Gauge Symmetry from Coherent Comparison

Gauge symmetry is usually introduced as a postulate: physical laws are assumed to be invariant under certain internal transformations.
While empirically successful, this leaves open a foundational question: why should such internal symmetries exist at all, and why do they take the specific form observed in nature?

In this work we derive gauge symmetry from a minimal, non-physical requirement.
We assume only that a system capable of persistence must be able to compare the outcome of a transformation against a reference.
Coherent comparison necessarily produces a quantitative deviation, represented in an internal complex vector space.
Comparison fixes magnitudes and relations but leaves internal orientation undetermined, introducing an unavoidable redundancy of description.

By analyzing the distinct ways in which internal orientation remains unfixed, we show that coherent comparison forces exactly three independent and non-redundant continuous symmetries:
a global phase freedom yielding $U(1)$,
an unbiased mixing of identity-like and difference-like components yielding $SU(2)$,
and a basis redundancy of a triadic deviation arena under collective neutrality yielding $SU(3)$.
These symmetries arise sequentially and exhaust the continuous internal freedom compatible with coherent comparison.

When comparison is required between logically independent acts, internal symmetry becomes local.
Preserving coherent comparison under local reorientation necessitates a connection that translates internal frames, and its curvature measures the obstruction to consistent global alignment.
In physical interpretations, these structures correspond to gauge fields and field strength.

No spacetime, dynamics, particles, or empirical input are assumed.
The resulting gauge group and connection structure coincide with those of the Standard Model, not by hypothesis, but as a structural consequence of coherent comparison.