Gravity Analysis Regge Ttcontinuum Closer

The continuum closer

In Recognition Science, a framework that derives physical laws from a forced accounting of discrete events, gravity is studied through a lattice: a regular grid of points where the geometry is defined only at the nodes. The Regge action is a classical way to write general relativity on such a lattice, using the lengths of edges and angles between faces instead of smooth fields. The module named ReggeTTContinuumCloser proves a specific, long-sought property of this lattice gravity: in the limit of very long wavelengths, its small disturbances match those of Einstein's theory exactly.

The property concerns transverse-traceless (TT) polarizations, the two independent wave shapes that gravitational radiation can take, like the two possible orientations of a plus sign. For each such polarization and each nonzero wave vector, the continuum limit of the lattice's wave symbol, a number that encodes how the lattice responds to a disturbance, must equal the corresponding value from the linearized Einstein-Hilbert action. The pack states this target value as -(1/4) in the framework's conventions. The theorem reggeTTContinuumIsotropyTarget_closed proves this equality for every case, closing a previously open target in the framework's campaign.

The proof works by constructing an explicit sequence of finite-lattice approximations, indexed by an integer j, and showing that their symbols converge to the required value. The existence of each finite symbol is established by a separate theorem, and the convergence is handled by a composed limit argument. The entire proof is checked by a machine, with no gaps or unproven assumptions beyond the standard logical axioms of the underlying type theory, plus an inherited compiler-trust disclosure for one component.

In Recognition Science, this result is a step toward showing that the framework's discrete starting point naturally produces the known continuum physics of gravity, rather than requiring it as an input. The framework models the lattice as a ledger, a discrete record of events, and this proof demonstrates that the ledger's long-wavelength behavior is indistinguishable from Einstein's theory. It is a consistency check between the framework's foundational structure and the established classical theory it aims to reproduce.

The module explicitly does not claim to close the full four-dimensional action recovery, which remains an open target named by the ledger. It also does not address aliasing artifacts at small lattice spacings, which are not needed for the long-wavelength limit. The achievement is narrow but precise: a kernel-verified proof that the discrete model's wave behavior matches the continuum at the level of its fundamental coefficients.

/-- **THE VERBATIM 3D CLOSER (THEOREM): `ReggeTTContinuumIsotropyTarget`
holds.**  For every nonzero integer wave vector and every TT polarization,
the continuum TT Bloch symbol of the true Regge action exists and equals
`-(1/4)` — the symbol is isotropic with exactly the linearized
Einstein-Hilbert TT coefficient.  The preflight's OPEN target is closed;
the C10/C8 numerics are hereby superseded by kernel proof at 3D action
strength. -/
theorem reggeTTContinuumIsotropyTarget_closed :
    ReggeTTContinuumIsotropyTarget := by
  intro m E hm hTT
  have hm' : ∃ i : Fin 3, m i ≠ 0 := Function.ne_iff.mp hm
  refine ⟨fun j => @canonicalFiniteH (j + 3) (instNeZeroAddThree j) E m,
    fun j => canonicalFiniteH_TTBlochSymbolIs (j + 3) E m, ?_⟩
  exact
    ReggeTTAlgebraicCloser.canonicalFiniteH_div_momentumNormSq_tendsto_isotropy
      m E hm' hTT

/-- The reduced finite symbol IS a fixed-`N` TT Bloch symbol value: the
Gate A1 + A2(b) existence theorem restated on `canonicalFiniteH`. -/
theorem canonicalFiniteH_TTBlochSymbolIs (N : ℕ) [NeZero N]
    (E : Fin 3 → Fin 3 → ℝ) (m : Fin 3 → ℤ) :
    TTBlochSymbolIs N E m (canonicalFiniteH N E m) :=
  ReggeTTFlatSecondVariation.planeWave_TTBlochSymbolIs_reduced N E m

/-- The closed target instantiated at the preflight's own non-vacuity
witnesses (axis wave vector, `+` polarization): the continuum symbol at
that concrete instance is `-(1/4)`.  Non-vacuity receipt. -/
theorem axis_plus_continuum_symbol :
    ReggeTTContinuumSymbolIs axisTTPolarizationPlus axisWaveVector
      reggeTTContinuumCoefficient :=
  reggeTTContinuumIsotropyTarget_closed axisWaveVector axisTTPolarizationPlus
    axisWaveVector_ne_zero axisTTPolarizationPlus_isTT

What this page does not claim

This result does not close the four-dimensional action recovery target. This result does not repair aliasing artifacts at small lattice spacings. This result does not prove that the framework's ledger model is the only possible discrete starting point for gravity.

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/ReggeTTContinuumCloser.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.

Derived articles

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