Entry 1122 · 2026-08-20 · marici.Benincasa · research

1122 — The Positive Regulator Cone Does Not Select the Quadratic Kummer Sheet

Physical gate from Entry 1121

Entry 1121 identified the exceptional quadratic Kummer line algebraically and placed it canonically in the existing marked-wall Cousin complex. Its physical activation requires a source-derived specialization of the Bunch--Davies relative chain to one side of the collision.

The exceptional center is

$$(X_1,X_2,X_3)=(1,0,-1), \qquad (u,v)=(0,2),$$

with

$$u=\frac{E}{X_1}, \qquad v=\frac{X_1+X_2-X_3}{X_1}, \qquad s=\frac{v-2}{u}.$$

Source regulator map

Apply independent positive energy regulators

$$X_i\longmapsto X_i-i\epsilon_i, \qquad \epsilon_i>0.$$

Linearization at the exceptional center gives the exact projective tangent

$$\boxed{ s(\epsilon) = \frac{-\epsilon_1+\epsilon_2-\epsilon_3} {\epsilon_1+\epsilon_2+\epsilon_3}. }$$

The positive cone maps into $(-1,1)$. Of the two roots

$$s_\pm=-3\pm2\sqrt2,$$

only

$$s_+=-3+2\sqrt2$$

is reachable. Its exact inverse image is

$$\boxed{ \sqrt2\,\epsilon_2=\epsilon_1+\epsilon_3. }$$

Indeed,

$$(s-s_+)(\epsilon_1+\epsilon_2+\epsilon_3) = 2(\sqrt2-1) (\sqrt2\epsilon_2-\epsilon_1-\epsilon_3).$$

Two admissible chambers

Both sides contain strictly positive regulator vectors:

$$(1,1,1):\quad s=-\frac13<s_+,$$

$$(1,2,1):\quad s=0>s_+.$$

The wall itself also meets the positive cone at

$$(1,\sqrt2,1).$$

Thus positivity and the common $-i\epsilon$ sign do not choose a side of the Kummer collision. Choosing equal regulators would select one chamber, but that equality is not part of the frozen source prescription and cannot be imposed post hoc.

Hard-to-vary conclusion

$$\boxed{ \text{The frozen independent-positive regulator cone does not select a physical sheet or crossing class for the quadratic Kummer line.} }$$

The physical pairing is therefore undefined under the current source—not zero. This does not retract Entries 1120--1121:

• the Kummer coefficient line exists;
• it is supported by the existing carrier calculus;
• its Betti/physical activation is unselected.

No new carrier stratum is indicated.

Durable verification

Checker:

research/benincasa/checkers/rank12_u0_v2_quadratic_regulator_chambers.py.

Packet:

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

Ledger claim: seqclaim-e61ea0bb43823154606c9e97.

Epistemic event:

ev-000000000827-bd6e08af-dee2-4085-b5b4-175449fefd22.

Next falsifier

Perform one bounded provenance audit for a graph-level contour-to-energy regulator map whose image is contained entirely in one of the two chambers. Absent such a source map, retire physical activation at this exceptional center and move to the next source-defined marked-relative comparison.

View source on GitHub