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PureOne
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eve-symmetry-resolved-krylov-complexity-proof

id string	title string	statement_latex string	scope string	assumptions list	proof_status string	proof_location string	dependencies list	limitations string	version string
KRYLOV-T1	Krylov-index tail domination	\sum_{j\ge n}\|a_j(t)\|^2\ge\sum_qp_q\sum_{j\ge n}\|a_{q,j}(t)\|^2	All real times and integer n >= 1	[ "A is self-adjoint on a finite-dimensional Hilbert space", "Sectors are mutually orthogonal reducing subspaces", "The global seed and nonzero sector seeds are normalized", "Sector weights are squared projection norms; zero weights are omitted" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md section 2; PDF Theorem 1	[]	No requirement that each individual sector complexity is below the global complexity.	1.0.0
KRYLOV-T2	All-time symmetry-resolved complexity inequality	C(t)\ge\sum_qp_qC_q(t)	All real times in finite dimension	[ "A is self-adjoint on a finite-dimensional Hilbert space", "Sectors are mutually orthogonal reducing subspaces", "The global seed and nonzero sector seeds are normalized", "Sector weights are squared projection norms; zero weights are omitted" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md sections 2 and 3	[ "KRYLOV-T1" ]	Does not establish a universal diagnostic of quantum chaos.	1.0.0
KRYLOV-I1	Exact positive projection gap	E_n(\mu,t)-\sum_qp_qE_n(\mu_q,t)=\\|(R_n-P_n)v(t)\\|^2	Every Krylov tail	[ "A is self-adjoint on a finite-dimensional Hilbert space", "Sectors are mutually orthogonal reducing subspaces", "The global seed and nonzero sector seeds are normalized", "Sector weights are squared projection norms; zero weights are omitted" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md section 2 equation (3)	[ "KRYLOV-T1" ]	Nested projections must refer to global and resolved polynomial spaces of the same degree.	1.0.0
KRYLOV-I2	Exact Gram coefficient gap	E_n(\mu,t)-\sum_qp_qE_n(\mu_q,t)=\sum_qp_q(c-c_q)^*G_{n,q}(c-c_q)	Finite measures and necessary moments	[ "Weights are nonnegative and sum to one", "c and c_q are exact least-squares minimizers", "G matrices are moment Gram matrices; singular matrices are allowed" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md section 3 equation (9)	[ "KRYLOV-I1" ]	Fourier-weighted moments are needed; six ordinary moments are not enough for exact all-time evaluation.	1.0.0
KRYLOV-C1	Increasing index costs	\sum_jh(j)\|a_j\|^2\ge\sum_qp_q\sum_jh(j)\|a_{q,j}\|^2	All nondecreasing real functions h on nonnegative integers	[ "A is self-adjoint on a finite-dimensional Hilbert space", "Sectors are mutually orthogonal reducing subspaces", "The global seed and nonzero sector seeds are normalized", "Sector weights are squared projection norms; zero weights are omitted", "h is nondecreasing" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md section 4 Corollary 2	[ "KRYLOV-T1" ]	Finite-dimensional statement; expectations in infinite dimension need appropriate extended-value interpretation.	1.0.0
KRYLOV-E1	All-time equality criterion	[C(t)=\sum_qp_qC_q(t)\ \forall t]\iff[\mu_q=\mu_{q\prime}\ \forall q,q\prime\text{ with positive weight}]	Finite dimension	[ "A is self-adjoint on a finite-dimensional Hilbert space", "Sectors are mutually orthogonal reducing subspaces", "The global seed and nonzero sector seeds are normalized", "Sector weights are squared projection norms; zero weights are omitted" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md section 4 Proposition 3	[ "KRYLOV-I1" ]	Equality at one isolated time does not imply equal measures.	1.0.0
KRYLOV-X1	Infinite-dimensional extension	C(t)\ge\sum_qp_qC_q(t)\quad\text{in }[0,\infty]	Countably many reducing sectors	[ "Self-adjoint generator with reducing sector projections", "Seed lies in the domains of required powers", "Evolved global and sector vectors lie in their closed polynomial Krylov spaces" ]	analytic proof supplied; internally reviewed; no independent certification	MANUSCRIPT.md section 5	[ "KRYLOV-T1" ]	Power-domain and polynomial-completeness assumptions are essential to the proof; an unrestricted unbounded theorem is not claimed.	1.0.0

All-Time Symmetry-Resolved Krylov Complexity Inequality

Proof of the Caputa-Di Giulio-Loc conjecture in finite dimensions

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Maintainer: Maciej Nowicki / PureOne
Release: 1.0.0 | Date: 27 September 2026

This standalone mathematical-physics release supplies an analytic proof that total Krylov complexity is at least the probability-weighted average of its symmetry-resolved complexities at every time:

$C(t)\geq\sum_q p_q C_q(t).$

The theorem applies to all finite-dimensional Hermitian generators with orthogonal invariant sectors, including repeated eigenvalues and terminating Lanczos chains. It treats quantum-state spread complexity and symmetry-invariant operator Krylov complexity. The stronger result orders every tail of the Krylov-index distribution, yielding inequalities for all increasing index costs.

Start with the PDF manuscript, the complete Markdown proof, or the expert review guide. All proof ingredients are included; the motivating earlier manuscript is not needed to follow the argument.

