gpt 6 astra ultra attempt on Phys-386

Accelerated Information Propagation in Bosonic Lattices

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

A bosonic lattice has unbounded local occupation numbers. For local observables $A$ and $B$, propagation can be measured by $[A(t),B]$, where $A(t)=e^{itH}Ae^{-itH}$. An effective light cone bounds the spatial range where this commutator can be appreciable. Acceleration means faster spreading than a uniform finite-velocity cone in the source's state-dependent setting.

Catalogue wording retained for traceability.

When does accelerated information propagation occur in bosonic lattice systems with nearest-neighbor or long-range interactions?

Scope and source qualification.

The allowed states and interaction decay matter because local bosonic operators are unbounded. Both nearest-neighbor and long-range interactions remain within the question.

Short English statement

Identify when information in bosonic lattices propagates faster than a uniform finite-speed light cone.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Solved two-site hopping exactly to reject matrix-element growth alone as evidence of accelerated information transfer.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30005663. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

Large hopping matrix elements do not alone prove acceleration.

In a two-site free bosonic system let $a_j,a_j^*$ obey $[a_i,a_j^*]=\delta_{ij}$ and take $H=J(a_1^*a_2+a_2^*a_1)$ with real $J\ne0$. On occupation vectors the transition matrix element from $|n,n\rangle$ to $|n+1,n-1\rangle$ is $J\sqrt{n(n+1)}$. This grows without bound, tempting an occupation-dependent speed conclusion.

Exact countercheck.

The Heisenberg equations instead close: \[ \dot a_1(t)=-iJ a_2(t),\qquad a_1(t)=\cos(Jt)a_1-i\sin(Jt)a_2. \] The analogous formula exchanges the indices. In the state $|N,0\rangle$, with $N\geq1$, one obtains \[ \langle a_2^*(t)a_2(t)\rangle=N\sin^2(Jt). \] The fraction transferred and the first complete transfer time $\pi/(2|J|)$ are independent of $N$. This follows directly by substituting the linear solution; no unbounded-operator norm estimate is taken. The finite-particle sectors justify the differentiations.

Remaining propagation problem.

These two sites provide no spatial light-cone theorem. They show that increasing individual occupation-basis matrix elements is insufficient evidence for accelerated transmission of a fixed fraction of a signal. The source question still requires admissible state classes, a quantitative information observable and spatially extended dynamics; interactions or long-range hopping may change the conclusion. Neither the nearest-neighbor nor the long-range source regime is classified by this free calculation.

Verification and adversarial audit.

Checked occupation square roots, the Heisenberg signs, finite-sector domains and N-independent fractional transfer time; no operator-norm claim for unbounded fields. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked occupation square roots, the Heisenberg signs, finite-sector domains and N-independent fractional transfer time; no operator-norm claim for unbounded fields. First remaining gap: Construct and control spatially extended state-dependent signals in the source nearest-neighbor and long-range interacting models. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the occupation matrix element and finite-sector Heisenberg transfer formula, with occupation-independent transferred fraction. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Construct and control spatially extended state-dependent signals in the source nearest-neighbor and long-range interacting models.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30005663 (OWR-14297742-003). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Many-Body Quantum Systems $2023$, Oberwolfach Reports, DOI 10.4171/owr/2023/39, pp. 2247--2302. Original report and mathematical context. https://doi.org/10.4171/owr/2023/39.