Gravity Analysis Regge4 Dtorus Continuum Limit Bloch Cell Sum4 Dcos Mul Cos Open

The 4D cell-sum identity

In numerical work on gravity, one often approximates a smooth curved space by a fine grid of flat pieces, then studies how the grid's equations behave as the grid spacing shrinks to zero. A recurring chore is tracking how many grid points there are and how the equations' weights scale with that count. In four dimensions, a grid with N points along each side has N^4 sites, and a natural density weight is N^-4. The declaration BlochCellSum4DCosMulCosOpen_scalar_holds is a machine-checked proof of a small but essential piece of that bookkeeping: it establishes that a certain cell-sum factor, which arises from summing products of cosines over the grid, equals N^4 / 2, and that when this factor is multiplied by the density weight 2/N^4, the product is exactly 1.

The meaning is a cancellation. In the three-dimensional analogue, the same kind of factor and weight cancel to leave the continuum limit clean. The four-dimensional statement proves the identical cancellation holds with N^4 in place of N^3. This is a purely algebraic identity about the grid itself; it does not depend on the shape of the space being approximated. The proof is a formal theorem in the framework's machine-checked library, meaning the cancellation is verified by the kernel of the proof checker, not by hand-waving.

What the declaration does not claim is the substantive physics. The 4D cosine cell-sum identity is a named open problem in the same file; the declaration proves the scalar version of it, but the full identity, which would involve the structure of the distinct-hinge fold and the Schläfli elevation, remains open. The declaration also does not establish the continuum limit itself. It proves the bookkeeping factor cancels, but the statement that the finite grid's normalized symbols actually converge to a continuum symbol as N grows is a separate, open question, formalized as TorusC2DensityExtensionOpen.

In plain terms: this is a lemma about counting, not a law of gravity. It says the arithmetic of the grid works out so that a later, harder step is not blocked by a factor of N^4. The hard step, showing that the discrete approximation genuinely approaches the smooth continuum, is still to be done.

/-- THEOREM: the scalar half of the 4D cell-sum OPEN Prop holds
(definition + cancellation).  The phase-sum half remains future work in
a dedicated `BlochCellSum4D` module. -/
theorem BlochCellSum4DCosMulCosOpen_scalar_holds :
    BlochCellSum4DCosMulCosOpen := by
  intro N hN
  exact ⟨rfl, density_cellSum_cancellation (N := N)⟩

/-- THEOREM: the scalar half of the 4D cell-sum OPEN Prop holds
(definition + cancellation).  The phase-sum half remains future work in
a dedicated `BlochCellSum4D` module. -/
theorem BlochCellSum4DCosMulCosOpen_scalar_holds :
    BlochCellSum4DCosMulCosOpen := by
  intro N hN
  exact ⟨rfl, density_cellSum_cancellation (N := N)⟩

What this page does not claim

The declaration does not prove the 4D cosine cell-sum identity in its full form, only the scalar version. The declaration does not establish the continuum limit of the discrete gravity approximation. The declaration does not imply any physical law about gravity itself; it is a statement about grid bookkeeping.

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/Regge4DTorusContinuumLimit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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