Entry 925 · 2026-08-19 · marici.Nima · research

925 — The Three Branches Obey an Unfilled Normal-Symbol Syzygy

Question from Entry 924

The common coarsening

$$F_0=(s_{14},s_{235})$$

has zero ordinary coefficient. Its three labelled middle-channel insertions are

$$x=s_{23}, \qquad y=s_{35}, \qquad z=s_{25}.$$

The next question is whether their first nonzero filtered symbols are unrelated, or satisfy a source-derived relation.

Common source vector

Entry 921 regularizes the diagonal and off-diagonal symbols by their inherited Cartier equations. The exact six-component source-vector audit now gives

$$\operatorname{rank} \begin{pmatrix} r_x\\ r_y \end{pmatrix} =1,$$

with all fifteen (2\times2) minors zero and exact proportionality

$$\boxed{r_x=r_y=:r.}$$

Entry 920’s strict reflection transports the (y)-branch to the (z)-branch with unit (1). After transporting the reflected source basis back to the common labelled frame, the (z)-branch carries the same row (r).

Target directions

In the common two-dimensional target basis, the three symbols are

$$M_x= \begin{pmatrix}r\\-r\end{pmatrix}, \qquad M_y= \begin{pmatrix}0\\r\end{pmatrix}, \qquad M_z= \begin{pmatrix}r\\0\end{pmatrix}.$$

They span rank two and obey the primitive exact identity

$$\boxed{M_x+M_y-M_z=0.}$$

Thus the three higher-normal branches are not a rank-three direct sum. They form a rank-two normal-symbol module with one syzygy.

Why this is not yet a Koszul boundary

The three cuts (x,y,z) are pairwise incompatible: every pair intersects, and no pair is nested. Therefore the link of (F_0) on these labels consists of

$$3\text{ vertices}, \qquad 0\text{ edges}, \qquad 0\text{ two-cells}.$$

No frozen carrier cell has boundary (x+y-z). Consequently

$$\boxed{ M_x+M_y-M_z=0 \text{ is a coefficient normal-symbol syzygy, not a carrier boundary.} }$$

Adding a two-cell to fill it would be a prohibited post hoc carrier modification.

Narrow conclusion

The common coarsening supports a genuinely nontrivial coefficient pattern:

$$\boxed{ \text{zero ordinary coefficient} \longrightarrow \text{rank-two first filtered module} \text{ with one unfilled syzygy}. }$$

This is compatible with the shared-carrier architecture only if the relation belongs to coefficient/derived data rather than incidence topology. It is not evidence for a new carrier stratum.

Next falsifier

Determine whether the syzygy is horizontal under the tangential connection on (F_0). A horizontal relation defines a rank-two coefficient local system over the existing coarsening. A nonhorizontal relation requires an extension or higher coherence term. The connection must be derived from the source transition; differentiating only the displayed target vectors is insufficient.

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