MultiplicativeL4
plain-language theorem explainer
MultiplicativeL4 defines the multiplicative form of composition consistency (L4) for any MultiplicativeRecognizer m on positive reals. It requires existence of a combiner P such that the sum of costs on product and quotient equals P applied to the separate costs. Researchers deriving automatic satisfaction of (L4) from recognition structures cite this definition when moving from abstract hypotheses to the multiplicative carrier. The definition is a direct existential encoding with no additional proof steps.
Claim. Let $m$ be a multiplicative recognizer. Then there exists a function $P : ℝ → ℝ → ℝ$ such that for all $x, y > 0$, cost$_m(xy) +$ cost$_m(x/y) = P($cost$_m x$, cost$_m y)$, where cost$_m$ denotes the cost induced by the continuous comparator of $m$ that satisfies the Laws of Logic.
background
A MultiplicativeRecognizer is a structure pairing a geometric recognizer onto the positive reals with a continuous ComparisonOperator and a proof that the operator satisfies the Laws of Logic (four Aristotelian conditions plus scale invariance and non-triviality). Its cost is the derivedCost of that comparator, as defined in the sibling cost declaration. The module document states that the default equality-induced cost on (ℝ>0, ·) fails (L4), but the conditional theorem holds when the comparator is continuous and Law-of-Logic-satisfying. Upstream, LogicAsFunctionalEquation.RouteIndependence supplies the d'Alembert functional equation that this definition encodes, while DAlembert.LedgerFactorization.of calibrates the J-cost on the multiplicative group.
proof idea
This is a definition, not a derived theorem. It directly transcribes the existential claim of the d'Alembert form of route-independence onto the cost function of a MultiplicativeRecognizer. No lemmas are invoked and no tactics are used; the body is the statement of the required combiner P.
why it matters
The definition is the target of the downstream theorem multiplicativeRecognizer_satisfies_L4, which concludes that (L4) holds automatically for any multiplicative recognizer. It closes the gap noted in the module document between the abstract RecognizerComposition hypothesis and the derived result under the multiplicative event space, referencing the companion paper RS_Recognition_Geometry_Logic_Unification.tex. It supports the Recognition framework's derivation of composition laws from the single functional equation via the d'Alembert inevitability route on positive ratios.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.