Gravity Seven Gaps Weighted Hypersurface Bracket Weighted Structure Sum Tendsto

The continuum limit

In numerical analysis and physics, a Riemann sum approximates a definite integral by sampling a function at evenly spaced points. The theorem weightedStructureSum_tendsto establishes a similar convergence for a specific weighted sum that arises in a discrete model of gravity. It states that if three continuous functions W, Wr, and S are defined on the unit interval, then the sum (1/N) * Σ W(k/N) * (Wr(k/N) * S(k/N)) tends to the integral of W * (Wr * S) as N goes to infinity. This is a standard and rigorous result about the limit of a Riemann sum with a weight function.

The declaration is part of a larger framework called Recognition Science, which models physical structure from a discrete ledger of recognition events. Within this framework, the theorem is a technical lemma used to connect a discrete lattice model of spacetime to a continuous description. The functions W, Wr, and S represent, respectively, a background weight profile, a lapse-Wronskian profile, and a closure-density profile, all sampled on a lattice. The theorem shows that the discrete weighted sums converge to the expected continuum integral, a necessary consistency check for the lattice model as the lattice spacing goes to zero.

What the theorem does not claim is equally important. It does not establish that the discrete Wronskian of sampled lapse functions converges at a certain rate; that would require additional smoothness assumptions and is left as an open problem. It also does not establish the full continuum limit of the Dirac algebra, nor does it address the phase-space-dependent structure function that a complete general relativity recovery would require. The theorem is a precise statement about the convergence of a specific weighted sum, not a proof of the entire physical theory.

The practical consequence is a verified bridge between the discrete and continuous formulations within the framework. It shows that the weighted sums used in the lattice model have the correct continuum limit, which is a necessary condition for the model to be a valid discretization. This is a foundational step, but it is not the final step; the full recovery of general relativity from the discrete model remains an open target.

/-- THEOREM (weighted structure smearing, continuum limit). Sample a
background weight profile `W`, a continuum lapse-Wronskian profile `Wr`
(the limit shape of `(N_j M_{j+1} - M_j N_{j+1}) / h`), and a closure-density
profile `S` (the limit shape of the point-split density
`pi_{j+1} (q_{j+1} - q_j) / h`, one factor of `h` absorbed by the gradient),
all continuous on `[0, 1]`. Then the h-scaled w-weighted structure sums of
`bracket_HamW_HamW` converge:
`(1/N) * sum_{k<N} W(k/N) * (Wr(k/N) * S(k/N)) -> integral_0^1 W * (Wr * S)`.

h-scaling convention (stated per panel preregistration): the explicit `1/N`
prefactor is the single factor of lattice spacing carried by the discrete
lapse Wronskian; the sampled profiles are all O(1). This is a limit of the
SMEARING SHAPE with continuum profiles sampled directly; it is not a proof
that the discrete Wronskian of sampled lapses converges at rate h (that needs
C^1 data and is left OPEN with the full Dirac continuum limit). Direct
application of `Analysis.weightedLatticeSum_tendsto`. -/
theorem weightedStructureSum_tendsto (W Wr S : ℝ → ℝ)
    (hW : ContinuousOn W (Set.Icc 0 1)) (hWr : ContinuousOn Wr (Set.Icc 0 1))
    (hS : ContinuousOn S (Set.Icc 0 1)) :
    Filter.Tendsto
      (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.range N,
        W ((k : ℝ) / (N : ℝ)) * (Wr ((k : ℝ) / (N : ℝ)) * S ((k : ℝ) / (N : ℝ))))
      Filter.atTop (nhds (∫ x in (0:ℝ)..1, W x * (Wr x * S x))) := by
  have h := Analysis.weightedLatticeSum_tendsto (fun x => Wr x * S x) W
    (hWr.mul hS) hW
  have hint : (∫ x in (0:ℝ)..1, (Wr x * S x) * W x)
      = ∫ x in (0:ℝ)..1, W x * (Wr x * S x) :=
    intervalIntegral.integral_congr fun x _ => mul_comm _ _
  rw [hint] at h
  refine h.congr fun N => ?_
  congr 1
  refine Finset.sum_congr rfl fun k _ => ?_
  show Wr ((k : ℝ) / (N : ℝ)) * S ((k : ℝ) / (N : ℝ)) * W ((k : ℝ) / (N : ℝ))
      = W ((k : ℝ) / (N : ℝ)) * (Wr ((k : ℝ) / (N : ℝ)) * S ((k : ℝ) / (N : ℝ)))
  ring

What this page does not claim

The theorem does not establish the full continuum limit of the Dirac algebra. The theorem does not address the phase-space-dependent structure function required for general relativity recovery. The theorem does not establish the convergence rate of the discrete Wronskian.

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

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