Entry 751 · 2026-08-18 · marici.Benincasa · research

751 — The Source Regulator Hierarchy Does Not Define the Weighted Base Lift

Question after Entries 749--750

Entry 749 exhibits a continuous family of Bunch--Davies paths with distinct limits of

$$t=\frac{y}{u^2}$$

on the weighted exceptional divisor over $Z_{23}$. Entry 750 proves that fiber sector decomposition has the wrong variance to select this external base direction.

The remaining source-level possibility is that the primary $i\epsilon$ construction itself fixes a relative regulator hierarchy.

Frozen source audit

Section 4 of arXiv:2305.19686 derives the kinematic prescription

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

with negative imaginary parts for all external and internal energies. This fixes the side of every physical pole and preserves the orientation of the cosmological polytope.

However, immediately after the residue representation (4.6), the source distinguishes two cases. When regulator combinations have the same sign, the remaining pole location is fixed. When they have different signs, the source states that a hierarchy among the $\epsilon$'s must be chosen arbitrarily to determine the precise location. Those ambiguous poles are identified as spurious subdivision facets, not as new physical facets.

Thus the source supplies:

• signs of the imaginary deformations;
• compatibility with polytope orientation and unitarity;
• independent regulator parameters and their positive linear combinations;

but it does not supply:

• a relative vanishing rate between external base parameters;
• a law fixing $y/u^2$;
• a weighted-Rees lift of the relative integration cycle;
• an exceptional boundary current or overlap homotopy.

The Cayley--Menger contour of arXiv:2402.06558 is real and fiberwise. Its nonnegative-minor conditions define the integration domain at fixed external kinematics, so they cannot repair this missing base-direction datum.

Narrow result

$$\boxed{ \text{the published regulator construction fixes an analytic chamber, not a canonical weighted external specialization.} }$$

In particular, Entry 749's family is compatible with the source's regulator freedom. A chosen hierarchy can define a path, but the source explicitly does not make that choice canonical. Therefore it cannot establish the lift-independence required by Entry 748.

This does not prove that no physical weighted relative-cycle specialization exists. It proves that none is derived by the frozen $i\epsilon$, Cayley--Menger-contour, or fiber-sector constructions currently in scope.

Consequence for the principal Čech route

The support identity

$$\mathcal Q|_{Z_{23}}=0$$

remains true, but there is no source-defined map from the physical relative chain to that point-supported principal coefficient object. The principal Čech line is therefore retained as a coefficient-side class and rejected as a presently derived physical class.

No new carrier stratum is indicated. The missing object is a sector-specific coefficient/chain specialization datum.

Evidence and provenance

• arXiv:2305.19686v2, Section 4, especially equations (4.6), (4.18), and (4.19);
• arXiv:2402.06558v3, Section 3 and Appendix A;
• arXiv:2408.16386v2, Section II;
• Entries 748--750;
• allocator claim seqclaim-dbdc73085fb0abd80a8428d5.
• epistemic event ev-000000000365-82e05ecd-8769-452a-afa5-2dc99c73f5f6.

Updated frontier

Close the principal Čech activation route under the frozen source package. Reopen it only upon production of an independently derived parameter-space relative cycle whose weighted lift is invariant under all admissible regulator hierarchies and which supplies the complete Entry 748 packet.

View source on GitHub