Gauge symmetry is often celebrated as one of modern physics’ deepest insights. Entire theoretical edifices—electroweak unification, quantum chromodynamics, the Standard Model itself—derive their authority from the claim that certain mathematical “redundancies” correspond to real symmetries of nature.
But the triumph has always carried a quiet absurdity: the formalism includes more structure than the phenomenon presents, and physics treats this surplus as a feature of the universe rather than a feature of the modelling choice.
After the over-openness of divergence (Post 3), we now encounter the inverted pathology: under-openness, where the construal lacks sufficient differentiation and then misinterprets its own indeterminacy as a physical redundancy. Gauge freedom, in this light, is the illusion that arises when a model fails to specify orientation and then mistakes that absence for a deep symmetry.
The miracle evaporates once we expose the relational mechanics at work.
1. The classic story: redundancy elevated into ontology
A gauge theory begins by presenting a field with apparently more degrees of freedom than physical measurements can detect. Electromagnetism’s vector potential can be transformed without altering the electric and magnetic fields; non-Abelian theories add layers of group structure that appear to generate arbitrarily many equivalent representations of the “same” physical configuration.
Physics tells the story like this:
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The equations contain redundant variables.
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The redundancy expresses a genuine symmetry of nature.
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Therefore: nature must operate in a gauge-invariant way.
But this conflates two things that should never be conflated:
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the under-specification of the mathematical construal, and
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the structure of the phenomenon.
2. Under-openness: when the construal fails to differentiate
Where Post 3 dealt with inclination that over-extends (too many degrees of freedom), gauge freedom emerges when the model under-determines its own perspectives. It does not complete the cut; it leaves a zone of ambiguity, an area it refuses to articulate.
This is under-openness: a failure of inclination to make sufficient distinction.
In simpler relational terms:
A gauge symmetry is not an extra feature of reality.It is a space left unspecified by the construal.
3. The mirage of surplus structure
Once the under-specification is baked into the mathematics, two illusions emerge:
Illusion 1: a vast “redundancy space” exists
Physics treats the family of gauge-equivalent descriptions as if nature somehow “contains” them all. But the multiplicity resides only in the formalism. The phenomenon does not multiply itself to accommodate alternative potentials or sections of fibre bundles.
The multiplicity is an artefact of under-differentiation inside the cut, not a shadow cast by the phenomenon.
Illusion 2: gauge symmetry reveals deep forces or fields
Often the narrative flips: instead of redundancy, gauge symmetry becomes a source of physical law. “Local gauge invariance requires the existence of interactions,” the textbooks say.
But again, this treats a constraint imposed by the mathematics as an ontological decree. The under-openness of the construal is then retroactively declared to be a truth about nature.
In both illusions, mathematics overreads its own ambiguity as metaphysics.
4. The relational correction: symmetry as a property of the cut, not the cosmos
The relational viewpoint clarifies the situation by re-locating the symmetry:
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It is not in the phenomenon.
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It is not in the world.
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It is in the orientation of the model itself.
Gauge symmetry is the result of a cut that preserves a region of indifference—a set of distinctions the model chooses not to make. To then interpret that indifference as a property of nature is to misread the behaviour of a construal as the behaviour of the universe.
The symmetry does not reflect ontology; it reflects the projective geometry of the construal.
5. Why this matters: the cost of mistaking under-specification for revelation
Because gauge symmetry is treated as ontologically deep, entire research programmes are built around it: elaborate unifications, symmetry-breakings, expanded groups, compactifications, and so on.
But if gauge freedom is simply the shadow of under-openness:
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the metaphysical stakes collapse,
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the “redundancy” is exposed as methodological rather than cosmic,
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and the sense of mystery shrinks to a habitable proportion.
Physics gains not less but more conceptual clarity by recognising the mirage for what it is.
6. Toward a more disciplined use of mathematics
Just as renormalisation becomes legible once we understand over-openness and counter-inclination, gauge freedom becomes legible when we recognise under-openness as a choice, not a revelation. The symmetry is the model’s own artefact, generated by its selective orientation.
The phenomenon does not wear the construal’s confusions.
Gauge freedom, in the relational frame, is best understood as:
the formal trace of a construal that has left part of its orientation uncut.
Next in the series
Post 5 will step into the quantum arena, where a different kind of confusion dominates: the temptation to read the wavefunction as an ontic state rather than a modelling disposition. If gauge freedom arises from under-openness, wavefunction metaphysics arises from over-closure—a premature fixation of the cut.
The pattern persists; only the pathology shifts.
We cut again soon.
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