A connection or a sequence of unitary links determines path transport. Fixed-frame variation is pure gauge, while a changing embedded carrier can support genuine projected curvature under additional assumptions.
Transport is more than moving an object
Two local records can carry states with the same number of components but no preferred identification between their bases. Parallel transport supplies that identification along a selected path. It can describe internal orientation or phase comparison even when no material particle is literally moved from one record to the next.
In SMM/DTT this provides a candidate bridge between generation and spatial relations. The source would produce both the completed carriers and the operations used to compare them. A common source does not automatically make every comparison path independent. Whether different routes agree is a calculable property of the supplied transport law.
Compose links in their actual order
For successive records v₀,v₁,…,vn, assign unitary links Uvk+1vk. The path map is their ordered product, with the earliest action on the right. Under ψ′v=hvψv, intermediate frame changes cancel:
U′γ=hvₙUγhv₀−1.
Only the endpoint frames remain. Reversing the path gives Uㆠwhen each link is reversed by its adjoint. This is the discrete version of a norm-preserving connection. Its derivation does not assume a metric length, a travel time, or a propagation speed for the comparison operation.
The differential transport equation
Along a smooth path x(s), write As=Aμ(x(s)) dxμ/ds. A parallel component satisfies dψ/ds=−Asψ. When As†=−As, differentiation gives d(ψ†ψ)/ds=0, so the comparison preserves the Hermitian norm.
If all matrices along the path commute, ordinary exponentiation suffices. Otherwise path ordering retains the order of the small operations. The connection transforms as A′=hAh−1−(dh)h−1 in the stated component convention, reproducing the endpoint law above.
An Abelian phase comparison
For a one-component carrier choose A=−ia, with a a real one-form. Transport gives Uγ=exp(i∫γa). Under h=exp(iχ), the real potential changes as a′=a+dχ. The path integral then changes by χ at the arrival point minus χ at the departure point, exactly compensating the endpoint components.
For a straight interval of coordinate length L and constant a=k dx, the transport phase is kL. The combination is dimensionless; k has inverse-length units when x is a length. A different physical implementation might parameterize the comparison by a dimensionless record index instead. The conversion to a measured length must be established before assigning k a physical interpretation.
A small noncommuting example
Consider a two-component carrier and two short operations U=exp(iεσx) and V=exp(iεσy). Since [σx,σy]=2iσz, reversing their order produces a difference at second order in ε. Their commutator loop is:
The first-order contributions cancel, leaving the ordered mismatch. For sufficiently small ε this makes the origin of curvature-like behavior transparent. The matrices and operations are supplied in this example. A source model would need to show why these operations are admissible and why their loop result reaches a completed detector rather than remaining in an inaccessible sector.
Transport inside a changing protected carrier
Suppose a smoothly varying orthogonal projector P(x) selects an isolated carrier subspace from a larger Hilbert space. For a section s satisfying Ps=s, projected differentiation defines ∇s=P ds. Its curvature on the carrier is P(dP∧dP)P. A changing embedded subspace can therefore produce geometric transport even though the surrounding Hilbert space has a fixed basis.
By contrast, if P is constant and only a basis inside its range changes, the curvature vanishes. Actual slow evolution also acquires dynamical phases from energy splittings. Extracting the geometric comparison may require a degenerate band, a controlled echo, or subtraction of those phases. The existence of a protected band alone does not specify which comparison a laboratory protocol measures.
Connect transport to generated records
The constructive DTT question is which nested operations create the carrier and which certify its comparison. The outer completion should retain a result of the actual ordered process. An independently chosen connection appended to a source diagram is mathematically usable, but it leaves the source-selection question unanswered.
A strong finite example would specify the full autonomous update, the carrier projector or links, the preparation, and a final interference readout. It would then calculate path dependence and an error bound for the approximation used. The holonomy guide closes the path and distinguishes a nontrivial structural loop from a loop effect that an experiment can actually access.
Continue with the articles
- Relational transport and gauge freedom
Explain why independently unframed record fibers require a rule for comparison. Introduce gauge transport as a conditional reconstruction idea, distinct from the draft’s exact Hamiltonian results.
- Invisible holonomy in inaccessible sectors
Explain the draft’s classification of residual parallel monodromy invisible to the prepared interface. Nontrivial structure can survive outside the behavior an experiment can access.
- How a protocol can reveal dual-time structure
Outline ways to move beyond the finite prepared linear interface: protocol-dependent access, noncommuting transport, or richer memory. Each route must be defined as an observable procedure.