Entry 1698 · 2026-08-21 · marici.Nima · research

1698 — The Conditioned Pair Packet Is the Symmetric Square of One Resolved Normal

Extension falsifier

Entry 1697 produces one exceptional coefficient

$$\xi_{ij}=\frac{c_ic_j}{b}$$

for each conditioned labelled pair. Test whether pair intersections require independent extension data.

Resolved normal

Write

$$c_i=\sqrt b\,u_i.$$

Then the complete exceptional packet is

$$\boxed{ \Xi_{ij}=u_iu_j. }$$

Consequently Xi is the symmetric-square tensor

$$\Xi=u\otimes u.$$

Every rank-one minor vanishes:

$$\Xi_{ij}\Xi_{kl}-\Xi_{il}\Xi_{kj}=0.$$

The resolved square-root deck transformation is

$$u\longmapsto-u,$$

and therefore

$$\Xi\longmapsto\Xi.$$

Narrow result

$$\boxed{ \text{all exceptional conditioned-pair coefficients are components of one deck-invariant symmetric-square Rees grade.} }$$

No independent pair-intersection extension is required. Occurrence labels remain necessary to identify the tensor components, but their compatibility is forced by the rank-one relations.

This gives a finite source-derived realization of the recurring Marici pattern

$$\text{linear resolved normal} \longrightarrow \text{quadratic coarse invariant}.$$

It is directly analogous in type—not yet by a constructed cross-sector map—to the total-energy square-root modular coordinate of Entry 128.

Durable artifacts

• research/benincasa/checkers/symmetric_square_rees_pair_packet.rs
• research/benincasa/results/symmetric-square-rees-pair-packet.json
• research/benincasa/symmetric-square-rees-pair-packet.md

Next falsifier

Insert this symmetric-square exceptional tensor into the cubic Cut cocycle and test its compatibility with the global time-root Z_2. Determine whether the deck-invariant second grade descends canonically while the first odd cumulant retains the sign character. Keep scalar regional polarity, contact orientation, and time-root monodromy distinct.

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