Assumptions, results and research status
A guide to the conditional gauge research programme.
Conventions used throughout the section
U′yx = gyUyxgx−1; A′ = gAg−1 − (dg)g−1.
F = dA + A ∧ A; W = I − F × oriented area + higher terms.
A is anti-Hermitian and products act from right to left. For the matter ledger all fermions are written as left-handed fields and q = 6Y. For the singlet model Δ is the positive penalty and the explorer’s energy unit. Every use of “gap” should name the operator and the parameters over which its bound is uniform.
Frames, records and comparison
- Carrier
A vector space carrying the states or representations being compared. Its dimension, scalar field, metric and retained tensors are inputs to a stabilizer calculation; naming a carrier does not specify a field Hamiltonian. Detailed guide →
- Local frame
An ordered orthonormal basis for a carrier, chosen separately at each point or record. A change of frame changes components. It represents the same underlying comparison when link maps transform at both endpoints. Detailed guide →
- Gauge link
A transport map Uᵧₓ from Vₓ to Vᵧ. The target index comes first throughout this section. Its active-frame law is U′ᵧₓ = gᵧUᵧₓgₓ⁻¹. Path products act from right to left. Detailed guide →
- Connection
A rule for differentiating and comparing sections of neighboring carriers. With D = d + A and ψ′ = gψ, this section uses A′ = gAg⁻¹ − (dg)g⁻¹ and an anti-Hermitian A. Detailed guide →
- Curvature
The two-form F = dA + A ∧ A describing infinitesimal failure of transport to be path independent. Curvature transforms by conjugation. Its existence and transformation law do not specify its field equation. Detailed guide →
- Holonomy
The transport obtained by following a closed path. Changing the frame at its base point conjugates the holonomy matrix. Its trace and conjugacy class can be invariant even though its entries depend on the frame. Detailed guide →
- Wilson loop
A trace of holonomy in a specified representation, often normalized by its dimension. Different representations and normalizations give different observables. A real trace, an amplitude and the squared modulus of that amplitude must be distinguished. Detailed guide →
- Completed record
The output of a completion map applied to a generative history. Its sufficiency is assessed against a declared set of future tests; histories yielding different admitted future statistics cannot be merged by an exactly sufficient record. Detailed guide →
- Predictive sufficiency
The property that all probabilities or amplitudes required by the chosen future-test family can be determined from the record. It depends on what tests are admitted and on whether coherent comparisons with references are available. Detailed guide →
- Relational comparator
A faithful comparison between separately framed carriers. Independently given endpoint structures cannot naturally select a nonzero comparator invariant under all their independent frame changes; relational data or a reference must supply the comparison. Detailed guide →
- Reference resource
A shared state, orientation, phase or tensor that enables comparisons otherwise unavailable from invariant local data. Forgetting a local selector is distinct from erasing a separately prepared shared reference. Detailed guide →
- Completely positive map
A positive linear operation that remains positive when extended by an arbitrary ancillary matrix algebra. The joint-reference argument uses a unital completely positive map fixing full local carrier algebras, a stronger condition than preserving selected expectation values. Detailed guide →
- Selector completion
A channel that removes a declared selector factor while retaining the permitted carrier data. The explicit conditional-expectation semigroup approaches exact erasure asymptotically; reversible conjugation cannot implement the same erasure. Detailed guide →
Algebra, phase and retained structure
- Real commutant
The algebra of real linear operators commuting with a given real representation or dynamics. The commutant depends on the complete represented data. A subsequently selected complex structure reduces which commuting operations remain admissible. Detailed guide →
- Complex structure
A real linear map J satisfying J² = −I. With the appropriate metric compatibility it permits a real vector space to be treated as complex. Its selection and the survival of that selection under a cut require explicit hypotheses. Detailed guide →
- Polar phase selection
A construction using the polar part of a suitable real skew generator to define J on a nonzero-frequency sector. Zero modes and sign conventions must be handled separately; the resulting phase structure does not itself select all particle representations. Detailed guide →
- Sevenfold recurrence
The specified recurrence used in the source carrier construction, with six inner generative roles and a seventh completion. Its mathematical realization includes source, minimality, cut and scalar-extension rules. It is distinct from an arbitrary numerical cycle length N. Detailed guide →
- Marked cut
A specified removal or distinction of a recurrence position used in the carrier construction. A real cut need not commute with an already chosen complex structure, so a desired complex subcarrier cannot be assumed without checking compatibility. Detailed guide →
- Scalar extension
