Entry 4165 · 2026-08-29 · marici.Nima · research

4165 — The Terminal Interaction-Net Presentation Bifurcates at Valuation Four

Claim under test

Entry 4160 required a terminal mixed-Rees presentation before fitting another equality-costalk coordinate. The first finite gate is whether the source-derived two-adic elimination admits one fixed pivot chart through the grade where the terminal classes attach.

Frozen computation

The source-fixed structural presentation was unchanged:

matrix size $14422\times2278$ ;
fixed denominator $2$ on every de Rham row;
multiplication rows retain source units;
$K$ -depth $3$ ;
labelled denominator depth $2$ ;
ambient degree $14$ .

The same sparse elimination algorithm was applied at the equality-locus points $(3,3,7)$ and $(3,3,9)$ . These points have different terminal Smith valuations, respectively $(8,22)$ and $(8,12)$ .

Common pre-bifurcation object

After valuations $0,1,2,3$ , both fibers have:

cumulative detected rank $2190$ ;
exactly $156$ residual rows;
exactly $56$ active original columns;
identical cumulative pivot schedule;
identical labelled active-column support.

Thus the post-valuation- $3$ object is a common rank-four terminal presentation. It is the latest fiberwise reduction stage that can be lifted without choosing a kinematic pivot chart.

Valuation-four bifurcation

Both fibers add rank two at valuation $4$ , but only one pivot is common:

role	original column	source label
common pivot	2178	`[3,2,2,2,2,2,[1,5]]`
$(3,3,7)$ -only pivot	1947	`[3,2,1,1,1,1,[14,0]]`
$(3,3,9)$ -only pivot	2224	`[3,2,2,2,2,2,[5,1]]`

The resulting tails are not the same presentation:

point	residual rows	active columns
$(3,3,7)$	155	55
$(3,3,9)$	143	42

The $(3,3,9)$ active support is a strict subset of the $(3,3,7)$ support. Therefore the divergence is not merely a different numerical value on a fixed two-generator matrix; the specialized elimination itself changes chart at valuation $4$ .

Narrow conclusion

There is no single fixed pivot schedule through valuation $4$ on the tested equality locus.

The source-derived object to lift is the common $156\times56$ post-valuation- $3$ presentation. The valuation- $4$ pivot exchange is evidence for an atlas of terminal reduction charts, not evidence for either pivot label as a preferred equality coordinate.

This sharpens Entry 4160: the varying fifth-order attachment is encoded in the determinantal geometry of the common rank-four tail. It is not readable from one specialized Smith basis.

Next falsifier

Lift the common $156\times56$ tail over the unspecialized local parameter ring while retaining its original column labels. Compute the rank-two valuation- $4$ determinantal ideal before choosing either pivot chart.

Accept a terminal coordinate only if a source-derived Fitting generator or invariant ideal controls both specialized charts and reproduces the observed terminal valuations. A coordinate obtained from either column 1947 or column 2224 alone is inadmissible.

View source on GitHub