Encyclopedia Gravity Gravity Seven Gaps Ledger Bridge No Go Bridge Forces Nonneg Geometric Deficit

ARTICLE 3 claims 2 theorems 1 model

Gravity Seven Gaps Ledger Bridge No Go Bridge Forces Nonneg Geometric Deficit

A machine-checked theorem shows that a proposed bridge between a recognition ledger and a geometric hinge deficit forces all such deficits to be nonnegative, ruling out a whole class of gravity models.

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The sign obstruction

In the Recognition Science framework, a ledger is a discrete record of recognition events, and the cost of each recognition is forced to be a specific nonnegative function. A proposed bridge (`LedgerToHingeBridge`) would equate the ledger's deficit at each cell with a raw geometric hinge deficit, a signed quantity that can be negative. The declaration `bridge_forces_nonneg_geometricDeficit` establishes that any such bridge forces the geometric deficit to be nonnegative at every hinge in the image of the comparison map. This is a direct consequence of the fact that ledger deficits are sums of nonnegative costs.

The immediate consequence is a sign obstruction: no bridge can exist for any specification that places a strictly negative deficit in the image of the comparison map. This is formalized as `no_bridge_matches_negative_deficit_spec`. The framework's library of formal theorems proves this for any recognition ledger whatsoever, with no exceptions. The declaration does not, however, claim that negative geometric deficits actually occur in physical weak-field deformation classes. That is a separate geometric premise, stated but not formalized in the same file.

The declaration's significance is that it rules out the assumed raw-deficit bridge form on two-sided weak-field deformation classes, provided the geometric premise holds. The corrected bridge target is a nonnegative curvature-quadratic energy, not the raw signed deficit. This is a structural result about what kind of bridge can be built, not a statement about the actual values of physical constants.

THEOREM bridge_forces_nonneg_geometricDeficit · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean
bridge_forces_nonneg_geometricDeficit · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean:68 
/-- **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 no_bridge_matches_negative_deficit_spec · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean
no_bridge_matches_negative_deficit_spec · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean:81 
/-- **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)
MODEL LedgerBridgeNoGoStatus · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean
LedgerBridgeNoGoStatus · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean:259 
/-- 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

Negative geometric deficits actually occur in physical weak-field deformation classes. The assumed bridge form is unsatisfiable without the geometric premise. The theorem applies to all possible bridge specifications, only those with a negative deficit in the image.

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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