FeedbackControlModesCert
plain-language theorem explainer
A certificate structure asserts that the feedback control modes form a finite set of cardinality exactly five, fixing the configuration dimension at five in the cybernetics layer. Control theorists working inside the Recognition Science framework would cite it when anchoring PID extensions plus feedforward and adaptive channels. The declaration is realized as a direct structure that encodes the cardinality via the Fintype instance on the enumerated inductive type.
Claim. The certificate that the set of feedback control modes has cardinality five: $ |M| = 5 $, where $ M = $ {proportional, integral, derivative, feedforward, adaptive}.
background
The module identifies five canonical feedback-control modes with configuration dimension D = 5. These consist of the proportional, integral, and derivative modes that form the PID axes, together with feedforward for model-based control and adaptive for learning-based channels. The modes are introduced as an inductive type whose five constructors derive DecidableEq, Repr, BEq, and Fintype, allowing direct cardinality computation without further axioms.
proof idea
This is a structure definition whose single field directly records the equality Fintype.card of the mode type equals five. It relies on the Fintype instance already derived from the inductive enumeration of the five constructors.
why it matters
The certificate fixes the mode count at five and thereby anchors the cybernetics depth of the Recognition Science framework. It is consumed by the concrete instance definition that supplies the cardinality value. The module records zero sorry and zero axiom, closing the enumeration for downstream control applications and aligning with the configuration dimension that extends the spatial D = 3 from the forcing chain.
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