Local frames and the relational origin of gauge freedom

What a local frame describes

Let V_x be the carrier attached to a completed record x. A vector in V_x is an abstract state amplitude; a frame supplies its component column ψ_x. An orthonormal frame makes the Hermitian norm look like ψ_x†ψ_x. Choosing another orthonormal frame changes the components by a unitary matrix. Throughout these guides the component convention is ψ′_x=g_xψ_x. If u_x is the matrix of embedded basis vectors, its corresponding change is u′_x=u_xg_x⁻¹, so the represented vector u_xψ_x stays the same.

The allowed group depends on what the record preserves. A complex inner product by itself admits U(m) for a rank-m carrier. A splitting, named volume tensor, real structure, or fixed polarization can reduce that group. The statement “there are three components” does not by itself select SU(3), and a chosen complex phase does not fix the energy spectrum. The source manuscripts keep these inputs separate. Completed records, §§3–5, pp.5–10.

A frame choice is therefore a choice of coordinates for physical data. It does not identify all vectors, density matrices, or measurements. Distinct states remain distinguishable when the admitted observables distinguish them. This elementary distinction becomes essential when a completion channel forgets a microscopic label while retaining the full carrier algebra.

Four operational assumptions make the problem precise

The record interpretation is organized by four assumptions. Each contributes a different step, and the fourth is unnecessary for the finite graph statements.

These are the R1–R4 interfaces of the operational manuscript. They make a conditional theorem possible: if the chosen completion has this structure, its comparisons require gauge-covariant transport. Naming inner generation and outer completion does not establish the assumptions. A model must show how its actual preparations, records, and permitted experiments realize them. Completed records, §5.1, p.9.

Why separate endpoint structures do not choose a comparison

Suppose someone tries to construct a map C:V_x→V_y using only the separate endpoint metrics, phases, and structural tensors. A natural construction must respect every automorphism that leaves those inputs unchanged. Independent endpoint automorphisms would therefore require C=g_yCg_x⁻¹ for every allowed pair. Set g_x=I. If the defining representation has no invariant vector, the only possible C is zero.

The determinant-linked 3+2 carrier gives an especially short proof. On its triplet block, choose g_y=(ζI₃,I₂) with ζ=e^2πi/3. This preserves the product-volume tensor and implies C₃=ζC₃, hence C₃=0. On its doublet block, choose (I₃,−I₂); then C₂=−C₂, hence C₂=0. No faithful invertible comparator can be constructed this way. Theorem 5.1, pp.9–10.

The conclusion concerns the available input. A common ambient embedding can supply overlaps; a calibrated instrument can supply a transport map; a prepared joint state can supply a relational reference. Those are legitimate additions. Once supplied, they belong in the record specification and must transform together with its local presentations.

Deriving the endpoint transformation law

Supply a norm-preserving complex-linear map U_yx:V_x→V_y. The polarization identity turns norm preservation into preservation of the inner product, so an invertible such map is unitary. To describe the same comparison in new components, its action must satisfy U′_yxψ′_x=g_yU_yxψ_x for every input. Substituting ψ′_x=g_xψ_x gives

The two endpoint matrices have different jobs: the right factor removes the old departure coordinates and the left factor installs the new arrival coordinates. When links compose, all intermediate changes cancel. Preservation of extra tensors restricts the link to their stabilizer group. The transformation identity itself neither specifies those tensors nor chooses a Hamiltonian. Theorem 5.2, p.10.

For a one-component carrier write U_yx=e^iθyx and g_x=e^iχx. Then θ′_yx=θ_yx+χ_y−χ_x, modulo 2π. Even this simplest example needs a relational phase: the separate endpoint component phases do not define an invariant difference by themselves.

