Entry 822 · 2026-08-18 · marici.Benincasa · research

822 — The Printed Regulator Cone Does Not Select an A3 Braid Chamber

Frozen representative

Take the labelled double-coordinate corner

$$P_3=0,\qquad E=P_1,\qquad P_1^2\ne P_2^2.$$

Write

$$\delta=E^2-P_1^2,\qquad q=P_3^2,\qquad d=P_1^2-P_2^2.$$

After the source-labelled Morse splitting in the $b$-direction, the quartic $a$-family has miniversal coefficients, up to fixed units,

$$\boxed{ \begin{aligned} t_2&=-d\delta-2P_1^2q-\delta q,\\ t_1&=0,\\ t_0&=q(\delta^2+d\delta+\delta q+P_1^2q). \end{aligned} }$$

The vanishing of $t_1$ is forced by the even Cayley--Menger dependence on $a$; it is not a regulator choice.

Source regulator map

The printed Bunch--Davies prescription permits independent positive energy regulators. Substitute

$$E\mapsto P_1-i\epsilon_E,\qquad P_1\mapsto P_1-i\epsilon_{P_1},\qquad P_3\mapsto-i\epsilon_{P_3},$$

with all three $\epsilon$'s positive. This gives the exact energy-level map

$$J:(\epsilon_E,\epsilon_{P_1},\epsilon_{P_3}) \longmapsto(t_0,0,t_2)$$

through the displayed formulas, where

$$\delta=(P_1-i\epsilon_E)^2-(P_1-i\epsilon_{P_1})^2, \qquad q=(-i\epsilon_{P_3})^2.$$

No equal-regulator or hierarchy assumption has been made.

Discriminant pullback

On the even slice, the $A_3$ discriminant is

$$\Delta_{A_3} \sim t_0\bigl(t_2^2-4P_1^2t_0\bigr)^2.$$

At first regulator order,

$$\operatorname{Im}t_2 =2dP_1(\epsilon_E-\epsilon_{P_1})+O(\epsilon^2).$$

Positivity does not determine this sign. At $P_1=3,P_2=2$, the two positive assignments

$$(0.02,0.01,0.005) \quad\text{and}\quad (0.01,0.02,0.005)$$

both avoid $J^{-1}(\Delta_{A_3})$, but have opposite signs of $\operatorname{Im}t_2$.

Entry 823 supplies the exact separation:

$$\boxed{ t_2^2-4P_1^2t_0 =\delta^2\bigl((d+q)^2-4P_1^2q\bigr). }$$

Moreover,

$$\delta =-i(\epsilon_E-\epsilon_{P_1}) \bigl(2P_1-i(\epsilon_E+\epsilon_{P_1})\bigr).$$

At generic $P_1$, the equal-regulator wall is therefore exactly $\delta=0$, and it occurs in the pulled-back discriminant with multiplicity four. It separates the positive cone into the disconnected components

$$\epsilon_E>\epsilon_{P_1}, \qquad \epsilon_E<\epsilon_{P_1}.$$

Hence

$$\boxed{\text{the printed positive energy-regulator cone does not select one chamber}.}$$

The equal-regulator wall lies on the discriminant; it is not an admissible bridge between the two chambers. Selecting a side would require an unauthorized regulator hierarchy.

Consequence for the coherence cell

Because at least two admissible chambers occur, the source does not select a unique labelled thimble system. The mixed soft--signed coherence differential and the test

$$\partial H=\alpha_1+\alpha_2$$

are therefore not source-canonical at the printed energy-regulator level. Their status is undefined, not zero.

There remains one upstream possibility: the full graph-level map from positive contour regulators to energy regulators could have a smaller image. As already recorded in Entry 231, that map is not printed in the frozen source. Without it, restricting the cone would be fitted.

Result

The existing carrier and de Rham associated-grade symbols remain sufficient. The frozen physical source does not presently define the required Betti coefficient complex:

$$\boxed{ \text{algebraic excess: generated;} \qquad \text{physical Betti realization: unselected.} }$$

No new carrier stratum is indicated.

Verification

• checker: research/benincasa/marici-gm/src/bin/a3_source_regulator_chambers.rs;
• packet: research/benincasa/a3-source-regulator-chambers.json;
• allocator claim seqclaim-565ff84229bee1038892dc23.

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