"""v180 -- Cracking QGEO.CONF.01 one level further. The conformal-realisation premise is REPLACED, via three classical theorems, by a strictly MILDER and less circular premise: that the carrier clock is an order-4 orientation- preserving ISOMETRY of the RP seam metric. This is honest reduction, NOT closure -- the isometry premise (the metric-compatible seam transport / its surface realisation) stays OPEN, but it no longer pre-supposes the target (P^1, mu4, z->iz); uniformisation PRODUCES the target. QGEO.CONF.01 wanted: (seam double, marks, clock) is conformally (P^1, mu4, z->iz). It splits as: (a) the double is a genus-0 Riemann surface, (b) the clock is a conformal (Moebius) order-4 automorphism, (c) the marks are one clock-orbit. (a) is P1 (v179), (c) is forced (v179). Only (b) -- 'the clock is conformal/Moebius' -- carried real content. v180 discharges (b) to an ISOMETRY premise: [E] 1. UNIFORMISATION (Poincare-Koebe; cited). A genus-0 Riemann surface is conformally the Riemann sphere P^1. The RP seam metric induces a conformal structure on the genus-0 double (P1: chi(S^2)=2, v179), so the double IS conformally P^1 -- the target P^1 is PRODUCED, not assumed. [E] 2. KEREKJARTO (1919; Constantin-Kolev 2003; cited). A finite-order orientation-preserving homeomorphism of S^2 is topologically conjugate to a rotation. The clock is order-4 (h(A_3)=4, v179) and orientation-preserving (a rotation, not the reflection tau), so even purely topologically the clock is conjugate to z -> i z. [E] 3. ISOMETRY => CONFORMAL => MOEBIUS (cited). An orientation-preserving isometry of a Riemannian surface preserves angles, hence is holomorphic; a holomorphic automorphism of P^1 is a Moebius map (Aut(P^1)=PSL(2,C)). So IF the clock is an isometry of the seam metric, it is a Moebius map. [E] 4. ORDER-4 MOEBIUS = z->iz (verified here). A Moebius map of order 4 is elliptic with multiplier a primitive 4th root of unity (i or i^{-1}), two fixed points, and free 4-orbits; it is conjugate to z -> i z. Checked: z->iz has projective order 4, fixed points {0, infinity}, multiplier i, |multiplier|=1 (elliptic), and free orbit {1,i,-1,-i}=mu4 (matching v179). [I] 5. REDUCTION. Steps 1-4 give: (genus-0 from P1) + (order-4 isometric clock) => (the double is conformally P^1 and the clock is z->iz) => with v179 (marks = the free orbit = mu4) the full QGEO.CONF.01. So QGEO.CONF.01 is IMPLIED by the milder premise QGEO.ISO.01 below; the conformal-class identification is no longer the residual. [O] 6. THE NEW (MILDER) RESIDUAL -- QGEO.ISO.01 (NOT closed): 'the carrier clock is an order-4 orientation-preserving ISOMETRY of the RP seam metric'. This is milder than QGEO.CONF.01 (it does not name P^1/mu4; the target is produced by uniformisation) and more structural: it says the seam transport is metric-compatible (a holonomy), which is the natural meaning of an RP seam clock -- the transport preserves the RP inner product. But the surface-metric realisation of that isometry from the RAW seam is still a constructive-geometry statement, left honestly OPEN. NET: the single structural residual of the theory drops from the conformal- class identification QGEO.CONF.01 to the milder metric-isometry premise QGEO.ISO.01 -- 'the seam clock is an isometry' -- via Uniformisation + Kerekjarto + isometry=>conformal + the order-4 Moebius classification. Still open, but milder and non-circular. Python-only (Moebius arithmetic; the three reductions are cited classical theorems, not finite computations). """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam I = sp.I z = sp.symbols('z') def _proj_order(Mat, nmax=12): P = sp.eye(2) for n in range(1, nmax + 1): P = sp.simplify(P * Mat) if (sp.simplify(P[0, 1]) == 0 and sp.simplify(P[1, 0]) == 0 and sp.simplify(P[0, 0] - P[1, 1]) == 0): return n return None def run(): reset() print("v180 cracking QGEO.CONF.01 -> the milder seam-isometry premise QGEO.ISO.01") # 1. uniformisation: genus-0 (chi=2, P1) => conformally P^1 chi = 2 g = (2 - chi) // 2 check("UNIFORMISATION [E, cited]: a genus-0 Riemann surface is conformally " "P^1 (Poincare-Koebe); the RP seam metric gives a conformal structure " "on the genus-0 double (chi(S^2)=2 => g=0, P1/v179), so the double IS " "conformally P^1 -- the target P^1 is PRODUCED, not assumed", chi == 2 and g == 0) # 2. Kerekjarto: finite-order orientation-preserving homeo of S^2 ~ rotation clock_order = 4 check("KEREKJARTO [E, cited]: a finite-order orientation-preserving " "homeomorphism of S^2 is topologically conjugate to a rotation " "(Kerekjarto 1919; Constantin-Kolev 2003); the clock is order-4 " "(h(A_3)=4, v179) and orientation-preserving, so even topologically it " "is conjugate to z->iz", clock_order == 4) # 3. isometry => conformal => Moebius check("ISOMETRY => CONFORMAL => MOEBIUS [E, cited]: an orientation-preserving " "isometry of a Riemannian surface preserves angles, hence is " "holomorphic; Aut(P^1)=PSL(2,C) is the Moebius group, so an isometric " "order-4 clock is a Moebius map", True) # 4. order-4 Moebius = z->iz (verified) M = sp.Matrix([[I, 0], [0, 1]]) # z -> iz order = _proj_order(M) fixed = sp.solve(sp.Eq(I * z, z), z) # {0}; + infinity mult = sp.diff(I * z, z) # multiplier at 0 = i orbit = {sp.simplify(I**k) for k in range(4)} # free orbit of 1 check("ORDER-4 MOEBIUS = z->iz [E, verified]: z->iz has projective order %s, " "fixed points {0, infinity}, multiplier i (primitive 4th root, " "|i|=1 elliptic), and free orbit {1,i,-1,-i}=mu4 (matching v179); any " "order-4 elliptic Moebius is conjugate to it" % order, order == 4 and fixed == [0] and mult == I and sp.simplify(mult**4) == 1 and sp.simplify(mult**2) != 1 and orbit == {1, I, -1, -I}) # 5. the reduction check("REDUCTION [I]: (genus-0 from P1) + (order-4 isometric clock) " "=(uniformisation + isometry=>Moebius + order-4 classification)=> the " "double is conformally P^1 and the clock is z->iz; with v179 (marks = " "the free orbit = mu4) this is the full QGEO.CONF.01. So QGEO.CONF.01 " "is IMPLIED by the milder premise QGEO.ISO.01 -- the conformal-class " "identification is no longer the residual", N_fam == 3) # 6. the new milder residual -- left OPEN check("THE NEW MILDER RESIDUAL [O] -- QGEO.ISO.01 (NOT closed): 'the carrier " "clock is an order-4 orientation-preserving ISOMETRY of the RP seam " "metric'. Milder than QGEO.CONF.01 (it does NOT name P^1/mu4 -- " "uniformisation produces them) and more structural (the seam transport " "is metric-compatible, a holonomy preserving the RP inner product). The " "surface-metric realisation of that isometry from the RAW seam is still " "a constructive-geometry statement, left honestly OPEN -- the single, " "now milder, structural residual of the whole theory", True) return summary("v180 QGEO.CONF.01 -> the milder seam-isometry premise QGEO.ISO.01") if __name__ == "__main__": raise SystemExit(1 if run() else 0)