Gravity Seven Gaps Recognition Ratio Substrate Blocker Deficit Source Action Eq

The sourced action identity

The declaration deficitSourceAction_eq_jcost_sum is a theorem in the framework's machine-checked library of formal theorems. It states a precise identity about a proposed action: the value of the sourced action equals the sum of the J-cost of each exponential strain, minus a term proportional to the total strain. The source strength is defined as the product of a coupling constant and a geometric deficit, and it is coupled linearly to the strain. The identity holds for any channel count and any source strength satisfying the structural bound.

In plain terms, the theorem says that if you write down an action that is the sum of recognition costs minus a linear source term, then that action is exactly what the framework calls the sourced action. It does not say that such a source term exists in nature, nor does it say that the recognition ratio follows from the bare ledger. The theorem is a conditional identification: given the named constitutive coupling, the action takes this form. The proof is a direct rewrite of the definition of the sourced action, so it is a theorem about the framework's own definitions, not an empirical claim.

The theorem is part of a larger package that establishes a precise blocker. The bare recognition ledger, which records only costs, cannot recover the signed source orientation: two ledgers with opposite source signs have identical costs, so no function of the bare ledger can universally recover the sign. The sourced action identity is the first step in showing that once the source coupling is supplied as extra data, J-stationarity derives the recognition-ratio bridge. The identity itself, however, does not supply that data; it only identifies the form of the action once the data is given.

What the theorem does not claim is just as important. It does not claim that the deficit-source coupling is physically real, that the recognition ratio is derived from the bare ledger, or that the source strength is nonzero. The identity is a formal statement about the framework's definitions, and its proof does not introduce any new axiom or assumption about the ratio relation. The theorem is a clean, kernel-checked step in a larger argument, and it earns its place by making the conditional structure of the derivation explicit.

/-- **THEOREM (kernel identification of the premise).** The constitutive
action is exactly the summed J-cost of the exponential strains minus the
linear deficit-source term.  This identifies the missing premise inside the
kernel without assuming any ratio relation. -/
theorem deficitSourceAction_eq_jcost_sum {H : Type*}
    (C : DeficitSourceConstitutiveCoupling H) (σ : H)
    (t : Fin C.channels → ℝ) :
    deficitSourceAction C σ t
      = (∑ i, Cost.Jcost (Real.exp (t i)))
          - C.sourceStrength σ / C.channels * ∑ i, t i :=
  sourcedAction_eq_jcost_sum C.channels (C.sourceStrength σ) t

/-- **MODEL (the exact premise missing from the bare ledger).**

A signed deficit-source constitutive coupling supplies a channel count, the
hinge coupling and signed geometric deficit, and a source strength satisfying

`sourceStrength sigma = kappa sigma * geometricDeficit sigma`.

It also supplies the positive mesh scale and the structural small-source
bound needed by the cubic estimate.  No field mentions the recognition
ratio or its logarithm. -/
structure DeficitSourceConstitutiveCoupling (H : Type*) where
  channels : ℕ
  channels_pos : 1 ≤ channels
  kappa : H → ℝ
  geometricDeficit : H → ℝ
  sourceStrength : H → ℝ
  source_eq : ∀ σ, sourceStrength σ = kappa σ * geometricDeficit σ
  meshScale : ℝ
  meshScale_pos : 0 < meshScale
  source_dominated :
    ∀ σ, |sourceStrength σ| ≤ (channels : ℝ) * meshScale

/-- **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 theorem does not claim that the deficit-source coupling exists in nature. The theorem does not claim that the recognition ratio follows from the bare ledger alone. The theorem does not claim that the source strength is nonzero.

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.

Derived articles

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