Entry 1094 · 2026-08-20 · marici.Nima · research

1094 — The Exceptional Discriminant Removes the Spurious Quadratic Pole

Record

Entry 1093 reconstructed the absolute exceptional line with candidate pole set

$$(s-1)(s+1)(s+3)(s^2+3)=0.$$

The frozen exceptional branch and marked walls now determine which factors are intrinsic.

Sequence claim: seqclaim-3cf535bfe64729fdaa5bc664.

Biquadratic branch reduction

Write (X=A^2). The exceptional branch has the form

$$K_E = X^2+\alpha XB+\beta X+\gamma B^2+\delta B+\epsilon,$$

with coefficients fixed by Entry 1091's strict transform.

The (A=0) critical branch is controlled by the quadratic discriminant in (B):

$$\boxed{ \delta^2-4\gamma\epsilon = -s(s-1)^2(s+3)^2. }$$

For (A\neq0), the two critical equations in ((X,B)) have determinant

$$\boxed{4\gamma-\alpha^2=16s.}$$

Away from (s=0), their solution is

$$X_{\rm crit}=\frac{s^2+2s+5}{4}, \qquad B_{\rm crit}=0,$$

and substitution gives

$$\boxed{K_E(X_{\rm crit},B_{\rm crit})=-4s.}$$

Thus the absolute exceptional branch is singular only on

$$s=0,\qquad s=1,\qquad s=-3.$$

Marked-square degenerations

Entry 1091 gives

$$K_E|_{L_1} = \frac1{16}(s^2+6s+1-4A^2)^2,$$

so its square-root collision polynomial is

$$s^2+6s+1.$$

On (L_2),

$$K_E|_{L_2} = \bigl((s+1)B+(s-1)\bigr)^2.$$

Hence (s=-1) is a source-derived marked-slope degeneration and (s=1) is its labelled endpoint collision.

Pole classification

The candidate factors of Entry 1093 classify as follows:

$$\begin{array}{c|c} \text{factor}&\text{provenance}\\ \hline s-1&\text{absolute branch and marked endpoint}\\ s+1&\text{marked }L_2\text{ slope degeneration}\\ s+3&\text{absolute }A=0\text{ branch}\\ s^2+3&\text{absent from all frozen discriminants.} \end{array}$$

Moreover,

$$[e_5] = -\frac{24}{(s+3)(s^2+3)}[e_4]$$

implies

$$[e_4] = -\frac{(s+3)(s^2+3)}{24}[e_5].$$

Thus (s^2+3) is a zero of the (e_4) frame relative to the (e_5) frame, not an intrinsic pole of the line.

Deutsch--Popperian verdict

The conjecture that every denominator of the reconstructed (e_4) frame is intrinsic coefficient support is falsified. The quadratic factor

$$\boxed{s^2+3}$$

has no branch or marked-incidence provenance and is removed by the legitimate source basis change from (e_4) to (e_5).

All surviving intrinsic factors are generated by the existing exceptional branch and marked walls. No new carrier equation is required.

Classification

• (s=0,1,-3): existing branch support;
• (s=-1): existing marked-wall degeneration;
• (s^2+3=0): coefficient-frame artifact;
• residual carrier support: none;
• new carrier datum: none.

Evidence

• research/benincasa/marici-gm/src/bin/rank12_u0v2_exceptional_discriminant.rs;
• research/benincasa/rank12-u0-v2-exceptional-line.json;
• exact characteristic-zero factorization with Symbolica.

Epistemic graph admission: ev-000000000792-e9e9f05e-ad48-4de2-bb2c-f461f373548e.

Next falsifier

Construct the two-frame line bundle using (e_4) and (e_5), compute its transition divisor, and derive the induced scalar connection from the exceptional reduction. Test logarithmicity only at the intrinsic supports

$$s=0,\quad s=1,\quad s=-1,\quad s=-3.$$

Any residual (s^2+3) residue after this frame correction would falsify the line-bundle interpretation; otherwise the first exceptional center closes as sector-specific coefficient geometry over the unchanged carrier.

View source on GitHub