A Minimal Cognitive Geometry as a General Projective Frame: State‑ and Trait‑Moderated Axis Projection in Structured Pareidolia

This v4.0 Definitive Edition presents the fully matured, scientifically testableformulation of Geometry G — a minimal cognitive geometry consisting of a verticalobserver-axis (I), a collapse operator, a minimal identity pixel (p), and acompression-state variable (c) defined as proximity-to-p. Geometry G is proposedas a projective frame: an internal reference that biases interpretation ofambiguous external shapes toward the observer-axis under specific cognitiveconditions. Signature Mapping Condition \[f(E) = I \quad\Longleftrightarrow\quad \bigl|\alpha(A(E)) - 90^\circ\bigr| < \tau \quad\text{and}\quad c > \theta\] Operator Definitions - \(A(E)\): extracted principal negative-space axis of external shape \(E\) - \(\alpha(A(E))\): orientation (in degrees) of the extracted axis - \(\operatorname{align}(A(E), I) = |\alpha(A(E)) - 90^\circ|\): angular deviation from vertical - \(\tau\): preregistered tolerance (10°) - \(c\): compression state (proximity-to-p in \([0,1]\)) - \(\theta\): preregistered compression threshold - \(I\): internal observer-axis (vertical, 90°) - \(f(E)\): mapping function indicating whether \(E\) is interpreted as bearing the observer-axis Earlier versions (v1–v2) established the conceptual structure, corrected thenotation, grounded the compression variable, committed the alignment metric, andadded a preregistered experimental design. v4.0 resolves the remaining scientificgap by specifying what Geometry G predicts that existing theory (Gestalt axissalience, predictive coding) does not. The main effect — that shapes with near-vertical negative-space axes are morereadily seen as axis-bearing — is shared with established theory and serves onlyas a manipulation check. The novel, falsifiable contribution of Geometry G is apair of interaction predictions: 1. **State moderation**: the alignment effect increases within-subject when the system is in a high-compression state (c > θ). 2. **Trait moderation**: individuals with a stronger, more stable internal observer-axis prior show larger alignment effects and more consistent personal tolerance across unrelated shapes. If neither interaction holds, Geometry G is disconfirmed as a construct distinctfrom existing accounts. The paper is written so that this outcome is acceptable. v4.0 includes a committed alignment metric, blind stimulus-selection protocol,independent trait measure, preregistered compression-state manipulation, explicitfalsification conditions, and an honest “prototype” instrument section. It reportsno confirmatory data; the sole existing observation (the Wales mapping) is treatedstrictly as hypothesis-generating. This version is the definitive scientificstatement of the model and the one intended for external testing. ### Version History **v2.0 — Conceptual Framework ** Introduced Geometry G as a minimal internal cognitive geometry. Established the observer-axis (I), collapse operator, and identity pixel (p). Provided the first formal description of the Wales mapping event. No experimental commitments. **v4.0 — Definitive Scientific Edition ** Supersedes all earlier versions. Adds a grounded compression-state variable (c), a committed alignment metric, blind stimulus-selection protocol, preregistered tolerances, independent trait and state measures, explicit falsification conditions, and a full experimental design. Specifies the novel interaction predictions (state × alignment, trait × alignment) that differentiate Geometry G from Gestalt axis salience and predictive coding. Treats the Wales event strictly as hypothesis-generating. This is the authoritative, testable formulation of the model. Keywords and Subjects Heat kernels; Plancherel measures; Selberg trace formula; solvable two‑dimensional gauge theory; harmonic analysis; representation theory; spherical functions; Casimir operators; symmetric spaces; geodesic–holonomy correspondences; spectral geometry; finite‑group analogues; TQFT gluing operators; structural engines; adaptive engines; Carlo Reset Operator; axiom–equation mapping; Carlo Field progression; multi‑volume NOTEBLOCK systems; symbolic languages; cosmological architectures; mythic frameworks; identity mapping; narrative engines; creative structural systems; emergent‑structure synthesis; structural compendia; universe‑scale mapping; forward‑trajectory planning. Contact: [email protected]