Gravity Analysis Regge Bloch Transported All Orbit M2 Eval4 D M2 Transported All

The distinct-hinge moment

In the Recognition Science framework's study of four-dimensional spacetime geometry, a recognition event is a discrete record of a physical interaction. The declaration m2TransportedAllOrbitMomentDistinctHinge_axisTTCrossNormalized_symbolDir concerns a specific weighted average, called the distinct-hinge moment, over a family of orbit slices in a four-dimensional space. The result it establishes is a number: for the normalized cross-axis configuration on the symbol direction, this moment equals -1/8.

The calculation is not a free-floating numerical coincidence. It is a theorem in the framework's machine-checked library of formal theorems, meaning every step from definitions to the final equality is verified by a computer. The value -1/8 emerges from summing integer certificates over 24 orbit slices and 10 time steps, a process the library checks exhaustively. This matters because it closes a specific evaluation: the raw all-orbit moment on the cross-axis is zero, but when weighted by the distinct-hinge factor (a term proportional to 1/r_τ), the result becomes -1/4; after normalization, it becomes -1/8.

The declaration also establishes a comparison between two axis configurations. On the plus-axis, the same normalized distinct-hinge moment on the symbol direction also equals -1/8. So the two configurations, plus and cross, agree on this particular direction. This agreement is a small but concrete isotropy result: for the symbol direction, the weighted moment does not distinguish between the two axis choices.

What the declaration does not claim is broader isotropy. It does not establish that all directions behave alike. In fact, the library explicitly leaves open the question of whether the full four-dimensional continuum is isotropic on the axis mode. The declaration also does not claim that the full two-jet extension repairs earlier anomalies: the full cosine two-jet, written A0*K2 + A2*K0, leaves the e0 anisotropy and plus-axis vanishing unrepaired, with the relevant restoration properties marked false. Finally, it does not flip the gap action recovery status, which remains false.

The practical consequence is a sharper map of where the framework's geometry is well-behaved and where it is not. The symbol direction now has a certified value, and the plus and cross axes agree there. The open question of full isotropy, and the unrepaired anomalies, define the next targets for the framework's analysis.

abbrev Mat4 := Matrix (Fin 4) (Fin 4) ℝ
-- decoyGauge lives in ReggeEdgeStencil4D (already opened above)

abbrev Mat4 := Matrix (Fin 4) (Fin 4) ℝ
-- decoyGauge lives in ReggeEdgeStencil4D (already opened above)

theorem sum_m2SlotCertZ_cross :
    (∑ s : Fin 24, ∑ t : Fin 10, m2SlotCertZ axisTTCrossCoeffZ s t) =
      (0 : ℤ) := by
  decide

set_option maxRecDepth 12000 in
set_option maxHeartbeats 8000000 in

/-- Named restoration claim: full two-jet makes `axisTTPlus` on `e0Dir`
leave zero.  Status false (plus stays `0` because `A2*K0` vanishes). -/
def Regge4DFullTwoJetRestoresE0PlusVanishing : Prop :=
  m2TransportedAllOrbitMomentDistinctHingeFull axisTTPlus e0Dir ≠ (0 : ℝ)

What this page does not claim

The declaration does not establish full isotropy across all directions in four-dimensional space. It does not claim the full two-jet extension repairs the e0 anisotropy or the plus-axis vanishing. It does not claim any change to the gap action recovery status.

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/ReggeBlochTransportedAllOrbitM2Eval4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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