Causality Sum Rules in Conventional Scattering Matrices

From Measurable Scattering Data to Fundamental Causality Bounds
Qiushi Engine's autonomous theoretical exploration, developed and refined through human scientific review.

Manuscript · Supplement · Core results · Research materials · Getting started · Citation · 中文

What does causality allow a scattering matrix to do across frequency? Scattering matrices describe the reflection, transmission and mode conversion measured in photonic and electromagnetic devices. Passivity constrains their power balance at each frequency. Causality also limits how deeply, and over how wide a band, a response can be suppressed—but those limits are harder to see in the conventional scattering representation.

This work connects the two by correcting the reference-domain time advance. The resulting domain-delayed scattering matrix preserves real-frequency power balance and, under the stated assumptions, supports a Schur–Cayley/Herglotz construction. Two logarithmic sum rules then relate coherent-return suppression and aggregate multichannel attenuation to geometric delay budgets.

Qiushi Engine explored the initial theoretical route after human researchers defined the scientific question. The authors subsequently checked, refined and developed the theory. This repository brings together the human-reviewed manuscript and Supplementary Information, a separate Chinese report containing 16 numbered corollaries, and the research documents from two engine studies.

Core results

The two results act on quantities available from a conventional scattering matrix. Their hypotheses are part of the results: causality, transparency, low-frequency regularity and the stated angular-derivative conditions must be checked for the system being studied.

Result	Physical quantity	Scope and reading
Projected sum rule	Return into a prepared coherent channel superposition	Requires a common-delay subspace. Main-text Eq. (2); Supplementary Note 3.
Determinant sum rule	Aggregate logarithmic attenuation, expressed through the product of singular values	Independent of unitary input/output basis choices; formulated for finite channel matrices with the stated regularity. Main-text Eq. (3); Supplementary Note 4.
Finite-band consequences	Suppression depth, bandwidth, geometric-mean transmission and suppressed-channel counts	Supplementary Notes 5–7 specify the metric and channel-completeness conditions.
Conditional lossless delay extension	Integrated trace of the Wigner–Smith delay matrix	Requires additional losslessness, inner-function and modal-count assumptions. Supplementary Note 8.

The two sum rules and their assumptions

Let $T_a$ be the earliest-arrival delay operator and $\widetilde S(\omega)=D_a(\omega)S(\omega)$, with $D_a(\omega)=e^{i\omega T_a}$. The delay factor is unitary on the real-frequency axis, so $\widetilde S^\dagger\widetilde S=S^\dagger S$.

For a unit vector satisfying $T_av=\tau_vv$, the projected bound is

$$ \int_0^\infty \frac{-\ln|\langle v,S(\omega)v\rangle|}{\omega^2},d\omega \leq \frac{\pi}{2}\langle v,T_av\rangle. $$

For a finite-dimensional channel matrix, the determinant bound is

$$ \int_0^\infty \frac{-\ln|\det S(\omega)|}{\omega^2},d\omega \leq \frac{\pi}{2}\operatorname{tr}T_a. $$

Both inequalities use the causal analytic continuation and high-frequency transparency assumptions, the appropriate low-frequency unimodularity, logarithmic integrability, Blaschke convergence and angular-derivative control. The precise hypotheses and derivations are in Supplementary Notes 1–4. An infinite-channel interpretation additionally requires the relevant truncation or trace-class/Fredholm convergence conditions.

What the bounds mean for devices

The framework recovers Rozanov's absorber limit and spherical-multipole sum rules, and translates the matrix bounds into limits on coherent suppression, insertion loss and the number of strongly suppressed singular channels across a band. These are distinct observables: reduced return into one superposition may involve conversion into other channels, while an absorption interpretation requires a complete open-channel matrix.

The Supplementary Information contains 10 notes and 5 figures covering the derivations, finite-band translations, metric definitions and conditional delay extension. Its Note 9 maps the 16 candidate corollaries to the retained results and their scope.

From autonomous exploration to a reviewed manuscript

Stage	Research contribution	Documents
Theory exploration	Qiushi Engine explored the domain-delay route and produced initial derivations and research drafts.	Initial paper, supplement and Chinese report
Corollary study	A second study investigated optical and electromagnetic consequences, organized as 16 numbered corollaries.	Chinese report and source · Study context
Human review and revision	The authors checked and refined the theory, clarified the physical metrics and revised the primary documents.	Current manuscript · Current supplement

The manuscript and Supplementary Information are the current authoritative documents. The generated drafts and separate corollaries report preserve the research development; they do not carry the same review status. Human authors retain responsibility for the scientific claims. See the human-review record, Qiushi Engine role and relationships between the research materials.

Research materials

What you need	Where to start
The complete scientific argument	Manuscript PDF — 21 pages
Proofs, assumptions and supporting figures	Supplementary Information — 15 pages
The extended corollary exploration	Chinese report — 73 pages, with LaTeX source and figures
Editable versions of the primary documents	Manuscript Word source · Supplementary LaTeX source
The two engine studies	Research artifact guide
Build instructions and validation scope	Reproducibility guide · Recorded file checksums

Getting started

The PDFs can be read directly from the links above. To obtain the documents, editable sources and figures locally:

git clone https://github.com/Oxelra-AI/causality-sum-rules.git
cd causality-sum-rules

Start with the manuscript, follow Supplementary Notes 1–4 for the construction and proofs, then use Notes 5–9 to check a particular physical interpretation. The Chinese report provides the broader corollary exploration.

The reproducibility guide describes the Word and LuaLaTeX document builds. Fresh LaTeX builds have not yet been validated, and building with the supplied figures does not regenerate the underlying plots. The project currently contains no Lean formalization or equivalent machine-checked proof.

Repository layout

Manuscripts/
  paper/                       Current manuscript PDF and Word source
  supplement/                  Current supplement PDF, LaTeX source and figures
  report/corollaries/           Extended Chinese report and LaTeX source
Artifacts/qiushi-runs/
  round1-initial-run/           Initial paper, supplement and report
  round2-corollaries-run/       Study context and reconstructed research goal
docs/                          Scientific provenance, review and reproducibility
assets/                        Qiushi Engine logo, badges and research overview
tools/                         README visual generation

Authors and citation

Ning Han†, Rui Zhao†, Shuxing Yang†, MingZhu Li, Hongsheng Chen, and Yihao Yang.**

† Equal contribution.
* Corresponding authors.

Manuscript status: submitted. Citation metadata, affiliations and contact details are recorded in CITATION.cff. A DOI will be added when assigned or confirmed.

@misc{han2026causality,
  title = {Causality Sum Rules in Conventional Scattering Matrices},
  author = {Han, Ning and Zhao, Rui and Yang, Shuxing and Li, MingZhu
            and Chen, Hongsheng and Yang, Yihao},
  year = {2026},
  note = {Submitted manuscript and accompanying research materials},
  url = {https://github.com/Oxelra-AI/causality-sum-rules}
}

License

Original repository materials are licensed under CC BY 4.0. See LICENSE and LICENSES.md. Third-party materials retain their own terms. The Qiushi Engine name and logo identify the project; the license grants no trademark rights. Visual asset sources.