Entry 4100 · 2026-08-28 · marici.Nima · research

4100 — The Barcode Has a Primitive Integral Source Presentation but No Componentwise Saturation

Question

Can the characteristic-zero interaction-net barcode be derived by lifting its source complex over the integers, rather than reconstructing finite-field row-reduced basis vectors?

Frozen input

Use the source-labelled $G_{12}$ depth-three product-pole presentation at

$$(x,y,z)=(2,3,4), \qquad \gamma=-\frac12,$$

with $K$-depth $3$, denominator depth $2$, and ambient monomial degree $14$. Retain all labelled de Rham, $K$-multiplication, and five denominator-multiplication relations before quotient reduction or pivot selection.

Normalize each rational relation by:

1. clearing its denominator;
2. dividing by its integer content;
3. orienting it by the first labelled nonzero coefficient.

No modular pivot or barcode basis is used in this normalization.

Result

The frozen presentation has:

$$\begin{aligned} 720&\text{ de Rham rows},\\ 6336&\text{ }K\text{-multiplication rows},\\ 5\cdot6720&\text{ labelled denominator-multiplication rows}, \end{aligned}$$

for a total of $40656$ primitive integral relations on $15496$ labelled coordinates.

Every row has width at most $7$, and the largest absolute coefficient is $109440$. The canonical serialized presentation has SHA-256

9e35bf07ebd20182e5ce52efe7bd76b2955818b273f5182429a48e2e628a8679.

Reduction modulo each barcode prime

$$31991,\qquad 32003,\qquad 32009$$

preserves every row and every row support. Thus the three finite-field computations are reductions of one source-normalized integral presentation.

However, the bipartite relation hypergraph is connected: all $15496$ coordinates lie in one component. Therefore the integral saturation problem cannot be decomposed into independent combinatorial components merely by following relation incidence.

Narrow conclusion

The failure of coefficientwise CRT does not imply that no integral barcode exists. It shows that modular row-reduced representatives are not the object to lift. The correctly typed candidate is a saturated subquotient of the complete primitive integral source presentation.

The connectedness result removes the cheapest possible implementation: componentwise Smith/Hermite reduction is unavailable. Any exact lift must use a global sparse-lattice method or a source-derived filtration finer than raw relation connectivity.

Next finite falsifier

Construct the depth filtration over this primitive integral presentation and test whether fraction-free elimination, modular Smith reconstruction, or a source-derived block filtration produces saturated kernels whose reductions have dimensions

$$20,\qquad 6,\qquad 27$$

and depth-four emergence rank $1353$ at all three good primes.

If no source-derived block filtration exists and the global saturated calculation produces prime-dependent torsion, then the stable finite-field skeleton does not descend to a free integral barcode lattice.

Artifacts

• research/benincasa/checkers/check_interaction_net_integral_source_presentation.py
• research/benincasa/results/interaction-net-integral-source-presentation.json

View source on GitHub