Gravity Analysis Srsconverges Eh4 D Typed Residual Midpoint Bloch Symbol Zero Cl

The zero-residual gate

In the Recognition Science framework's campaign to show that its discrete model of spacetime reproduces general relativity, one central object is a residual: a number that measures how far a discrete configuration is from satisfying a continuum equation. The theorem typedResidual_midpointBloch_symbolZero_closed establishes that for every flat 4x4 matrix H, the residual called exactMidpointBlochSymbolZero is exactly zero. In plain terms, it proves that a specific algebraic expression, built from the midpoint of a Bloch symbol, vanishes identically on flat space. This is a formal, machine-checked statement: the proof is a theorem in the framework's library, and its axiom audit is clean, resting only on the standard three axioms of the ambient type theory.

The significance is that this zero-residual fact is one of several gates that must be inhabited before the framework can claim its discrete action converges to the Einstein-Hilbert action in the weak-field limit. The declaration is a named closer: it is the specific theorem that closes the symbolZero face of the campaign. The framework's library shows that this zero-residual property, together with other typed residual faces, implies the bridge statement TypedResidual_discrete_torus_family_bridge, which in turn participates in the proof of the headline proposition S_RS_converges_EH_4d. That proposition is the claim that the discrete action converges to the Einstein-Hilbert action, but only in the weak-field quadratic regime, and only for transverse-traceless and pure-gauge configurations.

What the theorem does not claim is equally important. It does not prove the full Einstein field equations, nor does it establish convergence for arbitrary curvature or for the full nonlinear action. The library's own docstring is explicit: the scope is weak-field quadratic action convergence only. It is not a sourced Einstein equation, not continuum Ricci or stress, not horizon or coframe results, and not arbitrary-curvature general relativity. The zero-residual statement is a necessary algebraic identity on flat configurations, not a derivation of gravity itself. The gap-action recovery flag flips only when both named theorems are inhabited, with focused axiom audits and adversarial review; this theorem is one such inhabitant, not the whole recovery.

The practical consequence for the framework is that one discrete face of the bridge to continuum gravity is now closed by a kernel-checked proof. A reader can now see that the discrete model's flat-space residual is exactly zero, a fact that the framework treats as a precondition for the larger convergence claim. The remaining faces, such as the option-C faces and the Rayleigh algebraic face, are also inhabited, but the full physical bridge from recognition to linking remains an open target, not a proved theorem.

/-- **THEOREM (R2):** midpoint Bloch vanishes at zero momentum. -/
theorem typedResidual_midpointBloch_symbolZero_closed :
    TypedResidual_midpointBloch_symbolZero :=
  ReggeExactFlatHessianBlochSymbolZero4D.typedResidual_midpointBloch_symbolZero

theorem typedResidual_discrete_torus_family_bridge_of_symbolZero
    (hZ : TypedResidual_midpointBloch_symbolZero) :
    TypedResidual_discrete_torus_family_bridge :=
  discrete_torus_family_bridge_of_symbolZero hZ

theorem S_RS_converges_EH_4d_closed :
    S_RS_converges_EH_4d :=
  srs_converges_eh_4d_of_m2_optionC_faces typedResidual_m2_optionC_faces

What this page does not claim

This theorem does not prove the full Einstein field equations or any continuum Ricci or stress result. This theorem does not establish convergence for arbitrary curvature or the full nonlinear action. This theorem does not derive the physical bridge from recognition to linking, which remains open.

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/SRSConvergesEH4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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