Local frames and the relational origin of gauge freedom
Local gauge frames describe one physical configuration through different conventions, with transport connecting comparisons between neighbouring frames.
Read the guideComparison, transport, interaction
How do generated records support interactions?
Comparing internal states at different records requires a transport rule. The gauge programme studies how local frame freedom, retained tensors, and completion conditions organize such comparisons, then asks which interaction laws a common source could select.
From local frames to physical interactions
Start by distinguishing an observable from the frame used to describe it. Follow transport along a link, transport around a loop and the integer winding of a phase. The later guides explain gauge-invariant observables, singlet sectors, effective interactions and the additional structures required to select a gauge group.
SMM/DTT motivates asking how relations between local completions could produce transport data. The examples show exactly what follows once a link law or carrier space is supplied. They also identify the action, representation content and coupling normalization needed to turn those relations into a physical interaction.
8 connected guides
Parallel transport between completed records
Derive unitary path transport, work through Abelian and noncommuting links, and connect projected carrier geometry to SMM/DTT records.
Read the guideHolonomy: what a closed path remembers
Calculate gauge-invariant loop data, distinguish local curvature from global monodromy, and test whether a temporal loop is observable.
Read the guideWinding numbers, phase bundles, and charge normalization
Derive integer phase winding and a sphere flux example, then separate topological charge from a physical gauge coupling.
Read the guideGauge-invariant observables and neutral completion
Construct invariant record comparisons and singlet tests, distinguish invariant density matrices from neutral amplitudes, and identify observable limits.
Read the guideA singlet-bound loop with an exact energy response
Solve the finite singlet-loop secular equation, derive its protected gap, and distinguish static energy from neutral-return probability.
Read the guideHow eliminating auxiliary variables creates interactions
Derive an effective Hamiltonian and a mediator energy, track the loop coefficient, and state when the resulting approximation predicts dynamics.
Read the guideReconstructing a gauge group from retained structures
Derive S(U(3)×U(2)) from explicit carrier blocks and a determinant tensor, calculate its finite quotient, and identify the remaining physical work.
Read the guideThe detailed articles separate conditional carrier reconstruction from exact calculations in supplied Hamiltonians. Relational transport explains how descriptions change with local frames. Singlet-loop models make static response and controlled dynamics calculable. These are useful constructions, while the physical selection of carriers, couplings, and gauge-field dynamics remains an explicit part of the programme.
Gap claims receive the same careful treatment. A positive finite-volume gap, a locally stable minimum, and a uniform continuum mass gap are different results. Following the chain of assumptions reveals which bounds survive changes of volume or regulator and which additional proof would connect a controlled model with a full gauge theory.
Relational gauge freedom
See how maps between retained carriers transform under local frame changes, and what further comparison structures are needed before assigning a physical gauge interaction.
Holonomy and singlet response
Explore finite loop models in which transport phases, neutral sewing, and binding produce exact response formulas with clearly stated dynamical inputs.
Beyond a finite gap
Distinguish threshold calculations, local stability, and finite-volume spectra from the uniform bounds and controlled limits required for a complete mass-gap claim.
Continue with the articles
All 220 articles →Relational transport and gauge freedom
Unframed record carriers need relational transport, whose transformation law gives a conditional gauge-connection structure. Curvature, physical propagation, and force dynamics require additional mechanisms.
Read article →An exact singlet-loop response
A mobile carrier, stationary dual, and supplied singlet-binding Hamiltonian reduce to one exact resolvent equation. Its unique physical root determines the holonomy-dependent ground response and a conditional completion-cost candidate.
Read article →A protected gap in the finite singlet model
The finite singlet-loop Hamiltonian has a unique ground state and a uniform lower bound on its excitation gap. The derivation identifies the supplied penalty, carrier rank and limits that separate this result from a continuum gauge-theory mass gap.
Read article →What the holonomy Hessian does and does not contain
The singlet-loop response near identity holonomy is proportional to tr(X²), with no independent [tr(X)]² term at quadratic order. Explicit carrier weights then give a conditional relation among response coefficients, with physical coupling normalization still to be supplied.
Read article →Controlled effective gauge Hamiltonians
A controlled effective gauge Hamiltonian needs an operator remainder, a prepared subspace and a useful evolution-time bound. The finite-cell construction shows how folded terms, representation cutoffs and state preparation affect that control.
Read article →The conditional route to S(U(3) × U(2))
Retaining rank-three and rank-two Hermitian carriers together with a product determinant tensor selects S(U(3) × U(2)). The stabilizer calculation leaves carrier selection, chiral matter and empirical Standard-Model matching as separate requirements.
Read article →A charged threshold is not a full mass gap
A positive charged cycle threshold leaves neutral centralizer modes in the spectrum. The exact finite-operator calculation and projected inequality identify why compact recurrence alone does not establish a full physical mass gap.
Read article →The remaining gap-proof obligation
Finite-volume gaps, curvature masses and screening estimates do not yet establish a volume-uniform quantum gap. This audit identifies the connected-correlation estimate and the additional gauge and continuum matching steps left open in the cited manuscript.
Read article →Keep following the question.
The glossary explains the vocabulary; the reading paths connect the ideas; the source register points to the original arguments.