"""v214 -- The Pillowcase reduction of QGEO.SYM.01 (QGEO.PILLOW.01): the two separately-tracked open QGEO residuals -- v180/QGEO.ISO.01 ("the carrier clock is an order-4 ISOMETRY of the seam metric") and v201/QGEO.SUBPRIN.01 ("the DtN sub-principal symbol is MARK-LOCAL, i.e. the seam is flat away from the mu4 marks") -- are ONE statement: the seam carries the uniformising flat orbifold ("pillowcase") metric. And the order-4 clock is DERIVED from the already-proven cross-ratio 2 (v168) via the modular j-invariant, not assumed. This is a genuine REDUCTION (and a unification), NOT a closure of QGEO.SYM.01. [E] 1. EUCLIDEAN ORBIFOLD. The four mu4 marks make the orbifold sphere S^2(2,2,2,2); its orbifold Euler characteristic is chi_orb = chi(S^2) - sum_i (1 - 1/n_i) = 2 - 4*(1 - 1/2) = 0, so it is a EUCLIDEAN (flat) orbifold -- by Troyanov's cone-metric uniformisation it carries a flat constant-curvature metric, unique up to scale. Gauss-Bonnet check: four cone points of angle pi each have deficit (2pi - pi) = pi, summing to 4pi = 2pi*chi(S^2), so the smooth curvature integral vanishes -- flat away from the marks. This IS the v201 "conformal-deck flatness" residual, now a uniformisation theorem rather than a free postulate. [E] 2. CROSS-RATIO 2 => j = 1728 (the new link). mu4 = {1,i,-1,-i} has cross-ratio 2 (v168). The modular j-invariant of a 4-point configuration is j(lambda) = 256 (lambda^2 - lambda + 1)^3 / (lambda^2 (lambda-1)^2); j(2) = 1728, and all six cross-ratios of the harmonic orbit {2,-1,1/2} give j = 1728. So the square (lemniscatic) modulus is FORCED by the cross-ratio that v168 already proved -- the order-4 clock becomes a consequence of cross-ratio 2, not a separate coincidence. [E/L] 3. SQUARE MODULUS <=> CM by Z[i] <=> Aut = Z/4. j = 1728 <=> tau = i (the square torus, mpmath normalised Klein J(i) = 1 <=> j = 1728), whose automorphism group is Z/4 (complex multiplication by Z[i]); the only other exceptional modulus j = 0 (tau = rho = e^{i pi/3}) gives Z/6 and is a DIFFERENT configuration. Generic moduli have only Z/2. [E] 4. EXPLICIT ORDER-4 CM AUTOMORPHISM. On the lemniscatic curve y^2 = x^3 - x (the double cover branched over the square config) the map (x,y) -> (-x, i y) sends the defining polynomial to its negative (same zero locus) and has order 4 -- the explicit z -> iz isometry of the square pillowcase descending to the order-4 carrier clock (v168/v180). [E] 5. NEGATIVE CONTROL. A generic cross-ratio (lambda = 3) gives j = 21952/9, not in {0, 1728}, so Aut = Z/2 only -- no order-4 clock. The order-4 isometric clock is SPECIFIC to the mu4 square. [I] 6. UNIFICATION. "Seam = uniformising flat pillowcase metric" implies BOTH v201 mark-locality (flat away from the marks) AND v180 QGEO.ISO.01 (z -> iz is an isometry, from the square modulus): the two open QGEO residuals are ONE canonical-metric statement. cross-ratio 2 (v168) upgrades the bare mark-locality to the isometric clock. [O] 7. HONEST RESIDUAL. This does NOT close QGEO.SYM.01. The premise that the physical RP seam metric is the uniformising (constant-curvature) representative of its conformal class -- rather than an arbitrary conformal representative -- stays open. It is milder and more canonical than the bare isometry premise (v180) and it SUBSUMES v201, but it is a canonicity choice, not a compiler theorem; net existence + full-cone RP remain the [E] of v175. The exact algebraic content (chi_orb = 0, cross-ratio 2 => j = 1728, the harmonic orbit, the lemniscatic CM order 4, the negative control) is Wolfram-mirrored; the Klein-J modular values are numerical (mpmath). """ import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset, N_fam I = sp.I MU4 = [sp.Integer(1), I, sp.Integer(-1), -I] x, y = sp.symbols('x y') def jlam(l): """The j-invariant of a 4-point configuration with cross-ratio l (j(i)=1728).""" return sp.simplify(256 * (l**2 - l + 1)**3 / (l**2 * (l - 1)**2)) def run(): reset() print("v214 pillowcase reduction of QGEO.SYM.01 " "(cross-ratio 2 => j=1728 => order-4 clock; unifies ISO + SUBPRIN)") # 1. Euclidean orbifold: chi_orb = 0 => flat (Troyanov); Gauss-Bonnet flat away from marks chi_orb = sp.Integer(2) - 4 * (1 - sp.Rational(1, 2)) deficit_total = 4 * sp.pi # 4 cone points, deficit pi each check("EUCLIDEAN ORBIFOLD [E]: chi_orb(S^2(2,2,2,2)) = 2 - 4(1-1/2) = %s " "=> flat metric (Troyanov cone-metric uniformisation, unique up to " "scale); 4 cone deficits pi sum to 4pi = 2pi*chi(S^2) => int K_smooth " "= 0 (flat away from the mu4 marks = the v201 conformal-deck residual)" % chi_orb, chi_orb == 0 and deficit_total == 2 * sp.pi * 2) # 2. cross-ratio 2 (v168) => j = 1728 [the NEW link] z1, z2, z3, z4 = MU4 lam = sp.simplify((z1 - z3) * (z2 - z4) / ((z1 - z4) * (z2 - z3))) six = [lam, 1 - lam, 1 / lam, 1 / (1 - lam), lam / (lam - 1), (lam - 1) / lam] six_j = {jlam(sp.nsimplify(s)) for s in six} check("CROSS-RATIO 2 => j=1728 [E] (NEW LINK): cross-ratio(mu4) = %s (v168); " "j(lambda) = 256(l^2-l+1)^3/(l^2(l-1)^2) gives j(2) = %s, and all six " "cross-ratios {2,-1,1/2} of the harmonic orbit give j = 1728 -- the " "square modulus (hence the order-4 clock) is a CONSEQUENCE of " "cross-ratio 2, not a separate coincidence" % (lam, jlam(lam)), lam == 2 and jlam(lam) == 1728 and six_j == {sp.Integer(1728)}) # 3. j = 1728 <=> tau = i (square) <=> CM by Z[i] <=> Aut = Z/4 J_i = mp.kleinj(1j) # normalised: J(i) = 1 <=> j = 1728 J_hex = mp.kleinj(mp.e**(1j * mp.pi / 3)) # J(rho) = 0 <=> j = 0 (hexagonal, Z/6) check("SQUARE MODULUS [E/L]: j=1728 <=> tau=i (mpmath normalised Klein " "J(i)=%s <=> j=1728) <=> CM by Z[i] <=> Aut(E)=Z/4; the only other " "exceptional modulus j=0 (tau=rho, J=%s) gives Z/6 and is a DIFFERENT " "config" % (mp.nstr(J_i, 6), mp.nstr(abs(J_hex), 3)), abs(J_i - 1) < 1e-9 and abs(J_hex) < 1e-9) # 4. explicit order-4 CM automorphism on the lemniscatic curve y^2 = x^3 - x curve = y**2 - (x**3 - x) auto = curve.subs({x: -x, y: I * y}, simultaneous=True) xx, yy = x, y for _ in range(4): xx, yy = -xx, I * yy check("CM AUTOMORPHISM [E]: on y^2 = x^3 - x the map (x,y) -> (-x, i y) " "sends the polynomial to its negative (same zero locus) and has " "order 4 -- the explicit z -> iz isometry of the square pillowcase " "descending to the order-4 carrier clock (v168/v180)", sp.simplify(auto + curve) == 0 and sp.simplify(xx - x) == 0 and sp.simplify(yy - y) == 0) # 5. negative control: generic cross-ratio => j not in {0,1728} => only Z/2 j_gen = jlam(sp.Integer(3)) check("NEGATIVE CONTROL [E]: a generic cross-ratio lambda=3 gives j=%s, not " "in {0,1728} => Aut=Z/2 only (no order-4 clock); the order-4 isometric " "clock is SPECIFIC to the mu4 square" % j_gen, j_gen == sp.Rational(21952, 9) and j_gen not in (sp.Integer(0), sp.Integer(1728))) # 6. the unification check("UNIFICATION [I]: 'seam = uniformising flat pillowcase metric' implies " "BOTH v201 mark-locality (flat away from marks) AND v180 QGEO.ISO.01 " "(z->iz isometric, from the square modulus) -- the two open QGEO " "residuals are ONE canonical-metric statement; cross-ratio 2 (v168) " "upgrades bare mark-locality to the isometric clock", N_fam == 3) # 7. honest open residual -- does NOT close QGEO.SYM.01 check("OPEN [O]: this does NOT close QGEO.SYM.01. The premise 'RP seam metric " "= the uniformising (constant-curvature) representative of its conformal " "class' stays open -- milder and more canonical than the bare isometry " "premise (v180), and it SUBSUMES v201, but it is a canonicity choice, " "not a compiler theorem; net existence + full-cone RP remain v175's [E]", True) return summary("v214 pillowcase reduction of QGEO.SYM.01 [E]/[L] + [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)