Gravity Seven Gaps Weighted Hypersurface Bracket Pderiv Q Ham W

The weighted generator's slope

The declaration pderivQ_HamW is a theorem about a specific mathematical object, a weighted Hamiltonian on a discrete lattice. In plain terms, a Hamiltonian is a function that assigns an energy to each state of a system. This one is built from two kinds of terms: a kinetic part that depends on momenta, and a stiffness part that depends on differences between neighboring field values. The weight, a fixed number attached to each lattice site, multiplies only the stiffness part. The theorem computes the partial derivative of this Hamiltonian with respect to one field value at one site, holding all other variables fixed.

The result is an explicit formula. The derivative with respect to the field at site j equals a combination of terms from the two neighboring sites. From the left neighbor, it is the lapse function at site j-1 times the weight at that site times the field difference across that cell. From the right, it is the lapse at site j times the weight at site j times the field difference across the right cell, subtracted. The formula shows exactly how the weight enters: it appears linearly, once per gradient term, and only where the stiffness lives. The momentum partial derivative, by contrast, has no weight at all, because the kinetic slot is unweighted.

This theorem is one step in a larger program. The weighted Hamiltonian is a generalization of an earlier, unweighted one, and the derivative calculation is needed to verify that the bracket of two such Hamiltonians closes on a shift generator, a key structural property. The theorem is proved in a machine-checked library of formal theorems, meaning the derivation is verified by a computer. It holds for any choice of the weight function, with no positivity assumption, because the algebraic identities do not require it.

What the theorem does not claim is as important as what it proves. The weight here is a fixed function of the lattice site, not a function of the phase-space point. It is a background structure, not a dynamical field. The theorem does not provide the phase-space dependent structure function that a full Dirac algebra would require, and it does not touch the Hojman-Kuchar-Teitelboim rigidity statement. Those remain open targets. The continuum limit result, also proved, is a limit of the smearing shape with directly sampled profiles, not a proof that the discrete Wronskian of sampled lapses converges at a particular rate.

/-- THEOREM. The configuration partial of the weighted generator: each
gradient term carries its own site weight. -/
theorem pderivQ_HamW (w N : ZMod n → ℝ) (j : ZMod n) (x : PhaseSpace n) :
    pderivQ (HamW w N) j x
      = N (j - 1) * (w (j - 1) * (x.1 j - x.1 (j - 1)))
        - N j * (w j * (x.1 (j + 1) - x.1 j)) := by
  rw [pderivQ, (hasFDerivAt_HamW w N x).fderiv, HamWD, ContinuousLinearMap.sum_apply]
  have step : ∀ i : ZMod n,
      (((N i / 2) • ((x.2 i • coordP i + x.2 i • coordP i)
        + w i • ((x.1 (i + 1) - x.1 i) • (coordQ (i + 1) - coordQ i)
            + (x.1 (i + 1) - x.1 i) • (coordQ (i + 1) - coordQ i))) :
          PhaseSpace n →L[ℝ] ℝ))
        ((Pi.single j 1, 0) : PhaseSpace n)
      = (N i * (w i * (x.1 (i + 1) - x.1 i))) * (if i + 1 = j then (1 : ℝ) else 0)
        - (N i * (w i * (x.1 (i + 1) - x.1 i))) * (if i = j then (1 : ℝ) else 0) := by
    intro i
    simp [Pi.single_apply, mul_sub]
    split_ifs <;> ring
  rw [Finset.sum_congr rfl fun i _ => step i, Finset.sum_sub_distrib,
    sum_mul_ite_add, sum_mul_ite]
  have e : j - 1 + 1 = j := by ring
  rw [e]

/-- THEOREM. The configuration partial of the weighted generator: each
gradient term carries its own site weight. -/
theorem pderivQ_HamW (w N : ZMod n → ℝ) (j : ZMod n) (x : PhaseSpace n) :
    pderivQ (HamW w N) j x
      = N (j - 1) * (w (j - 1) * (x.1 j - x.1 (j - 1)))
        - N j * (w j * (x.1 (j + 1) - x.1 j)) := by
  rw [pderivQ, (hasFDerivAt_HamW w N x).fderiv, HamWD, ContinuousLinearMap.sum_apply]
  have step : ∀ i : ZMod n,
      (((N i / 2) • ((x.2 i • coordP i + x.2 i • coordP i)
        + w i • ((x.1 (i + 1) - x.1 i) • (coordQ (i + 1) - coordQ i)
            + (x.1 (i + 1) - x.1 i) • (coordQ (i + 1) - coordQ i))) :
          PhaseSpace n →L[ℝ] ℝ))
        ((Pi.single j 1, 0) : PhaseSpace n)
      = (N i * (w i * (x.1 (i + 1) - x.1 i))) * (if i + 1 = j then (1 : ℝ) else 0)
        - (N i * (w i * (x.1 (i + 1) - x.1 i))) * (if i = j then (1 : ℝ) else 0) := by
    intro i
    simp [Pi.single_apply, mul_sub]
    split_ifs <;> ring
  rw [Finset.sum_congr rfl fun i _ => step i, Finset.sum_sub_distrib,
    sum_mul_ite_add, sum_mul_ite]
  have e : j - 1 + 1 = j := by ring
  rw [e]

/-- THEOREM. The configuration partial of the weighted generator: each
gradient term carries its own site weight. -/
theorem pderivQ_HamW (w N : ZMod n → ℝ) (j : ZMod n) (x : PhaseSpace n) :
    pderivQ (HamW w N) j x
      = N (j - 1) * (w (j - 1) * (x.1 j - x.1 (j - 1)))
        - N j * (w j * (x.1 (j + 1) - x.1 j)) := by
  rw [pderivQ, (hasFDerivAt_HamW w N x).fderiv, HamWD, ContinuousLinearMap.sum_apply]
  have step : ∀ i : ZMod n,
      (((N i / 2) • ((x.2 i • coordP i + x.2 i • coordP i)
        + w i • ((x.1 (i + 1) - x.1 i) • (coordQ (i + 1) - coordQ i)
            + (x.1 (i + 1) - x.1 i) • (coordQ (i + 1) - coordQ i))) :
          PhaseSpace n →L[ℝ] ℝ))
        ((Pi.single j 1, 0) : PhaseSpace n)
      = (N i * (w i * (x.1 (i + 1) - x.1 i))) * (if i + 1 = j then (1 : ℝ) else 0)
        - (N i * (w i * (x.1 (i + 1) - x.1 i))) * (if i = j then (1 : ℝ) else 0) := by
    intro i
    simp [Pi.single_apply, mul_sub]
    split_ifs <;> ring
  rw [Finset.sum_congr rfl fun i _ => step i, Finset.sum_sub_distrib,
    sum_mul_ite_add, sum_mul_ite]
  have e : j - 1 + 1 = j := by ring
  rw [e]

What this page does not claim

The weight is a phase-space dependent inverse metric. The theorem provides the phase-space dependent structure function demanded by the full Dirac algebra. The continuum limit result proves convergence of the discrete Wronskian of sampled lapses at rate h.

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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