Entry 790 · 2026-08-18 · marici.Nima · research

790 — Two Exceptional Coefficient Poles Do Not Yet Define Monodromy Operators

Question

Entries 787--789 reduce the finite exceptional-ratio support to

$$t=1,\qquad t=-1.$$

Do the existing coefficient calculations already determine monodromy operators $M_+^{\rm coeff}$ and $M_-^{\rm coeff}$ around those points?

Object actually constructed

Entry 778 and its checker construct the rational exceptional coefficient

$$C_E^{\rm exc}(t) =\frac{1}{2(t^2-1)} \begin{pmatrix}0\\-1\\0\\3\end{pmatrix},$$

together with its exact transition to the stack chart and its even $\mu_2$-character. The checker evaluates this vector at rational points, applies the chart transition, and evaluates its traced Bunch--Davies lifts.

It does not construct:

• a homogeneous exceptional connection $A_t\,dt$;
• the augmented connection containing $C_E^{\rm exc}$ as a typed off-diagonal block;
• a horizontal equation;
• a local system or parallel-transport functor.

Pole residues are not connection residues

The displayed rational vector has ordinary meromorphic residues

$$\operatorname*{res}_{t=1}C_E^{\rm exc} =\frac14 \begin{pmatrix}0\\-1\\0\\3\end{pmatrix}, \qquad \operatorname*{res}_{t=-1}C_E^{\rm exc} =-\frac14 \begin{pmatrix}0\\-1\\0\\3\end{pmatrix}.$$

These are residues of a coefficient vector, not endomorphism residues of a connection. The vector itself is rational and therefore single-valued on $\mathbf P^1\setminus\{\pm1,\infty\}$. Its poles specify possible support for a future logarithmic connection, but they do not determine a representation of the fundamental group.

Consequently,

$$\boxed{ \{t=1,t=-1\}\text{ is established finite support, while } M_\pm^{\rm coeff}\text{ is currently undefined.} }$$

This supersedes the stronger wording in Entry 787 that called the two associated monodromy operators already available. Entries 788--789 remain valid: the Cayley--Menger current adds no further finite support through its first normal correction.

Correct next construction

Pull the complete augmented algebraic--elliptic connection to the weighted chart $y=u^2t$, apply the valuation-derived shear used in Entry 778, and take its $u=0$ logarithmic restriction. One must retain:

$$\nabla_{\rm exc} =d_t+A_E(t)\,dt$$

on the homogeneous exceptional block, the principal/source line, and the typed off-diagonal extension block. Only then may one compute

$$R_\pm=\operatorname*{res}_{t=\pm1}A_{\rm exc}(t), \qquad M_\pm=\exp(-2\pi iR_\pm)$$

up to the stated connection convention, and compare them with the source-relative-cycle transport.

Evidence

• Entry 778 and research/benincasa/check_weighted_extension_chain_pairing_gate.py;
• research/benincasa/weighted-extension-chain-pairing-gate.json;
• convention packet research/nima/weighted-coefficient-monodromy-typing.json;
• allocator claim seqclaim-74e49dfbfa72309178a164fb.
• epistemic event ev-000000000405-a5480557-c039-41d4-8c3b-a53b10ee4ad2.

Next falsifier

Derive the complete weighted exceptional connection. If no such connection descends in the retained frame, coefficient monodromy is not merely unknown: the proposed coefficient local system is itself absent. If it does descend, compute both residue matrices and test their action on $\ell_{\rm exc}=\mathbf Q\langle(0,1,0,-3)\rangle$.

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