Abstract
We study a finite relational model in which a two-state register is coupled to a
spin-1/2 Heisenberg chain restricted to its total-singlet sector. Every term of the
Hamiltonian is a nearest-neighbour bond operator tensored with a register operator,
so the model looks spatially local. That inference fails. Writing the model as
H = I⊗(D + V/2) + Z_reg⊗(V/2) + λ X_reg⊗A, the conditional term Z_reg⊗(V/2)
gives the single register simultaneous edges to every bond in the extended region
V. We prove that for an observable O = I⊗A at the register's attachment point
and a bond B_j anywhere in V, the commutator [O(t), B_j] vanishes to second
order in t and its third-order coefficient factorises into a local factor and a
remote factor with disjoint supports. Its norm is therefore independent of the
separation and of the system size: exactly 1/16 for an interior bond and 1/32
at the edge, at every size tested. We further show that the global constraint is
not responsible. Restriction to an invariant sector inherits any Lieb–Robinson
bound, and two controls — deleting the conditional term, or removing the register
entirely — both recover a clean linear cone with identical threshold-crossing
times. The moral is that checking that each Hamiltonian term acts on neighbouring
sites is not sufficient to establish locality when a low-dimensional ancilla
carries a conditional coupling.
1. Why this is worth writing down
It is common, in relational and observer-centric models, to attach a small "register" or "pointer" to a many-body system and then reason about the composite as though it were the original system with a local probe. The temptation is strengthened when the Hamiltonian is written as a sum of terms each of which acts on neighbouring sites: the model reads local.
We exhibit a concrete finite model where that reasoning fails, exactly and quantitatively. The failure is not exotic. It arises from a single conditional term in which a register operator multiplies a sum of spatially separated bond operators. In the interaction-graph picture the register becomes a hub with edges to an extended region, and the composite is a small-world graph rather than a line — even though no individual term couples distant sites to each other.
We also isolate what is not responsible. The model lives in a globally constrained sector (total spin zero), and it is tempting to blame the constraint, since a constrained sector does not factorise into independent site Hilbert spaces. That diagnosis is wrong, and we show why.
2. The model
Let K_ij = S_i · S_j on a chain of N = 2n spin-1/2 sites, and let
h_i = K_{i,i+1}. Split the chain at the midpoint and write
D = sum of bonds strictly inside the left region
V = sum of the remaining bonds (the cut bond and the right region)
A = h_1 (the register's attachment bond)
Restrict to the total-singlet sector S^2 = 0, of dimension the Catalan number
C_n (5, 14, 42, 132 for N = 6, 8, 10, 12). Adjoin a two-state register and set
H_λ = [[ D+V , λA ],
[ λA , D ]]
= I⊗(D + V/2) + Z_reg⊗(V/2) + λ X_reg⊗A .
The second form is an exact algebraic identity, verified in rational arithmetic.
Every term is a single bond operator tensored with a register operator, so the
matter interactions are nearest-neighbour. The register, however, appears in
Z_reg⊗(V/2) multiplying a sum over the whole region V.
3. The constraint is not the problem
Lemma (inheritance). Let P be an orthogonal projection with [P,H] = 0 and
[P,A] = [P,B] = 0, and let A_P(t) denote evolution generated by H_P = PHP on
the range of P. Then
[A_P(t), B_P] = P [A(t), B] P and hence ‖[A_P(t),B_P]‖ ≤ ‖[A(t),B]‖ .
Any Lieb–Robinson bound on the unconstrained system therefore descends to the
constrained one, with the same constants. This requires no tensor factorisation of
the constrained Hilbert space. For disjoint rotationally invariant bonds it gives
[K_12, K_45] = 0 ⟹ [P₀K_12P₀, P₀K_45P₀] = 0 — local commuting observables
survive the constraint intact.
The lemma gives ≤, never =. A four-spin counterexample: take
A = K_12 + K_13 + K_23 = (S²_{123} − (9/4)I)/2 and B = K_34. On the total
singlet, S_{123} = −S_4, so S²_{123} = 3/4 and A restricts to exactly
−(3/4)I, a multiple of the identity. The restricted commutator vanishes
identically while the full-space norm is √2/2. Equality of commutator norms,
threshold times, or optimal velocities does not follow from inheritance.
Numerically, for the plain chain H = Σ J_i h_i with A = h_1,
B_d = h_{d+1}, full-space and singlet-sector wavefronts agree on the tested grids
at N = 10 and 12, under uniform, mildly disordered, strongly disordered and
dimerised couplings, with the inheritance inequality satisfied to ~1e-15. The
observed near-equality is a property of this observable family, not a theorem.
4. The register is the problem: an exact short-time obstruction
Take O = I⊗A and B_j = I⊗h_j for h_j a bond in V. Then
[ad_H^k(O), B_j] = 0 for k = 0, 1, 2
[ad_H^3(O), B_j] = ZX ⊗ [A,[D,A]] · [V, h_j]
so that
[O(t), B_j] = (it)³/6 · ZX ⊗ [A,[D,A]]·[V,h_j] + O(t⁴).
