Foundation Pair Kernel Onsite Exclusion Mean Field Ledger Cost Shift Invariant

A limit of shift symmetry

A ledger, in the Recognition Science framework, is a discrete record of events, and its cost is a number assigned to a configuration of those events. One natural symmetry for such a cost is shift invariance: adding the same constant to every entry leaves the total cost unchanged. The declaration meanFieldLedgerCost_shift_invariant establishes that a particular cost, built from a mean-field weight graph, satisfies this symmetry. The mean-field graph couples every pair of sites with equal strength, including sites that are far apart.

The framework's library proves a stronger result, l1_onsite_forced_constant: under shift invariance, any per-site term in a general ledger cost must be a constant function. That is, no term of the form u^2 at each site can appear, because shifting all entries would change the total cost. The mean-field cost has no such per-site term at all, so it trivially satisfies the symmetry. The declaration meanFieldLedgerCost_shift_invariant confirms this in the machine-checked library.

The point of the mean-field example is negative. Shift invariance alone does not force the cost to be local, meaning it does not force interactions to depend only on nearby sites. The mean-field cost is shift-invariant yet couples every pair of sites, no matter how distant. Excluding such all-to-all coupling requires a separate locality hypothesis, which the framework has not yet formalized. The declaration therefore marks a boundary: it shows what shift symmetry can and cannot do.

In Recognition Science, this boundary is load-bearing. The framework's own cost primitive, exactJCostAction, has no per-site term by definition, so its shift invariance is a tautology. The mean-field example shows that a genuine hypothesis, shift invariance, rules out one class of terms but leaves another class open. The framework's library is honest about this residual gap, and the declaration records that honesty in formal form.

/-- **The honest negative.** The mean-field ledger cost is shift-invariant (it has no
    onsite term, so `ShiftInvariant` holds for the shape reason isolated in
    `shiftInvariant_iff_onsite_sum`, not because locality was ever assumed), yet its
    weight graph is non-local (`meanFieldWeight_full_support`). So `L1`
    (`l1_onsite_forced_constant`), even fully proved and even fully forced from
    first principles, does NOT by itself exclude a screened kernel: excluding the
    mean-field / all-to-all route needs a SEPARATE locality hypothesis `L0`
    (finite-range weights), which is not addressed in this module and, per the panel
    audit, is currently nowhere in the Lean surface. This is the scoped, honest residual
    of Door 2 / L1-hard: on-site exclusion is real and forced (conditional on
    `ShiftInvariant`), but on-site exclusion alone is not the whole story. -/
theorem meanFieldLedgerCost_shift_invariant (n : ℕ) :
    ShiftInvariant (meanFieldLedgerCost n) := by
  rw [shiftInvariant_iff_onsite_sum]
  intro ε c
  simp [meanFieldLedgerCost]

/-- **L1-FORCE (`l1_onsite_forced_constant`).** Under `ShiftInvariant` and `n ≥ 1`, the
    onsite part of an admissible `GeneralLedgerCost` is forced to be a constant
    function of its real argument — i.e. no on-site mass/absolute term is writable.

    Non-vacuity: this is FALSE without `ShiftInvariant`. Take `onsite u = u²` (the exact
    shape of the L2 decoy `+m² Σ φᵢ²`, which the numeric harness correctly flagged
    Yukawa): `yukawaOnsiteDecoy_not_shift_invariant` below shows this instance fails
    the hypothesis, so deleting `hR1` lets it stand as a live counterexample to the
    conclusion. -/
theorem l1_onsite_forced_constant {n : ℕ} (C : GeneralLedgerCost n) (hn : 0 < n)
    (hR1 : ShiftInvariant C) :
    ∃ k : ℝ, ∀ u : ℝ, C.onsite u = k := by
  refine ⟨C.onsite 0, fun u => ?_⟩
  have honsite := (shiftInvariant_iff_onsite_sum C).mp hR1 (fun _ => (0 : ℝ)) u
  simp only [zero_add] at honsite
  have hL : (∑ _i : Fin n, C.onsite u) = (n : ℝ) * C.onsite u := by
    rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
  have hR : (∑ _i : Fin n, C.onsite 0) = (n : ℝ) * C.onsite 0 := by
    rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
  rw [hL, hR] at honsite
  have hnr : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hn.ne'
  exact mul_left_cancel₀ hnr honsite

/-- The mean-field weight has full support: every pair, including every distinct pair,
    is coupled with nonzero weight. This is the formal shape of "not finite-range" — a
    genuinely local weight graph must vanish outside a bounded neighborhood as `n`
    grows, and this one never does. -/
theorem meanFieldWeight_full_support (n : ℕ) (i j : Fin n) :
    (meanFieldWeight n).weight i j ≠ 0 := by
  simp [meanFieldWeight]

What this page does not claim

The declaration does not prove that shift invariance alone excludes all non-local costs. The declaration does not establish that the framework's cost primitive is shift-invariant by construction. The declaration does not claim that the mean-field cost is the only admissible non-local cost.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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