Gravity Analysis Regge4 Dtensor Algebraic Closer Distinct Hinge Moment Form Axis

A computed ray value

In four-dimensional Regge calculus, a discrete model of spacetime built from flat four-simplices, the distinct-hinge moment is a number attached to a chosen hinge, the shared triangle where two tetrahedra meet, that measures how the geometry responds to a change in the metric. The declaration distinctHingeMomentForm_axisTTCross_symbolDir is a machine-checked theorem stating that for a particular polarization, called axisTTCross, and a particular direction vector, called symbolDir, this moment equals exactly minus one quarter.

To see what this means, consider the quadratic form that produces the moment. The form takes a 4 by 4 matrix E, representing a metric perturbation, and a direction vector dir, and returns a real number. The theorem evaluates this form at two specific inputs and records the output. It is one of several such evaluations in the framework's library, a machine-checked collection of formal theorems, that together probe how the moment behaves along special rays in the space of polarizations. A companion result shows that scaling the matrix E by a constant c scales the moment by c squared, a homogeneity property that holds for all inputs, not just the special ray.

The value minus one quarter is not an accident of the chosen ray. The same value appears for the plus polarization along the same direction, and after normalizing by the squared length of the direction vector, the continuum-facing coefficient becomes minus one sixteenth on both the plus and cross rays. These coincidences suggest a deeper structure, but the framework does not claim to have found it. A universal closed-form expression for the moment as a function of E and dir, valid for all transverse-traceless polarizations and nonzero directions, remains an open target. The arithmetic factor of four that separates the discrete minus one sixteenth from the Einstein-Hilbert continuum value of minus one quarter is likewise open, with no geometric explanation installed.

What the declaration does not claim is as important as what it proves. It does not establish the closed form, does not explain the factor of four, and does not touch the separate question of whether the discrete theory recovers the continuum action in a limit. It is a single computed point, banked as a theorem, in a larger search for structure that is still underway.

theorem distinctHingeMomentForm_axisTTCross_symbolDir :
    distinctHingeMomentForm axisTTCross symbolDir = (-1 / 4 : ℝ) :=
  m2TransportedAllOrbitMomentDistinctHinge_axisTTCross_symbolDir

theorem distinctHingeMomentForm_smul (c : ℝ) (E : Mat4) (dir : Fin 4 → ℝ) :
    distinctHingeMomentForm (c • E) dir =
      c ^ 2 * distinctHingeMomentForm E dir :=
  m2TransportedAllOrbitMomentDistinctHinge_smul c E dir

What this page does not claim

The declaration does not provide a closed-form expression for the moment valid on all polarizations. The declaration does not explain the factor of four separating the discrete and continuum coefficients. The declaration does not address whether the discrete theory recovers the continuum action in a limit.

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/Regge4DTensorAlgebraicCloser.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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