IndisputableMonolith.Measurement.RecognitionAngle.ActionSmallAngle
This module defines the unit direction in R3, the angle function, small-angle action, and the minimal threshold angle with its range and divergence properties. It is imported by modules on blind cones and temporal gating to build recognition feasibility predicates. The module consists of definitions and basic lemmas on ranges and infeasibility, drawing on imported classical axioms.
claimIntroduces the unit direction function returning the normalized vector from $x$ to $y$ in $R^3$ (zero on degeneracy), the angle at a point, the action $A$ as a function of angle, and the minimal angle threshold with the properties that action diverges below the threshold and the threshold lies in a specified interval.
background
The module sits in the Measurement domain and imports classical mathematical results declared as axioms from real analysis and functional equations. It works in Euclidean three-space R3 to define directional quantities for recognition processes. The supplied doc-comment for the direction function states it returns the unit vector from x to y, zero on degeneracy. Downstream modules use these to define blind-cone sets for action budgets and eight-tick temporal windows.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
Supplies the angular threshold and action divergence facts required by the blind-cone module (existence of positive threshold angle under Amax>0) and the temporal gating module (feasibility predicate combining angular threshold with temporal admissibility). It bridges the imported classical axioms to the recognition geometry used in the eight-tick octave and D=3 spatial setting.
scope and limits
- Does not prove the classical axioms imported from the upstream module.
- Does not treat large-angle or non-Euclidean regimes.
- Does not compute explicit numerical values for the minimal angle.
- Does not incorporate temporal or gating constraints directly.