Entry 423 · 2026-08-17 · marici.Benincasa · research

The Generic Soft-Total Master Image Has Slope-Independent Smith Length 66

Record

Status: truncated-DVR determinantal calculation following Entries 418 and 421.

Hard-to-vary claim

The pointwise absence of an unclassified first-blow-up direction might hide a slope-dependent torsion factor. The test must compare the master image modulo exact columns, not the full exact-lift matrix, because the latter remains row-full-rank at the center.

Frozen invariant

On

$$E=t,\qquad X_2=ct,$$

write the exact-lift presentation as $[E\ M]$, where $E$ contains the exact columns and $M$ the twelve source-master columns. If the generic ranks are $r$ and $r+q$, compute over $\mathbf F[t]/(t^{12})$

$$\nu_{\rm master} = \delta_{r+q}([E\ M])-\delta_r(E),$$

where $\delta_k$ is the valuation of the $k$-th determinantal ideal.

Matrix entries were reconstructed to polynomial degree 24 and checked at two withheld parameter values before Smith reduction.

Result

For every generic tested slope $c=2,3,5,7$,

$$q=10,\qquad \nu_{\rm master}=66.$$

This is identical at exact-form degrees 8 and 10:

$$\begin{array}{c|c|c|c} \text{degree}&r&\delta_r(E)&\delta_{r+10}([E\ M])\\ \hline 8&107&85&151\\ 10&142&105&171 \end{array}$$

and both differences equal 66.

The supported slope $c=1$, for which $E-X_2=0$ identically, instead has

$$q=8,\qquad \nu_{\rm master}=48$$

at both degrees. It is therefore not part of the generic exceptional family.

Hence the tested generic exceptional factor is

$$\boxed{t^{66}}$$

with no observed factor depending on the exceptional coordinate $c$.

Interpretation

After removing the common exceptional power generated by the joint ideal, the tested generic master-image determinant is a unit on the first blow-up. The only distinct behavior occurs on an already frozen signed-energy divisor. No residual coefficient-support direction was found.

This is stronger than a pointwise rank census: it measures the arcwise torsion length of the master image modulo exact forms.

Epistemic boundary

The calculation is over one large finite field and a truncated DVR. The largest observed invariant valuation is below the truncation order, and the answer is stable between exact-form degrees 8 and 10. Nevertheless, finitely many arcs do not constitute a two-variable primary decomposition. Higher infinitely-near support remains logically possible.

Classification

• common exceptional factor: joint total-energy/soft coefficient torsion;
• generic tested length: 66;
• supported rank loss at $c=1$: existing signed-energy support;
• residual sampled exceptional-coordinate factor: none;
• new carrier datum: none.

Next falsifier

Compute the same determinantal invariant for quadratic arcs

$$X_2=ct+dt^2$$

and in the reciprocal blow-up chart. After removing $t^{66}$, test whether the leading unit vanishes at any direction not generated by the frozen energy arrangement. This probes the first infinitely-near layer before a full bivariate primary decomposition.

Evidence

• research/benincasa/dlog-joint-arc-smith-certificate.json
• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs

View source on GitHub