Gravity Analysis Regge4 Dtransported Algebraic Closer Regge4 Dcontinuum Gauge Ze

A narrow gauge result

In the Recognition Science framework's study of gravity, a central question is how a discrete, combinatorial structure can approximate the continuous Einstein-Hilbert action of general relativity. This declaration, Regge4DContinuumGaugeZeroTargetLongitudinal_closed, is a small but rigorously established piece of that larger puzzle. It establishes that for a specific, specially chosen direction in the space of metric perturbations, a certain calculated quantity vanishes, and that a related limit exists.

The result concerns a "transported" version of a four-dimensional Regge calculus, where the discrete geometry is built from hinges and orbits. The theorem establishes two facts. First, a particular combination of terms, the "transported all-orbit moment" for a specific decoy gauge and a specific direction, equals zero. Second, a limit, called FoldAlongM2Tendsto, exists for this same configuration. The direction is called "longitudinal" and is defined by a vector with components (1,1,0,0). The result is a theorem in the framework's machine-checked library of formal theorems, with no unproven axioms beyond the standard three.

This result is deliberately narrow. It does not establish that the limit of the full, normalized transported fold approaches the Einstein-Hilbert value of -1/4. That target, Regge4DContinuumEHTarget, remains open. It also does not establish the broader statement that the gauge-zero target holds for every possible gauge vector. In fact, the framework's library explicitly records a counterexample: for the mode m=(1,1,0,0) and a specific vector v=e₂, the unrestricted claim is false. The theorem's power is in showing exactly what can be closed: a single, well-chosen direction, not the whole space.

The declaration also banks several supporting algebraic identities. It confirms that the continuum symbol sequence is a specific fold, that this fold is quadratically homogeneous, and that a one-orbit normalized coefficient is not equal to the frozen Einstein-Hilbert coefficient. These are all established facts. The broader status flags, such as whether the transported target is closed or whether the gap action is recovered, are set to false, meaning they are explicitly not established here.

What this means for the reader is a clear picture of the framework's method: it establishes what it can, marks precisely what it cannot, and refuses to conflate the two. The result is a concrete, verifiable step, and an honest statement of the boundary of that step.

theorem Regge4DContinuumGaugeZeroTargetLongitudinal_closed :
    Regge4DContinuumGaugeZeroTargetLongitudinal :=
  ⟨m2TransportedAllOrbitMomentDistinctHinge_decoyGauge_symbolDir,
    FoldAlongM2Tendsto_of_decoyGauge⟩

What this page does not claim

The theorem does not claim to establish the Einstein-Hilbert limit or the full gauge-zero target. The theorem does not claim to inhabit the broader convergence statement S_RS_converges_EH_4d. The theorem does not claim to flip the gap_action_recovery or transportedGaugeZeroClosed flags.

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/Regge4DTransportedAlgebraicCloser.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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