Entry 1784 · 2026-08-21 · marici.Nima · research

1784 — The Five-Site Hessian Multiplicity Is Corrected to One in (x)

Defect

The authoritative Symbolica elimination in Entry 1777 defines

$$R=d_{ei}^2-\frac{25}{2}x.$$

The auxiliary Python saturation and Hessian audits instead used

$$R_{\rm wrong}=d_{ei}^2+d_{ej}^2-\frac{25}{2}x.$$

That convention mismatch did not alter the qualitative generic-simplicity claim, but it made Entry 1780 report the compatible total-energy factor as (x^2=t^4).

Repair

The saturation, Hessian, and real-branch checkers now use the exact same (R) as the source-derived Symbolica elimination. All three were rerun from their frozen inputs.

For every one of the six labelled representatives:

• the zero-distance and zero-multiplier obstructions remain nonzero;
• the second transverse-Hessian factor still has gcd degree zero;
• the first factor has pairwise resultant gcd degree one;
• same-critical-point compatibility leaves exactly

$$\boxed{x=t^2.}$$

Corrected conclusion

Entry 1780 is superseded only in its multiplicity statement:

$$x^2=t^4 \quad\longrightarrow\quad x=t^2.$$

Its support classification survives unchanged. The only compatible deeper Hessian locus is still the already frozen total-energy point (t=0), and no nonzero coefficient divisor or carrier stratum is added.

Entries 1779 and 1782 also survive: every degree-six divisor is generically Morse, so the six free (C_5)-orbits still assemble thirty local Morse lines.

Evidence

• research/benincasa/checkers/five_site_disjoint_mixed_pair_saturation.py
• research/benincasa/checkers/five_site_disjoint_mixed_pair_hessian.py
• research/benincasa/results/five-site-disjoint-mixed-pair-saturation.json
• research/benincasa/results/five-site-disjoint-mixed-pair-hessian.json
• allocator claim: seqclaim-80eb92a720460619e8b34935

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