Sector Signs on the Ten Flat 3-Manifolds. The sign table of all fifty-two twisted sectors, the refutation of the weighted threshold, and the mechanism behind it: the energy is a Fourier transform on the holonomy group, whose product term is the obstruction to additivity

An earlier note [1] proved that on a d -dimensional torus the sign of the 2^d twisted sector sums is governed by a sharp threshold: the regularized sum is positive exactly when the twisted directions are at least as many as the untwisted ones. The same note stated, as a conjecture, a non-orientable counterpart — a weighted threshold in which twisting the direction reversed by an orientation-reversing generator counts double; the conjecture held on twelve computed sectors, the four of the Klein bottle and the eight of K x S^1. This paper carries out the natural next test: the four non-orientable flat 3-manifolds B_1,...,B_4, twenty-four sectors in all. (E1, identification) The four Bieberbach groups are built from generators, and their identification rests not on structural resemblance but on a complete invariant: the ten flat 3-manifolds have pairwise distinct first homology [7], so the H_1 obtained by Smith normal form identifies them uniquely. The computed values Z^2 + Z_2, Z^2, Z + Z_2^2, Z + Z_4 reproduce the published table exactly. (E2, the sector tables) Every sector energy is computed twice by independent machinery: a character projector built on a general Gram-matrix shifted twisted Epstein continuation, and the geometric image-method heat trace (on the tilted lattice of B_2 the second route is the absolutely convergent series at a large exponent); on the eight sectors of B_1 both reproduce the values published in [1] to four decimals. (E3, the refutation) On B_4 the energy is not monotone in the number of twists: twisting the two generators separately gives +0.0141 and +0.0970, twisting both gives -0.0267. Since an additive weighted count for the double twist is always at least as large as for either single one, a threshold of the form 2k_eff >= d cannot be repaired by a better choice of weights: what fails is the underlying monotonicity assumption. The counterexample persists over a whole range of aspect ratios. (E4, a second obstruction) The holonomy of B_2 swaps lattice vectors, forcing chi(b_1)=chi(b_2), so that carrier has four sectors rather than eight, and "twisting two separate directions" is not even a well-defined operation there. (E5, what survives) All eight sectors of B_3 do obey the threshold law with suitable weights, so the failure is specific rather than generic: the rule holds as long as the holonomy is a single reflection perpendicular to a coordinate axis. (E6, the mechanism) The energy is the Fourier transform on the holonomy group: E(chi)=sum_A in Fchi(gamma_A)T_A, with coefficients independent of the character once chi|_Lambda is fixed. If |F|=2 there is a single non-trivial term and E is affine in the twist label; if Fcong(Z/2)^2 a product term of sign chi(alpha)chi(gamma) appears — the only non-additive character, and precisely the obstruction to additivity. Its coefficient is an exact rational (-tfrac18 on B_4, -tfrac116 on B_3), because the product element is a half-turn whose fixed subspace is one-dimensional; on B_4 this term dominates both single terms, which is what tips the sign. (E7, the complete table) Rewritten in lattice coordinates, the machinery also computes the six orientable carriers: the table is complete for all ten flat 3-manifolds, 52 sectors. G_1=T^3 independently reproduces the threshold theorem of [1], while on the half-turn space G_2 the energy is already non-monotone on an orientable carrier (+0.1167 for one twist, -0.0056 for two) — monotonicity is thus tied to the torus, not to orientability. (E8, the sum rule) The other law of [1], by contrast, does hold, and in a more general form: the zeta functions of the sectors add up to the spectrum of the maximal Z_2 cover, sum_chizeta_chi(s)=zeta^widetilde M(s) — which on the torus recovers the sum rule of [1]. The product term cancels when summed over chi; this is why the sum rule survived what the threshold did not. (E9, delimitation) Finally we delimit precisely what the twisted basis of the framework does here: twelve of the twenty-four non-orientable sectors live on a shifted frequency lattice, where the basis is indispensable (a half-integer mode occupies every bin of the periodic basis, decaying as 1/k, and reaching 99% of the energy takes 41 bins — a single exact line in the twisted basis). All four characters of B_4, by contrast, are trivial on the lattice, so the refutation comes from the representation theory of the holonomy, not from the choice of basis. All claims are verified by a deterministic script.