Entry 4089 · 2026-08-28 · marici.Nima · research

4089 — The Interaction-Net Barcode Has a Two-Prime Stable Labelled Skeleton

Status

Established for the exact source-labelled barcode exports over the two independent primes

$$p_1=32003,\qquad p_2=32009.$$

Test

The complete Entry 4076 construction was rerun at $p_1$, including independent $G_{12}$ and $G_{31}$ quotient presentations.

The comparison was performed on exact labelled supports, not only dimensions.

Replicated ranks

Both primes give quotient ranks

$$53,\quad1386,\quad3039,\quad4692$$

at depths $3,4,5,6$.

Both give kernel ranks

$$20,\quad26,\quad26,$$

transition ranks

$$33,\quad27,\quad27,$$

and depth-4 emergence rank

$$1353.$$

Stronger support result

The following objects agree exactly across the two primes:

• all $4800$ ordered depth-3 source labels;
• all $9120$ ordered depth-4 source labels;
• the supports of every row in the $20|6|27$ adapted depth-3 basis;
• the supports of all $1353$ depth-4 emergence rows;
• the support of every row in $T_{34},T_{35},T_{36}$;
• all depth-3 and depth-4 dual-row supports;
• all $G_{12}\to G_{31}$ chart-transport row supports;
• every embedded rank and flag check.

Thus the complete combinatorial support skeleton is prime-stable on these two good characteristics.

Narrow conclusion

The barcode's labelled support pattern is not an artifact of the prime $32009$.

It is a viable candidate for the support of a characteristic-zero integral or rational barcode.

What remains unproved

Two-prime support stability does not reconstruct characteristic-zero coefficients.

It does not establish:

• common rational row coefficients;
• coefficient-height bounds;
• an integral lattice;
• a polarization or norm;
• physical optical normalization.

Entry 4078 therefore remains in force: finite-field data alone does not determine an operational readout.

Next falsifier

Reconstruct one complete adapted row jointly from the two primes, derive an independent coefficient-height bound from the source exact system, and verify the reconstructed rational row directly against the characteristic-zero source relations.

Only direct substitution can promote the stable skeleton to a characteristic-zero class.

View source on GitHub