gpt 6 astra ultra attempt on Phys-99

Short-Range Spin Glasses

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.

Definitions and notation.

In the Edwards--Anderson model, $\sigma_x\in\{-1,1\}$ and $H_\Lambda(\sigma)=-\sum_{\langle x,y\rangle}J_{xy}\sigma_x\sigma_y$ at zero external field. The nearest-neighbour $J_{xy}$ are independent and identically distributed, with mean zero and finite variance. Infinite-volume Gibbs measures satisfy the conditional Gibbs laws; pure states are their extreme points. Spin flip sends all spins to their negatives. A metastate is a probability measure on Gibbs measures, encoding possible volume-dependent limiting states.

Catalogue wording retained for traceability.

For almost every coupling realization on $\mathbb Z^d$, prove or disprove a thermodynamic transition above a lower critical dimension $d_c$, and determine $d_c$ if a transition exists. Decide whether some low-temperature phase breaks spin-flip symmetry. For each applicable $d,T$, determine the number of pure-state pairs. If there are infinitely many, establish or refute the source’s proposed metastate picture: infinitely many mixed states, each the equal-weight mixture of a single spin-flip pair, with the visible pair changing with volume. If multiple Gibbs states exist, determine whether that phase persists under a sufficiently small external magnetic field.

Scope and source qualification.

The final small-field question and the almost-sure quantifier are restored from the original source; they were omitted from the short catalogue summary.

Short English statement

Determine whether short-range spin glasses have a phase transition, how many pure-state pairs occur, and how those pairs appear in large volumes.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Reconstructed almost-sure uniqueness for the one-dimensional zero-field iid chain at positive temperature. Higher-dimensional transition, lower critical dimension, pure-state/metastate structure and small-field persistence remain open here.

Sources and literature check.

Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.

A one-dimensional boundary-influence calculation.

At zero field on a finite chain, write $t_i=\tanh(\beta J_{i,i+1})$, with $0<\beta<\infty$. Summing the Boltzmann weight over one spin shows that its conditional transition kernel with a free forward endpoint is $K_i(s,s')=e^{\beta J_i ss'}/(2\cosh\beta J_i)=(1+ss't_i)/2$. Multiplication of these two-by-two kernels gives \[\mathbb E(\sigma_n\mid\sigma_0=s)=s\prod_{i=0}^{n-1}t_i.\] Thus the effect of a single fixed endpoint decays with the absolute product. With both endpoints fixed, the same expansion gives, for an interior spin $\sigma_k$, \[\mathbb E(\sigma_k\mid\sigma_0=s,\sigma_n=u)= \frac{s\prod_{i<k}t_i+u\prod_{i\ge k}t_i}{1+su\prod_{i<n}t_i}. \] The analogous finite-block conditional probabilities follow by multiplying the kernels through each exterior segment.

Almost-sure disappearance of the boundaries.

For any real-valued iid coupling law, choose finite $M$ with $\mathbb P(|J|\le M)>0$. Almost surely infinitely many bonds in either half-line satisfy this bound: independence makes the probability of no such bond in an arbitrarily long tail zero. At those bonds $|t_i|\le r=\tanh(\beta M)<1$, while all other factors are at most one. Therefore each long product tends to zero. The displayed endpoint formulas and their block versions then show that all finite-volume boundary choices converge to the same local law. Any infinite-volume Gibbs measure obeys these finite-interval conditional laws, so its finite-dimensional marginals equal that limit. Hence the one-dimensional zero-field Gibbs state is unique at every positive temperature under the stated iid assumptions.

Limits of the attempt.

This is an elementary known one-dimensional test, not a determination of the lower critical dimension. The limit $\beta=\infty$ invalidates $r<1$, and higher-dimensional loops prevent the chain-product argument. No low-temperature transition, pure-state multiplicity, metastate organization or persistence in a small magnetic field in the dimensions where multiple states might occur is established. All of those restored source questions remain part of the original target.

Verification and adversarial audit.

Checked arbitrary signed finite couplings, rare unbounded bonds, both endpoints, finite-block uniqueness, almost-sure quantifier and exclusion of zero temperature. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked arbitrary signed finite couplings, rare unbounded bonds, both endpoints, finite-block uniqueness, almost-sure quantifier and exclusion of zero temperature. Scope remains limited: Replace the one-dimensional product contraction by control of higher-dimensional frustrated loops at low temperature.

Coordinator checked the two-endpoint chain formula, almost-sure decay through infinitely many bounded couplings and finite-temperature restriction. The higher-dimensional spin-glass questions remain open here. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. 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. Replace the one-dimensional product contraction by control of higher-dimensional frustrated loops at low temperature.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 7800011 (AMR-077-0011). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 C. M. Newman and D. L. Stein, original contribution to Open Problems in Mathematical Physics $1998$, edited by Michael Aizenman. https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9803.SpinGlass.html.