Arithmetic Berry Phase: A Conceptual Map of the abc Conjecture with Uniformity Barrier and Suggestive Holonomy Interpretation

This paper presents a complete conceptual reduction of the abc conjecture 
to a single uniformity barrier, together with a suggestive geometric 
interpretation via "arithmetic Berry phase."

The trace E = log(c/rad(abc)) is identified as an arithmetic holonomy—
the unavoidable residual arising from the attempted gluing of additive 
and multiplicative structures. We prove an elementary CRT-layer bound 
on the powerful part P(c), localize potential exceptions to a single 
arithmetic progression, and show that the uniformity property (UNIFORM) 
is the sole barrier beyond elementary methods.

A five-line closure argument is provided: the negation of UNIFORM 
produces infinitely many S-units in an arithmetic progression, which 
contradicts Baker–Matveev lower bounds for linear forms in logarithms.

Appendix B records the "arithmetic Berry phase" interpretation—viewing E 
as a holonomy of the non-commutative loop Add → Mult → Add—as a 
suggestive perspective for future research. No new proof is claimed there.

This paper serves as the author's final statement on the abc conjecture. 
The conceptual map is complete; further pursuit is left to others.

Keywords: abc conjecture, trace, arithmetic Berry phase, holonomy, 
uniformity barrier, S-units, arithmetic progression, linear forms 
in logarithms, Baker theory, CRT layer decomposition

Keywords: 
- abc conjecture
- trace
- arithmetic Berry phase
- holonomy
- uniformity barrier
- S-units
- arithmetic progression
- linear forms in logarithms
- Baker theory
- CRT layer decomposition

Related: https://zenodo.org/records/17989916 (Trace Theory v4)

Final statement:
"I'm 70 years old and I don't have time for just ABC.
 The answer is out. The proof is for others."
 --- H. Sasaki, December 2025