Combine links and states to obtain invariant comparisons

A scalar comparison is obtained by transporting the departure state into the arrival carrier before taking an inner product:

Its transformed expression contains g_y†g_y and g_x⁻¹g_x; both cancel. A supplied hopping energy can therefore use E_hop=−t(c_yx+c_yx*), with real t. If ψ_v=r_ve^iαv, the Abelian form is −2tr_xr_ycos(θ_yx+α_x−α_y). The invariant phase combination contains the link and both endpoint states.

This verifies covariance of the chosen energy; it does not derive that energy or its coefficient. There can be many invariant functions of the same data. In particular, the existence of local frame freedom is logically earlier than a force law or an electric-field kinetic term. The singlet-loop construction supplies one explicit Hamiltonian whose loop response can be calculated.

Passive changes transform states, observables, and generators together. An active rotation of a state while holding the detector fixed may change measured probabilities. Treating these operations as interchangeable would erase physical information that the record channel was supposed to preserve. Remark 4.5 and §5.2, pp.7–10.

Local forgetting does not remove an existing shared reference

The standard-form completion channel separates a left selector from a right carrier. It can forget which left line was marked while fixing every right observable. This does not imply that it erases a prepared relation between right carriers at different records. A unital completely positive joint channel that fixes both full local carrier algebras pointwise also fixes their generated joint algebra. The mathematical reason is that each fixed algebra lies in the channel’s multiplicative domain, so products of observables from the two endpoints remain fixed.

For example, a prepared maximally entangled pairing |Ω〉=m^−1/2Σ_a|a〉_x⊗|a〉_y retains its correlations under independent selector-forgetting channels on separate left factors. Its chosen pairing is additional joint state data. Changing local coordinates changes the component expression for that state; it does not destroy the state. Proposition 4.7, p.8.

The operational no-comparator theorem therefore needs a precise preparation statement. It excludes a comparator from separate structural data alone. It cannot be strengthened to say that a common source never prepares shared references. The completion-channel guide gives the exact preservation argument and its consequence for source models.

A complicated collection of links can still be flat

Take arbitrary vertex matrices h_x and set U_yx=h_yh_x⁻¹. Along a path x→y→z, the product is h_zh_x⁻¹. Around every closed loop it is I. The frame change g_x=h_x⁻¹ sets every link to identity. Matrix variation by itself therefore does not establish a nontrivial comparison field.

More generally, on a finite connected graph one may choose a root and set all spanning-tree links to identity. For a tree edge from parent x to child y, use g_y=g_xU_yx⁻¹. The remaining |E|−|V|+1 edges close fundamental cycles. Their based holonomies transform together by conjugation at the root. This counting concerns internal graph comparison; fixed boundary frames or external charged probes supply further data and must be included in the experiment. Proposition 5.3, p.11.

The source manuscript exhibits a flat realization with the same seven-state source, doubled defect, channel, and retained tensor at every record. It satisfies those local constructions with identity transport and no autonomous field. A stronger dynamics claim must explain what extra mechanism excludes this allowed realization. Proposition 9.2, p.18.

The smooth limit and the remaining source-level task

Assume records admit a smooth chart and short comparisons have U_x+dx,x=I−A_μ(x)dx^μ+O(|dx|²). Expanding the endpoint law gives the connection convention used throughout this section:

The derivative term compensates the variation of the coordinate change itself. Curvature is F=dA+A∧A and transforms homogeneously, F′=gFg⁻¹. Smoothness and local path correspondence are additional assumptions; a finite internal source does not by itself construct the chart, its dimension, or causal structure. Corollary 2.4 and Theorem 5.2, pp.5 and 10.

Within the SMM/DTT interpretation, six nested inner generative levels and a seventh outer completion describe dependent generative and observable roles. They should not be identified with independent time axes, gauge generators, or spacetime components. A concrete source law must determine what each level contributes to the final carrier and comparison record. The next mathematical steps are path transport, closed-loop comparison, and selection of the retained frame group.

Continue reading

• Parallel transport between completed records
• Completion channels and retained references
• Reconstructing the gauge group from retained structure