Gravity Analysis Regge Bloch M2 Tendsto4 D Tendsto Transported Slot Term Div Sq

The transported slot limit

In the Recognition Science framework's study of gravity, a slot is a discrete unit of geometric data on a hinged lattice, and a transported slot term measures how that data changes when the lattice is shifted along a chosen direction. The declaration tendsto_transportedSlotTerm_div_sq proves a precise limit statement: as a momentum parameter μ approaches zero, the transported slot term divided by μ² converges to a fixed coefficient called m2SlotCoeff. This is a continuum limit result, showing that the discrete lattice expression has a well-defined quadratic behavior in the small-momentum regime.

The theorem's proof rests on two established facts. First, the cosine two-jet expansion shows that the deficit kernel, a measure of curvature violation, vanishes to second order at zero momentum. Second, the zero-momentum value of the kernel is already known to vanish for two specific polarizations: the axis TT+ configuration and the decoy gauge. The theorem then assembles these ingredients to show the transported slot term's ratio has a finite limit, not a divergence, as μ goes to zero. The coefficient m2SlotCoeff itself is defined as the product of the area term and the second-jet coefficient of the kernel, but only when a certain T1 condition holds; otherwise it is zero.

What this theorem does not claim is broader. It does not establish the limit for a general matrix H; the proof requires the hypothesis that the kernel vanishes at zero momentum, which is verified only for the two named polarizations. It does not show that the full fold-along term converges for arbitrary configurations, only that the transported slot term does under the stated condition. The general case, where the kernel does not vanish at zero, remains an open target in the framework's library. The theorem is a building block, not a complete theory of the continuum limit.

In practice, this result matters because it gives a controlled way to take the discrete lattice expression to a smooth limit, a step needed for connecting the framework's combinatorial gravity to classical continuum physics. The convergence is proven in the machine-checked library, so the statement carries the weight of a formal theorem rather than a heuristic approximation. The two polarizations covered, axis TT+ and decoy gauge, are the cases where the zero-momentum condition is known to hold, and the theorem's scope is exactly that: a precise limit for those cases, with the general case left as a named open problem.

theorem tendsto_transportedSlotTerm_div_sq (H : Mat4) (s : Fin 24) (t : Fin 10)
    (h0 : kerAlong H s t 0 = 0) :
    Tendsto
      (fun μ : ℝ =>
        transportedSlotTerm H (fun i => μ * symbolDir i) s t / μ ^ 2)
      (𝓝[≠] (0 : ℝ)) (𝓝 (m2SlotCoeff H s t)) := by
  by_cases ht : isT11 s t
  · have hterm :
        (fun μ : ℝ =>
            transportedSlotTerm H (fun i => μ * symbolDir i) s t / μ ^ 2) =
          fun μ : ℝ => areaAlong H s t μ * kerAlong H s t μ / μ ^ 2 := by
      funext μ
      simp [transportedSlotTerm, ht, areaAlong, kerAlong]
    rw [hterm, m2SlotCoeff_eq_area_kerM2 H s t, if_pos ht]
    exact tendsto_slot_product H s t h0
  · have hterm :
        (fun μ : ℝ =>
            transportedSlotTerm H (fun i => μ * symbolDir i) s t / μ ^ 2) =
          fun _ : ℝ => (0 : ℝ) := by
      funext μ
      simp [transportedSlotTerm, ht]
    rw [hterm, m2SlotCoeff_eq_area_kerM2 H s t, if_neg ht]
    exact tendsto_const_nhds

theorem kerAlong_axis_zero (s : Fin 24) (t : Fin 10) :
    kerAlong axisTTPlus s t 0 = 0 := by
  rw [kerAlong_zero]
  simpa [classDot] using classDot_slotDeficitKer_axis s t

theorem kerAlong_gauge_zero (s : Fin 24) (t : Fin 10) :
    kerAlong decoyGauge s t 0 = 0 := by
  rw [kerAlong_zero]
  simpa [classDot] using classDot_slotDeficitKer_gauge s t

theorem m2SlotCoeff_eq_area_kerM2 (H : Mat4) (s : Fin 24) (t : Fin 10) :
    m2SlotCoeff H s t =
      (if isT11 s t then areaAlong H s t 0 * kerM2Coeff H s t else 0) := by
  unfold m2SlotCoeff kerM2Coeff
  by_cases ht : isT11 s t
  · simp [ht, areaAlong_zero]
  · simp [ht]

What this page does not claim

The theorem does not prove the limit for a general matrix H without the zero-momentum kernel condition. The theorem does not establish convergence of the full fold-along term for arbitrary configurations. The theorem does not cover the case where the kernel does not vanish at zero momentum.

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/ReggeBlochM2Tendsto4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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