Foundation Pair Kernel Event Metric Pricing S16 Spatial Temporal Only Does Not R

The spatial-temporal-only configuration

In the Recognition Science framework, a posting event is a discrete record of a recognition: a line in the ledger that says something happened, when it happened, and what its local configuration was. The configuration of an event is the list of independent facts that the event carries. One such configuration, called spatial-temporal-only, assigns to every event a fixed degree count: the spatial dimension D plus one for the temporal successor, so D plus 1 in total.

The framework's library of machine-checked formal theorems proves that this spatial-temporal-only configuration does not read parent degrees. Reading parent degrees means that the configuration's degree count is derived from the actual number of parent configurations that the event references. The theorem states that the spatial-temporal-only configuration and the parent-derived configuration are not the same function. The spatial-temporal-only configuration is a definitional choice, a model, not a derived consequence.

What the theorem does not claim is that the spatial-temporal-only configuration is impossible or inconsistent. It is a perfectly valid configuration within the framework. The theorem only establishes a separation: a configuration that ignores parent degrees cannot be the one that the framework's forcing chain uses to derive three spatial dimensions. The parent-derived configuration, which reads parent degrees, is the one that carries the forcing result.

This separation is what keeps the framework's three-dimensional result honest. The forcing chain that derives three dimensions depends on a configuration that reads the actual parent structure. A configuration that only tracks space and time, without reading the parent degrees, is a different object and does not inherit that result. The theorem is a boundary marker, not a construction.

theorem ledgerCostEnergy_does_not_read_recognitionPrice :
    ¬ PostingEnergyReadsRecognitionConfigurationPrice3
      canonicalChannel
      (@parentDerivedPostingConfigurationDimension3 3 _)
      (ledgerCostEventKinematics3 3) := by
  intro hread
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 :=
    ⟨pair, hpair⟩
  have h := hread event
  change
    Cost.Jlog 1 =
      recognitionConfigurationPrice canonicalChannel
        (parentDerivedPostingConfigurationDimension3 event) at h
  rw [recognitionConfigurationPrice_eq_phi_zpow_neg] at h
  rw [show
      parentDerivedPostingConfigurationDimension3 event =
        GapDerivation.configDim GapDerivation.D by
          exact
            parentDerivedConfiguration_has_forcedDimension
              event] at h
  exact ledgerJlogUnitCost_ne_constants_hbar
    (h.trans GapDerivation.hbar_exponent_eq_configDim.symm)

/-- Configuration countermodel: omitting the balance degree leaves the
spatial plus temporal count `D+1`, while every event still has all committed
parent facts. -/
def spatialTemporalOnlyConfigurationDimension3
    {N : ℕ} [NeZero N] :
    PostingConfigurationDimension3 N :=
  fun _ => GapDerivation.D + 1

What this page does not claim

The spatial-temporal-only configuration is inconsistent or impossible. The theorem proves that space and time are the only relevant degrees of freedom. The theorem establishes the three-dimensional result for the spatial-temporal-only configuration.

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/PairKernelEventMetricPricingS16.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: