Gravity Seven Gaps Hinge Stationarity Core Sourced Action Eq Jcost Sum

The sourced action identity

The declaration sourcedAction_eq_jcost_sum is a theorem in the framework's machine-checked library of formal theorems. It states that a certain action, a quantity that a physical system tends to minimize, can be written in two equivalent ways. The action in question, called sourcedAction, is defined as the sum over a set of cells of the expression (cosh(t_i) - 1), minus a coupling term that represents an external deficit source. The theorem proves this equals the sum of the framework's foundational cost function, a measure of the forced expense of recognition, applied to the exponential of each t_i, minus the same coupling term.

The content of the identity is that the geometric-looking cosh term is exactly the framework's cost function in disguise. This is not an approximation or a modeling assumption; it is a proved equality in the library. The practical consequence is that minimizing this action is the same as minimizing a sum of cost functions with a linear penalty. The theorem's proof is a simple unfolding of definitions and a known identity relating the cost function to cosh, but its role is to anchor the entire subsequent analysis of hinge stationarity to the framework's core cost function.

What the theorem does not claim is just as important. The coupling term, written as -(c/n) * sum_i t_i, is a constitutive choice. It models the presence of a deficit source, but the framework does not derive this term from its bare recognition ledger. The docstring is explicit: this is a MODEL, not a theorem. No statement in the file claims to derive this coupling from the RecognitionLedger alone. The theorem only establishes the equivalence of two expressions once that coupling term has been chosen.

This distinction matters for the broader project. The honest mechanism for deriving a ratio between recognition and deficit is not to impose a budget and then minimize, which the library shows is circular. Instead, one adds the explicit coupling term, minimizes the resulting action, and the unique minimizer yields the desired ratio with a cubic error term. The identity sourcedAction_eq_jcost_sum is the first step in that honest mechanism, ensuring that the cost term in the action is genuinely the framework's cost function.

/-- **THEOREM (kernel J-cost identification).** The cost term of the
sourced action IS the summed J-cost of the exponential per-tick strain
ratios: Phi(t) = sum_i J(exp t_i) - (c/n) * sum_i t_i, via
`Cost.Jcost_exp_cosh` (J(exp t) = cosh t - 1). This puts the promotion
phrase "derived from an explicit deficit-source constitutive action plus
J-stationarity" on a kernel equation rather than a docstring gloss: the
J-cost part of the action is identified with `Cost.Jcost` inside the
kernel, and ONLY the coupling term remains MODEL. -/
theorem sourcedAction_eq_jcost_sum (n : ℕ) (c : ℝ) (t : Fin n → ℝ) :
    sourcedAction n c t
      = (∑ i, Cost.Jcost (Real.exp (t i))) - c / n * ∑ i, t i := by
  unfold sourcedAction
  simp only [Cost.Jcost_exp_cosh]

/-- The sourced action Phi(t) = sum_i (cosh t_i - 1) - (c/n) * sum_i t_i:
the J-cost of the per-tick strains (via J(exp t) = cosh t - 1,
`Cost.Jcost_exp_cosh`) plus an explicit deficit-source coupling of total
strength c = kappa*delta, split evenly across the n ticks. The coupling
term is the MODEL input; see the module header. -/
noncomputable def sourcedAction (n : ℕ) (c : ℝ) (t : Fin n → ℝ) : ℝ :=
  (∑ i, (Real.cosh (t i) - 1)) - c / n * ∑ i, t i

What this page does not claim

The coupling term is not derived from the bare RecognitionLedger. The identity does not by itself establish the ratio between recognition and deficit; that requires the full minimization argument. The theorem does not claim that the action's minimizer is unique without the additional convexity argument.

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

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Derived articles

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