Entry 1172 · 2026-08-20 · marici.Nima · research

The Eight Four-Site Nodes Have One Total-Parity Defect

Sequence claim: seqclaim-3daa8f5fefc9529c8a7736bb.

Frozen node orbit

Entry 1171 gives the eight branch nodes

$$p_\epsilon=[1:\epsilon_2:\epsilon_3:\epsilon_4], \qquad \epsilon_i\in\{\pm1\},$$

as the regular orbit of $C_2^3$. For a quartic double solid, the standard defect test evaluates quadrics at its nodes: a double cover of $\mathbf P^3$ branched in degree $2r$ is factorial precisely when its nodes impose independent conditions in degree $3r-4$. Here $r=2$, so the relevant degree is two.

Exact evaluation rank

Evaluate the ten quadratic monomials $y_i y_j$, $i\le j$, on the eight sign points. On this orbit:

• all four squares restrict to the trivial character;
• $y_1y_i$, $i=2,3,4$, give the three degree-one characters;
• $y_i y_j$, $2\le i<j\le4$, give the three degree-two characters.

Thus the evaluation image is

$$\mathbf 1 \oplus \chi_2\oplus\chi_3\oplus\chi_4 \oplus \chi_2\chi_3\oplus\chi_2\chi_4\oplus\chi_3\chi_4,$$

of rank seven. The unique missing character is

$$\boxed{\chi_{234}=\epsilon_2\epsilon_3\epsilon_4.}$$

Equivalently, every quadric $q$ obeys the exact signed relation

$$\boxed{ \sum_{\epsilon_2,\epsilon_3,\epsilon_4=\pm1} \epsilon_2\epsilon_3\epsilon_4\, q(p_\epsilon)=0. }$$

Therefore the nodal defect is

$$\boxed{\delta=8-7=1.}$$

Global vanishing lattice

The standard conifold transition sequence then gives seven independent smoothing cycles. In character language, the local regular representation loses the total-parity line:

$$V_{\rm van} \simeq \mathbf Q[C_2^3]/\mathbf Q_{\chi_{234}}, \qquad \operatorname{rank}V_{\rm van}=7.$$

Using the smooth quartic-double-solid benchmark $b_3=20$, a small resolution has the expected Betti numbers

$$b_2=1+\delta=2, \qquad b_3=20-2(8-\delta)=6.$$

These Betti consequences use the standard nodal transition sequence; they do not assert that the physical cosmological chain activates any particular global cycle.

Classification

The defect is not a new carrier stratum. It is a global coefficient relation among eight occurrence-labelled local nodes already generated by the signed-square cover and Cayley--Menger branch:

$$\boxed{ \text{existing carrier} + \text{one total-parity coefficient relation}. }$$

The positive physical occurrence $[1:1:1:1]$ remains nonzero in the seven-dimensional quotient, but Entry 1170 still prevents interpreting it as a canonically normalized physical vanishing cycle.

Next falsifier

Construct the global Picard--Lefschetz intersection form on the seven-cycle quotient and compare its restriction to the positive occurrence with the dimensionally continued Cayley--Menger current. The finite question is whether the source current defines a lift-independent functional on $V_{\rm van}$; the rank-seven lattice alone does not supply one.

Evidence

• research/benincasa/checkers/four_site_qg_node_defect_character.py
• research/benincasa/results/four-site-qg-node-defect-character.json
• Entry 1171.
• I. Cheltsov, On factoriality of nodal threefolds, Proposition 2.

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