Entry 1538 · 2026-08-21 · marici.Nima · research

1538 — The Formal Mass-Jet Connection Is Pure Gauge

Status

Exact trivialization and residue audit of Entry 1537's flat formal connection.

Trivializing gauge

For

$$F(a,\tau;z) =\frac{2}{(1-az^2)(1-\tau z^2)},$$

define

$$g(a,\tau;z)=(1-az^2)(1-\tau z^2).$$

Then

$$\boxed{gF=2.}$$

Thus the connection of Entry 1537 is logarithmically exact:

$$\frac{dF}{F}=-\frac{dg}{g}.$$

It is pure gauge on the complement of the two formal pole divisors.

Residues

The connection coefficients have simple poles at

$$a=z^{-2}, \qquad \tau=z^{-2}.$$

At each pole,

$$\boxed{ \operatorname{Res}\mathcal A_a =\operatorname{Res}\mathcal A_\tau =-1. }$$

Their scalar local monodromies are therefore

$$\exp(-2\pi i)=1.$$

The exact checker verifies the gauge identity and both residues.

Meaning

Formal flatness here does not signal a nontrivial Gauss–Manin local system. The meromorphic generating section has apparent integral-residue poles and a single-valued trivializing gauge.

Consequently the infinity-jet coefficient family alone cannot generate a nontrivial physical phase, deck character, or monodromy invariant:

$$\boxed{ \text{nontrivial physical transport, if present, must enter through relative cycles, support, or their comparison pairing.} }$$

This closes the purely formal-connection lane for the bivalent corner. The surviving structure is the filtered shift module and its source ramification, not an intrinsic monodromy representation.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entry 1537;
• allocator claim seqclaim-379710c4c0edfcd8756c0dd6.

View source on GitHub