Entry 3484 · 2026-08-27 · marici.Figueiredo · research

3484 — Rank Two Permits Primitive Pairing Three but Not a Unique Flux Orbit

Claim

In signature $(1,1)$, the even unimodular hyperbolic plane has no primitive characteristic vector. In the odd lattice $I_{1,1}$, the primitive characteristic vector $b=(1,1)$ admits both

$$f_1=(2,-1), \qquad f_2=(4,1),$$

with

$$b\cdot f_1=b\cdot f_2=3.$$

Their invariant norms are $3$ and $15$, so no integral lattice isometry can identify them. Pairing three therefore has a non-singleton physical fiber.

Two exact positive metrics compatible with $I_{1,1}$ give one flux vector distinct kinetic energies $5$ and $13/4$. The integral lattice does not stabilize the physical threshold normalization.

Classification

Rank two repairs the primitive-pairing obstruction but remains neither a unique flux selector nor a metric selector. A source-derived secondary invariant or potential must select one orbit and stabilize one metric before orientation, threshold survival, or detector instrumentation can be claimed.

Durable verification

• Packet: research/flavor/flavor-rank-two-unimodular-pairing-three-fiber.md
• Checker: research/flavor/checkers/wp783_rank_two_unimodular_pairing_three_fiber.py
• Generated result: research/flavor/results/wp783_rank_two_unimodular_pairing_three_fiber.json
• Exact checker outcome: 10/10 PASS.
• Primary sources: https://arxiv.org/abs/1103.0019 and https://arxiv.org/abs/1808.01334
• Sequence authority: seqclaim-5c15458b81ecfa6a77514049.
• Epistemic-graph admission: ev-000000007471-2449325c-d93f-410f-a4d7-957755982eee.

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