"""Independent linear-generator geometry and scalar-accounting examples.
Pure standard library. No neural training, accuracy estimate or runtime measurement.
"""
import json,math
rs=list(range(5));target=[1.,1.,1.,1.];errors=[]
for r in rs:
 projected=[target[i]if i<r else 0 for i in range(4)]
 residual=[x-y for x,y in zip(target,projected)]
 err=math.sqrt(sum(x*x for x in residual));assert math.isclose(err,math.sqrt(4-r),abs_tol=1e-12)
 for j in range(r):assert residual[j]==0 # orthogonal residual proves least-squares projection
 errors.append(err)
# A one-coordinate family c*I can have a rank-two target matrix.
c=1.;w=[[c,0],[0,c]];assert w[0][0]*w[1][1]-w[0][1]*w[1][0]!=0
# A diagonal family cannot represent the fixed off-diagonal target.
off=[0.,1.,1.,0.];diag_projection=[off[0],0,0,off[3]]
assert sum((a-b)**2 for a,b in zip(off,diag_projection))==2
P=1000;r=4;ts=[1,4,16,32,64,100];separate=[t*P for t in ts];shared=[r*P+t*r for t in ts]
for t,a,b in zip(ts,separate,shared):assert(b<a)==(t>r*P/(P-r))
assert shared[0]>separate[0]and shared[-1]<separate[-1]
figures=[dict(id='basis-projection-error',x=rs,y=errors),dict(id='task-storage-crossover',x=ts,series=[dict(name='Independent targets: T×1000',y=separate),dict(name='Shared 4×1000 basis + 4 coefficients/task',y=shared)])]
print(json.dumps(dict(status='PASS',scope='Exact constructed basis projection, rank counterexamples and scalar counts; not neural model quality or runtime performance.',figures=figures),indent=2))