# Source extract: FA-QD-ST-01.docx

Objectivity under Superposed Interaction Structures: A Survival Theorem for Quantum Darwinism

Common-pointer preservation and redundant relational records under hidden-controlled channel families

Scott Muniz

Dr. Ware Technology Services  ·  scott@drware.com

Submission manuscript · Revised 1

Abstract

Quantum Darwinism explains operational classical objectivity through the redundant proliferation of pointer information into a system’s environment. Existing formulations, including the two-body Hamiltonian classification of Doucet and Deffner, assume a fixed system–environment interaction structure. We ask what remains possible when that structure is indefinite: a classical mixture or a hidden-controlled quantum superposition of distinct coupling channels. We prove a two-part survival criterion. First, a common conserved rank-one pointer decomposition is necessary and sufficient for the defined channelwise exact nondemolition-preservation property. Second, redundant objectivity requires that distinct pointer values generate sufficiently informative records in many disjoint environmental fragments. Thus, commutation makes a record distinction survivable, while conditional-record distinguishability makes it broadcast. For an initially uncorrelated orthogonal control register and block-controlled evolution, tracing out the inaccessible register reduces the observer-accessible assignment to an incoherent channel mixture, and the criterion extends from finite mixtures to measurable, measure-weighted families. Within the Theory of Emergent Timeless Potentiality (ETP), quantum states and density operators are interpreted as mathematical encodings of relational potentialities rather than observer-independent physical waves or intrinsic conditions. On that interpretation, the theorem supplies algebraic and information-theoretic conditions for stable physical record formation without construing the partial trace or state reduction as objective wave-function collapse. This interpretive consequence is not part of the theorem’s proof and does not derive the Born rule or a unique first-person outcome. Exact computations illustrate a practical asymmetry: redundancy degrades continuously as records weaken, whereas pointer misalignment can rapidly destroy a fixed-tolerance plateau. We also derive a Lieb–Robinson envelope for finite superpositions of interaction graphs and state its continuum extension under uniform control of the sectorwise bounds. Together, the results identify separate algebraic and information-theoretic conditions under which the defined rank-one records and causal bounds survive a hidden-controlled interaction family.

Truth-First Basement and Record-Survival Interface

Authority and role. This successor is synchronized with Core Synthesis Draft-07, Basement Provenance and Instantiation Protocol Version 03, Local Relational Time Version 56, and the truth-first Local Relational Geometry architecture. This manuscript remains a bounded Layer 1 and Target III theorem on channelwise record preservation and operational redundancy over supplied interaction families. It does not select a basement, derive physical instantiation, or derive definite events from a merely formal structure.

• Relational potentiality remains a candidate basement, not an established premise or conclusion of this manuscript.

• Physical systems, controls, channels, Hamiltonians, fragments, pointer alternatives, instruments, access structures, outcome registers, probabilities, and records are supplied physical or model inputs whose provenance must be declared.

• Formal states, process matrices, commutators, entropies, mutual informations, Holevo quantities, causal witnesses, postselection identities, and numerical simulations do not establish physical instantiation.

• Objectivity, common-record preservation, redundancy, recoverability, accessibility, causal nonseparability, operational boundaries, topology, and geometry remain distinct claims.

• Target I concerns basement instantiation. Target II concerns definite-event generation. This paper addresses a bounded Target III reconstruction or certification question and does not establish Targets I or II.

• Every conclusion remains indexed to its candidate class, complete carrier, witness or instrument family, selected output, access structure, comparator, tolerance, and analytic or computational status.

Inherited Input Certificate and Claim Ceiling

1. Introduction

Operational objectivity in Quantum Darwinism means that multiple observers can independently recover the same classical information about a system from disjoint environmental fragments without directly perturbing the system. Redundancy is commonly diagnosed through partial-information plots, for which averaging over fragments of equal size is important when the environment is not permutation symmetric [11]. Quantum mutual information, accessible classical information, strong Quantum Darwinism, and spectrum broadcast structure are nevertheless distinct criteria. This manuscript uses a Holevo threshold for operational pointer information and invokes a mutual-information plateau only for the class of assignments used in the finite model.

