Gravity Analysis Regge Bloch Transported All Orbit4 D Regge Bloch Transported Al

The all-orbit Bloch fold

In the Recognition Science framework, a ledger (a discrete record of events) for gravity assigns to each hinge of a 4D cell complex two kinds of seed data: an area covector and a deficit kernel. The declaration reggeBlochTransportedAllOrbit4DStatus_flags establishes a construction called the all-orbit Bloch fold: it transports these seeds across every orbit type using a covering permutation, then sums the results. The construction is a definition, not a theorem about physics; what is proved is that on one particular slice, the (1,1) orbit, the transported data exactly recovers the previously defined single-orbit Bloch fold, its area covector, and its deficit kernel (anchors: slotOrbitAreaCov_t11_eq, transportedOrbitDeficit_t11). The docstring also records a measured raw value of -5/2 for the all-orbit m² moment along a specific direction, with a distinct-hinge weighted variant giving -1/4.

What the declaration does not claim is as important as what it proves. The all-orbit m² moment's limit behavior (Tendsto) remains open, meaning the construction has not been shown to converge in the continuum limit. The continuum Einstein-Hilbert isotropy is also open, requiring a residual 3D-style cell-sum dictionary before any continuum limit can be established. The declaration explicitly states it does not flip gap_action_recovery, meaning it does not alter a separate recovery mechanism for the action gap. These are targets for future work, not failures of the existing proofs.

The proofs that do exist are narrow but solid. The construction respects quadratic homogeneity: scaling the 4x4 matrix H by a constant c scales the fold by c² (anchor: blochFoldAll_smul). The zero matrix gives zero fold (anchor: blochFoldAll_zero). The pushforward identities for class dot products are proved (anchors: classDot_pushforward, phasedClassDot_pushforward), and the orbit star sizes are all positive (anchor: orbitStarSize_pos). These are the load-bearing checks that the construction is internally consistent on the slices where it is evaluated.

For a reader, the practical takeaway is that the framework has a working, machine-checked construction for transporting gravitational seed data across all orbit types, with exact recovery on the (1,1) slice and a measured raw moment value on a specific direction. The open items are precisely stated: no continuum limit, no full isotropy, no action-gap recovery. This is a status report on a formal construction, not a physical prediction.

/-- Full transported multi-orbit Bloch fold (incidence × full-star). -/
def blochFoldAll (H : Mat4) (m : Fin 4 → ℝ) : ℝ :=
  ∑ ty : HingeOrbitType, blochFoldOrbit ty H m

/-- Compatibility wrapper (audit / older callers): same as `slotOrbitAreaCov_t11`. -/
theorem slotOrbitAreaCov_t11_eq (s : Fin 24) (t : Fin 10) (h : isT11 s t) :
    slotOrbitAreaCov .t11 s t = slotAreaCov s t :=
  slotOrbitAreaCov_t11 s t h

theorem transportedOrbitDeficit_t11 (p : Fin 24) :
    transportedOrbitDeficit .t11 p = transportedDeficit p := by
  funext d
  simp [transportedOrbitDeficit, pushforwardClass, transportedDeficit,
    orbitSeedKernel_t11]

/-- Formerly OPEN: raw all-orbit moment on axisTTPlus / symbolDir equals `-5/2`
(orbit slices t11=-3, t12=+2, t13=-3/2).  After `/|symbolDir|²` this is
`-5/4`.  Closed in `ReggeBlochTransportedAllOrbitM2Eval4D`.
Distinct-hinge weight `1/r_τ` upgrades the raw axis value to `-1/4`
(path A; residual continuum dictionary still needed for EH Tendsto). -/
def M2TransportedAllOrbitAxisSymbolDirEvalOpen : Prop :=
  m2TransportedAllOrbitMoment axisTTPlus symbolDir = (-5 / 2 : ℝ)

What this page does not claim

The all-orbit Bloch fold has been shown to converge in the continuum limit. The construction establishes continuum Einstein-Hilbert isotropy. The declaration changes or recovers the gap action.

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/ReggeBlochTransportedAllOrbit4D.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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