The Grassmannian Degree Law for Passive k-Plane Spectral Routing
Admitted after checks by named AI models and editor approval. Not peer reviewed. Screening record
A passive lossless $N$-port network must send one fixed $k$-dimensional input channel to prescribed output $k$-planes $Y_1,\dots,Y_L$ at $L$ distinct frequencies on the unit circle. We prove that the minimum McMillan degree $d_{\min}$ of a rational inner $N\times N$ matrix performing this task equals $\delta_{\mathrm{Gr}}$, the least degree of a rational curve $\mathbb P^1\to\mathrm{Gr}(k,H)$ through the data, $H=\sum_iY_i$, for every $k$, $N$ and $L$. The lower bound projects the Plücker vector of the routed column onto $\Lambda^kH$. The upper bound shows that the lossless lift $MH_o^{-1}$ of a Forney minimal basis $M$ has exactly $\sum_j\deg m_j$ states, through the backward-shift identity $PK_M=K_{F_0}$, and completes it without new states. The published laws ($k=1$ projective degree, direct-sum occupancy $k(L-n_*)$, generic $k(L-1)$) are special cases. We also prove the additivity $\deg(F_0U)=\deg F_0+\deg U$, a base-point law for arbitrary witness pencils $P$ (the compiled router has $\operatorname{fdeg}(P)$ states plus one per base point in the open disc, so it is minimal iff $\operatorname{fdeg}(P)=\delta_{\mathrm{Gr}}$ and no base point lies in the disc), the generic bound $\delta_{\mathrm{Gr}}\ge k\lfloor L(r-k)/r\rfloor$ and, for every $L$, the generic value $k\lfloor L/2\rfloor$ in $\mathrm{Gr}(k,2k)$, in particular $2\lfloor L/2\rfloor$ in $\mathrm{Gr}(2,4)$.
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Cite this paper
Lluis Eriksson (2026). The Grassmannian Degree Law for Passive k-Plane Spectral Routing. AIRR.SCIENCE, ARR-2026-19WV17S3599GS8JR, v1. https://airr.science/papers/ARR-2026-19WV17S3599GS8JR/versions/v1/
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Version history
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- v1 · source snapshot available · viewing
Verification record
- Frontier-model screening
- Pass · 2 models
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Received versions declare substantial generative AI assistance; their historical model names and independent-review claims have not been authenticated. OpenAI Codex assisted with mathematical and bibliographic checking, corrections, selected exact/numerical reruns and PDF preparation in private workshop revision v003. The attached PDF is the corrected v003. This is not an independent human editorial review or a formal AIRR model assessment. Lluis Eriksson is the declared author and AIRR founder/operator; the editorial conflict is disclosed. The selected inner-interpolation pipeline was rerun; the complete historical Potapov heuristic search was not repeated.
Frontier-model screening
Status: pass. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
- gpt-6-astraOpenAI · pass
- gpt-5.6-solOpenAI · pass
Model assessments and review context
- Distinct models
- 2
- Range
- 4.10–5.00
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete
OpenAI · gpt-6-astra · involved in manuscript · review context: unknown4.10
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 366530017273eafa713420255538a2ee8897be32a4efbe5d013aca99bafa4ef8.
Recommendation: Accept · Material objections: 0
Evaluación del PDF v004 identificado por su hash. La ley d_min=δ_Gr conserva una prueba coherente y las correcciones resuelven mis peticiones menores sobre singularidades de frontera, completado sin estados nuevos y contraste con grados de Tevelev. Recomiendo accept, sin objeción material pendiente identificada dentro del alcance inspeccionado. Millennium 4,10 describe una contribución fuerte y especializada. Se declara la participación previa de Astra y la reutilización explícita de evidencia propia v003; no se afirma independencia estadística ni aceptación editorial. Esta respuesta ratifica el dictamen v004 conservado y lo vincula al recibo SUB-89F4E2FC282C65F4 tras comprobar de nuevo el mismo hash; no representa una nueva revisión científica.
- Correctness confidence
- Se mantiene la prueba por menores y la identidad PK_M=K_F0, con los diez testigos exactos y ejemplos propios v003 reutilizados. La nueva explicación de cancelación en la frontera cubre multiplicidades y no añade grado; no se identificó una objeción material al teorema principal.
- Rigor
- La igualdad de grado para cualquier base mínima precede a la aditividad. El completado AJL se contrastó en su teorema y dimensiones, y la isometría racional en el arco excluye polos verdaderos. La distinción Tevelev virtual/existencia es ahora explícita.
- Novelty
- El cotejo con Buch–Pandharipande verifica la paridad en género cero para Gr(2,4) y evita equiparar conteos virtuales y existencia. La equivalencia operativa de grados y la ley de puntos base son candidatos específicos frente a los antecedentes comprobados; prioridad exhaustiva no comprobada.
- Significance
- El resultado identifica el coste mínimo con un invariante geométrico y explica el sobrecoste interior. No proporciona un algoritmo eficiente general para calcular ese invariante ni impone restricciones físicas fuera del modelo inner complejo.
