Selectivity at Three Geometric Calibration, Cortical Periodicity, and the Architecture of Embodied Verification

Two distinct enumeration logics produce comparable counts at small parameter values: the Boolean lattice count (2ⁿ − 1, the number of non-empty subsets of an n-element set) and the partition-refinement-pair count (OEIS A265947, the number of ordered pairs of integer partitions of n related by refinement). The two sequences agree exactly at n = 1 and n = 2, differ by exactly one at n = 3 and n = 4, and diverge for n ≥ 5. The unit gap at n = 3 is closed by a single bifurcation in the partition-refinement structure, corresponding to the geometric distinction between cubic and trigonal symmetry in three-dimensional crystallography. At n = 3, partition-refinement structure plus the crystallographic restriction theorem plus the orthogonality bifurcation uniquely determines a realized lattice geometry per refinement-comparable pair — a selectivity property that fails at n ≥ 4. Three is the unique parameter value at which abstract conceptual distinction and geometric realizability calibrate to within a single observable bifurcation. Independently, cortical tissue exhibits a graded decline in translational periodicity from V1’s two-dimensional hypercolumnar lattice to the Default Mode Network’s non-periodic checkered interdigitation, with the body operating throughout as a piezoelectric crystalline measurement instrument that converts mechanical force into gene expression through a fully mapped molecular pathway. The cortical periodicity gradient correlates with the Inverse Constraint Hypothesis: language’s conditioning power over cognition varies inversely with the body’s capacity for sensory verification of the relevant domain. This paper argues that these results are two presentations of one architecture. The unifying principle is selectivity-as-geometric: where periodic or geometric structure constrains, selection occurs; where it does not, selection is absent. The principle has three architecture-matched instances — condensed-matter band structure, the n = 3 crystallographic selectivity property, and cortical lattice filtering — with preserved internal decomposition independent of the cardinality at which each operates. The joint claim: the cortical periodicity gradient is the tissue-level implementation of the dimensional selectivity gradient. V1 and adjacent crystalline tissue process inputs whose conceptual distinctions are uniquely anchored under the n = 3 selectivity property; the Default Mode Network and adjacent mosaic tissue process constructions whose distinctions sit in the n ≥ 4 regime, where geometric anchoring fails. Chronic linguistic load on mosaic-end constructions produces measurable tissue damage through the documented matrix-metalloproteinase pathway that injurious mechanical loading also activates. Healing operates by relocating constructions from the n ≥ 4 regime to the n = 3 regime, with physiological signatures distinct from interventions that reframe within the n ≥ 4 regime. The architecture is falsifiable through three distinct experimental routes using existing measurement technology.