Cosmology Recognition Equilibrium Conjugate Birth Charge Sum

The conserved charge

In the Recognition Science framework, a ledger (a discrete record of events) tracks numerical levels assigned to regions of a system. A theorem named conjugateBirth_chargeSum establishes a conservation law for these levels. It states that when a new pair of regions is born with levels +u and -u, the total sum of all levels across the system does not change. In symbols, the sum of the list (u :: xs) ++ [-u] equals the sum of xs, where xs is the pre-existing list of levels. This is a formal, machine-checked result in the framework's library of theorems.

This conservation law is the open-system counterpart to a closed-system result. In a closed system, a forced resolution between two coupled regions replaces both levels with their average. A separate theorem, pairResolve_levelSum, proves this averaging move also conserves the total level sum. The new theorem extends this conservation to the driven case, where the system grows by adding conjugate pairs. The framework models this as a birth of two regions with opposite levels at the horizon each cycle. The theorem proves that any number of such births, not just one, conserves the sum. This is the forced open input that keeps non-homogenizing structure alive, preventing the system from simply settling into a uniform state.

The theorem does not claim that the individual levels are conserved, nor that the sum is conserved under any other kind of operation. It applies specifically to the addition of conjugate pairs, not to arbitrary changes. It also does not claim that the system reaches a particular state, only that this one quantity, the total level sum, remains invariant. The result is a precise statement about a specific operation within the framework's dynamics, not a general principle about all possible evolutions.

/-- **A conjugate birth conserves sigma.** Inserting `+u` at the fine frontier and `-u` at
the coarse frontier leaves the total charge unchanged: the net of the born pair is zero. -/
theorem conjugateBirth_chargeSum (u : ℝ) (xs : List ℝ) :
    ((u :: xs) ++ [-u]).sum = xs.sum := by
  simp only [List.sum_append, List.sum_cons, List.sum_nil]
  ring

/-- **Sigma is conserved.** Resolving a pair leaves the total level unchanged. -/
theorem pairResolve_levelSum {n : ℕ} (x : Fin n → ℝ) {i j : Fin n} (h : i ≠ j) :
    levelSum (pairResolve x i j) = levelSum x := by
  unfold levelSum
  have hagree : ∀ k, k ≠ i → k ≠ j → pairResolve x i j k = x k :=
    fun k hi hj => pairResolve_other x hi hj
  have hsp := sum_split_pair (pairResolve x i j) x h hagree
  rw [pairResolve_at_i, pairResolve_at_j] at hsp
  -- hsp : (∑ resolved) - (∑ x) = ((xi+xj)/2 + (xi+xj)/2) - (x i + x j)
  have hzero : ((x i + x j) / 2 + (x i + x j) / 2) - (x i + x j) = 0 := by ring
  rw [hzero] at hsp
  linarith [hsp]

/-- **Any number of conjugate births conserves sigma.** After `k` cadence cycles of
expansion the ladder carries `k` extra `+u` charges at the fine frontier and `k` extra `-u`
charges at the coarse frontier; the total charge is still the initial total. So sigma = 0 is
preserved through the entire driven run. -/
theorem manyBirths_chargeSum (k : ℕ) (u : ℝ) (xs : List ℝ) :
    (List.replicate k u ++ xs ++ List.replicate k (-u)).sum = xs.sum := by
  simp only [List.sum_append, List.sum_replicate, nsmul_eq_mul]
  ring

What this page does not claim

The theorem does not claim that individual recognition levels are conserved, only their sum. It does not claim that the sum is conserved under any operation other than the addition of conjugate pairs. It does not claim that the system reaches a particular state, only that this one quantity remains invariant.

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

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