Gravity Seven Gaps Hypersurface Deformation Bracket Dgen Sym Ham One

The translation symmetry

In the canonical formulation of general relativity, the Hamiltonian and momentum constraints generate deformations of a spatial hypersurface. The Poisson bracket of a momentum constraint with a Hamiltonian constraint encodes how a time evolution followed by a spatial shift differs from the reverse order. In the continuum, when the shift is constant, this bracket vanishes, reflecting translation invariance.

On a periodic one-dimensional lattice, the Recognition Science framework constructs a finite-dimensional phase space with configuration q and conjugate momentum π, and defines a Poisson bracket from honest derivatives. The momentum generator DgenSym (a discretized momentum constraint using a symmetric difference) and the Hamiltonian generator Ham (a discretized Hamiltonian constraint) are quadratic polynomials in the field variables. The theorem bracket_DgenSym_Ham_one proves that for a constant lapse function, the bracket of these two generators is exactly zero: {DgenSym, Ham[1]} = 0.

This result is not a numerical approximation. It is an identity in the machine-checked library of formal theorems, derived from the definitions with no additional axioms. The proof follows from a more general relation, bracket_DgenSym_Ham, which gives the bracket for an arbitrary lapse as a point-split smearing of the Hamiltonian density by the lattice derivative of the lapse. Setting the lapse to the constant function 1 makes that derivative vanish, and the bracket collapses to zero.

The significance is that the symmetric difference is the correct lattice discretization for preserving translation symmetry. The naive one-sided difference, in contrast, does not commute with the Hamiltonian for constant lapse; the obstruction is an explicit second-order lattice artifact that vanishes only in the continuum limit. The symmetric version closes exactly, a necessary property for a consistent discrete constraint algebra.

This theorem does not establish full discrete gravity. The system is one polarization of a linearized field on a 1D lattice, with no metric degrees of freedom. The full Dirac algebra, including the Jacobi identity and the continuum limit, remains open. The theorem is a single rung in a larger program, not the whole ladder.

/-- THEOREM. Constant lapse: the symmetric-difference momentum generator
commutes exactly with the Hamiltonian, `{Dsym_a, H[1]} = 0` (exact discrete
translation invariance). -/
theorem bracket_DgenSym_Ham_one (a : ZMod n) (x : PhaseSpace n) :
    bracket (DgenSym a) (Ham (fun _ => 1)) x = 0 := by
  rw [bracket_DgenSym_Ham]
  simp

/-- THEOREM. Constant lapse: the symmetric-difference momentum generator
commutes exactly with the Hamiltonian, `{Dsym_a, H[1]} = 0` (exact discrete
translation invariance). -/
theorem bracket_DgenSym_Ham_one (a : ZMod n) (x : PhaseSpace n) :
    bracket (DgenSym a) (Ham (fun _ => 1)) x = 0 := by
  rw [bracket_DgenSym_Ham]
  simp

/-- THEOREM. Constant lapse: the symmetric-difference momentum generator
commutes exactly with the Hamiltonian, `{Dsym_a, H[1]} = 0` (exact discrete
translation invariance). -/
theorem bracket_DgenSym_Ham_one (a : ZMod n) (x : PhaseSpace n) :
    bracket (DgenSym a) (Ham (fun _ => 1)) x = 0 := by
  rw [bracket_DgenSym_Ham]
  simp

What this page does not claim

This theorem does not prove full discrete gravity, only a single polarization on a 1D lattice. It does not establish the Jacobi identity or the continuum limit of the constraint algebra. It does not claim that the one-sided difference is a valid discretization for any purpose.

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

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