The passage from a real vector space to a complexified construction by a stated rule. In the doubled-cut argument it is an additional step, rather than an automatic consequence of selecting a raw real cut. Detailed guide →
- Stabilizer
The group preserving exactly the declared tensors or other structural data. Retaining additional tensors typically reduces the group. The result depends on which data are preserved separately and which are preserved only in combination. Detailed guide →
- Parallel product volume
A global nonzero tensor in the product determinant line, retained and parallel under the connection. This reduces U(3) × U(2) to the special product group. Vanishing trace curvature is a weaker local condition and need not eliminate global flat holonomy. Detailed guide →
- Global gauge group
The full group, including discrete quotient identifications, rather than only its Lie algebra. Representations of SU(3) × SU(2) × U(1) descend through the stated ℤ₆ quotient only when q + 2t₃ + 3t₂ is divisible by six. Detailed guide →
- Color triality
The integer modulo three describing the action of the center of SU(3) on a representation. A fundamental triplet has triality one and its conjugate has triality two. This enters the global quotient rule alongside weak parity and Abelian charge. Detailed guide →
- Winding
An integer topological count for a specified map or loop. It can constrain a representation character or phase under an explicit correspondence. It becomes a physical electric charge only after a normalization and observable coupling are supplied. Detailed guide →
- Singlet
A vector invariant under the specified simultaneous group action. For a carrier and its dual, the normalized invariant pairing defines a rank-one projector. A singlet in one tensor product should not be confused with every invariant density operator. Detailed guide →
Singlets and exact models
- Coherent Haar sewing
Averaging representation amplitudes with normalized Haar measure, P = ∫R(g)dg. For a unitary compact-group representation this is the orthogonal projector onto invariant vectors. The operation differs from conjugation averaging of a density matrix. Detailed guide →
- Density-matrix twirling
The channel ρ ↦ ∫R(g)ρR(g)†dg. It removes frame-dependent density-matrix components but can retain weight in nontrivial irreducible sectors. It is not generally the projector onto the singlet-vector subspace. Detailed guide →
- Secular root
The scalar root of a resolvent equation that determines an eigenvalue after a rank-one perturbation. In the singlet model the physical root lies above the largest free-cycle eigenvalue; a bracket selects it without ambiguous polynomial branches. Detailed guide →
- Holonomy Hessian
The quadratic response of the ground energy near identity holonomy. Here ΔE = ½κ̂ Tr X² + O(X⁴) for W = exp(iX) in the declared model. It is a response coefficient whose physical normalization must still be specified. Detailed guide →
- Effective Hamiltonian
An operator describing a chosen retained sector after eliminated degrees of freedom have been controlled. Its validity includes the expansion parameter, the retained projector, the operator norm or form used, and a remainder or dynamical error estimate. Detailed guide →
Spectral geometry and matter
- Spectral projector
An orthogonal projection onto a specified spectral set of a self-adjoint operator, or an isolated cluster defined by a resolvent contour under the required hypotheses. A smooth cutoff weight is generally not idempotent and cannot simply replace it in bundle identities. Detailed guide →
- Protected band
An isolated spectral cluster whose separating gap and regularity remain controlled over a declared parameter range. Fixed rank permits a vector-bundle description. Protection at one regulator does not establish uniform protection across a continuum limit. Detailed guide →
- Projected or Berry connection
For a section s = Ps, the connection ∇s = P ds. In a local orthonormal frame u it has A = u†du, and the invariant curvature is P(dP ∧ dP)P. The identity P(dP)P = 0 does not make this connection vanish. Detailed guide →
- Geometric phase
The phase accumulated by the specified parallel transport around a parameter loop, distinct from a chosen dynamical phase. In a finite projection sequence its interpretation includes the survival probability and the possibility of a zero overlap. Detailed guide →
- Spectral triple
An algebra, its represented Hilbert space and a Dirac-type operator, with the analytic and algebraic properties required by the chosen framework. A physically relevant triple must be constructed or explicitly assumed, not inferred just from a desired list of action terms. Detailed guide →
- Spectral action
An action constructed from spectral data, often through a trace of a function of a Dirac operator and a scale. Its asymptotic expansion depends on regularity, heat-kernel, cutoff and dimensional assumptions, plus finite-geometry and matter choices. Detailed guide →
- Gauge anomaly
A failure of a classical gauge symmetry to survive the quantum construction. The local ledger tests the displayed polynomial coefficients for fixed chiral representations. Global quotient descent and global anomalies require separate questions and hypotheses. Detailed guide →
- Chirality
The left- or right-handed structure of a spacetime fermion representation. Counting an internal module with familiar charges is distinct from constructing a chiral quantum field, its kinetic operator and its statistics. Detailed guide →
