and_cost
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N)
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    ∃ A B : LedgerPostingAdjacency.LedgerState 3,
      PostingStep A B ∧
        LegalAtomicTick A B ∧
        JMinimalPostingStep A B ∧
        ledgerL1Cost A B = 1 ∧
        ledgerJlogCost A B = Cost.Jlog 1 ∧
        (∑ k : Fin (TorusCard3 N),
          divF (elementaryPosting event.1 event.2) k) = 0 := by
  obtain ⟨A, B, hpost, hmin, hdiv⟩ :=
    realizedPrimitivePostingPair_has_ledger_event event hevent
  have hlegal : LegalAtomicTick A B :=
    postingStep_implies_legalAtomicTick hpost
  refine ⟨A, B, hpost, hlegal, hmin, hlegal.2, ?_, hdiv⟩
  exact ledgerJlogCost_eq_Jlog1_of_postingStep hpost
THEOREM realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:59 
/-- A realized primitive spatial event is an actual legal atomic ledger tick,
has the exact ledger cost `Jlog 1`, and is double-entry conserved. This is the
strongest committed ancestry before physical energy and duration are added. -/
theorem realizedPrimitivePostingPair_has_legalAtomicTick_and_cost
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N)
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    ∃ A B : LedgerPostingAdjacency.LedgerState 3,
      PostingStep A B ∧
        LegalAtomicTick A B ∧
        JMinimalPostingStep A B ∧
        ledgerL1Cost A B = 1 ∧
        ledgerJlogCost A B = Cost.Jlog 1 ∧
        (∑ k : Fin (TorusCard3 N),
          divF (elementaryPosting event.1 event.2) k) = 0 := by
  obtain ⟨A, B, hpost, hmin, hdiv⟩ :=
    realizedPrimitivePostingPair_has_ledger_event event hevent
  have hlegal : LegalAtomicTick A B :=
    postingStep_implies_legalAtomicTick hpost
  refine ⟨A, B, hpost, hlegal, hmin, hlegal.2, ?_, hdiv⟩
  exact ledgerJlogCost_eq_Jlog1_of_postingStep hpost

What this page does not claim

The theorem does not establish that any physical energy or time is attached to the posting step. The theorem does not mention or derive the value of hbar. The theorem does not select between the identity and ledger-normalized branches for mapping exact-J cost to physical action.

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/PairKernelEventActionAncestryS14.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:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND