"""Reproduce independent checks of positional endpoint-distance formulas. Run: python3 verify_positional.py Quadrature checks the endpoint formula independently; it is not a proof. Opaque examples have separate elementary convex-hull proofs stated below. No sampled line-intersection count is promoted to a rigorous global bound. """ from pathlib import Path import json import math import platform import numpy as np import scipy from scipy.integrate import quad from positional_overlap import pair_overlap,positional_certificate ROOT=Path(__file__).resolve().parent def projection_overlap(theta,first,second): n=np.array([math.cos(theta),math.sin(theta)]) u=np.asarray(first,dtype=float)@n v=np.asarray(second,dtype=float)@n return max(0.,min(float(u.max()),float(v.max()))- max(float(u.min()),float(v.min()))) def independent_quadrature(first,second): points=np.vstack([first,second]).astype(float) critical=[0.,math.pi] for i in range(4): for j in range(i+1,4): difference=points[j]-points[i] if np.linalg.norm(difference)>0: critical.append((math.atan2(difference[1],difference[0])+math.pi/2)%math.pi) critical=sorted(set(critical)) values=[] errors=[] for a,b in zip(critical[:-1],critical[1:]): if b-a<=1e-15: continue v,e=quad(lambda theta:projection_overlap(theta,first,second),a,b, epsabs=1e-12,epsrel=1e-12) values.append(v) errors.append(e) return math.fsum(values),math.fsum(errors) def checks(): analytic=[ ("parallel unit segments, separation 1",[[0,0],[1,0]],[[0,1],[1,1]],2*(math.sqrt(2)-1)), ("crossing orthogonal diameters of length 2",[[-1,0],[1,0]],[[0,-1],[0,1]],8-4*math.sqrt(2)), ("right-triangle adjacent legs",[[0,0],[1,0]],[[0,0],[0,1]],2-math.sqrt(2)), ("collinear shared endpoint",[[0,0],[1,0]],[[1,0],[2,0]],0.), ("collinear disjoint segments",[[0,0],[1,0]],[[2,0],[3,0]],0.), ("degenerate isolated point",[[0,0],[0,0]],[[1,0],[2,1]],0.)] analytic_records=[] for label,first,second,expected in analytic: actual=pair_overlap(first,second) quadrature,error=independent_quadrature(first,second) assert abs(actual-expected)<1e-10 assert abs(actual-quadrature)<1e-10 analytic_records.append({"case":label,"expected":expected,"endpoint_formula":actual, "independent_quadrature":quadrature, "quadrature_reported_error":error}) rejected=False try: pair_overlap([[0,0],[2,0]],[[1,0],[3,0]]) except ValueError: rejected=True assert rejected rng=np.random.default_rng(20261029) random_records=[] for number in range(100): first=rng.uniform(-3,3,size=(2,2)) second=rng.uniform(-3,3,size=(2,2)) actual=pair_overlap(first,second) quadrature,error=independent_quadrature(first,second) symmetry=max(abs(actual-pair_overlap(second,first)), abs(actual-pair_overlap(first[::-1],second)), abs(actual-pair_overlap(first,second[::-1]))) assert abs(actual-quadrature)<2e-9 assert symmetry<2e-9 random_records.append({"index":number,"first":first.tolist(),"second":second.tolist(), "endpoint_formula":actual,"independent_quadrature":quadrature, "absolute_discrepancy":abs(actual-quadrature), "quadrature_reported_error":error,"symmetry_error":symmetry}) square=[([-1,-1],[1,-1]),([1,-1],[1,1]),([1,1],[-1,1]),([-1,1],[-1,-1])] square_record=positional_certificate(square,certified_max_hits=2) square_record["independent_geometric_facts"]=( "The square boundary surrounds the unit disk, so every line meeting the disk meets the boundary. " "A line not containing an edge and not through a vertex meets a convex polygon boundary at most twice; the exceptional lines have zero line measure.") root2=math.sqrt(2) cross=[([-root2,0],[root2,0]),([0,-root2],[0,root2])] cross_record=positional_certificate(cross,certified_max_hits=2) cross_record["independent_geometric_facts"]=( "The cross is connected. Its convex hull is the diamond |x|+|y|<=sqrt(2), which contains the unit disk. " "Every linear projection of a connected set is an interval, so containment of the disk in its convex hull makes it opaque. Two segments give the automatic almost-everywhere hit bound 2.") triangle_vertices=np.array([[0.,2.],[-math.sqrt(3),-1.],[math.sqrt(3),-1.]]) triangle=[(triangle_vertices[i],triangle_vertices[(i+1)%3]) for i in range(3)] triangle_record=positional_certificate(triangle,certified_max_hits=2) triangle_record["independent_geometric_facts"]=( "The equilateral triangle is circumscribed about the unit disk; its boundary is opaque and a generic line hits it at most twice.") for record in [square_record,cross_record,triangle_record]: assert record["forest_necessary_lower_bound"]<=record["total_segment_length"]+1e-10 assert record["conditional_pair_necessary_lower_bound"]<=record["total_segment_length"]+1e-10 assert abs(square_record["pair_energy"]-8)<1e-10 assert abs(triangle_record["pair_energy"]-6*math.sqrt(3))<1e-10 assert abs(cross_record["pair_energy"]-(8*math.sqrt(2)-8))<1e-10 fractional=[] for n in [3,4,6,12,24,96,384]: half_perimeter=n*math.tan(math.pi/n) fractional.append({"circumscribed_regular_polygon_sides":n, "half_density_cost":half_perimeter, "excess_above_pi":half_perimeter-math.pi}) assert all(fractional[i]["half_density_cost"]>fractional[i+1]["half_density_cost"] for i in range(len(fractional)-1)) assert fractional[-1]["excess_above_pi"]<1e-4 return {"scope":"Necessary positional certificates, not an opacity decision procedure and not a solution of the minimum opaque-barrier problem.", "arithmetic":"Exact formulas evaluated in float64; numerical tests do not replace the analytic proofs.", "runtime":{"python":platform.python_version(),"numpy":np.__version__,"scipy":scipy.__version__}, "summary":{"all_assertions_passed":True,"analytic_cases":len(analytic_records), "random_cases":len(random_records),"positive_collinear_overlap_rejected":rejected, "max_random_quadrature_discrepancy":max(r["absolute_discrepancy"] for r in random_records), "max_random_symmetry_discrepancy":max(r["symmetry_error"] for r in random_records), "actual_opaque_examples":3}, "analytic_cases":analytic_records,"random_cases":random_records, "actual_opaque_examples":{"square_boundary":square_record,"cross":cross_record,"triangle_boundary":triangle_record}, "fractional_no_go":{"analytic_statement": "Half of the unit-circle boundary measure has cost pi and weighted intersection count 1 on every non-tangent line through the disk. Consequently line-count covering constraints allowing fractional density cannot imply a bound strictly greater than pi. For a regular circumscribed N-gon, half boundary density has cost N*tan(pi/N), tending to pi. This is a fractional relaxation, not a physical barrier.", "polygon_diagnostics":fractional}} if __name__=="__main__": result=checks() (ROOT/"POSITIONAL_VERIFICATION.json").write_text(json.dumps(result,indent=2)+"\n") print(json.dumps(result["summary"],indent=2))