Which microscopic dynamics can produce this redundancy was characterized by Doucet and Deffner (2024), who classified two-body system–environment Hamiltonians that support Quantum Darwinism: redundant proliferation of a pointer requires that the system-side operators entering the interaction commute with the pointer observable and with one another. That classification, like much of the standard Quantum Darwinism literature, assumes a fixed system–environment interaction structure.

Many settings motivate a family of interaction structures rather than one fixed structure, including controlled open dynamics, switching environments, and topology-dependent models. Tracing out an inaccessible orthogonal control register produces a channel mixture for measurements restricted to the system and accessible environment. We identify the pointer decompositions preserved by every nonzero-weight channel and state a separate fragment-information condition for operational redundancy.

The result bears on the measurement problem only conditionally. ETP currently treats relational potentiality as a candidate basement and quantum states as mathematical assignments rather than observer-independent physical waves. This paper does not prove that interpretation or physical instantiation. Its narrower contribution is to give record-stability and redundancy conditions after physical systems, interactions, fragments, records, probabilities, and access structures are already admitted.

Interpretive convention used in this manuscript. Unless otherwise specified, “state,” “state vector,” and “density operator” mean mathematical assignments on a stated Hilbert space or observable algebra. Descriptions of a state as evolving, decohering, being reduced, being traced, or being conditioned concern transformations or restrictions of represented probabilities, coherences, and correlations under physical dynamics. They do not assert that an additional physical state-object undergoes those processes. “Pointer state” is retained only where it is standard terminology; “pointer sector,” “pointer value,” or “stable record distinction” is preferred.

The distinction is substantive but does not alter the theorem. Common-pointer conservation answers which relational record distinctions can remain nondemolished in every channel. Conditional fragment information answers whether physical environmental fragments carry those distinctions redundantly. Algebraic restriction to accessible observables answers which assignment is appropriate to a specified observer. None of these operations is identified with objective wave-function collapse.

2. Setup, Definitions, and Representational Status

Let a finite-dimensional system S interact with an environment E composed of N fragments {E_k}. A control register T selects among M channels. In channel a, the full Hamiltonian is

H⁽ᵃ⁾ = H_S⁽ᵃ⁾ ⊗ I_E + I_S ⊗ H_E⁽ᵃ⁾ + Σ_{k,α} A⁽ᵃ⁾_{αk} ⊗ B⁽ᵃ⁾_{αk}.

Here H_S⁽ᵃ⁾ and H_E⁽ᵃ⁾ are the free system and environment Hamiltonians, and A⁽ᵃ⁾_{αk} and B⁽ᵃ⁾_{αk} are Hermitian coupling operators, with fragment k containing the support of B⁽ᵃ⁾_{αk}. Let U⁽ᵃ⁾(t)=exp(−iH⁽ᵃ⁾t). Let Π be a nondegenerate candidate pointer observable with rank-one spectral projectors P_s=|s⟩⟨s|. For F⊆E, I(S:F)=S(ρ_S)+S(ρ_F)−S(ρ_SF) is the quantum mutual information. The Holevo quantity χ({p_s,ρ_{F|s}}) separately measures accessible classical information in F about the pointer label s. Operational Holevo redundancy at deficit δ means that many disjoint fragments satisfy χ≥(1−δ)H({p_s}). A conventional mutual-information plateau is a distinct diagnostic and requires additional state-structure assumptions.

For the conditional-record formula and numerical results below, assume an initially uncorrelated density assignment ρ_0=ρ_S⊗ρ_E, with the control register also initially uncorrelated with SE. The control basis assignments are orthonormal and the joint unitary is block diagonal in that basis. More general initially correlated SE assignments require conditional fragment operators defined directly from the evolved joint density operator and are outside the present theorem’s record formula. Define m_δ as the minimum fragment size |F| for which χ({p_s,ρ_{F|s}})≥(1−δ)H({p_s}); the redundancy density is R_δ:=N/m_δ, which need not be an integer, while the number of guaranteed disjoint equal-size fragments is floor(N/m_δ).

In the ETP interpretation, P_s labels a candidate stable record distinction, not an absolute physical condition that S possesses independently of interaction. The density operators ρ_S, ρ_E, and ρ_{F|s} encode relational probabilities and correlations for their stated algebras. This interpretive convention leaves all operator identities, commutator conditions, entropy definitions, and numerical calculations unchanged.

Notation used in this manuscript

3. Reduction: An Inaccessible Register Yields an Observer-Accessible Channel Mixture

The control register T carries information about which interaction structure labels the controlled dynamics. In the settings of interest, including a hidden control qubit, a fluctuating coupling, or the connectivity label of a quantum-gravitational representation, T is not among the environmental fragments accessible to the observer. Its inaccessibility has a precise algebraic consequence.

Lemma (superposition reduces to mixture for the restricted assignment). Let T be assigned an arbitrary vector |τ⟩=Σ_a c_a|a⟩ over orthonormal channel basis vectors, let the initially uncorrelated joint density operator be ρ_T⊗ρ_SE with ρ_T=|τ⟩⟨τ|, and let the controlled unitary be Σ_a|a⟩⟨a|⊗U⁽ᵃ⁾. Tracing out T gives the observer-accessible channel mixture

ρ′_SE = Σ_a p_a U⁽ᵃ⁾ ρ_SE U⁽ᵃ⁾†,      p_a = |c_a|².

Proof. The evolved density operator is Σ_{a,b} c_a c_b* |a⟩⟨b|⊗U⁽ᵃ⁾ρ_SEU⁽ᵇ⁾†. Taking the partial trace over T and using ⟨b|a⟩=δ_ab removes the off-diagonal control terms from the restricted density operator and gives ρ′_SE=Σ_a|c_a|²U⁽ᵃ⁾ρ_SEU⁽ᵃ⁾†. ∎

Thus, relative to measurements restricted to S and E, the observer-accessible assignment is the displayed incoherent mixture. The global controlled representation can retain coherence relative to observables that include T. The partial trace therefore must not be read as an observer-independent physical selection of one channel, destruction of the other amplitudes, or collapse of a wave physically spread across interaction structures. It is an algebraic restriction to the observables available in the specified relational context.

The reduction relies on an initially uncorrelated orthogonal control register and a unitary controlled in that basis. Nonorthogonal control assignments, initial T–SE correlations, non-block-diagonal evolution, or later access to T-interference observables require a different reduced description. The new interpretation does not relax any of these mathematical hypotheses.

4. The Survival Theorem

The theorem separates two questions that are sometimes conflated. The first is whether every relevant channel preserves the same rank-one pointer decomposition without transferring amplitude between its sectors. The second is whether the integrated observer-accessible channel creates sufficiently distinguishable physical records in many disjoint fragments. The first is algebraic compatibility; the second is information-theoretic broadcasting.

Theorem (channelwise rank-one pointer preservation and redundant-record survival). Assume ρ_0=ρ_S⊗ρ_E and consider ρ_SE(t)=Σ_a p_aU⁽ᵃ⁾(t)ρ_0U⁽ᵃ⁾(t)† with nondegenerate pointer projectors P_s=|s⟩⟨s|. (i) For the defined channelwise exact rank-one nondemolition property, every nonzero-weight channel preserves every pointer sector for all t in an interval containing zero if and only if [H⁽ᵃ⁾,P_s⊗I_E]=0 for every a and s. If the environment-side operators in a chosen decomposition are linearly independent after the identity component is separated, this is equivalent to [H_S⁽ᵃ⁾,Π]=0 and [A⁽ᵃ⁾_{αk},Π]=0 for every a, α, and k. (ii) Nondemolition compatibility yields operational Holevo redundancy only when many disjoint fragments F_r of the integrated channel satisfy χ({p_s,ρ_{F_r|s}})≥(1−δ)H({p_s}). A plateau of I(S:F) additionally requires state-structure assumptions connecting I and χ, as in the numerical model. The first condition determines which record distinction can survive; the second determines whether that distinction is redundantly recorded.

Proof of sufficiency. The commutators [H⁽ᵃ⁾,P_s⊗I_E] vanish if and only if U⁽ᵃ⁾(t) preserves each rank-one pointer sector. Rank one then gives U⁽ᵃ⁾(t)=Σ_sP_s⊗V_s⁽ᵃ⁾(t). This controlled form preserves pointer populations but neither diagonalizes the observer-relative reduced density operator nor creates informative records by itself. Under the assumed product initial assignment, define for a fragment F

ρ_{F|s} = Σ_a p_a Tr_{E\F}[V_s⁽ᵃ⁾ ρ_E V_s⁽ᵃ⁾†].

If many disjoint fragments satisfy the Holevo threshold, each carries all but a fraction δ of the classical pointer information, so operational Holevo redundancy follows. In the class of assignments used in Section 5, this also produces the displayed mutual-information plateau; no general equivalence or full spectrum-broadcast product structure is claimed. ∎

Proof of necessity. Suppose every nonzero-weight channel preserves every pointer sector for all t in an interval containing zero. Differentiating U⁽ᵃ⁾(t)†(P_s⊗I_E)U⁽ᵃ⁾(t)=P_s⊗I_E at t=0 gives i[H⁽ᵃ⁾,P_s⊗I_E]=0. If this fails for any channel, that channel transfers amplitude between pointer sectors at sufficiently small times, preventing a channel-independent nondemolition record. After separating the identity component, linear independence of the environment operators gives the stated termwise conditions, including [H_S⁽ᵃ⁾,Π]=0. Informative conditional fragment operators remain separately necessary for redundancy. ∎

The theorem concerns exact channelwise preservation of a shared nondegenerate rank-one decomposition. It is not a classification of approximate objectivity, noncommuting single-channel pointer selection, or degenerate record algebras. For degenerate boundary or charge sectors, the natural extension is preservation of a common commutative algebra generated by higher-rank projectors, with records broadcasting the sector label rather than a unique vector inside each block; that extension is left for future work.

4.1 Interpretive consequence within ETP

Within ETP, the theorem describes conditions for stable physical records without objective wave-function collapse. The operator P_s labels a possible relational record distinction. The commutator condition says that every nonzero-weight channel preserves that distinction without demolition. The Holevo condition says that many accessible physical fragments carry nearly all the classical information about the same distinction. A measurement record is therefore established through physical interaction and redundant correlation, while the observer-relative density operator is conditioned or restricted to the applicable record context. No additional physical state-object is required to contract.

This is an interpretive consequence, not a theorem corollary in the deductive sense. The survival theorem does not prove that quantum states are non-ontic, derive the Born rule, identify a dynamical mechanism selecting a unique first-person outcome, or establish full spectrum broadcast structure. Its contribution to the measurement problem is narrower: once the state is treated as a relational mathematical assignment, the theorem supplies exact stability conditions and a separate operational record criterion without adding collapse dynamics.

5. Robustness and Numerical Results

We illustrate the theorem and characterize the approach to the plateau in an exact simulation of one system qubit and seven environment qubits. Each selected channel imprints A⁽ᵃ⁾=cosφ_a σ_z+sinφ_a σ_x through a controlled rotation of angle θ, and the channels are mixed with equal weights. The state vectors and density operators in the calculation are the standard mathematical assignments used to compute the fragment information. The calculation is a model illustration, not a proof of the general theorem or of ETP’s interpretive postulate.

Figure 1. (a) Partial-information plots. A shared pointer produces the classical plateau at H_S from the smallest fragment for a perfect copy, or after a few fragments when the channels write through different environment operators. Conjugate pointers produce no plateau, with the information remaining below H_S until essentially the entire environment is collected. (b) Redundancy R_δ for δ=0.1. As the coupling weakens, redundancy declines continuously; as the channels’ pointer axes misalign, it collapses across a narrow interval near 30°. The plots show properties of the observer-accessible assignments and fragment records, not physical branches collapsing.

The calculations illustrate the theorem’s distinct algebraic and information-theoretic mechanisms in the selected finite model. With a shared pointer, a single fragment carries the full bit, giving R_δ=7 for every M from one to four channels: the plateau is unchanged by the number of controlled structures. When the channels share the pointer but write through different environment operators, the plateau still reaches H_S, though only by three-fragment groups, giving R_δ=2.33. Weakening the coupling lowers R_δ smoothly through 7.0, 3.5, 2.33, and 1.75. When the channels broadcast conjugate observables, the plateau is absent and R_δ=1 for every M. A continuous tilt of one pointer away from the other drives R_δ from 7 to 1 across a narrow window near 30°. Objectivity is therefore robust against weak or heterogeneous record writing only while the integrated fragments remain informative, and sharply sensitive to pointer disagreement because common-pointer conservation is an exact algebraic requirement.

The phrase “the plateau collapses” in this numerical discussion, when used, refers only to disappearance of the information-theoretic plateau as parameters change. It does not denote physical wave-function collapse. Likewise, “pointer misalignment destroys redundancy” means that the selected fixed-tolerance diagnostic is no longer satisfied in the computed assignment.

6. Continuum Limit: A Measure of Channels

The channel-mixture construction extends from a finite sum to a measurable direct integral when the unitary field, pointer projectors, and conditional density-operator fields are measurable and the integrals are well defined. Exact channelwise conservation is imposed μ-almost everywhere. Operational redundancy is evaluated from the integrated conditional fragment operators, not from an arithmetic average of sectorwise redundancies:

ρ_SE = ∫ dμ(a) U⁽ᵃ⁾ ρ_0 U⁽ᵃ⁾†.

The finite reduction extends to a measurable channel family when a common pointer is conserved for μ-almost every channel and the operator fields and conditional assignments are measurable. Exact nondemolition compatibility then holds almost everywhere. Redundant objectivity additionally requires the integrated conditional fragment operators

ρ_{F|s} = ∫ dμ(a) ρ_{F|s,a}

to satisfy the fragment-information threshold. Because R_δ is defined through a nonlinear threshold, the redundancy of the mixture is not generally the μ-average of the sectorwise redundancies. A positive-measure misaligned component prevents exact common-pointer conservation, although an approximate plateau may persist when its weight and misalignment remain below a tolerance-dependent threshold. The sampling study illustrates this approximate regime but does not replace the measure-theoretic assumptions.

Within the ETP convention, μ weights the represented channel family in the selected relational assignment. The integral does not by itself assert that one observer-independent physical geometry is secretly selected, nor that a physical wave simultaneously occupies a collection of classical geometries. Those ontological questions are outside the measure-theoretic theorem.

7. Application: Causal Bounds under Superposed Interaction Graphs

The same register structure has a second consequence for information propagation. For a finite family of interaction graphs, observables that act trivially on the register evolve block diagonally, and the norm of the resulting decomposable commutator is the maximum of the block norms:

‖[A(t),B]‖ = max_a ‖[A⁽ᵃ⁾(t),B]‖.

Each block obeys a graph-specific Lieb–Robinson estimate. Under common prefactor and decay-length bounds, relationally identified supports, a common distance lower bound, and uniform interaction-norm and graph-growth controls, the finite family has a uniform envelope obtained from the maximum standardized sector velocity. For a measurable continuum, the corresponding norm is the essential supremum. Essential boundedness is sufficient with the other uniform controls; essential unboundedness is an obstruction only on the companion paper’s common-time lower-front class. Bounded degree alone is not sufficient. The result is a propagation-envelope parameter, not automatically an operational signaling speed or physical c. Globally, each exponential estimate is understood with the trivial cap 2‖A‖‖B‖.

These norm statements concern represented observables and their physical influence bounds. They neither introduce collapse dynamics nor identify the decomposable operator as a physically propagating wave function.

8. Discussion

The result extends the analysis of Quantum Darwinism from one fixed interaction to mixtures and hidden-controlled families. In controlled open dynamics, switching environments, and topology-dependent couplings, a common conserved pointer is required for channel-independent nondemolition recording. Variations in coupling strength or environment operators are not automatically harmless: they preserve operational objectivity only when the integrated conditional records remain sufficiently informative at the selected tolerance. Pointer-axis variation is especially damaging because it directly violates the common conservation condition.

The plateau characterized here is an operational redundancy signature compatible with an approach to spectrum-broadcast-like structure under additional state-structure conditions [5]. The criterion says which channel mixtures admit the defined stable rank-one records; it does not prove full spectrum broadcast structure. The theorem also complements work on Quantum Darwinism for noncommuting evolutions [7]: that work addresses noncommutation within an interaction, whereas this paper asks whether a single conserved pointer is shared across members of a hidden-controlled family.

Reducing the controlled superposition to a mixture uses orthogonality, initial T–SE independence, block-controlled evolution, and inaccessibility of T. The same algebraic mechanism is compatible with relational or timeless conditional descriptions [8], but it should not be read as proof that one channel becomes objectively actual. Relative to S and E, interference terms involving T are absent from the restricted density operator; globally, the controlled representation may remain coherent.

Within the current ETP candidate interpretation, this separation clarifies roles only after the physical carrier is admitted. Interactions write records, common-pointer conservation stabilizes a distinguished record across the declared channel family, fragment distinguishability supports repeated access, and partial tracing restricts the mathematical assignment to an access domain. None of these operations explains why the carrier, interaction, event, probability law, or record is physically instantiated.

The interpretation remains deliberately bounded. The theorem does not derive the Born weights p_s, explain every aspect of first-person definiteness, choose the observer decomposition in arbitrary graph-changing gauge systems, or make ETP empirically distinct from all neighboring relational interpretations. These are open foundations questions rather than hidden steps in the proof.

OIRP Conditional-Closure Interface

Program status. The core Observer-Invariant Reconstruction Program (OIRP) is conditionally closed as a restricted truth-first constitution for conditional reconstruction. This closure does not establish a substrate theory, physical-instantiation bridge, definite-event selector, universal ontological core, or positive empirical ETP result. Physical actuality remains admitted rather than derived, ultimate provenance remains unresolved and reopenable, and no downstream result in this manuscript may retroactively justify the admitted systems, events, carriers, controls, records, probabilities, access structures, or equivalences on which it depends.

Manuscript effect. The theorem statements, numerical results, figures, and venue-specific claims of this manuscript retain their existing scopes. OIRP closure adds a governing claim discipline; it does not upgrade any analytic or computational result into physical implementation, laboratory evidence, intrinsic observer or object structure, intrinsic boundary or topology, Lorentzian geometry, or ultimate provenance.

Inherited OIRP Constitutional Kernel

K1. Admitted actuality: physical actuality, events, and carriers are admitted for reconstruction and are not derived from downstream success.

K2. Typed target and scope: every claim remains indexed to its target, domain, task, regime, and exact or approximate status.

K3. Complete carrier and boundary: retained and discarded systems, environments, branches, controls, references, resources, boundaries, correlations, and relevant continuations must be declared.

K4. Witness and access: measurements, testers, readouts, access limits, and witness-completeness status must be explicit.

K5. Equivalence before identity: operational, representational, gauge, and physical equivalence must be typed before promoting nullity, identity, objectivity, or invariance.

K6. Transformations, composition, and history: operations, ordering, interfaces, feedback, composition laws, and relevant histories must be declared; certification of constituents does not certify the composite.

K7. Criteria and uncertainty: metrics, tolerances, calibration, uncertainty, error propagation, and resource bounds must be explicit.

K8. Provenance, rivals, falsifiers, and claim ceilings: inputs, alternatives, falsifiers, evidence class, reopening triggers, and nonretroactivity must be preserved.

K9. Plural and null admissibility: empty, null, quotient-valued, plural, nonunique, Pareto, restricted, and underdetermined outcomes remain legitimate scientific results.

Reopening and Successor-Research Control

This manuscript should be reopened or promoted only by qualifying evidence, including a new physical bridge, a noncircular event selector, a witness-completeness theorem or counterexample, a newly relevant carrier or branch, a composition theorem or failure, a revised physical-equivalence result, an approximation failure or stronger error bound, measured rival discrimination, a theorem contradiction, or an artifact-integrity failure. Otherwise future work proceeds through restricted successor programs for recoverability completeness, composition and boundary closure, physical realization and empirical discrimination, or separate provenance and definite-event research.

Publication statement. OIRP provides the governing methodology for this manuscript’s future revisions: truth-first, carrier-complete, witness-explicit, equivalence-controlled, composition-aware, provenance-preserving, falsifiable, and nonretroactive.

9. Conclusion

Indefinite interaction structure imposes two separate requirements. A common conserved rank-one pointer decomposition is necessary and sufficient for the defined channelwise exact nondemolition-preservation property across all channels of nonzero weight. Redundant objectivity follows only when the resulting integrated conditional fragment operators carry sufficiently large classical information in many disjoint environmental fragments. The controlled-register reduction extends these statements from finite mixtures to measurable families, with exact claims stated almost everywhere and approximate redundancy evaluated at finite tolerance. The same block structure yields a finite-family Lieb–Robinson envelope and motivates a continuum extension under explicit uniformity assumptions.

In ETP’s non-ontic relational-state interpretation, the central lesson is not that a physical branch collapses or survives. It is that a mathematically represented record distinction must possess common algebraic stability and must be physically encoded in informative environmental fragments. State reduction is then conditioning or restriction of the relational assignment after record formation, not objective wave-function collapse. This interpretation strengthens the measurement-theoretic significance of the theorem without changing its mathematical content or elevating the interpretive postulate into a proved result.

Data Availability Statement and Declarations

This is a theoretical study; no experimental datasets were generated. The finite calculations use Python 3, NumPy, deterministic finite-dimensional matrix operations, and double-precision linear algebra; no random seed is required for the reported calculations. Parameters are one system qubit, seven environment qubits, the controlled rotations and angles stated in the text and figure-source data, equal channel weights where specified, and δ=0.1. The complete scripts, exact package versions, figure-source tables, and a stable repository DOI or URL must accompany the submission package before journal submission. S.M. is the sole author and declares no competing interests. A large language model assisted with drafting, editing, bibliographic checking, numerical code, and critical review; the author takes responsibility for the manuscript and its claims.

References

[1] Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715.

[2] Ollivier, H., Poulin, D., & Zurek, W. H. (2004). Objective Properties from Subjective Quantum States: Environment as a Witness. Phys. Rev. Lett. 93, 220401.

[3] Blume-Kohout, R., & Zurek, W. H. (2006). Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information. Phys. Rev. A 73, 062310.

[4] Zurek, W. H. (2009). Quantum Darwinism. Nat. Phys. 5, 181–188.

[5] Korbicz, J. K., Horodecki, P., & Horodecki, R. (2014). Objectivity in a Noisy Photonic Environment through Quantum State Information Broadcasting. Phys. Rev. Lett. 112, 120402.

[6] Doucet, E., & Deffner, S. (2024). Classifying Two-Body Hamiltonians for Quantum Darwinism. Phys. Rev. X 14, 041064. arXiv:2405.00805.

[7] Chisholm, D. A., Palma, G. M., & Innocenti, L. (2026). Emergence of quantum Darwinism and pointer states for noncommuting evolutions. APS Open Sci. 1, 000024.

[8] Page, D. N., & Wootters, W. K. (1983). Evolution without evolution: dynamics described by stationary observables. Phys. Rev. D 27, 2885.

[9] Rovelli, C., & Smolin, L. (1995). Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593.

[10] Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

[11] Chisholm, D. A., Innocenti, L., & Palma, G. M. (2024). Importance of using the averaged mutual information when quantifying quantum objectivity. Phys. Rev. A 110, 012218.

[12] Muniz, S. (2026). The Theory of Emergent Timeless Potentiality. Companion manuscript preprint.

OIRP-S2.MS1 Corpus Synchronization Addendum

Status: SUCCESSOR CANDIDATE. Scientific cutoff: verified record through HHLWBUILD2F Level 5. HHLW Level 6 excluded because no completed raw result was available.

Authority and scope

This checksum-distinct manuscript successor is synchronized through OIRP-S2.MS1 using the verified OIRP-S2 record available on September 18, 2026. The core OIRP program remains conditionally closed only as a truth-first constitution for conditional reconstruction. OIRP-S2 remains open. Physical actuality is admitted rather than derived, ultimate provenance remains unresolved and reopenable, and no downstream reconstruction may retroactively justify its admitted physical inputs.

OIRP-S2 composition update

The current successor program separates constituent certification from composite certification. Channel, instrument, process, comb, superchannel, adaptive-network, controller-inclusive, and memory-bearing claims require their own complete carriers, interfaces, histories, witnesses, equivalence relations, metrics, tolerances, and composition laws. Equality or recoverability under restricted access does not establish complete physical equivalence, and a finite-dimensional computation does not establish physical realization or a nonfinite theorem.

Current CP4.1 numerical and theorem status

CP4.1 has established a restricted finite-dimensional CPTP telescoping theorem for compatible channel sequences, genuine diamond-norm semidefinite-program executions on declared finite-dimensional cases, noncommuting-channel tests, dimension-changing examples, and an ancilla-advantage benchmark. These results do not establish a universal physical composition law, arbitrary process composition, unrestricted adaptive equivalence, nonfinite closure, or OIRP-S2 closure.

HHLWBUILD2F status firewall

The patched repository hierarchy record is L1 = 0.9268200938227155, L2 = 0.9998804689772093, L3 = 1.0000370777829688, L4 = 1.00015286168964, and L5 = 1.0000000197695689. Values slightly above one are treated only as solver-tolerance-compatible with probability one, not as literal probabilities. Level 6 was not available for this synchronization. Formal patch correctness, formal convergence, exact probability one, equality with the strict-classical optimum, equality with the Definition-23 factor-affine hierarchy, CP4.1 closure, and CP4.2 entry are not established.

External relationship and prohibited inference

The external constrained-separability literature formulates limited-memory and classically adaptive channel discrimination using separable variables with additional affine constraints and discusses PPT and constrained symmetric-extension outer approximations. That framework supports keeping the repository patched hierarchy distinct from the full factor-affine constrained hierarchy unless an explicit equivalence and implementation audit is completed. No external source is used here to replace proof, raw execution, or authority-grade numerical certification.

Evidence and release classification

This addendum is a manuscript-synchronization action. It changes cross-program status language and interfaces but does not alter previously proved theorem statements, raw calculations, figures, datasets, or experimental status except where this addendum explicitly narrows a claim. No physical-device experiment, definite-event selector, substrate ontology, universal witness completeness, or ultimate-provenance result is claimed.

Record-survival composition boundary

Channelwise common-pointer preservation and fragment-information sufficiency remain separate. Family-level survival does not by itself establish survival under sequential composition, adaptive control, retained memory, or changed instruments. The theorem remains a bounded Target III result over supplied systems, channels, fragments, probabilities, and access structures.

Source note

Internal authority: OIRP_LEDGER.md and the applicable OIRP/OIRP-S1/OIRP-S2 reproducibility and correction packages in the Recovery corpus. External relationship checked against T.-A. Ohst, S. Zhang, H. C. Nguyen, M. Plavala, and M. T. Quintino, “Characterising memory in quantum channel discrimination via constrained separability problems,” arXiv:2411.08110. Publication status must be verified independently before journal submission.

OIRP-S2 CP4.1-CP4.3 Interface Addendum

This survival-theorem manuscript retains its existing channel-family and record-redundancy scope. CP4.1-CP4.3 prevent promotion from channelwise preservation to unrestricted instruments, adaptive testers, higher-order composites, physical realization, or empirical confirmation without new typed evidence.

OIRP-S2 remains open. These closures do not establish physical realization, laboratory confirmation, empirical confirmation, indefinite-causal-order closure, a universal process-matrix composition law, a universal operational-memory formula, definite-event generation, substrate ontology, or ultimate physical provenance.

Physical actuality, events, carriers, interactions, records, instruments, calibrations, and empirical inputs remain admitted rather than derived. Ultimate provenance remains unresolved and reopenable. No downstream result may retroactively justify an admitted input.

Restricted closure status

• CP4.1 is closed at a restricted channel-level and HHLW claim ceiling. The exact strict-classical optimum, complete arbitrary-history factor-affine hierarchy, physical realization, and empirical confirmation remain excluded.

• CP4.2 is closed at a restricted instrument, comb, superchannel, adaptive-memory, postselection, tomography, identifiability, implementation, and empirical-preregistration ceiling.

• CP4.3 is closed at a restricted operational-quotient, extension, tensor-descent, operational-memory, fixed-context composite-certification, and general-process no-go ceiling.

Composite-certification firewall

Certified constituents do not automatically certify a composite. Composite use requires matching typed interfaces, complete carriers, causal order, controllers, interleavings, memory and environment access, branch and history treatment, tester class, normalization, success floors, equivalence or gauge, uncertainty model, and implementation class.

The restricted fixed-context hybrid theorem does not imply a universal composition rule for arbitrary process matrices or indefinite causal order. Direct operational tensors must be defined on declared operator-system domains or proved extension independent.

Operational-memory firewall

Operational memory is defined only relative to a fixed realization class by minimizing over all admitted realizations of one operational quotient. Selected ambient-extension Choi rank, selected Stinespring multiplicity, and nonminimal dilation size are not automatically operational-memory invariants.

Authority and reopening

Detailed theorem, numerical, execution, dataset, figure, and empirical authority remains package specific. This manuscript successor does not replace the certified research packages or the canonical OIRP program ledger.

The affected branch reopens if a new carrier, environment, branch, controller, tester, causal architecture, tensor structure, gauge, identifiability result, correction, retraction, counterexample, hardware result, or empirical dataset changes the restricted claim ceiling.