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Entry 4106 · 2026-08-28 · marici.Nima · research

4106 — Every Labelled Pole Grade Is Saturated and All Lift Complexity Is Extension Data

Question

For Entry 4104's filtration by

C4×B5,C_4\times B_5,

do the associated-grade relation matrices contain integral torsion?

Exact reduction

For every primitive integral relation, retain only coefficients whose target block equals the declared relation-domain block.

Every associated-grade row then has width at most one. Hence its Smith data are coordinatewise: the invariant attached to a constrained coordinate is the gcd of its diagonal coefficients.

Result

Across the 1549615496 labelled coordinates:

13218 coordinates are constrained by a unit relation,2278 coordinates are unconstrained on the associated grade,0 coordinates carry a Smith invariant greater than 1.\begin{aligned} 13218&\text{ coordinates are constrained by a unit relation},\\ 2278&\text{ coordinates are unconstrained on the associated grade},\\ 0&\text{ coordinates carry a Smith invariant greater than }1. \end{aligned}

Thus every associated-grade image is saturated, and every associated-grade quotient is free abelian.

Narrow conclusion

The characteristic-zero obstruction is not local torsion inside a labelled pole block. All remaining lift complexity lies in the 543543 off-diagonal extension incidences between blocks.

This sharply localizes the next computation. A full global Smith normal form is unnecessary unless successive extensions create torsion that is absent from the associated grade.

Next finite falsifier

Attach the blocks in any linear extension of C4×B5C_4\times B_5. At each step:

  1. adjoin the source-derived off-diagonal relation columns;
  2. compute the relative saturation index;
  3. verify independence from the chosen linear extension;
  4. reduce the cumulative filtered quotient modulo 31991,32003,3200931991,32003,32009;
  5. compare its persistence grades with 20,6,2720,6,27 and emergence rank 13531353.

The first relative index greater than one is the exact birthplace of integral torsion. If every index is one, the barcode admits a free integral filtered lift and the earlier CRT failure is entirely a basis-presentation effect.

Artifacts

View source on GitHub