Gravity Analysis Regge4 Dflat Second Variation Schlaefli Elevation To Candidate

The missing face

In four-dimensional spacetime, Einstein's theory of gravity describes how matter curves space and time. A simpler approximation, Regge calculus, builds spacetime from flat four-dimensional blocks, like a curved surface made from flat triangles. The Recognition Science framework's machine-checked library of formal theorems studies how these two descriptions connect, and one of its results is a precise negative finding about a proposed bridge between them.

The declaration schlaefli_elevation_to_candidate_misses_eh_face proves that a number called the Einstein-Hilbert coefficient, which is -1/4 in the framework's units, is not equal to -1/16, the value that a candidate formula produces on a specific test direction. This is a theorem, meaning it is a proved result in the machine-checked library, not a conjecture or a numerical approximation. The theorem is a consequence of a broader result, candidate_face_ne_eh, which states both that -1/16 differs from -1/4 and that a related dictionary factor equals 1.

The proof works by evaluating a particular mathematical object, the distinct-hinge moment form, along a chosen direction called symbolDir. On that direction, the candidate produces -1/16, while the target Einstein-Hilbert value is -1/4. The theorem does not say the candidate is wrong everywhere; it says that on this one specific face of the candidate's behavior, the two disagree. The result is a precise, local statement about a single component of a larger comparison.

What the theorem does not claim is broader. It does not prove that the candidate formula is completely incorrect, nor does it prove that the full elevation from the nonlinear Regge action to the candidate is impossible. That full elevation remains an open problem in the framework's library. The theorem also does not claim that the candidate cannot be repaired or that no other bridge between Regge calculus and Einstein's theory exists. It establishes one fact: on this particular test, the candidate misses the target.

In Recognition Science, this result is part of a status report that tracks which pieces of a larger proof are complete and which are not. The library records that the candidate is identified, its face is evaluated, and the flat Freudenthal Schläfli summand table is present. But the pathwise Schläfli identity, which would be needed for the full elevation, is absent. The theorem about the missing face is a checkpoint, not a conclusion: it tells researchers exactly where the gap is, so the next step is clear.

theorem schlaefli_elevation_to_candidate_misses_eh_face :
    (-1 / 16 : ℝ) ≠ einsteinHilbertTTCoefficient4D :=
  candidate_face_ne_eh.1

/-- **THEOREM (falsifier arithmetic).** The candidate's continuum face on
normalized TT at `symbolDir` is `-1/16`, which is not the frozen EH
coefficient `-1/4`.  Density dictionary survivor is already `1`. -/
theorem candidate_face_ne_eh :
    (-1 / 16 : ℝ) ≠ einsteinHilbertTTCoefficient4D ∧
      survivingDictionaryFactor4D = 1 :=
  ⟨by rw [einsteinHilbertTTCoefficient4D_eq]; norm_num, rfl⟩

What this page does not claim

The candidate formula is incorrect in all directions. The full elevation from the nonlinear Regge action to the candidate is impossible. The theorem establishes any property of the candidate beyond the single evaluated face.

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/Regge4DFlatSecondVariation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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