#!/usr/bin/env python3 # SPDX-License-Identifier: MIT # Copyright (c) 2026 Pablo Nogueira Grossi / G6 LLC """ book26/nielsen-lens-determinant.py -- checklist item C13. The preprint asserts ln det' B_n = L(n) - zeta(3)*n^2 + O(1) for B = *d on coexact 1-forms of S^3 restricted to the Z_n-invariant sector (the lens space L(n,1)), and uses exp(-zeta(3) n^2) as the Gaussian suppression in its mass formula. This computes the determinant exactly and finds the coefficient is zeta(3)/(4*pi^2), not bare zeta(3) -- smaller by 4*pi^2 = 39.478 in the exponent. METHOD. On the unit S^3 the coexact 1-form eigenvalues of *d are +-N for N >= 2, in the SU(2)_L x SU(2)_R representations (N/2, (N-2)/2) and its swap. The Z_n is the Hopf U(1) in SU(2)_L, so the invariant sector keeps weights 2m_L = 0 (mod n), giving multiplicity f(N) = (N-1)*A(N,n) + (N+1)*A(N-2,n), A(M,n) = #{t = n*j : |t| <= M, t = M (mod 2)}. f is quasi-polynomial of period 2n, so the spectral zeta is a FINITE sum of Hurwitz zeta functions and ln det' = -zeta'(0) is exact. There is no cutoff and no truncation. WHY THE NUMBER IS TRUSTWORTHY, three ways: (1) At n = 1 the sector is all of S^3 and zeta(s) = 2[zeta_R(s-2) - zeta_R(s)], so ln det' = zeta(3)/(2 pi^2) - ln 2pi = -1.776980152292558942. The code reproduces this to 31 digits. This is also WHERE THE 4 pi^2 COMES FROM: zeta_R'(-2) = -zeta(3)/(4 pi^2) exactly, and the n^2 term inherits it. (2) The quasi-polynomial multiplicities are checked against a direct enumeration of SU(2) weights -- no closed form -- for n in {1,2,3,5,7,8} and every N < 200. (3) The coefficient is not fitted against a target. It CONVERGES as the fit window moves up (6e-7 at n=8..64, 5e-8 at n=32..64) and a local three-point solve with the target absent from the basis gives -0.030448471 at n = 48. LIMITS, stated. Trivial character only; other chi_n sectors untested. Unit radius and one zeta normalisation; sensitivity untested. The ln n coefficient ~ 4/3 is FITTED, not derived. This shows the asymptotic, not that it follows from the preprint's cited route through Nash-O'Connor and Cheeger-Mueller. "+ O(1)" in the preprint is too strong: there is a growing ln n term. Requires mpmath. Usage: python3 book26/nielsen-lens-determinant.py [--full] """ import math, sys try: from mpmath import mp, zeta, diff, mpf, log, pi except ImportError: print("SKIP: mpmath not installed (pip install mpmath). C13 is UNCHECKED here."); sys.exit(0) FAIL = [] def check(label, ok, detail=""): print(("PASS " if ok else "FAIL ") + label + (f" -- {detail}" if detail else "")) if not ok: FAIL.append(label) def A(M, n): if M < 0: return 0 c, j = 0, -(M // n) while n * j <= M: if (n * j - M) % 2 == 0: c += 1 j += 1 return c def mult(N, n): return (N - 1) * A(N, n) + (N + 1) * A(N - 2, n) def mult_direct(N, n): tot = 0 for twojL, twojR in ((N, N - 2), (N - 2, N)): if twojL < 0 or twojR < 0: continue for twom in range(-twojL, twojL + 1, 2): if twom % n == 0: tot += twojR + 1 return tot def _solve3(M, y): def det3(a): return (a[0][0]*(a[1][1]*a[2][2]-a[1][2]*a[2][1]) - a[0][1]*(a[1][0]*a[2][2]-a[1][2]*a[2][0]) + a[0][2]*(a[1][0]*a[2][1]-a[1][1]*a[2][0])) d = det3(M); out = [] for col in range(3): Mx = [r[:] for r in M] for i in range(3): Mx[i][col] = y[i] out.append(det3(Mx) / d) return out def lndet(n, dps=25): mp.dps = dps P, terms = 2 * n, [] for r in range(P): m0 = 0 while P * m0 + r < 2: m0 += 1 Ns = [P * (m0 + i) + r for i in range(3)] a, b, c = _solve3([[mpf(x)**2, mpf(x), mpf(1)] for x in Ns], [mpf(mult(x, n)) for x in Ns]) for i in (3, 4, 5): # the quasi-polynomial must hold beyond its fit x = P * (m0 + i) + r assert abs(a*x*x + b*x + c - mult(x, n)) < mpf(10)**(-20), (n, r, x) terms.append((a, b, c, mpf(m0) + mpf(r)/P, P)) def Z(s): return sum(a * mpf(P)**(2-s) * zeta(s-2, o) + b * mpf(P)**(1-s) * zeta(s-1, o) + c * mpf(P)**(-s) * zeta(s, o) for a, b, c, o, P in terms) return -diff(Z, 0) def main(): mp.dps = 30 target = zeta(3) / (4 * pi**2) check("zeta_R'(-2) = -zeta(3)/(4 pi^2) [this is where the 4 pi^2 comes from]", abs(diff(lambda s: zeta(s), -2) + target) < mpf(10)**(-25)) closed = zeta(3)/(2*pi**2) - log(2*pi) check("n=1 reproduces the closed form zeta(3)/(2 pi^2) - ln 2pi", abs(lndet(1, 30) - closed) < mpf(10)**(-25), mp.nstr(closed, 19)) bad = [(N, n) for n in (1, 2, 3, 5, 7, 8) for N in range(2, 200) if mult(N, n) != mult_direct(N, n)] check("quasi-polynomial multiplicities match direct SU(2) weight enumeration", not bad, "n in {1,2,3,5,7,8}, N < 200, %d mismatches" % len(bad)) hi = 64 if "--full" in sys.argv else 40 vals = {n: lndet(n) for n in range(2, hi + 1)} P = [hi - 16, hi - 8, hi] # local 3-point: the target is NOT in the basis a, _, _ = _solve3([[mpf(x)**2, log(mpf(x)), mpf(1)] for x in P], [vals[x] for x in P]) check("the n^2 coefficient is -zeta(3)/(4 pi^2), not -zeta(3)", abs(a + target) < mpf("1e-6"), "local solve at n=%d: %s vs %s" % (hi, mp.nstr(a, 10), mp.nstr(-target, 10))) check("bare zeta(3) is excluded", abs(a + zeta(3)) > mpf("1"), "bare would be %s, off by 4 pi^2 = %s" % (mp.nstr(-zeta(3), 7), mp.nstr(4*pi**2, 7))) g_bare, g_true = math.exp(-3*float(zeta(3))), math.exp(-3*float(target)) check("consequence: the Gaussian factor in m2/m1 is ~0.91, not ~0.027", abs(g_true - 0.913) < 0.01 and abs(g_bare - 0.027) < 0.001, "bare %.4f vs corrected %.4f" % (g_bare, g_true)) print("\n n | ln det' B_n") for n in (2, 3, 4, 5, 6, 8, 12, 16, 24, 32, 40): if n in vals: print("%3d | %s" % (n, mp.nstr(vals[n], 8))) print("\n[HONESTY] Trivial character only; unit radius; one zeta normalisation. The ln n") print(" coefficient (~4/3) is fitted, not derived. This establishes the asymptotic,") print(" not that it follows from Nash-O'Connor with Cheeger-Mueller.") if FAIL: print("\nFAILED: %d check(s)" % len(FAIL)); sys.exit(1) print("\nALL PASS") if __name__ == "__main__": main()