recognitionThetaConvergenceAttackSurface
plain-language theorem explainer
The definition packages the majorant-based convergence theorem and its geometric specialization into the RecognitionThetaConvergenceAttackSurface structure. Researchers addressing sub-conjecture A.1 cite it to mark that the comparison step is discharged once a summable majorant is available. It is assembled as a direct structure literal that routes the two upstream theorems to the required fields.
Claim. The attack surface structure supplies the map sending any majorant package for the Recognition Theta terms to a proof of summability for every $t > 0$, together with the map sending any geometric majorant to the same summability result.
background
The module tracks sub-conjecture A.1, which requires summability of the Recognition Theta term for every positive $t$. The attack surface structure records that the general comparison theorem has been proved, leaving only the construction of a summable nonnegative majorant as the remaining input, ideally geometric after finitely many terms. Upstream results establish that a majorant package yields convergence by applying norm-bounded summability to the majorant at each $t$, while the geometric version reduces to the general case via conversion to a standard majorant.
proof idea
This definition is a direct structure construction. It assigns the theorem that any majorant package proves Recognition Theta convergence to the comparison field and the theorem that any geometric majorant proves the same result to the geometric comparison field.
why it matters
This declaration completes the comparison step for sub-conjecture A.1 in the Recognition Science framework. It supports the summability argument needed for the Recognition Theta term, which aligns with the discrete eight-tick octave and J-cost selection in the RS model as opposed to continuous relaxation in axion approaches. The open question it leaves is the explicit construction of the summable majorant from lower bounds on the cost spectrum values and upper bounds on the phi-rung weights.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.