IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.CertifiedAnalyticProtocols
Countable registry of certified analytic protocol ingredients for the primitive recognition calculus. Constant protocols and unary transformers are indexed by naturals; binary structure stays with the native additive operations already available on DeltaReal. Downstream analysis and strong-closure modules import this registry so every evaluated real sits under a certified protocol. The module is definitional scaffolding plus closure lemmas, not a deep existence proof.
claimA countable registry of certified analytic protocols: constants $c_k$ and unary transformers $U_k$ indexed by $k \in \mathbb{N}$, with evaluation of protocol expressions built from rationals and native additive operations on $\Delta\mathbb{R}$. The set of values is countable, and every value admits a protocol witness. Binary analytic operations beyond addition are deferred to unary transforms on paired protocols.
background
Primitive recognition calculus works over a real-like carrier DeltaReal that already carries certified additive structure. Analytic claims in the Recognition stack must not float free: each constant and each unary transform used in estimates must be registered so that evaluation stays inside a countable, inspectable class.
This module introduces that registry. Expressions are built from registered constants, registered unary transformers, and the native additive operations (addition, negation, subtraction) already proved for DeltaReal. Evaluation maps an expression to a real value; the image of evaluation is the set of protocol values.
The design choice is deliberate: keep binary analytic operations out of the core registry and encode them later as unary maps on paired protocols if needed. That keeps the inductive structure thin and reuses DeltaReal's additive theorems rather than re-proving a full field of analytic combinators.
proof idea
Definition module with supporting lemmas, not a single deep proof. The registry and expression type are inductive data; evaluation is a recursive interpreter. Countability of the value set follows from countability of the expression language. Closure under rationals and under native addition, negation, and subtraction is by construction of the evaluators. A covering lemma states that every value in the image has some protocol expression witnessing it. Transcendental-protocol closure packages the registry as a closed class for later analytic transformers.
why it matters in Recognition Science
Certified analytic work in the foundation layer needs a single place that names which constants and unary maps are allowed. This module is that place. It is imported by CertifiedAnalyticTransformers (which builds further protocol maps on top of the registry), by DeltaNativeAnalysis (native analytic estimates over protocol values), and by DeltaNativeStrongClosure (strong closure of the native DeltaReal fragment under certified operations).
Without a countable certified registry, later claims that every relevant real arising in recognition estimates is protocol-backed would be informal. The module therefore sits under the analytic half of PrimitiveRecognitionCalculus and feeds the path from DeltaReal additive structure toward closed native analysis used in the broader forcing and recognition chain.
scope and limits
- Does not define binary analytic operations beyond native addition, negation, and subtraction.
- Does not prove physical or Recognition-forcing theorems; only registry and evaluation infrastructure.
- Does not certify arbitrary Mathlib analytic functions unless registered as const or unary entries.
- Does not establish uniqueness of protocols; multiple expressions may evaluate to the same value.
- Does not replace DeltaReal; it only layers countable protocol witnesses on top of it.