Gravity Seven Gaps Recognition Ratio Substrate Blocker Deficit Source Coupling L

The strain receipt

The theorem deficitSourceCoupling_logRatio_eq_minimizer_strain is a receipt inside the Recognition Science framework's machine-checked library of formal theorems. It states that once a signed deficit-source constitutive coupling is supplied, the logarithm of the recognition ratio equals the total strain of the unique global minimizer of the sourced action. In plainer terms: the framework models a ledger, a discrete record of recognition events, and this result says that when the ledger is augmented with a source that couples linearly to total strain, the derived ratio is exactly the exponential of that strain. The equality is not an assumption; it is proved from stationarity of the action.

The result is conditional. It does not claim that the bare ledger, without the coupling, yields the ratio. In fact, the pack proves the opposite: a bare two-cell ledger cannot distinguish between opposite source orientations, so no function of the bare ledger can universally recover the signed source. The coupling is extra constitutive data, not information contained in the ledger itself. The theorem's role is to show that once that data is named, the ratio follows from J-stationarity with an explicit remainder bound, and the minimizer's strain is the bridge.

The theorem also does not claim that the recognition ratio is derived from nothing. It depends on the named premise, and the pack's headline theorem states this exactly: the missing premise is the signed deficit-source constitutive coupling, and without it the ratio is not derivable from the bare ledger. The positive result is a conditional derivation, not an unconditional one. The pack also shows the sourced construction is populated by a genuine family, not a zero-source or fixed-mesh witness, so the conditional route is not vacuous.

/-- **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

/-- **THEOREM (same bare ledger, opposite signed source).** Reversing the
source orientation leaves every J-cost, hence the entire bare recognition
ledger, unchanged. -/
theorem signBlindBareLedger_neg_eq (d : ℝ) :
    signBlindBareLedger (-d) = signBlindBareLedger d := by
  apply recognitionLedger_cost_ext
  funext i j
  rw [show (signBlindBareLedger (-d)).cost i j
      = Cost.Jcost ((twoHingeWitnessBridge (-d)).xRatio i
          / (twoHingeWitnessBridge (-d)).xRatio j) from
        ratioBridgeLedger_cost (twoHingeWitnessBridge (-d)) i j]
  rw [show (signBlindBareLedger d).cost i j
      = Cost.Jcost ((twoHingeWitnessBridge d).xRatio i
          / (twoHingeWitnessBridge d).xRatio j) from
        ratioBridgeLedger_cost (twoHingeWitnessBridge d) i j]
  have hratio :
      (twoHingeWitnessBridge (-d)).xRatio i
          / (twoHingeWitnessBridge (-d)).xRatio j
        = ((twoHingeWitnessBridge d).xRatio i
            / (twoHingeWitnessBridge d).xRatio j)⁻¹ := by
    rw [twoHingeWitnessBridge_xRatio_neg d i,
      twoHingeWitnessBridge_xRatio_neg d j, inv_div_inv, inv_div]
  rw [hratio]
  exact (Cost.Jcost_symm
    (div_pos ((twoHingeWitnessBridge d).xRatio_pos i)
      ((twoHingeWitnessBridge d).xRatio_pos j))).symm

What this page does not claim

The theorem does not claim the recognition ratio is derivable from the bare ledger without the coupling. The theorem does not claim the signed source orientation is information contained in the bare ledger. The theorem does not claim the sourced construction is unique or that the family is the only admissible one.

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)

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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