Foundation Pair Kernel Discrete Gauss Const Flow Breaks Conservation

The conservation null test

In the Recognition Science framework, a recognition current (a flow of value between accounts on a finite lattice) obeys a conservation law only when it has double-entry structure: the flow from account i to account j must be the negative of the flow from j back to i. This antisymmetry is the formal meaning of "every debit has a matching credit." The declaration constFlow_breaks_conservation proves that this condition is not decorative: remove it, and conservation fails.

The proof works by exhibiting a decoy current, constFlow, which sends exactly 1 unit from every account to every other account, including itself. This uniform flow is not antisymmetric, because F i j = 1 while F j i = 1, not -1. Its total divergence, the sum of net outflows over all accounts, equals n², where n is the number of accounts. For any positive n, that sum is nonzero, so global conservation fails. The theorem constFlow_breaks_conservation states this failure precisely: ∑ divF (constFlow n) ≠ 0 for all n > 0.

This null test, built before the positive theorems, gives the discrete Gauss law its teeth. In contrast, an antisymmetric current always sums to zero over any region, and its total divergence over the whole lattice is zero. A source that equals the divergence of such a current is globally neutral. The contrast shows that antisymmetry, not any particular formula for the flow, is the load-bearing assumption.

In Recognition Science, this is the discrete analogue of Gauss's law: the net source in a region equals the flux through its boundary, and the total source over the whole lattice is zero. The framework models a recognition event as a single posting from account a to account b, which is antisymmetric and conserves globally, carrying a +1 divergence at the source and -1 at the sink. The uniform flow, by breaking antisymmetry, shows what happens when double-entry structure is absent.

/-- **Null test passes.** A non-antisymmetric current breaks global conservation
    (`∑ divF ≠ 0`). So the double-entry hypothesis is load-bearing in the Gauss law. -/
theorem constFlow_breaks_conservation (n : ℕ) (hn : 0 < n) :
    ∑ i : Fin n, divF (constFlow n) i ≠ 0 := by
  rw [constFlow_sum_div]
  have hpos : (0 : ℝ) < (n : ℝ) := by exact_mod_cast hn
  exact (mul_pos hpos hpos).ne'

/-- Over any region `S`, an antisymmetric current sums to zero: the double-entry cancellation.
    This is the engine of both the global and regional Gauss laws. -/
theorem antisym_sum_finset_zero {n : ℕ} {F : Fin n → Fin n → ℝ} (h : IsAntisym F)
    (S : Finset (Fin n)) : ∑ i ∈ S, ∑ j ∈ S, F i j = 0 := by
  have hcomm : (∑ i ∈ S, ∑ j ∈ S, F j i) = ∑ i ∈ S, ∑ j ∈ S, F i j := Finset.sum_comm
  have hzero : (∑ i ∈ S, ∑ j ∈ S, (F i j + F j i)) = 0 := by
    apply Finset.sum_eq_zero
    intro i _
    apply Finset.sum_eq_zero
    intro j _
    have := h i j
    linarith
  have hsplit : (∑ i ∈ S, ∑ j ∈ S, (F i j + F j i))
      = (∑ i ∈ S, ∑ j ∈ S, F i j) + (∑ i ∈ S, ∑ j ∈ S, F j i) := by
    rw [← Finset.sum_add_distrib]
    apply Finset.sum_congr rfl
    intro i _
    rw [Finset.sum_add_distrib]
  rw [hsplit, hcomm] at hzero
  linarith

/-- **Continuity ⇒ global neutrality.** If a source `sigma` is the divergence of an antisymmetric
    current (`divF F = sigma`, the Gauss law "site-divergence = sigma-imbalance"), then the total
    source is zero. This is the sigma = 0 conservation law stated on the source, with no `sigma :=
    Δφ` definitional shortcut. -/
theorem sigma_sum_zero_of_continuity {n : ℕ} {F : Fin n → Fin n → ℝ} {sigma : Fin n → ℝ}
    (h : IsAntisym F) (hcont : ∀ i, divF F i = sigma i) :
    ∑ i : Fin n, sigma i = 0 := by
  have : ∑ i : Fin n, sigma i = ∑ i : Fin n, divF F i := by
    apply Finset.sum_congr rfl
    intro i _
    exact (hcont i).symm
  rw [this, sum_divF_zero h]

theorem elementaryPosting_antisym {n : ℕ} (a b : Fin n) : IsAntisym (elementaryPosting a b) := by
  intro i j
  unfold elementaryPosting
  have c1 : (j = b ∧ i = a) ↔ (i = a ∧ j = b) := and_comm
  have c2 : (j = a ∧ i = b) ↔ (i = b ∧ j = a) := and_comm
  simp only [c1, c2]
  ring

/-- A single posting conserves globally (it is antisymmetric). -/
theorem elementaryPosting_sum_div_zero {n : ℕ} (a b : Fin n) :
    ∑ i : Fin n, divF (elementaryPosting a b) i = 0 :=
  sum_divF_zero (elementaryPosting_antisym a b)

What this page does not claim

This does not claim that the recognition current is a gradient of a potential. This does not claim that any antisymmetric current arises from a single posting. This does not claim that the uniform flow is physically realizable, only that it is a mathematical counterexample.

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