- Physical matching
The map from model variables and coefficients to a declared physical observable, unit system and renormalization convention. It identifies the scale and any corrections used in comparing a model coefficient with measured couplings or spectra. Detailed guide →
Lattice, vacuum and continuum
- Auxiliary cycle
A compact discrete direction added in the specified regulated model. Its length, spacing and twist affect a fiber Laplacian. It is not automatically a new observable spatial or temporal coordinate, and its spectrum is not automatically a physical mass spectrum. Detailed guide →
- Centralizer sector
A sector commuting with a chosen holonomy. In an adjoint auxiliary problem it can remain untwisted and possess zero auxiliary modes. This explains why a charged-mode estimate does not alone control every sector. Detailed guide →
- Transfer operator
A positive operator encoding a chosen time step, often T = exp(−aH) after the vacuum energy has been normalized. Passing from its contraction bounds to Hamiltonian lower bounds requires the correct invariant subspace and physical time scale. Detailed guide →
- Vacuum
A lowest-energy state of the declared physical Hamiltonian. Its projected component can lie inside a chosen low window and obstruct a positive excitation estimate there. A finite periodic-time thermal trace is a different state from the two-sided zero-temperature vacuum process. Detailed guide →
- Osterwalder–Schrader reconstruction
The route from Euclidean correlation data satisfying the required properties to a physical Hilbert space and time evolution. A regulator-level reflection or positivity argument must be accompanied by the other conditions needed in the intended limit. Detailed guide →
- Feshbach or Schur reduction
Elimination of a controlled high block to obtain an effective operator or quadratic form on a low block. For a two-by-two matrix F(z) = a − z − b²/(d − z); the positivity equivalence uses d > z. Detailed guide →
- Punctured low window
The low subspace after removing its projected vacuum direction, using P° = P − |PΩ⟩⟨PΩ|/||PΩ||² when PΩ is nonzero. Its rank and domain must be checked; in the smallest two-state example it is the zero space. Detailed guide →
- Coercivity
A quantitative positive lower bound for an operator or quadratic form on a specified subspace. The constant, norm, excluded vacuum and all regulator dependencies are part of the statement. A formal positive expression without uniform control may not survive a limit. Detailed guide →
- Uniform estimate
A bound whose constant remains controlled over the stated family of volumes, cutoffs, twists or other parameters. Each parameter must be named. A sequence of positive finite-regulator gaps can converge to zero. Detailed guide →
- Continuum identification
The construction and identification of the limiting physical theory, including observables, locality, covariance, positivity, nontriviality and the intended spectrum. Discrete link laws or convergent isolated coefficients establish only part of this task. Detailed guide →
- Physical mass gap
A strictly positive threshold above the vacuum in the physical theory’s energy-momentum spectrum under the required quantum-field assumptions. The four-dimensional Yang–Mills target is distinct from any one finite singlet, auxiliary-band or fixed-volume gap. Detailed guide →
Reference depolarization
A declared channel acting on a quantum reference carrier, with a retained coefficient s multiplying its nontrivial components. It acts on a different subsystem from erasure of an independent classical label.
Partial transpose
Transposition on one specified tensor factor of a bipartite matrix. In the Bell-mixture example its negative eigenvalue gives a directly checkable entanglement diagnostic.
Negativity
Half the excess trace norm of the partial transpose above one. For the specified Bell mixture it is max[0,(3s−1)/4]; its vanishing threshold is not the time of exact channel erasure.
Loop tomography
Reconstruction of specified holonomy invariants from a declared collection of readouts and reference settings. Its completeness depends on which amplitudes or probabilities the protocol can access.
Weyl alcove
A choice of fundamental domain for conjugacy classes after permutations and lattice identifications of torus phases. A two-angle SU(3) calculation must state how boundaries and identifications are handled.
Kato transport
For a differentiable orthogonal projector P, the anti-Hermitian generator K=[Ṗ,P] defines Ẇ=KW and intertwines the projector family. Its relation to a physical evolution needs further assumptions.
Compression defect
The difference between multiplying compressed operators and compressing their product. Intermediate transitions through the complementary subspace can make that difference nonzero.
Norm-resolvent convergence
Convergence of suitable resolvents in operator norm, with any comparison between varying Hilbert spaces explicitly specified. It is stronger than convergence tested on each fixed vector alone.
Sources for this guide
- Completed records, v21
- Temporal completion and exact loops, v20
- Sevenfold reconstruction, v18
- Vacuum obstruction and form Feshbach, v2
- Conditional interactions and unification, v1
Source manuscripts, version notes, and standard references are listed in the reference library.