Same local spectra, different quantum gaps

Summary

Can two quantum chains agree in every two-site and three-site energy measurement, yet differ when the chain becomes long? This candidate gives an exact family where they do.

The spectral gap is the energy needed to leave the ground-state space. A uniformly gapped chain retains a positive threshold as its length grows. A gapless chain can have excitations at progressively smaller energies. Here the familiar short-chain spectra, the pair's entanglement probabilities and even the number of ground states at every length all agree—but they do not determine which behaviour occurs.

The paper also proposes a complete classification within a precise class of interactions. Its dividing condition is geometric: how the two support planes of a forbidden neighbouring-pair state meet, and whether an extremal overlap can propagate along the chain.

The banner compares the common three-site energy levels with the family’s long-chain behaviour. The teal curve is a proved lower bound, not a plot of the actual gap. The gold endpoint has a different, length-dependent exact formula.

Summary for specialists

Let $H_N=\sum_{i=1}^{N-1}|\psi\rangle\langle\psi|_{i,i+1}$ on an open chain, with no boundary penalties, $N\ge2$, and finite local dimension $d\ge2$. The normalised forbidden vector has Schmidt probabilities $(1/2,1/2,0,\ldots)$. Local projector rank is one; Schmidt rank is two.

The candidate theorem says the chain is gapless precisely when, up to phase,

$$\psi=(u\otimes w-w\otimes v)/\sqrt2,$$

where $u,v,w$ are unit vectors and $w\perp u,v$. Every such balanced-marker interaction has the exact gap

$$\gamma_N=1-\cos(\pi/N).$$

All other interactions in this class are uniformly gapped. If the two Schmidt supports intersect in the line $\mathbb Cw$, let $t<1$ be the nontrivial principal-angle cosine and $\tau=2|\langle w,w|\psi\rangle|^2$. For $\tau>0$,

$$\gamma_N\ge\frac{3\tau(2-t)(1-t)}{1024}>0.$$

Coincident supports are gapless; disjoint supports have the positive bound $1-\|\Pi_L\Pi_R\|$.

Technical account

The complete qutrit short-spectrum fibre has the on-site-unitary normal form

$$\psi_\theta=(\cos\theta\,|00\rangle+\sin\theta\,|01\rangle-|12\rangle)/\sqrt2,$$

with $0\le\theta\le\pi/2$. Before the endpoint, $\gamma_N\ge\cos^2\theta/6$. At the endpoint, the balanced-marker formula applies. Throughout the family, the two-site spectrum is $0^{(8)},1$, the three-site spectrum is $0^{(21)},1/2,1^{(4)},3/2$, and the ground-space dimension is $F_{2N+2}$, using $F_0=0,F_1=1$.

Four overlapping sites supply information absent from those spectral lists. A range-Gram decomposition establishes the finite-size inequality used in the fibre bound. One negative direction of the Gram difference lies in the kernel of the concatenated range map; the argument does not claim positivity of the whole matrix. The revised appendix gives a basis construction covering the full parameter interval, including both endpoints.

For the general classification, a quantitative obstruction to simultaneous overlap saturation separates the gapped case from compatible marker propagation. The open-boundary argument connects local windows to the full Hamiltonian. Complementary dimer, rank-one perturbation and opposing-bias results retain their own hypotheses; the opposing-bias result is an exponential upper bound, not a matching asymptotic estimate.

Evidence, assurance and limitations

The archive includes analytic proofs, seven exact full-four-site positivity certificates, 43 exact test groups, 146 full-Hamiltonian numerical cases and 24 complex relative-form checks. All five historical checking programs were replayed. Separately supplied referee code also passed its full-Gram and numerical checks. These are finite producer-side checks, not a proof of every parameter and length or an authenticated independent reproduction.

The supplied review exposed a real software boundary: generic symbolic ranks can hide exceptional parameter values. The executable classifier now accepts only parameter-free exact inputs and rejects free symbols. Tests cover the reviewer's gapless and gapped specializations; a deliberately broken classifier must fail, both normally and with Python assertions disabled.

The manuscript remains unrefereed and unformalised. This is not a classification of unequal Schmidt probabilities, arbitrary rank-two projectors, periodic chains or infinite-volume GNS gaps. Equality of the stated short-chain spectra does not imply equality of full spectra at larger lengths.

Relationship to earlier work

Bravyi–Gosset's qubit classification is an explicit predecessor. The ground-space recurrence appears in Movassagh and collaborators' earlier work; the unbiased one-particle hopping gap is also present in the Motzkin-chain literature. The revision supplies precise locators for both. Those ingredients alone are not the proposed novelty.

The contribution offered for scrutiny is the complete flat-Schmidt-rank-two dichotomy, its quantitative saturation argument, and persistence of the gap distinction across the complete specified short-spectrum fibre. The bounded source search does not establish historical priority.

Who should care, and why

Why the problem matters

Small systems are easier to calculate than long chains. Agreement on small-system energy spectra can therefore look more informative than it is. This family isolates the missing information exactly: identical short spectral lists and ground-state counts coexist with different long-chain gaps. It also identifies a geometric mechanism that separates the two behaviours within the stated class.

How to inspect or reproduce the recorded checks

Start with the linked repository's AI_INDEX.md, STATUS.md and ASSURANCE.md. Install requirements-recorded.txt in Python 3.13 and check the manifest before running programs that regenerate reports:

python code/verify_manifest.py
sh reproduce.sh
python code/publication_controls.py
python -OO code/publication_controls.py

Use a fresh extraction. The final release receipt binds the archive and exact-commit Linux CI. Numerical diagnostics are distinct from exact certificates, and both are distinct from the written universal proof. The classifier requires exact constant entries; specialise symbolic parameters first.

The most valuable next projects

An unaffiliated audit of the saturation inequality, the normal-form completeness argument and the full range-Gram basis would most improve assurance. A separately implemented exact classifier would challenge more than a replay of the same code. Unequal Schmidt probabilities and different boundary conditions are separate research questions, not included extensions.

What is in the evidence package

Original prose and data are CC0-1.0; original code is MIT. Preserved third-party material retains its own rights.

@misc{flatschmidtchains2026,
  title        = {Same local spectra, different quantum gaps},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23024569},
  url          = {https://doi.org/10.5281/zenodo.23024569},
  version      = {1.0.1-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/flat-schmidt-chains/}
}