From Gauge Invariance to Categorical Coherence: A Coherence-First Framework for the Strong-CP Problem, the Higgs Mechanism, Emergent Symmetry, and Condensed-Matter Phenomena

This work proposes a “coherence-first” reformulation of gauge theory in which the fundamental organizing principle is not gauge invariance itself, but the requirement that many local descriptions can be glued into a globally consistent physical account. In the standard symmetry-first paradigm, one begins by postulating a symmetry group and then introduces gauge fields as compensators that preserve local invariance. Here the logic is inverted: the primary requirement is global consistency of local descriptions, and what is traditionally called “gauge symmetry” is reinterpreted as the first visible shadow of a deeper hierarchical coherence structure.

The core conceptual shift is that physical consistency is governed by a ladder of compatibility conditions. At the lowest rung, ordinary gauge connections enforce comparability of local frames; at the next rung, higher-form fields enforce comparability of those comparisons; and so on, producing a hierarchy that naturally belongs to categorical and higher-categorical geometry. In this view, curvature is not merely a field strength but the measurable residue of “non-glueability”: the obstruction that cannot be removed by reparametrizing local descriptions. The paper formalizes this hierarchy in a concrete field-theoretic framework and develops a variational principle that drives systems toward maximal coherence across multiple levels.

A central technical element is the introduction of a structure field that dynamically selects the local “symmetry type” (more precisely, the local grammar of gluing). Instead of assuming a fixed gauge structure everywhere, the theory allows the type of redundancy to vary over spacetime, enabling a unified description of symmetry emergence, symmetry-type crossovers, and intermediate “proto-gauge” regimes in which familiar gauge notions are only approximately realized. These proto-gauge regimes are treated as physically meaningful, controlled departures from strict first-level gauge invariance that remain consistent because higher coherence conditions continue to hold.

Within this coherence-first setting, the manuscript develops several connected contributions:

1) A variational principle of maximal coherence

The paper formulates an action principle in which the fundamental “energy” is a measure of coherence defect across multiple categorical levels. Ordinary Yang–Mills theory emerges as a limiting corner in which only the first level of coherence is active and the higher coherence structure is frozen or trivial. More general stationary points correspond to higher-gauge dynamics in which multiple compensator fields coexist and exchange “charge” between levels through generalized Bianchi/Ward-type identities. This establishes a single conceptual and technical framework that unifies standard gauge dynamics, higher-form extensions, and topological sectors.

2) Two coherence-based mechanisms for the strong-CP problem

The manuscript presents two routes by which the effective strong-CP angle becomes unobservable without relying on an imposed Peccei–Quinn global symmetry or requiring an ultralight axion.

• Cohomological trivialization route: in an extended coherence complex, the CP-odd density that controls strong-CP sensitivity becomes exact in the extended descent structure, so the would-be CP-violating dependence is reduced to a boundary artifact (or eliminated under appropriate conditions). The physical interpretation is that the strong-CP sensitivity is a residue of incomplete gluing at lower coherence level and can disappear once higher coherence is properly accounted for.

• Structural relaxion route: the structure field couples in such a way that minimizing the total coherence defect dynamically drives the effective strong-CP angle to zero. Importantly, the field responsible is not postulated as a conventional axion with an assumed shift symmetry; it is a structural field controlling the local coherence grammar. Its mass and couplings are set by coherence stiffness parameters rather than by an axion decay constant.

These mechanisms are framed to be compatible with standard anomaly physics and are accompanied by concrete diagnostic directions (for example, via lattice-style toy models that incorporate the extended coherence sector and test flattening of the strong-CP dependence as coherence penalties are strengthened).

3) Higgs mechanism reinterpreted as a coherence lift

A major reinterpretation offered here is that the Higgs phenomenon can be understood as a coherence transmutation rather than simply “spontaneous symmetry breaking.” In the coherence-first view, a condensate reorganizes where consistency lives in the hierarchy: strict first-level coherence is traded for controlled higher-level coherence. Vector-boson masses acquire a geometric meaning as the energetic cost of maintaining higher coherence; the Higgs mass becomes a stiffness parameter of the higher-coherence condition. This perspective is designed to preserve consistency constraints (including generalized Ward identities) and to clarify why gauge–Higgs systems remain coherent even when the usual “symmetry breaking” narrative is conceptually misleading (because gauge redundancy is not an observable symmetry).

4) Emergent symmetry and proto-gauge regimes

The manuscript formalizes emergent symmetry as the limit of near-perfect coherence: when coherence defects are small, approximate conservation laws and selection rules become quantitatively controlled by the size of the defect. This makes the common statement “symmetry emerges in the infrared” more operational: it becomes a statement about how quickly coherence defects decay under coarse-graining. The paper also introduces “proto-gauge phases,” transitional regimes in which familiar gauge closures fail slightly, yet the overall higher coherence remains intact. These regimes are argued to be common across both high-energy and condensed-matter settings and to admit measurable signatures.

5) Condensed-matter applications and predictions

A distinctive feature of the manuscript is that it does not treat higher coherence as purely philosophical. It exports the formalism to several condensed-matter contexts where “gluing problems” and emergent gauge structures are already physically central:

• Topological insulators / axion electrodynamics: the coherence-first approach reframes bulk axion-like terms as artifacts of extended exactness while emphasizing quantized interfacial responses. Spatial textures of the structure field act as programmable domain walls where quantized transport steps and protected channels can arise.

• U(1) to Z2 spin-liquid crossovers: the structure field controls the effective gauge type, producing a controlled crossover between gapless and gapped gauge regimes, with predictions for spectral reweighting and domain-wall bound modes.

• Fracton elasticity duality: the higher-form coherence language is connected to defect compatibility and constrained mobility. Tuning coherence stiffness provides a principled route to mobility crossovers and defect suppression patterns.

• Hydrodynamic electron flow: the manuscript proposes a new collective excitation (“naturon”) associated with coherence fluctuations between microscopic and effective transport descriptions, along with signatures in nonlocal transport patterns and tunable response inversions.

Across these examples, the emphasis is on testable signatures (spectral features, quantized steps, crossover scales, interface modes) that would be difficult to motivate from a fixed-symmetry viewpoint.

6) Discrete/numerical pathway

To bridge formalism and data, the paper provides a discrete exterior calculus and lattice-style formulation in which the relevant coherence constraints and generalized Bianchi identities hold exactly at finite discretization. This is intended as an algorithmic route toward simulation of symmetry-type dynamics, domain walls in the structure field, and quantitative tests of the proposed coherence-based mechanisms in controlled models.

Intended audience and positioning

This manuscript is written as a foundational and unifying research document intended for readers interested in:

• gauge theory and higher gauge theory,

• categorical and higher-categorical structures in physics,

• anomalies and topological response,

• emergent symmetry and effective field theory,

• strongly correlated matter, topological phases, and fracton physics,

• computational formulations of generalized gauge structures.

While the paper touches multiple domains, the unifying thread is operational: physical consistency is treated as coherent gluing of local descriptions, and the hierarchy of compensators and residual obstructions is used as the common language across high-energy and condensed-matter contexts.