Gravity Seven Gaps Hinge Stationarity Core Budget Implies Ratio Without Stationa

The budget trap

In the Recognition Science framework, a central question is how a ratio of physical quantities, such as a strain ratio at a hinge, relates to a budget, a sum of contributions that must add up to a fixed total. A naive approach seems attractive: impose the budget as a constraint, minimize some cost function, and conclude that the ratio must equal the budget. The declaration budget_implies_ratio_without_stationarity shows that this approach is circular. It proves that if you assume the budget sum equals a target value, then the ratio trivially equals that same value, and the minimization step contributes nothing. The proof term is the budget hypothesis itself, which is the whole point.

The theorem is a formal statement in the framework's machine-checked library of formal theorems, with zero unproved axioms. It does not say that the ratio cannot be derived from a budget; it says that the naive formulation does not derive it. The budget hypothesis already contains the conclusion. This is a kill record: it eliminates a proposed route, not the goal. The honest mechanism, developed separately in the same module, adds an explicit coupling term to the cost function and then minimizes. That construction, with the coupling term as a constitutive model choice, yields a unique minimizer and a ratio with a controlled cubic error term.

What the theorem does not claim is as important as what it proves. It does not claim that stationarity is useless; it claims that stationarity adds nothing in the naive formulation. It does not claim that the ratio is unattainable; it claims that the budget hypothesis alone is insufficient. It does not claim that the coupling term is derived from the bare recognition ledger; that term is an explicit model choice. The declaration is a precise negative result, a guardrail that redirects the derivation to a sound path.

/-- **THEOREM (kill record raw 1b, the budget circularity).** If the
holonomy budget sum_i t_i = kappa*delta is IMPOSED, then the "conclusion"
log ratio = kappa*delta holds with NO optimization used: the proof term is
the budget hypothesis itself. This records, kernel-checked, that the naive
1b formulation (impose budget, minimize, conclude ratio) is CIRCULAR: the
budget already contains the conclusion; stationarity added nothing. -/
theorem budget_implies_ratio_without_stationarity {n : ℕ} (t : Fin n → ℝ)
    (kappa delta : ℝ) (hbudget : ∑ i, t i = kappa * delta) :
    naiveLogRatio n t = kappa * delta := hbudget

What this page does not claim

The theorem does not claim that stationarity is useless in general. The theorem does not claim that the ratio cannot be derived from a budget at all. The theorem does not claim that the coupling term is derived from the bare recognition ledger.

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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