R3
The three-dimensional Euclidean space serves as the ambient manifold for point triples in angular separation calculations. Researchers constructing blind regions under action budgets would cite this when typing reference points and deriving feasibility predicates. The declaration is a direct type abbreviation with no lemmas or reductions required.
claimThe three-dimensional Euclidean space over the reals, denoted here as the standard manifold in which points reside for angle computations.
background
The module supplies definitions for blind-cone sets of point pairs whose separation angle at a reference point lies below a budget-dependent threshold. The ambient space for these points is the standard three-dimensional Euclidean space, matching the spatial dimension fixed by the unified forcing chain. Upstream, the minimal admissible angle is realized as the arcsine of the exponential of the negated budget, guaranteeing a strictly positive threshold whenever the budget is positive. Identical space abbreviations appear in the sibling ActionSmallAngle and TemporalGating modules to maintain uniform typing across the measurement layer.
proof idea
This is a one-line abbreviation that directly equates the local name for three-dimensional space to the Mathlib EuclideanSpace constructor applied to the three-element finite index set. No tactics or upstream lemmas are invoked; the declaration is purely notational.
why it matters in Recognition Science
The abbreviation anchors all geometric constructions in the blind-cone module and directly feeds the blindCone set definition together with the angleAt function. These in turn support the angleOK predicate and the feasibility theorem exists_feasible_if_angleOK_and_time_slot in TemporalGating. Within the Recognition framework it supplies the concrete realization of the three spatial dimensions required by T8 of the forcing chain, enabling the transition from abstract action budgets to concrete angular constraints.
scope and limits
- Does not introduce any Recognition-specific inner product or scaling.
- Does not restrict points to the unit sphere or impose normalization.
- Does not encode temporal gating or action budgets inside the space itself.
- Does not connect to the phi-ladder, mass formulas, or constants such as alpha.
formal statement (Lean)
19abbrev R3 := EuclideanSpace ℝ (Fin 3)
proof body
Definition body.
20
21/-- Budget-dependent minimal angle threshold. -/
22@[simp] def θmin (Amax : ℝ) : ℝ := thetaMin Amax
23
24/-- The blind cone at `x` with budget `Amax`: pairs `(y,z)` whose separation angle at `x`
25falls below `θmin(Amax)`. -/