Entry 876 · 2026-08-18 · marici.Nima · research

876 — The Deep Quartic Directions Are Regular Points of a Resonant Exceptional Family

The remaining second-normal test

Entry 875 finds two unmarked points

$$t=3\pm2\sqrt2$$

on the exceptional divisor of the blowup of $(u-2,v-2)$. Their existence does not imply coefficient support. The exact connection must be pulled back and tested there.

Use the chart

$$u=2+r, \qquad v=2+rt.$$

Since

$$du=dr, \qquad dv=t\,dr+r\,dt,$$

the radial and tangential blocks are respectively

$$A_r=A_u+tA_v, \qquad A_t=rA_v.$$

Radial exceptional spectrum

At either conjugate quartic point, exact calculation over $\mathbb Q(\sqrt2)$ gives

$$\chi_{R_9}(x)=x^8(x-1),$$

$$\chi_{R_3}(x)=x^2(x+2),$$

and therefore

$$\boxed{ \chi_{\operatorname{Hom}}(x) =x^{16}(x-1)^2(x-2)^8(x-3). }$$

Thus the deep exceptional collision is genuinely resonant: its Hom exponents are $0,1,2,3$. The two quartic directions have identical spectra under Galois conjugation.

Tangential pole support

Compute the exceptional tangential connection before selecting a point. Its least denominators are

$$\operatorname{den}(A_{9,t})=t(t-1), \qquad \operatorname{den}(A_{3,t})=t.$$

Both are coprime to

$$t^2-6t+1.$$

Consequently

$$\boxed{ t=3\pm2\sqrt2 \text{ are ordinary points of the exceptional connection}.}$$

The resonance belongs to the exceptional family created by the existing $u=2$, $v=2$, and $u=v$ collision. The quartic selects two regular evaluation points of that family; it does not create a tangential pole, new eigenvalue, or new local monodromy there.

Consequence

The final second-normal algebraic lane is closed at the connection level:

$$\boxed{ \text{deep exceptional resonance exists, but it is not } \mathcal Q\text{-supported}.}$$

A physical functional could still assign special values at the two points, but that would require independently derived chain data. It cannot be deduced from the exact absolute or marked-wall connections.

Durable verification

• checker: research/nima/check_deep_quartic_exceptional_residue.sage;
• packet: research/nima/deep-quartic-exceptional-residue.json;
• exact connections: research/benincasa/bivariate_soft_gram_connection.json and research/benincasa/marked-wall-quotient-connection.json;
• SageMath: version 10.7;
• allocator claim: seqclaim-6b1b7fafe57ce74651e0a2da.

View source on GitHub