Gravity Analysis Regge4 Dexact Action Symbol Exact Star Member Offsets Incomplet

A status flag, not a result

In a formal library, where every statement is checked by a computer, a declaration can sometimes be a bookkeeping flag rather than a mathematical discovery. The declaration exact_star_member_offsets_incomplete_closed is exactly that. It states that the proposition exact_star_member_offsets_incomplete is false. That proposition, in turn, is defined as the constant False. So the theorem proves, by definition, that a certain named goal has not been achieved: the star-member offsets for the exact flat cross-term symbol are incomplete.

The context is a specific piece of gravity research. The framework works with a discrete model of spacetime called a Regge action, where the smooth manifold of general relativity is replaced by a flat, piecewise-linear approximation. The work concerns the Hessian, the second derivative, of this action on a particular torus. The goal is to compute a symbol, a kind of spectrum, for this Hessian on plane-wave-like strain fields. The symbol depends on how you transport the origins of the star-shaped neighborhoods of the hinges, the edges of the triangulation. The declaration records that this transport, for the distinct-hinge orbits, is not yet correctly defined for all cases. The t11 and t12 orbit types have their offsets defined, but the others do not.

This is not a failure of the framework. It is an honest status report. The library has a structure, ExactActionSymbolStatus, with boolean fields for each part of the project. The field otherOrbitOffsetsIncomplete is set to false, meaning the offsets are indeed incomplete. The theorem exact_action_srs_still_open further records that the full recovery of the Einstein-Hilbert action from the discrete one remains open. These declarations are the library's way of saying: this specific piece of the puzzle is not done, and we are not pretending it is.

What the declaration does not claim is any mathematical content about the symbol itself. It does not prove that the symbol is wrong, or that it cannot be completed, or that the approach is invalid. It is a meta-statement about the state of the formalization. It is the difference between saying "this theorem is false" and saying "this theorem has not been proved yet." The declaration is the latter, encoded with the precision that a proof assistant demands.

theorem exact_star_member_offsets_incomplete_closed :
    exact_star_member_offsets_incomplete = False := rfl

/- Legacy fold retained for comparison; after `H_fold` it is not the
continuum symbol. Continuum Props bind to `finiteExactReggeSymbol`.
Former `fold_retained_as_legacy_only` concluded `True` from an inequality
hypothesis: that is a repository status note, not a math claim. Status is
recorded by `exactActionSymbolStatus.foldRetainedAsLegacy`. -/

/-- Closed: non-`t11` orbits now use per-edge transported origins
(`ReggeBlochStarEdgeOrigins4D`).  Residual e0 isotropy of the fold face
remains OPEN (MEASURED; not this flag). -/
def exact_star_member_offsets_incomplete : Prop := False

def exactActionSymbolStatus : ExactActionSymbolStatus where
  continuumReboundToExact := true
  foldRetainedAsLegacy := true
  t11t12StarOffsetsDefined := true
  otherOrbitOffsetsIncomplete := false
  edgeOriginsM2Banked := true
  srsInhabited := false
  gapActionRecovery := false

/-- Local status flags remain open; geometric ContinuumSymbolIs Tendsto
is the Preflight ledger gate.  Edge-origin m² decide-certs are banked
elsewhere and do not inhabit `S_RS`. -/
theorem exact_action_srs_still_open :
    exactActionSymbolStatus.srsInhabited = false ∧
      exactActionSymbolStatus.gapActionRecovery = false := by
  decide

What this page does not claim

The declaration does not claim that the exact flat cross-term symbol is incorrect. The declaration does not claim that the star-member offsets cannot be completed. The declaration does not claim that the full Einstein-Hilbert action recovery is impossible.

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/Analysis/Regge4DExactActionSymbol.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.

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