Completed records, selector forgetting, and retained references

Completion is defined by what future tests can predict

A completed record is a selected output that is sufficient for a declared family of future experiments. It need not store every microscopic variable. It must retain every distinction that an admitted future experiment can detect. This makes completion an operational statement, with concrete preparation and measurement assumptions, rather than a synonym for simply ending a calculation.

In the SMM/DTT interpretation, inner generation and outer observable completion have different roles. Six nested inner generative levels can contribute dependent operations to the seventh outer completion. They are not independent Lorentzian time coordinates, and a finite completion channel does not derive a spacetime metric from their count. The mathematical results below concern the explicitly supplied maps and algebras. §§1–2, pp.1–4.

The first design question is where the information to be forgotten lives. If a microscopic selector projector belongs to an observable algebra that completion fixes pointwise, it cannot be erased. A successful construction must specify distinct roles for the selector and the retained carrier, or relax the claim that the full carrier algebra is preserved.

Separate the selector from the carrier in standard form

For a complex rank-m space V, the Hilbert space of m×m matrices with its Hilbert–Schmidt inner product can be written L²(M_m,Tr)≅V_L⊗V̄_R. Left and right matrix actions commute. In the chosen record model, the left factor carries a microscopic selector and the right factor carries retained data. This is an explicit representation choice; it is not forced by the existence of a complex carrier.

Define the conditional expectation

For a right observable, E_m(I_L⊗b)=I_L⊗b. For any rank-one left selector p_a, E_m(p_a⊗I_R)=(I_L/m)⊗I_R. All designated left lines have the same record image, while right matrix units remain distinguishable. The map is unital, trace preserving, completely positive, and idempotent; as an operator on the Hilbert–Schmidt space, it is an orthogonal projection. Definition 4.2 and Theorem 4.3, p.7.

For the stipulated 3+2 construction, additional block dephasing removes off-diagonal intertwiners and leaves M₃⊕M₂, of complex dimension 9+4=13. This retained algebra still contains rank-one projectors and nontrivial states. Forgetting the designated left in/out label does not erase all right polarizations as possible observables.

A mixing semigroup approaches exact completion

For γ_m>0, define Φ_t^(m)=E_m+e^−γmt(id−E_m). Since E_m²=E_m, composition gives Φ_tΦ_s=Φ_t+s. Each positive-time map has precisely the right algebra as its fixed space. At t=0, Φ₀=id and every observable is fixed.

The norm is induced by the Hilbert–Schmidt norm. The equality gives an exact attenuation law for every unwanted-selector direction. A one-percent operator error requires γ_mt≥log100≈4.605. This numerical threshold is a consequence of the chosen semigroup, not a source-derived physical clock duration. Theorem 4.3, p.7.

Finite-time attenuation is not exact deletion. Φ_t is invertible as a linear superoperator for finite t, but its inverse multiplies the erased directions by e^γmt and is not a positive quantum channel. Exact completion is the limit t→∞ or the direct operational use of E_m. The model must say which interpretation it intends when a discrete completion event is proposed.

Why closed reversible steps cannot perform this erasure

Choose two orthogonal left rank-one projectors and let D=(p_a−p_b)⊗I_R. This is a nonzero selector distinction with E_m(D)=0. Normalize D to Hilbert–Schmidt norm one. Any reversible same-algebra unitary conjugation α preserves that norm, so

The same bound applies to any finite composition of reversible conjugations, since the composition is still reversible. Repeating a closed unitary clock on the same algebra therefore cannot implement exact selector-forgetting completion. Theorem 4.6, pp.7–8.

A larger reversible dilation remains possible. A swap can transfer selector information to an environment, and discarding that environment changes the accessible experiment family. The irreversibility is operationally located in that loss of access or in a stated projection. The theorem neither rules out unitary microscopic dynamics on a larger space nor identifies a unique reservoir, clock, or temporal ontology. A proposed source mechanism must account for its environment preparation and replenishment.

Preserving full local algebras preserves their joint reference

Let B_x=B(V_x)⊗I and B_y=I⊗B(V_y) be the endpoint carrier algebras. Suppose a unital completely positive joint channel Ψ fixes each pointwise. For b in either algebra, Ψ(b†b)=b†b=Ψ(b)†Ψ(b), and the analogous equality holds for bb†. These equalities put b in the multiplicative domain of Ψ. Hence Ψ(b_xb_y)=Ψ(b_x)Ψ(b_y)=b_xb_y. Such products span the full joint algebra. Proposition 4.7, p.8.

Independent left-selector channels therefore preserve a prepared right-carrier Bell pairing and all of its right–right correlation observables. A source-prepared common reference does not disappear merely because each local selector is mixed. It must be retained as joint data, excluded by an explicit preparation restriction, or removed by a different channel that ceases to fix some right observables.

The hypothesis is stronger than preserving the reduced density matrices of one chosen state. A channel can change a particular state’s correlations while leaving those particular marginals unchanged. That does not mean it fixes the full local observable algebras pointwise on the joint system. The theorem specifies this stronger operator-level preservation, which is exactly the claim made by the standard-form record construction.

Retain a product volume without dephasing physical carriers

A separate four-state label register has basis |00〉,|10〉,|01〉,|11〉 for vacuum, the two block-volume labels, and their product. Let Q_R=diag(0,−6,6,0) and L_φ(X)=−γ_φ[Q_R,[Q_R,X]]/2. Its exact matrix-unit action is

The separate-volume coherences |10〉〈00| and |01〉〈00| decay as e^−18γφt. The product coherence |11〉〈00| is fixed because both labels have charge zero. The fixed register algebra is M₂⊕ℂ⊕ℂ, of dimension six; it is not silently absent from the full instrument. Including it with a single 3+2 carrier algebra gives dimension 13×6=78. Theorem 6.1, pp.11–12.

The register is declared auxiliary tensor-label data. Applying the same phase averaging directly to physical carrier states would generally remove charged coherences and could contradict preservation of their full matrix algebra. The labeling rule, its correlated noise, and its interpretation are supplied parts of the model. The same product-retention channel can be built for other carrier ranks; it does not select 3+2 by itself.

Preserving a determinant reference does not prepare it

Let D=detV₃⊗detV₂. A standalone vacuum-to-product coherence between ℂ|0〉 and D is charged under the full U(3)×U(2) group. Covariant processing of an invariant initial register and invariant ancillas must have invariant output, so it cannot create that coherence.

An explicit preparation uses a dual reservoir S=ℂ|0_S〉⊕D*. The canonical tensor in D⊗D* is invariant. A covariant unitary swaps it with the joint vacuum and leaves the two other occupation sectors fixed. If the reservoir is prepared in (|0_S〉+|τ*〉)/√2, the unitary produces (|0〉+|τ〉)/√2 in the register and |τ*〉 in the reservoir. Coherence is transferred from a supplied reference. Proposition 6.7, p.14.

This bound follows from covariance and trace-norm contractivity, with T_in the full-group Haar twirl. The balanced pure output has coherence 1/2 and the explicit reservoir transfer attains the bound. The cost is phase asymmetry, not a thermodynamic energy estimate. The relative label-noise subgroup preserves the prepared product, while the larger block group diagnoses its resource cost. Local preparation still does not establish global patching or parallel transport of τ. Proposition 6.8, pp.14–15.

An explicit joint completion instrument retains calculable memory

For a supplied set of n records, let E_rec be the product of the selector and block-dephasing expectations. Its range is B_R=⊗_v(M₃⊕M₂) with dimension 13ⁿ. On each volume register, exact charge pinching D_φ can be implemented by averaging three phase conjugations. For a separate finite probe with projector P and Hamiltonian H, set D_P(X)=PXP+(I−P)X(I−P), U_τ=e^−iτH/ℏ, and

The unitary acts only on the probe. Since E is a Hilbert–Schmidt orthogonal projection, a fixed X must obey E(X)=X and [X,U_τ]=0: equality of norms forces the first condition, then the fixed-point equation gives the second. Therefore FixΦ=B_R⊗(M₂⊕ℂ⊕ℂ)^⊗n⊗{P,U_τ}′. Under 0<τ diamSpecH<2πℏ, the last factor is {P,H}′. Theorem 8.1, pp.16–17.

A fixed-algebra calculation is not a convergence theorem. Probe sectors can retain coherences rotating under unitary conjugation. Sequential probabilities use the actual Kraus products M_rₙ⋯M_r₁; resetting the probe after each result is a different protocol with another preparation resource. Coupling H directly to protected right carriers also changes the theorem’s assumptions.

Reduced memory, curvature, and physical completion remain distinct

Amplitude compression gives another exact way to see the limits of retained records. Let C:H→H_rec satisfy CC†=I, set P=C†C, and define T(t)=Ce^−itH/ℏC†. Then

The compressed family is a unitary group for all real t exactly when [P,H]=0. Otherwise amplitude leaves the retained subspace. Eliminating hidden components produces a memory kernel involving the hidden evolution and the two coupling blocks. This mechanism can occur without any spatial connection or curvature. A compressed amplitude map is also not automatically a trace-preserving quantum channel. Appendix B, pp.25–26.

The construction has therefore identified precise things a source theory must supply: a test family, a carrier/selector separation, a completion channel or dilation, reference preparation, and a policy for retained auxiliary information. Coherent Gauss sewing, spatial adjacency, electric dynamics, and matter selection remain additional interfaces. Deriving them from one rule is a research target; exact finite channel identities do not establish their physical realization.

Continue reading

• Which information may completion erase?
• Frames and shared references
• The determinant-linked stabilizer
• Coherent sewing and observables
• Completion channels: exact erasure, preserved algebras and residual memory
• Shared-reference protocols: what label erasure leaves observable
• Completion, marginals and a shared Bell reference