Status and completeness

Item	Status
Finite-dimensional all-time inequality	Complete analytic proof supplied
Stated finite-dimensional proof obligations	100% supplied, author assessment
Independent expert review	Not completed
Proof-assistant verification	Not performed
Numerical diagnostics	Passed on 103 mixtures at 81 times each
Bounded self-adjoint generators	Covered by the stated extension
Unbounded generators	Requires power-domain and polynomial-Krylov completeness assumptions
Worldwide novelty	Not certified; targeted search found no earlier general proof

The percentage describes supplied proof obligations, not a measured probability that the result is correct. Detailed machine-readable status is in STATUS.json. The original conjecture is identified in SOURCE_AUDIT.json, including its location in the published papers.

The central argument

The normalized spectral measure is a positive mixture, mu = sum_q p_q mu_q. The probability beyond the first n Krylov vectors is the best squared polynomial approximation error

$E_n(\mu,t)=\min_{\deg P<n}\int |e^{-itx}-P(x)|^2\,d\mu(x).$

Optimizing one polynomial across all sectors gives

$E_n\!\left(\sum_qp_q\mu_q,t\right)\geq\sum_qp_qE_n(\mu_q,t).$

Summing these tails proves the complexity inequality. The second proof uses the inclusion of the global degree-n-1 Krylov space in the direct sum of sector spaces. The difference of their nested orthogonal projections is positive.

The exact gap is a sum of squared norms. At fixed time, equality means that every degree admits a common best polynomial in all positive-weight sectors. In finite dimension, equality at every time holds exactly when all their normalized spectral measures coincide.

Why the result is useful

Sector calculations provide rigorous lower bounds on full-system complexity. The result clarifies which part of complexity can arise from combining symmetry sectors and does not assume chaotic dynamics. It also supplies spectral-measure concavity, tail domination, and a precise equality criterion for analysis and numerical diagnostics.

The motivating Cardano-Clocked Hypergraph Spectral Manifolds manuscript contributes positive spectral mixtures, moment Gram matrices, and the Lanczos interpretation. The argument is built from classical projection and approximation theory. Six moments alone do not determine exact all-time complexity, and this release makes no such claim.

Reproduce the diagnostics

Python 3.10 or newer is recommended. From the repository root:

python code/check_release.py
python -m pip install -r requirements.txt
python code/verify.py

The numerical checks cover random finite spectra, shared atoms, equal measures, zero-weight sectors, one-point sectors, an exact three-node example, and the singular-Gram gap identity. They do not replace the analytic proof. See REPRODUCIBILITY.md for details and recorded environment information.

Machine-readable research corpus

This dataset repository contains research artifacts and structured theorem/proof records, rather than trained model weights. Every theorem record includes its assumptions, proof status, location, dependencies and limitations.

from datasets import load_dataset
claims = load_dataset(
    "PureOne/eve-symmetry-resolved-krylov-complexity-proof",
    "theorems", split="claims"
)

Available configurations:

Configuration	Split	Content
`theorems`	`claims`	Theorems, identities and scope records
`proof_sections`	`sections`	Full Markdown sections for retrieval
`references`	`sources`	Primary-source audit and provenance

For AI agents, begin with llms.txt, CLAIM_LEDGER.json, and RESEARCH_METADATA.json. Cite the exact scope and distinguish the supplied proof from independent validation. No benchmark score or statistical generalization claim is attached to this small corpus.

Repository map

Path	Purpose
`manuscript/`	Typeset PDF and editable LaTeX source
`MANUSCRIPT.md`	Complete proof in searchable Markdown
`EXPERT_REVIEW.md`	Proof obligations, scope and review questions
`CLAIM_LEDGER.json`	Stable claim IDs and proof dependencies
`data/`	JSONL theorem, proof-section and reference records
`code/`	Reproducible numerical and package checks
`verification/`	Numerical results and environment record
`CITATION.cff`, `CITATION.bib`	Citation metadata
`STATUS.json`, `SOURCE_AUDIT.json`	Verification and literature status
`MANIFEST.json`, `SHA256SUMS.txt`	Payload integrity and upload allowlist

Published problem and related work

The conjecture is stated by P. Caputa, G. Di Giulio and T. Q. Loc in Physical Review Research 7, 043055 (2025), after Eq. (7), and in JHEP 02 (2026), 189, after Eq. (4.25). Their preprints are 2507.02033 and 2509.12992.

January and April 2026 follow-ups discuss the inequality as conjectural: 2601.08723 and 2604.05630. 2608.19346v2 studies a distinct approximate late-time relation. The targeted literature check is dated 27 September 2026 and is not an exhaustive priority certification.

Citation and licensing

Use CITATION.cff or CITATION.bib. Authorship follows the creator's requested attribution. The manuscript and code were generated with AI assistance; this is a research release seeking independent scrutiny.

Original documentation and structured records: CC BY 4.0. Original Python code: MIT, as specified in LICENSE-CODE. Referenced third-party works retain their own rights and are linked, not redistributed. No claim of exclusive rights over mathematical facts is intended.

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Papers for PureOne/eve-symmetry-resolved-krylov-complexity-proof

Quantum chaos and late-time equipartition of symmetry-resolved Krylov complexity

Paper • 2608.19346 • Published Sep 14