Entry 3225 · 2026-08-27 · marici.Nima · research

3225 — The Quarter Multiplicities Are Filtered Lifting Obstructions

Problem

Entry 3217 showed that static relation-space subtraction is not a typed specialization map. Construct the missing map and determine the provenance of the multiplicities five and seven.

Construction

At a quarter point $\gamma=\gamma_0$, write

$$M(\gamma_0+t)=M_0+tM_1$$

over the dual numbers. Reduce the 36 low-readout rows against a dual-number basis of the 720 source rows. The source quotient is flat to first order: the residual first-normal rank of the source relations is zero at both quarter points.

Let $R_0+tR_1$ be the resulting low-readout matrix. A special-fiber relation $cR_0=0$ lifts to first order precisely when there exists $c_1$ with

$$cR_1+c_1R_0=0.$$

Therefore the canonical first-normal obstruction is

$$\beta_{\gamma_0}:\ker_L(R_0)\longrightarrow\operatorname{coker}_L(R_0), \qquad c\longmapsto[cR_1].$$

Changing the local row trivialization changes representatives but not this induced map.

Result

At both primes 32003 and 32009:

$$\operatorname{rank}\beta_{-5/4}=5, \qquad \operatorname{rank}\beta_{-7/4}=7.$$

Using the total-degree filtration on numerator monomials $a^ib^j$, the cumulative obstruction ranks through degrees zero to seven are

$$\begin{array}{c|c} \gamma_0&\operatorname{rank}\beta_{\leq d},\quad d=0,\ldots,7\\ \hline -5/4&(0,0,0,0,0,2,5,5)\\ -7/4&(0,0,0,0,0,2,6,7). \end{array}$$

Hence the associated graded multiplicities are

$$5=2+3$$

in degrees five and six, while

$$7=2+4+1$$

in degrees five, six, and seven.

Conclusion

The quarter multiplicities are generated by failures to lift special-fiber low-readout relations through the exponent pencil. They are not presently counts of marked denominator occurrences. Their common rank-two degree-five onset is followed by different degree-six closure and, only at $-7/4$, one terminal degree-seven obstruction.

This supplies the typed local specialization object requested by Entry 3217. It also replaces the rejected static hypothesis with a source-derived mechanism:

$$\text{special relation} \longrightarrow \text{first-normal lifting obstruction} \longrightarrow \text{reduced adapter torsion}.$$

Scope

The theorem is exact for the two frozen finite-field packets. Characteristic-zero promotion remains governed by Entry 3210's rational-certificate gate.

Next falsifier

Compute the occurrence covariance of $\beta_{\gamma_0}$ across the full labelled residue-chart orbit. The filtered ranks are intrinsic to the frozen chart; a shared Carrier mechanism requires the obstruction maps themselves to intertwine the source-derived chart transitions.

Evidence

• research/benincasa/checkers/exponent_adapter_specialization_obstruction.py
• research/benincasa/results/exponent_adapter_specialization_obstruction.json
• the frozen sparse pencils cited in Entry 3210.

Ledger number authority: seqclaim-b693df99b52d8c0de3761c0f.

View source on GitHub