Gravity Seven Gaps Ledger Bridge No Go Ledger Bridge No Go Status Flags

Two proven obstructions

In the Recognition Science framework, a recognition ledger is a discrete record of events, each event carrying a cost. The framework's central theorem proves that this cost must be the function J(x) = (x + 1/x)/2 - 1. A ledger's deficit at a cell is the sum of these costs, and because each cost is nonnegative, the deficit itself can never be negative. This is a proved fact in the framework's machine-checked library of formal theorems.

The framework also models gravity through a geometric structure of hinges, where a hinge deficit is a signed quantity that can be positive or negative. A proposed bridge, called the ledger-to-hinge bridge, would equate the ledger deficit at each cell with a raw geometric hinge deficit. The declaration ledgerBridgeNoGoStatus_flags records two proved obstructions to this specific bridge form. First, since a ledger deficit is always nonnegative, any bridge of this assumed form would force the geometric deficit to be nonnegative on its image. No bridge can exist whose specified deficit assignment is negative anywhere on that image. Second, if a one-parameter family of ledgers is built from J-costs of comparison ratios that satisfy the natural parity r(-ε) = r(ε)⁻¹, then the induced deficit is an even function of the deformation parameter ε, with a leading term of order ε². A signed linear response of order ε, which is what a weak-field Regge deficit would have, cannot be matched by such an even function unless both vanish identically.

The declaration itself is a status record, not new mathematics. It sets three boolean flags to true, all forced by definitional equality: the sign obstruction is proved, the parity obstruction is proved, and the corrected bridge target is the nonnegative curvature-quadratic energy Σ_h A_h · δ_h², not the raw signed deficit Σ_h A_h · δ_h. The theorems behind these flags are what carry the mathematical content.

What the declaration does not claim is equally important. The geometric premise that two-sided weak-field deformation classes contain hinges of strictly negative deficit, and that the weak-field Regge response is odd at leading order, is not formalized in this file. It is a prose-tier assumption that motivates the hypotheses. The parity obstruction also has genuine scope: ratio families that violate the parity condition, such as r = exp(ε·s + ε²·t) with t ≠ 0, escape the theorem. The declaration proves an obstruction to a specific assumed bridge form, not a general impossibility of connecting ledgers to geometry.

/-- **THEOREM (sign obstruction, positive form).** Any bridge satisfying the
assumed deficit-matching condition forces the geometric deficit to be
nonnegative at every hinge in the image of the comparison map `x_sigma`.
Direct consequence of `RecognitionLedger.deficit_nonneg`: ledger deficits
are sums of nonnegative J-costs. -/
theorem bridge_forces_nonneg_geometricDeficit
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ] {H : Type*}
    (L : RecognitionLedger.RecognitionLedger Λ)
    (B : LedgerToHingeBridge H L) (i : Λ) :
    0 ≤ B.geometricDeficit (B.x_sigma i) := by
  rw [← B.bridge_assumed i]
  exact RecognitionLedger.deficit_nonneg L i

/-- **THEOREM (sign obstruction, nonexistence form).** Given any hinge
specification (comparison map `x` and deficit assignment `δ`) for which some
cell `i` sees a strictly negative geometric deficit `δ (x i) < 0`, there is
NO `LedgerToHingeBridge` realizing that specification, for any recognition
ledger `L` whatsoever. Combined with the geometric premise that two-sided
(signed) weak-field deformation classes place negative deficits in the image
of any faithful comparison map (prose tier, not formalized here), this
excludes the assumed bridge form on such classes. -/
theorem no_bridge_matches_negative_deficit_spec
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ] {H : Type*}
    (L : RecognitionLedger.RecognitionLedger Λ)
    (x : Λ → H) (δ : H → ℝ) (i : Λ) (hneg : δ (x i) < 0) :
    ¬ ∃ B : LedgerToHingeBridge H L, B.x_sigma = x ∧ B.geometricDeficit = δ := by
  rintro ⟨B, hx, hd⟩
  have h := bridge_forces_nonneg_geometricDeficit L B i
  rw [hx, hd] at h
  exact absurd h (not_le.mpr hneg)

/-- **THEOREM (even vs. odd exclusion).** An even function of ε can equal an
odd function of ε only if both vanish identically. -/
theorem even_and_odd_forces_zero (g d : ℝ → ℝ)
    (hg : ∀ ε, g (-ε) = g ε) (hd : ∀ ε, d (-ε) = - d ε)
    (hmatch : ∀ ε, g ε = d ε) (ε : ℝ) :
    g ε = 0 ∧ d ε = 0 := by
  have h1 : d ε = - d ε := by
    calc d ε = g ε := (hmatch ε).symm
      _ = g (-ε) := (hg ε).symm
      _ = d (-ε) := hmatch (-ε)
      _ = - d ε := hd ε
  have h2 : d ε = 0 := by linarith
  exact ⟨(hmatch ε).trans h2, h2⟩

/-- Status flags for the ledger-bridge no-go (documentation record; the
mathematics lives in the theorems above, not in these booleans).

What is PROVED: (1) sign obstruction, `bridge_forces_nonneg_geometricDeficit`
and `no_bridge_matches_negative_deficit_spec` (no bridge for any
negative-deficit-in-image specification); (2) parity obstruction,
`jRatioDeficit_even` and `no_ledger_family_linear_response` (no signed linear
response for parity-covariant J-ratio families). What is GEOMETRIC PREMISE
(prose tier): two-sided weak-field Regge classes carry negative image
deficits and odd leading response. Under that premise the assumed raw-deficit
bridge form is excluded on such classes, and the honest bridge target is the
nonnegative curvature-quadratic energy Σ_h A_h · δ_h² built in
`SevenGaps.LedgerEnergyBridge` (deliverable B of this lane). -/
structure LedgerBridgeNoGoStatus where
  /-- PROVED: no bridge exists for any specification with a strictly negative
  deficit in the image of the comparison map (sign obstruction). -/
  sign_nogo_proved_for_negative_image_specs : Bool
  /-- PROVED: parity-covariant J-ratio ledger families admit no signed
  linear-response deficit (parity obstruction). -/
  parity_nogo_proved_for_parity_covariant_families : Bool
  /-- The corrected bridge target is the curvature-quadratic energy
  Σ_h A_h · δ_h², not the raw signed deficit Σ_h A_h · δ_h. -/
  corrected_target_is_quadratic_energy : Bool

What this page does not claim

The geometric premise that two-sided weak-field deformation classes contain negative image deficits is not formalized in this file. The parity obstruction does not apply to ratio families that violate the parity condition, such as r = exp(ε·s + ε²·t). The declaration does not prove a general impossibility of connecting ledgers to geometry, only an obstruction to the specific assumed raw-deficit bridge form.

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

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