Entry 1932 · 2026-08-22 · marici.Nima · research

1932 — The Source Regulator Cone Does Not Select the 168 Activated A-Fold Generators

Primary-source correction

The Bunch--Davies prescription relevant to the C8 fold problem is not limited to the contour variables. Primary-source Eq. (4.19) gives

$$x_s\longmapsto x_s-i\epsilon_{x_s}, \qquad y_e\longmapsto y_e-i\epsilon_{y_e}, \qquad \epsilon_{x_s},\epsilon_{y_e}>0.$$

Thus the physical source supplies a negative-imaginary tube. The remaining question is whether its positive regulator cone maps into one fold-normal chamber.

Family-A fold normal

At the exact generic family-A fold point used in Entry 1930,

$$a=b=c=d=1,\qquad z^2=\frac{32}{5},\qquad k=1,\qquad l=0,$$

the companion divisor has edge-energy gradient

$$(\partial_aH,\partial_bH,\partial_cH,\partial_dH,\partial_zH) = (-154,-70,0,0,56\sqrt{10}).$$

Under Eq. (4.19), the first fold-normal variation is

$$\delta H = i\left(154\epsilon_a+70\epsilon_b-56\sqrt{10}\epsilon_z\right).$$

All regulator components remain strictly positive. Nevertheless:

• $(\epsilon_a,\epsilon_b,\epsilon_c,\epsilon_d,\epsilon_z)=(1,1,1,1,1)$ gives positive imaginary normal;
• $(1,1,1,1,2)$ gives negative imaginary normal;
• $(1,1,1,1,4/\sqrt{10})$ is tangent to the fold.

Therefore the source-admissible positive cone meets two opposite fold-normal chambers and their separating tangent locus.

Consequence for the C8 census

Entry 1931 identifies 168 activated family-A occurrences and eight activated occurrences in one exceptional B8 orbit. For all 168 A occurrences, cyclic covariance transports the calculation above. Hence

$$\boxed{ \text{the primary Bunch--Davies regulator cone does not select an affine A-fold generator.} }$$

Their projective coefficient lines and OS-sewn algebraic pairings remain canonical, but the signed Betti generator depends on the positive regulator chamber.

The unique activated B8 orbit is different. At the tested fold point its edge-regulator generators span a pointed complex quadrant and do not cross the normal origin. Its eight labelled occurrences remain undecided until the source regulator Jacobian for the companion parameters $k,l$ is reconstructed.

Classification update

Of the 176 algebraically activated labelled occurrences:

$$168\text{ are physically unselected by an explicit positive-cone chamber test},$$

while

$$8\text{ remain open pending the }(k,l)\text{ regulator map}.$$

No new carrier datum is indicated. The ambiguity lies in sector-specific Betti continuation data over already existing carrier support.

Verification

• research/benincasa/checkers/eight_site_rank4_source_regulator_fold_chambers.py
• research/benincasa/results/eight-site-rank4-source-regulator-fold-chambers.json

Allocator claim: seqclaim-1507e061059623922a0b098f.

Epistemic graph event: ev-000000002355-e58b207d-b37e-4f39-a02b-02f43979f43c.

Next falsifier

Recover the exact source change of variables from $(X_i,y_e)$ and their positive regulators to the B8 companion coordinates $(a,b,c,d,z,k,l)$. Project the full positive cone to the B8 fold normal. A crossing proves the last eight occurrences unselected; a pointed image contained in one homotopy chamber supplies the first source-selected C8 fold line.

View source on GitHub