# -*- coding: utf-8 -*- """ ROUND 3 — (a) verify the collision curve lies over the double fold line downstairs, (b) extend the forge to m = 4, 5 (does a SECOND specimen exist, or is m = 3 unique?), (c) hash all breakthrough artifacts for the Paper II draft. """ import sympy as sp import json, time, random, hashlib, os t0 = time.time() x, y, z, s = sp.symbols('x y z s') u = 1 + x*y random.seed(20260723) def check(label, ok): print(f"[{'PASS' if ok else 'FAIL'}] {label}", flush=True) return ok print("=== (a) the fold line downstairs ===", flush=True) # descended coordinates a = xy, b = x^2 z ; verified Jac_ab(Phi) = 2*(3a+b-2)^2 # collision curve C(s) = (s, -3/(2s), 13/(2s^2)): aC = sp.cancel((x*y).subs({x: s, y: -3/(2*s)})) bC = sp.cancel((x**2*z).subs({x: s, z: sp.Rational(13, 2)/s**2})) check(f"C(s) descends to the CONSTANT point (a,b) = ({aC}, {bC})", aC == sp.Rational(-3, 2) and bC == sp.Rational(13, 2)) check("that point lies EXACTLY on the fold line 3a + b - 2 = 0", sp.Rational(-9, 2) + sp.Rational(13, 2) - 2 == 0) check("the z-axis descends to (a,b) = (0,0), which is OFF the fold line (value -2)", (0 + 0 - 2) == -2) print(" => upstairs: 3:1 fold onto the invariant line;", flush=True) print(" downstairs: Jacobian 2(3a+b-2)^2 vanishes to order 2 on the line", flush=True) print(" the collision curve sits over. The singular fold of the plane map", flush=True) print(" is exactly what the torus suspension renders etale upstairs.", flush=True) print("\n=== (b) FORGE m = 4, 5 — is there a SECOND specimen? ===", flush=True) OUT = 'post-jc-program/breakthrough/forge_m45.json' RES = {"configs": []} def map_degree(F, tries=2, tag=""): outs = [] for tr in range(tries): t1, t2, t3 = [sp.Rational(random.randint(3, 60), random.randint(1, 7)) for _ in range(3)] zs = sp.solve(F[2] - t3, z)[0] G1 = sp.numer(sp.together(F[0].subs(z, zs) - t1)) G2 = sp.numer(sp.together(F[1].subs(z, zs) - t2)) Rr = sp.expand(sp.resultant(G1, G2, y)) if Rr == 0: outs.append(("degenerate", None)); continue px = sp.Poly(Rr, x) k = min(m_[0] for m_ in px.monoms()) expr = sp.expand(Rr / x**k) g = sp.gcd(expr, sp.diff(expr, x)) sf = sp.cancel(expr / g) outs.append((int(sp.degree(expr, x)), int(sp.degree(sf, x)))) print(f" [{tag}] eliminant (all, squarefree): {outs}", flush=True) return outs def forge(m, dP, dQ, dR): cfg = {"m": m, "caps": [dP, dQ, dR], "status": "started", "detail": []} a = x*y ps = sp.symbols(f'p0:{dP+1}'); qs = sp.symbols(f'q0:{dQ+1}'); rs = sp.symbols(f'r0:{dR+1}') U = list(ps) + list(qs) + list(rs) P = sum(c_*a**i for i, c_ in enumerate(ps)) Q = sum(c_*a**i for i, c_ in enumerate(qs)) R = sum(c_*a**i for i, c_ in enumerate(rs)) v = sp.Matrix([u**m, m*x*u**(m-1), -x**m]) A = sp.Matrix([y**2*P, y + x*y**2*Q, x*R]) Ax, Ay, vx, vy = A.diff(x), A.diff(y), v.diff(x), v.diff(y) assert sp.expand(sp.Matrix.hstack(vx, vy, v).det()) == 0 M1 = sp.expand(sp.Matrix.hstack(vx, Ay, v).det() + sp.Matrix.hstack(Ax, vy, v).det()) M0 = sp.expand(sp.Matrix.hstack(Ax, Ay, v).det()) eqs1 = sp.Poly(M1, x, y).coeffs() lin = sp.linsolve(eqs1, U) if lin == sp.EmptySet: cfg["status"] = "stage1-EMPTY (EXCLUDED: slope admits no completion at these caps)" print(f" m={m} caps=({dP},{dQ},{dR}): stage1 EMPTY -> EXCLUDED", flush=True) RES["configs"].append(cfg); json.dump(RES, open(OUT, 'w'), indent=2); return sol_t = list(lin)[0] sub1 = dict(zip(U, sol_t)) free1 = sorted({s_ for e in sol_t for s_ in e.free_symbols if s_ in U}, key=str) M0s = sp.expand(M0.subs(sub1, simultaneous=True)) p0 = sp.Poly(M0s, x, y) c_expr, eqs2 = sp.Integer(0), [] for mono, co in zip(p0.monoms(), p0.coeffs()): if mono == (0, 0): c_expr = co else: eqs2.append(co) branches = (sp.solve(eqs2, free1, dict=True) if eqs2 else [dict()]) if free1 \ else ([dict()] if all(sp.expand(e) == 0 for e in eqs2) else []) print(f" m={m} caps=({dP},{dQ},{dR}): stage1 free={len(free1)}, branches={len(branches)}", flush=True) for bi, br in enumerate(branches): cb = sp.expand(c_expr.subs(br)) fullsub = {k_: sp.expand(v_.subs(br)) for k_, v_ in sub1.items()} rem = sorted({s_ for e in list(fullsub.values()) + [cb] for s_ in e.free_symbols if s_ in U}, key=str) info = {"branch": bi, "c": str(cb), "free": [str(r_) for r_ in rem]} if cb == 0 and not rem: info["verdict"] = "c = 0: not Keller" print(f" branch {bi}: c=0 -> dead", flush=True) cfg["detail"].append(info); continue vals_list = [dict()] if not rem else \ [dict(zip(rem, tuple(random.choice([1, -1, 2, sp.Rational(1, 2)]) for _ in rem))) for _ in range(2)] for vals in vals_list: cval = sp.expand(cb.subs(vals)) if cval == 0: continue inst = {k_: sp.expand(v_.subs(vals)) for k_, v_ in fullsub.items()} Fi = [sp.expand((A[i] + z*v[i]).subs(inst, simultaneous=True)) for i in range(3)] dJ = sp.expand(sp.Matrix(Fi).jacobian([x, y, z]).det()) okk = (dJ == cval) and (dJ.free_symbols == set()) tot = max(sp.total_degree(f) for f in Fi) degs = map_degree(Fi, tag=f"m={m} b{bi}") if okk else None v_ = {"keller": bool(okk), "det": str(dJ), "total_degree": int(tot), "fibers": str(degs), "coeffs": {str(k2): str(v2) for k2, v2 in inst.items()}} if okk and degs: sf = {d_[1] for d_ in degs if d_[1] is not None} v_["verdict"] = ("AUTOMORPHISM (deg 1)" if sf == {1} else f"*** NEW SPECIMEN: degree {sf}, total_degree {tot} ***" if min(sf) >= 2 else "inconclusive") print(f" branch {bi} sample: {v_['verdict']}", flush=True) info.setdefault("samples", []).append(v_) cfg["detail"].append(info) cfg["status"] = "done" RES["configs"].append(cfg); json.dump(RES, open(OUT, 'w'), indent=2) forge(4, 2, 2, 2) forge(4, 3, 2, 2) forge(4, 3, 3, 3) forge(5, 3, 3, 3) forge(5, 4, 3, 3) print("\n=== (c) hash the breakthrough artifact set ===", flush=True) D = 'post-jc-program/breakthrough' hashes = {} for fn in sorted(os.listdir(D)): p = os.path.join(D, fn) if os.path.isfile(p) and not fn.endswith('.html'): hashes[fn] = hashlib.sha256(open(p, 'rb').read()).hexdigest() print(f" {hashes[fn][:16]}... {fn}", flush=True) json.dump(hashes, open(os.path.join(D, 'breakthrough_hashes.json'), 'w'), indent=2) print(f"\ndone ({round(time.time()-t0,2)}s)", flush=True)