Gravity Analysis Regge Bloch Transported All Orbit4 D M2 Transported All Orbit M

The full distinct-hinge moment

The declaration m2TransportedAllOrbitMomentDistinctHingeFull defines a single real number from a 4 by 4 matrix H and a direction vector dir. The number is the sum, over all six hinge orbit types, of a per-orbit moment multiplied by the reciprocal of that orbit's star size. The star size is the count of distinct hinges the orbit covers: 6 for the t11, t13, and t31 types, and 4 for the t12, t21, and t22 types. Multiplying by the reciprocal makes the total a plain average over hinges rather than a sum that would count some hinges more than once.

The per-orbit moment itself is the sum over 24 slots and 10 ticks of a coefficient built from the transported area covector and deficit kernel. Those two objects are pushed forward from a seed orbit by a covering permutation, so each slot carries its orbit's geometric data to a common 15-dimensional class space. The definition is purely formal: it assembles the pieces into one expression, and the machine-checked library proves that the assembly is quadratic in the matrix H. That means scaling H by a constant c scales the whole moment by c squared, a homogeneity property the library records as blochFoldAll_smul.

What the declaration does not claim is any numerical value. The docstring marks as open the evaluation of this moment along the axisTTPlus direction with symbolDir, where the raw all-orbit moment is measured at -5/2 and the distinct-hinge version at -1/4. Those are sibling definitions in the same file, not theorems about this one. The declaration also does not claim to recover the continuum Einstein-Hilbert action; that limit is open and still needs a residual 3D-style cell-sum dictionary. And it does not flip gap_action_recovery, whatever that property asserts elsewhere.

def m2TransportedAllOrbitMomentDistinctHingeFull (H : Mat4)
    (dir : Fin 4 → ℝ) : ℝ :=
  ∑ ty : HingeOrbitType,
    (orbitStarSize ty)⁻¹ * m2TransportedOrbitMomentFull ty H dir

def m2TransportedAllOrbitMomentDistinctHingeFull (H : Mat4)
    (dir : Fin 4 → ℝ) : ℝ :=
  ∑ ty : HingeOrbitType,
    (orbitStarSize ty)⁻¹ * m2TransportedOrbitMomentFull ty H dir

/-- Geometric star size `r_τ` per orbit type.  Theorem-level in the star
modules / paper table: `(1,1)→6`, `(1,2)→4`, `(2,1)→4`, `(1,3)→6`,
`(3,1)→6`, `(2,2)→4`. -/
def orbitStarSize : HingeOrbitType → ℝ
  | .t11 => 6
  | .t12 => 4
  | .t21 => 4
  | .t13 => 6
  | .t31 => 6
  | .t22 => 4

/-- Full-jet orbit / distinct-hinge aggregates. -/
def m2TransportedOrbitMomentFull (ty : HingeOrbitType) (H : Mat4)
    (dir : Fin 4 → ℝ) : ℝ :=
  ∑ s : Fin 24, ∑ t : Fin 10, m2TransportedOrbitSlotCoeffFull ty H dir s t

theorem blochFoldAll_smul (c : ℝ) (H : Mat4) (m : Fin 4 → ℝ) :
    blochFoldAll (c • H) m = c ^ 2 * blochFoldAll H m := by
  unfold blochFoldAll
  simp_rw [blochFoldOrbit_smul, ← Finset.mul_sum]

What this page does not claim

The declaration assigns no numerical value to the moment; evaluations along axisTTPlus with symbolDir are open or measured in sibling modules. The declaration does not prove recovery of the continuum Einstein-Hilbert action; that limit remains open. The declaration does not assert any property named gap_action_recovery.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: