# Source extract: FA-CB-02.docx

Uniform and Sector-Weighted Locality Diagnostics for Topology-Controlled Quantum Dynamics

A direct-integral classification of sector-family Lieb-Robinson diagnostics

Scott Muniz
Dr. Ware Technology Services
Research Article submitted for editorial consideration at Journal of Mathematical Physics

Abstract

This paper is a conditional classification framework, not a new fixed-sector Lieb-Robinson theorem. The mathematical engine is the standard direct-integral norm identity for decomposable operators, while the contribution is the separation of uniform, sector-weighted, and averaged locality diagnostics. Lieb-Robinson bounds furnish finite propagation envelopes for local quantum dynamics on a fixed interaction graph. In topology-controlled settings, where topology means graph connectivity or interaction structure rather than operator topology, a fixed-graph locality estimate does not by itself determine which sector-family diagnostic should be quoted. This paper studies decomposable Heisenberg dynamics over a standard Borel probability space of sectors, assuming measurable support-identification data for the observable fields under comparison. The diagnostics are C_infty, the essential-supremum decomposable norm; C_1, the sector-weighted L1 diagnostic; and C_bar, an averaged-commutator norm defined only after a common representation and Bochner integrability are supplied. A uniform sector-envelope proposition, a lower-front sector-obstruction proposition, and a sector-weighted leakage theorem organize the resulting taxonomy. Deterministic finite TFIM examples illustrate the diagnostic distinction without claiming asymptotic LR scaling, signaling capacity, or coherent graph-changing dynamics.

Claim-interface status. The Lieb-Robinson envelopes and sector-family diagnostics studied here are candidate propagation constraints within declared graph, support, measure, Hamiltonian, norm, and representation data. They do not derive an intrinsic topology, metric, spatial direction, Lorentzian signature, geometrical light cone, or universal invariant speed. Intrinsic S1 and intrinsic S2 remain open in the broader research program.

Methodological Scope and Input Boundary

Authority and role. This journal-facing successor is synchronized with ETP Final Synthesis Draft 13, Ultimate Provenance and Conditional Reconstruction Protocol Version 06, Local Relational Time Version 58, and the Local Relational Geometry Version 7 architecture. The manuscript remains a conditional classification of propagation diagnostics over a supplied measurable family of graph or interaction sectors. It does not select a basement, derive physical instantiation, derive the sector family, reconstruct an intrinsic topology, or produce spacetime geometry.

• The sector space, measure, fibre representations, support-identification maps, Hamiltonians, graph data, observables, norms, and comparison structure are inherited physical or model inputs.

• Topology in this paper means supplied graph connectivity or interaction structure, not reconstructed physical topology and not operator topology.

• C_infinity, C_1, and C_bar answer different declared diagnostic questions and cannot be used as instantiation, topology, distance, or causal-geometry witnesses without a separately proved bridge.

• A Lieb-Robinson velocity is a bound parameter under a declared fixed-sector convention, not a universal invariant speed or Lorentzian light cone.

• Averaging, essential suprema, weighting, and cancellation are operations on an admitted sector family. They do not create physical sectors or select a basement.

• Numerical TFIM examples are finite model illustrations, not empirical evidence, asymptotic LR theorems, or geometric reconstruction.

Three-Target and Layer Firewall

Target I concerns physical instantiation. Target II concerns definite-event generation. Target III concerns reconstruction and certification of effective operational and geometric structures. This manuscript occupies a bounded Layer 3 interface within Target III: it supplies propagation diagnostics that may constrain a later geometric reconstruction after Layer 0 instantiation, Layer 1 event structure, and Layer 2 temporal and comparison inputs are independently certified. It does not establish the earlier layers.

Revised Claim Ceiling

The direct-integral norm identity, piecewise uniform envelope criterion, lower-front obstruction principle, sector-weighted leakage theorem, velocity-tail examples, and deterministic TFIM diagnostics retain their stated mathematical scopes. The manuscript does not derive a physical sector measure, intrinsic adjacency, intrinsic distance, metric, topology, spatial direction, causal cone, Lorentzian signature, universal speed, or physical instantiation.

1. Introduction and motivating case

A practical mistake can occur in programmable-connectivity hardware, randomly gated quantum simulators, annealed graph ensembles, or disordered-chain samples. A low-probability high-connectivity topology may spread a commutator much faster than a typical sample. Quoting only a typical-sector diagnostic can hide that worst positive-measure sector; quoting only the worst-sector norm can also obscure useful typical or sector-weighted behavior. The taxonomy below is designed to prevent that confusion.

Concrete 1% use case. Suppose a programmable simulator normally realizes chain-like or ring-like connectivity, but calibration drift or randomized scheduling produces a star-like high-connectivity topology in 1% of runs. For certification, safety, or worst-case leakage guarantees, the rare topology is relevant to C_infty because it has positive measure. For average experimental behavior over runs, the sector-weighted C_1 diagnostic may be more informative. This paper formalizes that choice of diagnostic.

Relative to random-Hamiltonian concentration approaches, the direct-integral formulation is deterministic and measure-theoretic. Concentration results ask how likely it is that typical commutator growth deviates from an ensemble expectation. The present framework asks which deterministic sector-family diagnostic is being reported once the sector measure is fixed. Concentration estimates can be used inside this framework, but they do not replace the distinction between C_infty, C_1, and C_bar.

Different operational diagnostics should be interpreted independently unless a theorem establishes an equivalence between them. The role of the present framework is to identify which locality question a diagnostic actually answers, thereby preventing conclusions licensed by one sector-family measure from being transferred to another without proof.

1.1 Relation to prior work and venue positioning

The original Lieb-Robinson theorem and later refinements establish fixed-sector finite-speed locality estimates. The present work assumes those sectorwise estimates and asks how they assemble over a measured sector family. It is closest in venue and style to mathematical-physics work on quasi-locality and operator-algebraic dynamics. Recent PRX Quantum papers such as McDonough-Yin-Lucas-Zhang and Baldwin-Ehrenberg-Guo-Gorshkov provide genuine fixed-sector LR advances; the present paper instead provides an organizational and diagnostic layer. This is deliberate and is why Journal of Mathematical Physics is the natural target venue.

2. Setup, measurability, and support identification

Let X be a standard Borel probability space with probability measure mu. The fibre Hilbert spaces form a measurable field of separable Hilbert spaces. A decomposable observable is A=integral_X^oplus A_x dmu(x), and decomposable Heisenberg dynamics are assumed block diagonal over sector labels. The direct-integral norm identity is used in the standard operator-algebraic sense: the norm of a decomposable operator is the essential supremum of its fibre norms, as in Dixmier and Kadison-Ringrose.

Representation and carrier declaration. The task-relative carrier includes the measured sector family, fibre Hilbert spaces, observable fields, support-identification maps, sector measure, Hamiltonians, retained representation data, and any reference structure used to compare fibres. The sector label x is an index for the declared decomposition and is not, by itself, an emergent spatial coordinate or physical direction. Omitting coherent inter-sector structure restricts every conclusion to decomposable dynamics.

A support-identification datum for the pair (A,B) consists of measurable fields x -> A_x and x -> B_x, a measurable support-distance map x -> d_x(A_x,B_x), and measurable sector-LR data x -> (C_x, xi_x, v_x). For each fixed t, the map x -> ||[A_x(t),B_x]|| is assumed measurable. All bounds and equalities involving these data are understood modulo mu-null sets, and all essential suprema are taken relative to mu.

Definitions:

(1)  C_infty(t) = ess sup_x ||[A_x(t),B_x]||,     C_1(t)= integral_X ||[A_x(t),B_x]|| dmu(x).

(2)  C_bar(t)= || integral_X I_x([A_x(t),B_x]) dmu(x) ||_E.

Equation (2) is asserted only when a common Banach or operator space E, measurable identification maps I_x, and Bochner integrability are supplied. A sufficient condition for Bochner integrability is that x -> I_x([A_x(t),B_x]) be strongly measurable and that x -> ||I_x([A_x(t),B_x])||_E belong to L^1(X,mu). If the maps I_x are uniformly contractive, then C_bar(t)<=C_1(t) follows from the Bochner triangle inequality; otherwise no inequality between C_bar and C_1 is asserted. Incompatible representations, representation-dependent cancellations, and non-contractive embeddings are therefore outside any comparison claim involving C_bar.

Cancellation example. If two equally weighted sectors embed to +K and -K in E, then C_bar=0 while C_1=||K|| and C_infty=||K||. This is not a contradiction; these are different diagnostics.

LRG cancellation-trap control. Vanishing of C_bar under a chosen common embedding is not a completeness theorem and cannot establish geometric identity, absence of propagation, or physical nullity. Unless the admitted witness and representation family is complete for the proposed distinction, the result is representation-relative, RESTRICTED-NULL, or UNDERDETERMINED. C_1 and C_infinity retain nonzero sector information in this example.

The vanishing of one diagnostic therefore does not certify the absence of locality effects in every sector, just as an averaged observable need not determine worst-case positive-measure sector behavior.

Figure 1. Diagnostic taxonomy for decomposable sector families. C_bar requires a common representation and may show cancellations.

3. Uniform envelope and lower-front obstruction

Reader roadmap. Proposition 1 gives a sufficient condition for a uniform sector-family exterior-cone envelope. Proposition 2 gives a conditional lower-front obstruction. Theorem 1 provides a sector-weighted cutoff diagnostic when uniformity fails.

Assume that for mu-almost every x,

||[A_x(t),B_x]|| <= min{2||A_x||||B_x||, C_x||A_x||||B_x|| exp[-(d_x-v_x t)/xi_x]}.

Proposition 1. Piecewise uniform sector-envelope criterion

Suppose C_x<=C, xi_x<=xi, v_x<=v_ess, and d_x>=d for mu-almost every x, with ||A||_infty and ||B||_infty finite. Then

C_infty(t) <= C||A||_infty||B||_infty exp[-(d-v_ess t)/xi]   if d-v_ess t>0,

C_infty(t) <= 2||A||_infty||B||_infty   if d-v_ess t<=0.

Proof. By the standard direct-integral norm identity for decomposable operators, C_infty(t)=ess sup_x ||[A_x(t),B_x]||. On d-v_ess t>0, the hypotheses imply d_x-v_x t >= d-v_ess t>0 and hence exp[-(d_x-v_x t)/xi_x] <= exp[-(d-v_ess t)/xi_x] <= exp[-(d-v_ess t)/xi], because xi_x<=xi and the numerator is positive. Multiplying by C_x<=C and ||A_x||||B_x||<=||A||_infty||B||_infty, and then taking the essential supremum, gives the exterior estimate. Inside or on the cone, ||[A_x(t),B_x]||<=2||A_x||||B_x|| gives the cap after taking essential suprema. □

Proposition 2. Lower-front sector obstruction principle

Let v_0<infinity be a proposed envelope velocity. Suppose there exist kappa>0 and fixed Delta>=0 such that, for arbitrarily large V>v_0, there are sets S_V with mu(S_V)>0, times t_V>0, normalized observable fields, and identified supports satisfying d_x(A_x,B_x)>=Vt_V-Delta on S_V, while ||[A_x(t_V),B_x]||>=kappa on S_V. If (V-v_0)t_V-Delta -> infinity, no fixed finite C and xi yield a uniform exterior-cone exponential envelope with velocity v_0 for all those fields and spacetime points.

Warning. Unbounded upper-bound velocities alone do not imply lower-front propagation. This obstruction applies only when positive-measure sectors actually satisfy the stated lower-front hypotheses. Baldwin, Ehrenberg, Guo, and Gorshkov provide a concrete disordered one-dimensional context in which all-site and subsequence propagation distinctions motivate this kind of lower-front hypothesis.

Contradiction proof. If such a uniform exterior-cone envelope existed, then at the specified spacetime points the positive-measure lower-front sectors would imply kappa <= C exp[-((V-v_0)t_V-Delta)/xi]. The right-hand side tends to zero along the assumed sequence, contradicting fixed kappa>0. □

Worked Delta example. If chain supports are embedded into a common labelled chain only up to a bounded relabelling of two lattice spacings, then d_x>=r_x-2. Here Delta=2 is a fixed support-identification slack and cannot absorb the divergence of (V-v_0)t_V.

Atomic-sector example. Let X={1,2,...} with mu=sum_n 2^{-n}delta_n. Set V_n=n and t_n=n. If a fixed-sector construction supplies lower-front events with d_n>=V_n t_n-Delta and commutator at least kappa, then (V_n-v_0)t_n-Delta=(n-v_0)n-Delta -> infinity for every fixed v_0. The direct-integral part is positive-measure bookkeeping; the lower-front events must still come from the fixed-sector dynamics.

4. Sector-weighted leakage theorem

Theorem 1. Let S_slow={x:v_x<=v_c} and S_fast={x:v_x>v_c}. Assume that, for mu-almost every x in S_slow, C_x<=C, xi_x<=xi, d_x>=d, and v_x<=v_c. Assume also that ||A||_infty and ||B||_infty are finite on all sectors. Then

C_1(t) <= mu(S_slow) min{2||A||_infty||B||_infty, C||A||_infty||B||_infty exp[-(d-v_c t)/xi]} + 2||A||_infty||B||_infty mu(S_fast).

Proof. Split the integral defining C_1(t) over S_slow and S_fast. On S_slow, the explicitly quantified slow-sector assumptions imply that the sectorwise LR branch is bounded above by the displayed exponential term, and the trivial cap is bounded by 2||A||_infty||B||_infty. Integrating this pointwise minimum over S_slow gives the slow term. On S_fast, no LR data are needed: use only ||[A_x(t),B_x]||<=2||A_x||||B_x||<=2||A||_infty||B||_infty and integrate over S_fast. □

5. Convention for the fixed-sector Lieb-Robinson velocity

To avoid hiding convention-dependence inside a numerical constant, this paper states Corollary 1 in convention-agnostic form. Let v_LR denote the velocity supplied by a selected fixed-sector finite-range bounded-degree LR theorem under its stated norm, graph-counting, prefactor, and decay-length conventions. For example, one may instantiate v_LR using the finite-range bounded-degree Lieb-Robinson framework described by Nachtergaele, Sims, and Young, or by the earlier Nachtergaele-Sims/Hastings-Koma formulations, after fixing the corresponding interaction norm and graph-growth convention. In many nearest-neighbor normalizations one obtains a velocity proportional to 2e times the local interaction scale, and in bounded-degree graph-counting conventions the same role is played by a constant proportional to Jz. The exact constant must be read from the cited fixed-sector theorem and convention.

Corollary 1. If sectorwise LR estimates with common C_LR and xi_LR hold mu-almost everywhere and v_x<=v_LR mu-almost everywhere under the selected convention, then Proposition 1 applies with C=C_LR, xi=xi_LR, and v_ess<=v_LR. The direct-integral step adds no new velocity constant.

6. Representative velocity-tail examples

Compact support. If v_x<=V_max almost everywhere and the remaining uniform data are bounded, Proposition 1 gives a uniform piecewise envelope with v_ess<=V_max.

Exponential tail. If mu(v_x>v)<=K exp(-a v), choosing v_c=(1/a)log(2K||A||_infty||B||_infty/epsilon) makes the fast-sector tail term in Theorem 1 at most epsilon, whenever the logarithm is meaningful. The remaining tolerance is governed by the slow-sector exterior exponential. This illustrates that the optimal cutoff is tolerance-dependent.

Power-law tail. If mu(v_x>v)<=K v^{-alpha} for v>=1, then choosing v_c=(2K||A||_infty||B||_infty/epsilon)^{1/alpha} makes the fast-sector tail contribution at most epsilon. This is a sector-velocity distribution statement and should not be confused with a fixed-sector Hamiltonian having power-law spatial interactions.

7. Numerical illustration

The numerical material is illustrative and deterministic. It does not prove the theorems, estimate asymptotic LR velocities, or define a signaling capacity. The TFIM example uses H_G=-J sum_{(i,j) in E(G)} Z_i Z_j - h sum_i X_i with J=1 and h=0.7 on six qubits, with A=Z_0 and B=Z_5.

Figure 2. Finite-system TFIM commutator growth on chain, ring, and star sectors; not an asymptotic LR regime.

The rare-sector mixture uses weights mu_chain=0.49, mu_ring=0.50, and mu_star=0.01. The point is diagnostic: positive-measure rare sectors affect C_infty, while C_1 responds in proportion to sector weight.

Figure 3. Finite-size diagnostic norms only. Panel (a) compares C_infty and C_1 for weights 0.49, 0.50, 0.01; panel (b) contrasts raw maximum with essential supremum after the star sector is null; panel (c) sweeps the star-sector weight at t=2.

Figure 4. Averaged-operator cancellation example. Opposite embedded commutators can cancel in C_bar while C_1 and C_infty remain nonzero.

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, intrinsic topology, or positive empirical ETP result. Physical actuality remains admitted rather than derived, ultimate provenance remains unresolved and reopenable, and no downstream locality diagnostic may retroactively justify the admitted sector family, measure, graph data, support identification, Hamiltonians, representations, observables, or equivalence relations on which it depends.

Manuscript effect. The direct-integral identities, envelope and obstruction propositions, sector-weighted theorem, CPR5.2-validated reproduction artifacts, deterministic TFIM illustrations, figures, and venue-specific claims retain their existing mathematical scopes. OIRP closure adds governance only; it does not promote a sector-family bound into intrinsic geometry, an operational signaling capacity, a Lorentzian light cone, a universal invariant speed, or physical implementation.

Inherited OIRP Constitutional Kernel

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

K2. Typed target and scope: each diagnostic remains indexed to its sector family, measure, representation, support-identification data, norm, task, and exact or approximate status.

K3. Complete carrier and boundary: retained and discarded sectors, registers, representations, controls, references, environments, resources, and correlations must be declared.

K4. Witness and access: observable fields, comparison maps, operational access, and witness-completeness status must be explicit.

K5. Equivalence before identity: null sets, representation equivalence, support identification, gauge, and physical equivalence must be typed before geometric identity or nullity is promoted.

K6. Transformations, composition, and history: sector dynamics, inter-sector assumptions, composition, interfaces, and histories must be declared; fixed-sector certification does not certify a changing-topology composite.

K7. Criteria and uncertainty: constants, tolerances, tail assumptions, approximation error, numerical regime, and resource bounds must be explicit.

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

K9. Plural and null admissibility: uniform, sector-weighted, averaged, null, nonunique, restricted, and underdetermined results remain legitimate when correctly typed.

Reopening and Successor-Research Control

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

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

8. Practical workflow and scope

The results apply only to decomposable sector families. They do not cover coherent graph-changing Hamiltonians, quantum network dynamics, process matrices, quantum switches, or communication-capacity claims unless a physical reduction to block-diagonal sector dynamics has already occurred.

More generally, the framework illustrates a methodological principle for relational and operational quantum theory: distinct operational resources require independent certification. Uniform locality guarantees, sector-weighted behavior, and averaged observables provide different forms of information, and none should be substituted for another without an explicit theorem establishing the relevant equivalence.

Draft-07 and Layer 0-3 interface. Relative to the current ETP corpus, this paper supplies uniform, sector-weighted, and averaged propagation diagnostics relevant to later recoverability and causal-reconstruction tests. It is a bounded Layer 3 interface, not a geometric reconstruction map. A finite propagation envelope is not a derived spacetime light cone, and a Lieb-Robinson velocity is not a universal relativistic invariant. The controlling corpus authorities are ETP Final Synthesis Draft 13, Ultimate Provenance and Conditional Reconstruction Protocol Version 06, Local Relational Time Version 58, Local Relational Geometry Version 7, and Manuscript Authority Register Version 9.

9. Reproducibility and data availability

The verified reproduction package accompanying this submission is supplied as supplementary material. It contains pinned dependency versions, generator and test scripts, CSV outputs, figures, a README, an MIT license, a verification log, SHA-256 checksums, CITATION.cff, codemeta.json, and an optional Dockerfile pinned by image digest. CPR5.2 independently validated the archive checksums, executed the stated tests, and regenerated the listed data and figures byte-for-byte in the audit environment. No public repository DOI or URL is claimed for this submission-stage supplement.

Acknowledgments and author declarations

S.M. is the sole author and declares no competing interests. Microsoft 365 Copilot based on GPT-5 assisted with drafting, editing, bibliographic checking, numerical code, and critical review; the author verified all mathematical statements and accepts responsibility for the manuscript and its claims.

References

[1] Lieb, E. H., & Robinson, D. W. (1972). The finite group velocity of quantum spin systems. Communications in Mathematical Physics 28, 251-257. https://doi.org/10.1007/BF01645779

[2] Hastings, M. B., & Koma, T. (2006). Spectral gap and exponential decay of correlations. Communications in Mathematical Physics 265, 781-804. https://doi.org/10.1007/s00220-006-0030-4

[3] Nachtergaele, B., & Sims, R. (2006). Lieb-Robinson bounds and the exponential clustering theorem. Communications in Mathematical Physics 265, 119-130. https://doi.org/10.1007/s00220-006-1556-1

[4] Nachtergaele, B., Ogata, Y., & Sims, R. (2006). Propagation of correlations in quantum lattice systems. Journal of Statistical Physics 124, 1-13. https://doi.org/10.1007/s10955-006-9143-6

[5] Nachtergaele, B., Sims, R., & Young, A. (2019). Quasi-locality bounds for quantum lattice systems. Part I. Journal of Mathematical Physics 60, 061101. https://doi.org/10.1063/1.5095769

[6] Chen, C.-F. (2021). Concentration of OTOC and Lieb-Robinson velocity in random Hamiltonians. arXiv:2103.09186. https://doi.org/10.48550/arXiv.2103.09186

[7] Chen, C.-F., & Lucas, A. (2021). Operator growth bounds from graph theory. Communications in Mathematical Physics 385, 1273-1323. https://doi.org/10.1007/s00220-021-04138-2

[8] Baldwin, C. L., Ehrenberg, A., Guo, A. Y., & Gorshkov, A. V. (2023). Disordered Lieb-Robinson bounds in one dimension. PRX Quantum 4, 020349. https://doi.org/10.1103/PRXQuantum.4.020349

[9] McDonough, B. T., Yin, C., Lucas, A., & Zhang, C. (2025). Lieb-Robinson bounds with exponential-in-volume tails. PRX Quantum 6, 040322. https://doi.org/10.1103/n242-m1l2

[10] Wang, Z., & Hazzard, K. R. A. (2020). Tightening the Lieb-Robinson bound in locally interacting systems. PRX Quantum 1, 010303. https://doi.org/10.1103/PRXQuantum.1.010303

[11] Else, D. V., Machado, F., Nayak, C., & Yao, N. Y. (2020). Improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactions. Physical Review A 101, 022333. https://doi.org/10.1103/PhysRevA.101.022333

[12] Bru, J.-B., & de Siqueira Pedra, W. (2017). Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory. SpringerBriefs in Mathematical Physics 13. https://doi.org/10.1007/978-3-319-45784-0

[13] Kliesch, M., Gogolin, C., & Eisert, J. (2014). Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems. In Many-Electron Approaches in Physics, Chemistry and Mathematics. Springer. https://doi.org/10.1007/978-3-319-06379-9_17

[14] Dixmier, J. (1981). Von Neumann Algebras. North-Holland.

[15] Kadison, R. V., & Ringrose, J. R. (1997). Fundamentals of the Theory of Operator Algebras, Vol. II. American Mathematical Society.

[16] Takesaki, M. (2002). Theory of Operator Algebras I. Springer. https://doi.org/10.1007/978-1-4612-6188-9

[17] Bratteli, O., & Robinson, D. W. (1987). Operator Algebras and Quantum Statistical Mechanics I, 2nd ed. Springer. https://doi.org/10.1007/978-3-662-02520-8

[18] Mahoney, B. J., & Lent, C. S. (2024). Lieb-Robinson correlation function for the quantum transverse-field Ising model. Physical Review Research 6, 023286. https://doi.org/10.1103/PhysRevResearch.6.023286

[19] Beckman, D., Gottesman, D., Nielsen, M. A., & Preskill, J. (2001). Causal and localizable quantum operations. Physical Review A 64, 052309. https://doi.org/10.1103/PhysRevA.64.052309

[20] Bravyi, S., Hastings, M. B., & Verstraete, F. (2006). Lieb-Robinson bounds and the generation of correlations and topological quantum order. Physical Review Letters 97, 050401. https://doi.org/10.1103/PhysRevLett.97.050401

[21] Arrighi, P., Christodoulou, M., & Durbec, A. (2025). On quantum superpositions of graphs, no-signalling and covariance. Journal of Physics A 58, 155303. https://doi.org/10.1088/1751-8121/adc881

[22] Oreshkov, O., Costa, F., & Brukner, C. (2012). Quantum correlations with no causal order. Nature Communications 3, 1092. https://doi.org/10.1038/ncomms2076

[23] Chiribella, G., D'Ariano, G. M., Perinotti, P., & Valiron, B. (2013). Quantum computations without definite causal structure. Physical Review A 88, 022318. https://doi.org/10.1103/PhysRevA.88.022318

[24] Cassemiro, K., & Bartlett, S. (2025). Editorial: Clarifying PRX Quantum's acceptance criteria. PRX Quantum 6, 020001. https://doi.org/10.1103/PRXQuantum.6.020001

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.

Propagation and composition boundary

Sectorwise Lieb-Robinson bounds, essential-supremum diagnostics, weighted leakage, and averaged commutators remain distinct. A bound for constituents or sectors does not automatically certify a sequential, feedback-controlled, adaptive, or quotient-composed network. None of these diagnostics alone establishes signaling capacity, intrinsic topology, Lorentzian causal order, or a universal invariant speed.

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 topology-controlled propagation manuscript retains its existing direct-integral and locality-diagnostic scope. CP4.1-CP4.3 add only the constituent, interface, operational-quotient, and composite-certification firewalls needed when these results are used inside a larger controlled process.

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.