Encyclopedia Gravity Gravity Seven Gaps Ledger Bridge No Go No Ledger Family Linear Response

ARTICLE 3 claims 3 theorems

Gravity Seven Gaps Ledger Bridge No Go No Ledger Family Linear Response

A machine-checked theorem shows that a certain class of discrete ledger models cannot produce a specific signed response to deformation, reshaping what a gravity bridge can be.

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The parity obstruction

In the Recognition Science framework, a ledger is a discrete record of recognition events, and each event carries a forced cost. The theorem no_ledger_family_linear_response concerns a family of such ledgers, one for each value of a small deformation parameter ε. It proves that if the cost of comparing two cells is the J-cost of a ratio that obeys the natural parity condition r(-ε) = r(ε)⁻¹, then the ledger's deficit at any cell is an even function of ε. An even function cannot equal a nonzero linear function c·ε on any symmetric interval, so no such family can have a deficit that responds linearly to the deformation.

The result is a formal theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions. It relies on two earlier facts: the ledger deficit is always nonnegative, and the J-cost satisfies the symmetry J(x) = J(1/x). For a one-parameter family of ratios like the exponential strain r = exp(ε·s), the parity condition holds automatically, and the induced deficit is even. The theorem then excludes any signed linear response, meaning the leading term in ε is quadratic, not linear.

The theorem does not claim that no ledger family can respond to deformation at all. It only rules out a specific linear response for parity-covariant J-ratio families. Ratio families that violate the parity condition, such as r = exp(ε·s + ε²·t) with t ≠ 0, escape the theorem. The geometric premise that two-sided weak-field deformation classes contain hinges with strictly negative deficit and odd leading response is a prose-tier assumption, not formalized in the same file. The corrected bridge target is a nonnegative curvature-quadratic energy, not the raw signed deficit.

THEOREM ledger_family_deficit_even_of_ratio_parity · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean
ledger_family_deficit_even_of_ratio_parity · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean:186 
/-- **THEOREM (parity no-go, ledger-family form).** For any one-parameter
family of recognition ledgers whose costs are J-costs of PARITY-COVARIANT
positive ratios (r(-ε) = r(ε)⁻¹), the `RecognitionLedger.deficit` at every
cell is even in ε. Scope note: the parity hypothesis is genuine scope, not
decoration; ratio families violating it (e.g. r = exp(ε·s + ε²·t) with
t ≠ 0) escape this theorem. Exponential strain ratios r = exp(ε·s), the
natural first-order substrate deformations, satisfy it. -/
theorem ledger_family_deficit_even_of_ratio_parity
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : ℝ → RecognitionLedger.RecognitionLedger Λ)
    (r : ℝ → Λ → Λ → ℝ)
    (hpos : ∀ ε i j, 0 < r ε i j)
    (hcost : ∀ ε i j, (L ε).cost i j = Cost.Jcost (r ε i j))
    (hpar : ∀ ε i j, r (-ε) i j = (r ε i j)⁻¹)
    (ε : ℝ) (i : Λ) :
    RecognitionLedger.deficit (L (-ε)) i
      = RecognitionLedger.deficit (L ε) i := by
  unfold RecognitionLedger.deficit
  refine Finset.sum_congr rfl fun j _ => ?_
  rw [hcost (-ε) i j, hcost ε i j, hpar ε i j]
  exact (Cost.Jcost_symm (hpos ε i j)).symm
THEOREM no_ledger_family_linear_response · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean
no_ledger_family_linear_response · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean:208 
/-- **THEOREM (parity no-go, ledger-family linear-response form).** No
one-parameter family of recognition ledgers with parity-covariant J-ratio
costs can have a deficit matching a signed linear response c·ε (c ≠ 0) on
any symmetric interval. -/
theorem no_ledger_family_linear_response
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : ℝ → RecognitionLedger.RecognitionLedger Λ)
    (r : ℝ → Λ → Λ → ℝ)
    (hpos : ∀ ε i j, 0 < r ε i j)
    (hcost : ∀ ε i j, (L ε).cost i j = Cost.Jcost (r ε i j))
    (hpar : ∀ ε i j, r (-ε) i j = (r ε i j)⁻¹)
    (i : Λ) (c a : ℝ) (hc : c ≠ 0) (ha : 0 < a) :
    ¬ (∀ ε : ℝ, |ε| ≤ a → RecognitionLedger.deficit (L ε) i = c * ε) := by
  intro hmatch
  have hpa : RecognitionLedger.deficit (L a) i = c * a :=
    hmatch a (le_of_eq (abs_of_pos ha))
  have habs : |(-a)| = a := by
    rw [abs_neg]
    exact abs_of_pos ha
  have hna : RecognitionLedger.deficit (L (-a)) i = c * (-a) :=
    hmatch (-a) (le_of_eq habs)
  rw [ledger_family_deficit_even_of_ratio_parity L r hpos hcost hpar a i]
    at hna
  have hca : c * a = 0 := by linarith
  rcases mul_eq_zero.mp hca with h | h
  · exact hc h
  · exact absurd h (ne_of_gt ha)
THEOREM no_ledger_family_linear_response · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean
no_ledger_family_linear_response · IndisputableMonolith/Gravity/SevenGaps/LedgerBridgeNoGo.lean:208 
/-- **THEOREM (parity no-go, ledger-family linear-response form).** No
one-parameter family of recognition ledgers with parity-covariant J-ratio
costs can have a deficit matching a signed linear response c·ε (c ≠ 0) on
any symmetric interval. -/
theorem no_ledger_family_linear_response
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : ℝ → RecognitionLedger.RecognitionLedger Λ)
    (r : ℝ → Λ → Λ → ℝ)
    (hpos : ∀ ε i j, 0 < r ε i j)
    (hcost : ∀ ε i j, (L ε).cost i j = Cost.Jcost (r ε i j))
    (hpar : ∀ ε i j, r (-ε) i j = (r ε i j)⁻¹)
    (i : Λ) (c a : ℝ) (hc : c ≠ 0) (ha : 0 < a) :
    ¬ (∀ ε : ℝ, |ε| ≤ a → RecognitionLedger.deficit (L ε) i = c * ε) := by
  intro hmatch
  have hpa : RecognitionLedger.deficit (L a) i = c * a :=
    hmatch a (le_of_eq (abs_of_pos ha))
  have habs : |(-a)| = a := by
    rw [abs_neg]
    exact abs_of_pos ha
  have hna : RecognitionLedger.deficit (L (-a)) i = c * (-a) :=
    hmatch (-a) (le_of_eq habs)
  rw [ledger_family_deficit_even_of_ratio_parity L r hpos hcost hpar a i]
    at hna
  have hca : c * a = 0 := by linarith
  rcases mul_eq_zero.mp hca with h | h
  · exact hc h
  · exact absurd h (ne_of_gt ha)

What this page does not claim

No claim that all ledger families are even in the deformation parameter. No claim that the geometric premise about negative deficits is formally proved. No claim that the theorem applies to ratio families violating the parity condition.

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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