Gravity Seven Gaps Recognition Ratio Substrate Blocker Recognition Ratio Derived

The conditional bridge

In the Recognition Science framework, a ledger is a discrete record of recognition events, and the cost of each recognition is forced by a proved law. A central question is whether a particular ratio, the recognition ratio, can be derived from the ledger alone. The declaration recognition_ratio_derived_of_deficit_source_coupling answers that question with a precise conditional: the ratio follows, but only after a specific extra ingredient is added.

That ingredient is a deficit-source constitutive coupling: a rule that assigns to each event a source strength equal to a coupling constant times a geometric deficit, and couples that source linearly to the total strain in the cost action. The theorem proves that once this coupling is supplied, stationarity of the resulting action derives the bridge relation between the log of the recognition ratio and the product of the coupling and deficit, up to an explicit cubic remainder bounded by the mesh scale cubed.

The declaration is a theorem in the framework's machine-checked library of formal theorems. It is not a definition of the ratio, and it does not assume the ratio relation as a hypothesis. The proof shows the relation emerges from the named coupling and the already-proved cost law. The statement also includes a stationarity receipt: the log of the ratio equals the total strain of the unique minimizer of the sourced action.

What the declaration does not claim is just as important. It does not claim the ratio can be derived from a bare ledger. A separate theorem proves that no function of a bare two-cell ledger can universally recover the signed source, because reversing the source orientation leaves the entire ledger unchanged. The signed deficit-source orientation is extra constitutive data, not information contained in the ledger itself. The conditional theorem is the positive route: with that data supplied, the derivation closes.

/-- **THEOREM (the conditional `recognition_ratio_derived`).** Once the
named deficit-source constitutive coupling is supplied, J-stationarity
derives the bridge relation with explicit remainder constant `n / 6`.
No hypothesis states a fact about `xRatio` or `log xRatio`. -/
theorem recognition_ratio_derived_of_deficit_source_coupling {H : Type*}
    (C : DeficitSourceConstitutiveCoupling H) (σ : H) :
    |Real.log ((ratioBridgeFromDeficitSourceCoupling C).xRatio σ)
        - C.kappa σ * C.geometricDeficit σ|
      ≤ (C.channels : ℝ) / 6 * C.meshScale ^ 3 :=
  (ratioBridgeFromDeficitSourceCoupling C).ratio_relation σ

/-- **THEOREM (stationarity receipt).** The ratio in the conditional
derivation is the exponential of the total strain of the unique sourced
minimizer. -/
theorem deficitSourceCoupling_logRatio_eq_minimizer_strain {H : Type*}
    (C : DeficitSourceConstitutiveCoupling H) (σ : H) :
    Real.log ((ratioBridgeFromDeficitSourceCoupling C).xRatio σ)
      = ∑ i, sourcedMinimizer C.channels (C.sourceStrength σ) i := by
  rw [C.source_eq σ]
  exact ofStationarity_log_xRatio_eq_minimizer_strain C.channels
    C.channels_pos C.kappa C.geometricDeficit C.meshScale
    C.meshScale_pos
    (fun τ => by
      rw [← C.source_eq τ]
      exact C.source_dominated τ)
    σ

/-- **THEOREM (the exact bare-ledger blocker).** No function of a bare
`RecognitionLedger (Fin 2)` can universally recover the signed source of
the exact unit-coupled witness family.  The ledgers at sources `1` and `-1`
are equal, while the required outputs are different.  Therefore signed
deficit-source orientation is extra constitutive data, not information
contained in the bare ledger. -/
theorem no_bare_ledger_selector_recovers_signed_source :
    ¬ ∃ select : RecognitionLedger.RecognitionLedger (Fin 2) → ℝ,
      RecoversSignedSourceFromBareLedger select := by
  rintro ⟨select, hselect⟩
  have hneg := hselect (-1)
  have hpos := hselect 1
  rw [signBlindBareLedger_neg_eq 1] at hneg
  norm_num at hneg hpos
  linarith

What this page does not claim

The recognition ratio is not derived from a bare recognition ledger without extra input. The theorem does not identify which physical system realizes the deficit-source coupling. The declaration does not prove the ratio equals a specific numeric constant.

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

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