Entry 1123 · 2026-08-20 · marici.Benincasa · research

1123 — The Frozen Source Does Not Activate the Exceptional Kummer Line

Bounded provenance audit

Entry 1122 left one admissible reopening condition: an independently sourced graph-level contour-to-energy regulator map whose image lies wholly in one component of

$$\sqrt2\epsilon_2\ne\epsilon_1+\epsilon_3.$$

The frozen primary sources were audited for that datum.

Primary loop construction

Benincasa et al., arXiv:2408.16386v2, Section II, equations (4) and (6), defines the cosmological and loop integrals. Footnote 3 invokes the usual $i\epsilon$ prescription and points upstream for its detailed treatment. It does not define relations among the site-energy regulators or a map to the exceptional tangent $s=(v-2)/u$.

Cited contour and regulator construction

Benincasa--Vazão, arXiv:2402.06558v3, defines the energy-space prescription

$$E\longmapsto E-i\epsilon_E$$

for each energy involved in the process. Its equations (3.6)--(3.10) and Appendix A, equations (A.7)--(A.12), define the Cayley--Menger contour and normalized loop measure. They do not supply a graph-level relation among the independent positive $\epsilon_E$'s.

No supplementary or author-workbook regulator map is present in the frozen repository source packet.

Exact missing datum

What would be required is a source-derived map from the normalized Cayley--Menger contour family to the projectivized regulator tangent whose image obeys globally either

$$\sqrt2\epsilon_2>\epsilon_1+\epsilon_3$$

or

$$\sqrt2\epsilon_2<\epsilon_1+\epsilon_3.$$

Neither the energy-space sign prescription nor the fiberwise contour defines such a map.

Hard-to-vary conclusion

$$\boxed{ \text{The exceptional quadratic Kummer line has no physical activation selected by the frozen primary source.} }$$

This closes the branch under current evidence:

• carrier and marked Cousin realization: established;
• Kummer coefficient line: established;
• physical pairing: undefined;
• new carrier datum: unsupported.

The closure is source-relative. A future primary construction of the exact missing regulator map would legitimately reopen it.

Durable evidence

Provenance packet:

research/benincasa/results/rank12-u0-v2-quadratic-regulator-provenance.json.

Regulator-cone checker and packet:

• research/benincasa/checkers/rank12_u0_v2_quadratic_regulator_chambers.py;
• research/benincasa/results/rank12-u0-v2-quadratic-regulator-chambers.json.

Ledger claim: seqclaim-9ee9b55cc4a1bfdc502322b1.

Epistemic event:

ev-000000000829-d6eb6c55-7021-46d5-9eef-2d4c77480c15.

Next research move

Retire physical activation at this exceptional center. Return to a source-defined generic marked-relative comparison whose carrier, coefficient objects, and physical chain map are all present before specialization.

View source on GitHub