tests/test_geometry.py

from itertools import combinations, product
import sympy as sp
import pytest
from noulith import TwistedAlgebra, bits, partitions, ideal_generators, equal_polynomial_ideals
from noulith.geometry import *


def test_refinement_identity_symbolic_four_channels():
    pairs=list(combinations(range(4),2)); params=sp.symbols('q0:6')
    alg=TwistedAlgebra(4,dict(zip(pairs,params))); sh=alg.shuffle_table()
    for mask in range(1,16):
        for a in range(1,mask.bit_count()):
            total=0
            for sub in range(1,16):
                if sub & mask == sub and sub.bit_count()==a:
                    other=mask^sub
                    _,weight=alg.monomial_product(sub,other)
                    total+=weight*sh[sub]*sh[other]
            assert sp.expand(total-sh[mask])==0


def test_two_deficit_generator_formula():
    a,b,c,d,e,f=sp.symbols('a b c d e f')
    pairs=list(combinations(range(4),2))
    alg=TwistedAlgebra(4,{ij:-1+x for ij,x in zip(pairs,[a,b,c,d,e,f])})
    sh=alg.shuffle_table()
    full=15
    motifs=[sh[mask] for mask in range(16) if mask.bit_count()==3]
    motifs += [a*f,b*e,c*d]
    actual=ideal_generators(sh,full,2)
    assert equal_polynomial_ideals(actual,motifs,[a,b,c,d,e,f])

@pytest.mark.parametrize('m',range(2,9))
@pytest.mark.parametrize('seed',[1,2,3])
def test_pfaffian_tangent_theorem_exact(m,seed):
    t=sp.Symbol('t'); pairs=list(combinations(range(m),2))
    edges={pair:sp.Integer(((i+seed)*7)%13-6) for i,pair in enumerate(pairs)}
    alg=TwistedAlgebra(m,{pair:-1+t*v for pair,v in edges.items()})
    polynomial=sp.Poly(alg.shuffle_table()[-1],t)
    k=m//2
    assert all(polynomial.coeff_monomial(t**j)==0 for j in range(k))
    matrix=sp.zeros(m)
    for (i,j),v in edges.items(): matrix[i,j]=v;matrix[j,i]=-v
    assert sp.expand(polynomial.coeff_monomial(t**k)-leading_shuffle_form(matrix))==0


def test_formal_group_axioms_and_differential():
    x,y,z=sp.symbols('x y z');F=resonance_group
    assert sp.cancel(F(F(x,y),z)-F(x,F(y,z)))==0
    assert F(x,0)==x and sp.cancel(F(x,resonance_inverse(x)))==0
    assert sp.cancel(sp.diff(F(x,y),y).subs(y,0)-(1-x+x*x))==0


def test_flat_triangle_identity_symbolic():
    a,b,c=sp.symbols('a b c')
    A=flat_edge(a,b);B=flat_edge(a,c);C=flat_edge(b,c)
    assert sp.cancel(A*B*C-A*C+A-B+C)==0

@pytest.mark.parametrize('N',range(3,7))
def test_line_arrangement_generators_and_hilbert_function(N):
    z=sp.symbols(f'z1:{N+1}'); all_z=(sp.S.Zero,)+z
    motifs=[]
    edges=list(combinations(range(N+1),2))
    for a,e in enumerate(edges):
        for f in edges[a+1:]:
            if not set(e)&set(f):
                motifs.append((all_z[e[1]]-all_z[e[0]])*(all_z[f[1]]-all_z[f[0]]))
    generators=line_arrangement_generators(z)
    assert equal_polynomial_ideals(motifs,generators,z)
    groebner=sp.groebner(generators,*z)
    leading=[poly.LM(order=groebner.order).exponents for poly in groebner.polys]
    for degree in range(6):
        count=0
        for exponents in product(range(degree+1),repeat=N):
            if sum(exponents)==degree and not any(all(a>=b for a,b in zip(exponents,lead)) for lead in leading): count+=1
        assert count==([1,N][degree] if degree<2 else N+1)

@pytest.mark.parametrize('outlier',range(5))
def test_all_five_exact_resonance_branches(outlier):
    t=sp.Symbol('t');h=[0]*5;h[outlier]=t
    f={ij:flat_edge(h[ij[0]],h[ij[1]]) for ij in combinations(range(5),2)}
    for i,j,k in combinations(range(5),3):
        A,B,C=f[i,j],f[i,k],f[j,k]
        assert sp.cancel(A*B*C-A*C+A-B+C)==0
    for e,g in combinations(f,2):
        if not set(e)&set(g): assert sp.cancel(f[e]*f[g])==0


def test_triangle_flatness_alone_is_insufficient():
    t=sp.Symbol('t');h=[0,t,2*t,3*t]
    f={ij:flat_edge(h[ij[0]],h[ij[1]]) for ij in combinations(range(4),2)}
    for i,j,k in combinations(range(4),3):
        A,B,C=f[i,j],f[i,k],f[j,k]
        assert sp.cancel(A*B*C-A*C+A-B+C)==0
    assert sp.cancel(f[0,1]*f[2,3])!=0


@pytest.mark.parametrize('m',[4,5,6])
@pytest.mark.parametrize('stratum,defect',[
    ('generic',0),('single_star',1),('outlier_branch',2),('two_stars',2),('origin',None)])
def test_second_deficit_full_rank_strata(m,stratum,defect):
    from noulith.protocols import normalized_measurement_matrix
    f={pair:sp.S.Zero for pair in combinations(range(m),2)}
    if stratum=='generic':
        f={pair:sp.Rational(1,10) for pair in f}
    elif stratum=='single_star':
        for j in range(1,m):
            f[(0,j)]=sp.Rational(j,10*m)
    elif stratum=='outlier_branch':
        for j in range(1,m):
            f[(0,j)]=sp.Rational(1,10)
    elif stratum=='two_stars':
        f[(0,1)]=sp.Rational(1,10)
    algebra=TwistedAlgebra(m,{pair:-1+value for pair,value in f.items()})
    matrix=normalized_measurement_matrix(algebra.shuffle_table(),m-2)
    # The frame has disjoint one-coordinate row supports; column nonzeroness is
    # therefore its exact rank, without floating-point thresholding.
    rank=sum(any(value!=0 for value in matrix[:,i]) for i in range(2**m))
    expected=2**m-(m+1 if stratum=='origin' else defect)
    assert rank==expected
    if m==4:
        assert matrix.rank()==expected


@pytest.mark.parametrize('N',range(3,9))
def test_branch_carriers_and_normalization_conductor(N):
    z=sp.symbols(f'z0:{N}')
    t=sp.symbols('t')
    carriers=branch_carriers(z)
    branches=[{v:(t if j==i else 0) for j,v in enumerate(z)} for i in range(N)]
    branches.append({v:t for v in z})
    evaluations=sp.Matrix([[sp.expand(g.subs(branch)) for branch in branches] for g in carriers])
    assert evaluations==t*t*sp.eye(N+1)


@pytest.mark.parametrize('N',range(3,7))
def test_perfect_artinian_slice_pairing(N):
    z=sp.symbols(f'z0:{N}')
    G=sp.groebner(line_arrangement_generators(z)+[sum(z)],*z)
    h=z[0]*z[1]
    assert G.reduce(h)[1]!=0
    pairing=perfect_slice_matrix(N)
    for i in range(N-1):
        for j in range(N-1):
            assert G.reduce(z[i]*z[j]-pairing[i,j]*h)[1]==0
    assert sp.factor(pairing.det())==-(-N)**(N-2)