Gravity Regge Convergence Registry Ricci Convergence Faithful

The faithful projection

Regge calculus is a discrete approach to general relativity in which spacetime is approximated by flat pieces glued along hinges. A central question is whether the discrete theory converges to the continuous one as the pieces shrink. The ReggeConvergenceRegistry is a machine-checked library of formal theorems that bundles four external convergence results into one named structure, each field typed by the exact original proposition.

The declaration ricci_convergence_faithful is a theorem stating that the registry's ricci_convergence field, when projected out, recovers a proof of the original regge_ricci_convergence_axiom. In plain terms: if you have a registry value, you can extract from it the exact Regge Ricci-scalar convergence statement, with no weakening. The theorem is a projection, not a new result; it certifies that the registry is a faithful repackaging of the axiom it contains.

The registry itself is a structure with five fields: the Cheeger–Müller–Schrader measure bound (1984), a special quadratic action-convergence hypothesis, the Ricci-scalar convergence axiom, a Riemann or holonomy convergence axiom, and a provenance list documenting each as an external-mathematics result with a status tag. The ricci_convergence_faithful theorem is one of four faithful-projection theorems, each showing that a registry field recovers its original proposition. A round-trip theorem, mk_roundtrip, confirms that building a registry from the projections of a value recovers that value exactly.

In the Recognition Science framework, this declaration does not prove that Regge calculus converges to Einstein gravity. The convergence statements themselves are external hypotheses, axiomatized as inputs. The theorem's role is organizational: it guarantees that the registry does not alter, weaken, or strengthen the convergence content it stores. It is a bookkeeping certificate, not a physics result.

/-- The `ricci_convergence` field of a registry is a proof of the original
`regge_ricci_convergence_axiom` proposition. -/
theorem ricci_convergence_faithful (r : ReggeConvergenceRegistry) :
    regge_ricci_convergence_axiom := r.ricci_convergence

/-- A registry that consolidates the four external convergence Propositions
from `NonlinearConvergence` into a single named structure.

Each field is typed by the **exact** original proposition, so this is a
faithful repackaging, not a weakening.  The `provenance` field documents
each input as an external-mathematics result with a status tag. -/
structure ReggeConvergenceRegistry where
  /-- CMS Theorem 5.1 curvature-measure bound (Cheeger–Müller–Schrader 1984). -/
  cms_measure_bound : cms_theorem_5_1_measure_bound
  /-- Special-purpose `O(a²)` action-convergence hypothesis. -/
  special_quadratic : special_quadratic_regge_to_eh_convergence_hypothesis
  /-- Regge Ricci-scalar convergence axiom. -/
  ricci_convergence : regge_ricci_convergence_axiom
  /-- Regge Riemann / holonomy convergence axiom. -/
  riemann_convergence : regge_riemann_convergence_axiom
  /-- Documentation of each external-math result with a status tag. -/
  provenance : List String

/-- The default provenance list documenting each external-math result
with a status tag. -/
def defaultProvenance : List String :=
  [ "cms_measure_bound: Cheeger–Müller–Schrader (1984), Theorem 5.1 — curvature-measure convergence; status: external theorem, axiomatized"
  , "special_quadratic: stronger O(a²) action-convergence hypothesis; status: external hypothesis, axiomatized"
  , "ricci_convergence: Regge Ricci-scalar O(a²) convergence; status: external hypothesis, axiomatized"
  , "riemann_convergence: Regge holonomy / Riemann convergence; status: external hypothesis, axiomatized" ]

/-- Building a registry from the projections of `r` recovers `r`.
This confirms the registry is a faithful repackaging. -/
theorem mk_roundtrip (r : ReggeConvergenceRegistry) :
    mk r.cms_measure_bound r.special_quadratic r.ricci_convergence
       r.riemann_convergence r.provenance = r := rfl

What this page does not claim

The declaration does not prove that Regge calculus converges to Einstein gravity. The declaration does not derive the Ricci-scalar convergence axiom from first principles. The declaration does not establish the physical validity of the external convergence hypotheses.

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/Gravity/ReggeConvergenceRegistry.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

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