The key structural point. [V, h_j] is nonzero for every bond in V, however
distant, because V contains all of them. The third-order coefficient is a product
of a local factor [A,[D,A]] supported near the register's attachment and a remote
factor [V,h_j] supported near bond j. Their supports are disjoint, so the norm
is the product of the norms, and it neither decays with separation nor depends on
the system size:
‖[A,[D,A]]‖ = √3/4 , ‖[V,h_j]‖ = √3/2 (interior) or √3/4 (edge)
third-order coefficient norm = (1/6)(√3/4)(√3/2) = 1/16 (interior)
= (1/6)(√3/4)(√3/4) = 1/32 (far edge)
Verified exactly at N = 8 and N = 10, with orders 0–2 vanishing to 1e-15 and
the coefficient identical at both sizes.
5. Controls
Two controls, run in the same space so that only the register's role differs, both
recover a clean cone. Threshold-crossing times of ‖[O(t),B_d]‖ past 0.01 at
N = 12:
d 0 1 2 3 4 5 6 7 8 9 10
actual model .25 0 .25 .50 .75 .75 .75 .75 .75 .75 .75
register local .25 0 .25 .50 .75 1.25 1.75 2.25 2.75 3.25 3.75
no register .25 0 .25 .50 .75 1.25 1.75 2.25 2.75 3.25 3.75
The controls are linear with increments 0.5; the actual model flattens from d = 4
— precisely the bonds appearing in V. Far-end commutator norm at t = 1:
N = 8 0.0278374 vs 0.00124303 (~22×)
N = 10 0.0279394 vs 0.0000185944 (~1,500×)
N = 12 0.0279870 vs 0.000000161147 (~175,000×)
The control decays exponentially in distance while the actual model stays flat, so the ratio grows without bound with chain length.
A locally attached register preserves the cone but is not dynamically identical
to no register: the two differ already at second order, by
ad²_loc(O) − ad²_free(O) = X⊗[A,[D,A]], of norm √3/4.
The hub is avoidable. This matters for the moral, and is due to Sol. Take the locally attached register and add a single onsite register term:
H_loc(λ,ω) = I⊗Σ_i K_{i,i+1} + λ X_r⊗K_12 + ω Z_r⊗I
The register now touches only bond 12 and itself; there is no extended
Z_r⊗V coupling. At ω = 0 this model has exactly one operationally invisible
direction, N = X_r⊗I, which commutes with H and is killed by both register
outcome compressions, giving history-effect rank 99 of 100 — with
ρ_± = I/10 ± N/20 an explicit pair of distinct positive states with identical
declared history statistics. Switching on the onsite term breaks that conservation
and restores rank 100 of 100, certified by a nonzero 100×100 minor.
So a genuinely local interaction graph and a maximally complete declared history algebra can coexist. The extended conditional coupling was not needed for the process result it was carrying, which is what makes its cost worth reporting.
6. Related work
Locality inheritance under invariant-sector restriction is elementary and is not claimed as new; the Lieb–Robinson framework it inherits from is standard. Operational area-law bounds for arbitrary local instruments on finite-range lattice Hamiltonians are established by Kull, Allard Guérin & Brukner, A spacetime area law bound on quantum correlations, npj Quantum Information 5, 48 (2019), arXiv:1807.09187; note that such results assume locality in a sense the model here fails, so they do not apply to it unmodified. Measurement-assisted locality has its own literature (Friedman, Yin, Hong & Lucas, arXiv:2206.09929). Quasi-locality machinery is surveyed in Nachtergaele, Sims & Young (2019).
7. Claim ceiling and limitations
What is claimed: an exact, size-independent, third-order obstruction to locality in one specific finite model, together with an identification of the single term responsible and controls that remove it.
What is not claimed. This is not a new locality theorem; §3's lemma is
elementary. It is not an asymptotic result: a fixed Taylor coefficient does not by
itself establish a uniform lower bound at fixed positive time, since higher orders
may depend on system size, so this is a short-time statement and does not exclude
every possible chain-distance Lieb–Robinson bound. The measured wavefront
velocities are finite-size, finite-threshold, grid-dependent slopes and are not
asymptotic propagation speeds. Nothing here derives a metric, a Newton constant, an
entropy–area law, or Einstein dynamics, and nothing here bears on quantum gravity.
Sizes are N ≤ 12; couplings are uniform except where stated; norms are float64
except where exact rational arithmetic is noted.
8. Contributions and provenance
The model is from the BCRD lane of the physics-lane programme. The register
decomposition and the diagnosis that the conditional term, not the constraint, is
responsible are due to the Codex collaborator ("Astra"), who also independently
reproduced the constrained-sector numerics with a different sector construction.
The exact third-order coefficient of §4 and the counterexample of §3 are due to the
ChatGPT reviewer ("Sol"). The BCRD-Hamiltonian cone measurements of §5, and
independent verification of all of the above in separate implementations, are due
to Claude Opus 5. Nine earlier claims made during this work were retracted and are
recorded, with reasons, in README.md §2 and CORRECTIONS-EXTERNAL-REVIEW.md.
No external expert has reviewed this. AI cross-checking between independent implementations is not peer review.
9. Reproducibility
All scripts are in the directory named above. Python 3.9+, numpy, sympy.
python3 bcrd_cone.py 12 # §5 controls
python3 verify_hub_taylor.py # §4 exact third-order coefficient
python3 verify_sol_counterexample.py # §3 counterexample
python3 commnorm.py 10 # §3 numerics