ch-math-testbed

15 Mathematical Theory Construction

AGENTIC in Exact and Generated Worlds

Part II ended by composing causal diagnosis, executable skill construction, and sequential experiment choice into the AGENTIC meta-sketch. Part III changes the unit of comparison. Its subject is no longer the component workflow but the admission environment: what kind of evidence can admit a constructed artifact or a proposed package comparison in mathematics, simulation, scientific literature, or visual generation?

These environments are not ranked versions of one ground truth. Exact proof, executable simulation, source-bearing testimony, and perceptual observation expose different objects and different failure modes. Provenance, uncertainty, counter-witnesses, and abstention must therefore be calibrated anew in each chapter rather than inherited unchanged from the preceding testbed.

Each chapter states which AGENTIC packet fields are registered, estimated, constructed, or independently admitted; a missing component remains visibly missing rather than being supplied by the change of domain. In particular, the common packet schema organizes comparisons across testbeds, but it does not make their evidence interchangeable or imply that every testbed realizes the complete CLIC–OPTIC–RELIC loop.

Throughout Part III, the verbs reports, supports, establishes, and proves retain the meanings fixed by the evidence contracts in Section 10.4.2. A stronger reading requires a separately named admission environment.

Table 15.1 fixes those environments before the individual studies begin. Its last column is as important as its middle one: an internally decisive result may still leave the external scientific or creative question open.

Regime	Internal reference	Strongest internal verdict	Not established internally
Generated mathematics	Frozen axioms, model finder, proof checker, and withheld constructor	Formal correctness and recovery or construction relative to the generated theory	Historical importance, conceptual value, or a new mathematical field
Executable simulation	Private simulator state and registered intervention semantics	Mechanism or theory adequacy under held-out simulated regimes	Truth about the corresponding natural system
Scientific corpus	Source-bearing claims, contexts, and locked reports	Corpus-relative support, contradiction, prediction, and a testable proposal	Causal truth or success of the proposed external experiment
ARTISTIC visual assurance	Registered or estimated scene declaration and an independent structural observer	Constraint satisfaction and preservation relative to that declaration	A unique correct image, universal perceptual validity, or artistic invention
DILATE artistic extension	Withheld operations in controlled studies; reuse and critical observers in open settings	Recovery or admission of a reusable operation under registered criteria	Objective artistic value or a general theory of creative importance

Table 15.1 Evidence regimes in Part III. The internal reference is the testbed’s ground truth or admission authority; the final column blocks a stronger external claim.

Mathematics supplies the cleanest setting in which a proposed theory can be checked without confusing fluency for correctness. It also supplies historical examples in which a new concept generated a durable field.

For AGENTIC, mathematics is the sharpest admission environment and the most delicate interpretation of CLIC. Transforming an algebraic object or querying a countermodel is an intervention in a formal world, but it is not thereby a causal intervention. We use the name CLIC only when the generated world has a declared mechanism and intervention semantics. Otherwise the same position in the workflow is occupied by a structural diagnostician: it discovers invariants, dependencies, closure failures, and discriminating constructions without making a causal claim.

¹1. Chapter roadmap. The opening sections define the mathematical contract and study controlled construction. The DIAL specialization of Universal Reinforcement Learning (DIAL-URL) then asks whether sequential model classes can be reorganized coalgebraically, before the Planck episode separates pattern repair from causal explanation. ↩

The controlled evidence in this chapter reaches finite declaration and presentation construction inside frozen formal languages. Its strongest positive certificates concern exact recovery, countermodel rejection, conservative extension, and reuse in generated worlds; every positive, negative, or unresolved outcome is carried by a status-bearing admission record. These results do not establish autonomous field formation or the historical importance of a constructed concept. That unreached endpoint is why persistence appears as the final level of the hierarchy below rather than as a label attached to every successful recovery.