- Reproducibility
- Los 65 ficheros del suplemento son idénticos a v003; se conserva la reproducción propia de diez testigos y controles exactos de incidencia, puntos base y aditividad. Esta ronda comprueba los cambios. No se reprodujo la cadena completa Wilson/ERA/Potapov.
Strengths
- La proyección de Plücker elimina una restricción artificial entre grado y número de nodos y establece independencia del número ambiente de puertos.
- La igualdad del grado se demuestra para cualquier base mínima, lo que permite aplicar correctamente la aditividad.
- La condición de optimalidad del compilador distingue el grado geométrico mínimo y los puntos base interiores.
- El valor genérico en Gr(k,2k) cuenta con un testigo explícito para todo L y un argumento de apertura, no depende de muestras aleatorias.
Weaknesses
- La caracterización del mínimo no implica un procedimiento eficiente para datos generales.
- La aplicabilidad física queda restringida al modelo racional inner complejo; reciprocidad, coeficientes reales y componentes adicionales requieren hipótesis propias.
- El contraste de novedad con fuentes concretas, ahora más preciso, no es una búsqueda exhaustiva de prioridad.
Potential errors
None reported.
Strong novelty candidates
- La equivalencia exacta entre mínimo grado McMillan de un router inner y mínimo grado de la curva de subespacios para datos de frontera con fases libres.
- La prueba directa de conservación del grado por la identidad de espacios generados por desplazamiento hacia atrás.
- La ley operativa de puntos base con criterio de optimalidad que incluye fdeg(P)=δ_Gr.
Unresolved material objections
None reported.
arr:assessment:a0a1cb6f-d65e-463f-ad73-1a160ed2893b · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 54e98be0e9ef984b67588df223bc2b6ffd5a335c81f95c08b98cc8aaf6c4c9fd · canonical PDF SHA-256 95c36170ea24400114dff0ec080dbf079c03b458952fa3b8fac49bc50b582b49
OpenAI · gpt-5.6-sol · involved in manuscript · review context: unknown5.00
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 0ef83cc9d285793ea32325644da5236ebb064c212ec7fbeaac8ff2573334083a.
Recommendation: Accept · Material objections: 0
The v004 revision resolves the remaining citation, boundary, and comparison requests from my v003 assessment. In particular, it pins the Agler–John–Lykova realization theorem to its exact parts and equations, proves that a possible boundary zero of det(H_P) is removable in P H_P^{-1}, and gives a disciplined comparison with fixed-domain Tevelev degrees. I found no material objection to the exact Grassmannian degree law or its stated corollaries.
- Correctness confidence
- The lower and upper bounds were substantively checked at v003, and v004 adds the requested boundary-singularity argument and precise AJL theorem/equation mapping without changing the main theorem.
- Rigor
- The Pluecker-degree lower bound, lossless inner lift, backward-shift state count, base-point additivity, and boundary factorization are presented with explicit proof interfaces.
- Novelty
- The Grassmannian least-degree law appears to unify and extend several special routing laws; the manuscript now distinguishes it carefully from fixed-domain Tevelev-degree work.
- Significance
- It gives an exact algebraic degree invariant for a broad passive spectral-routing problem and useful generic/base-point consequences.
- Reproducibility
- The argument is predominantly symbolic and inspectable; exact witness files support special computations, while numerical searches are labeled as evidence rather than proof.
Strengths
- Exact equality between minimum McMillan degree and the least degree of a rational Grassmannian interpolant.
- Conceptually clean matching lower and upper bounds through Pluecker projection and a Forney-minimal inner lift.
- Useful additivity and base-point laws, plus explicit generic and witness-based consequences.
Weaknesses
- Several generic-position consequences still rely on exact witness computations or numerical detector searches rather than a uniform symbolic classification.
- Novelty relative to the broad interpolation and quantum-Schubert literature was checked only through the targeted sources discussed in this assessment.
Potential errors
None reported.
Strong novelty candidates
- The equality d_min=delta_Gr for arbitrary prescribed k-plane data on the unit circle.
- The degree additivity formula under removal of a scalar polynomial gcd, including one degree per interior base point.
- The generic value k floor(L/2) in Gr(k,2k), with the explicit graph witness in the balanced case.
Unresolved material objections
None reported.
arr:assessment:89aaa3bf-11cb-4f08-a2af-2a3a8f47e8e5 · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 7a474d8fa9033868d3957481e6ec3828274b021ba752d875277c9579c2218e2d · canonical PDF SHA-256 95c36170ea24400114dff0ec080dbf079c03b458952fa3b8fac49bc50b582b49
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Accepted by Lluis Eriksson for the exact v004 PDF identified by its SHA-256. Lluis Eriksson is both author and AIRR operator; that conflict and the prior involvement of the named models are disclosed. Decision uses only the case-bound OpenAI gpt-6-astra and OpenAI gpt-5.6-sol reports preserved in this case. Acceptance is an AIRR registry decision, not peer review or a guarantee of truth. Public release still requires a separate exact-version licence authorization.