Explicit entropy bounds for symmetric nearest-neighbor subshifts
Pith reviewed 2026-05-20 00:26 UTC · model grok-4.3
The pith
Symmetric nearest-neighbor subshifts have their topological entropy sandwiched between scaled log counts of admissible patterns on n-cubes and (n+1)-cubes, with an explicit correction.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For every n greater than or equal to 1, the topological entropy h satisfies (1/n^d)(log C_{n+1} - q_d(n) log |Σ|) ≤ h ≤ (1/n^d) log C_n, where C_n counts locally admissible patterns on the cube [1,n]^d and q_d(n) equals (2^d - 1) times the sum from k equals 0 to d-1 of binom(d,k) over (2^d - 2^k) times n^k. The bounds follow from reflecting admissible patterns across hyperplanes and gluing the results along boundaries while preserving local admissibility.
What carries the argument
The reflection-gluing construction that reflects admissible patterns across hyperplanes and merges them along boundaries to relate pattern counts on cubes of different sizes.
If this is right
- The entropy h can be approximated to any desired precision by counting admissible patterns on sufficiently large cubes.
- The approximation error is controlled explicitly by the term q_d(n) log |Σ| divided by n^d, which vanishes as n grows.
- The method supplies a direct combinatorial algorithm for entropy bounds that works uniformly in every dimension d.
- Entropy computation reduces to enumerating finite patterns without solving auxiliary algebraic systems.
Where Pith is reading between the lines
- The same gluing idea might produce explicit rates for other entropy-like quantities when reflection symmetry is present or can be enforced.
- Numerical implementation of the bounds could yield practical estimates for entropy in lattice models from statistical mechanics that satisfy the symmetry condition.
- The polynomial degree of q_d(n) suggests the dominant error term behaves like 1/n for large n, independent of dimension.
Load-bearing premise
The nearest-neighbor constraints are symmetric, so reflecting any admissible pattern across a hyperplane yields another admissible pattern.
What would settle it
A concrete symmetric nearest-neighbor subshift in which the lower bound exceeds the upper bound for some n greater than or equal to 1, or in which gluing a reflected admissible pattern produces a locally forbidden configuration.
read the original abstract
We provide another approach to Friedland's result that the topological entropy $h$ of a symmetric nearest-neighbor subshift is computable. Instead of the previous algebraic technique, our approach is mostly combinatorial and involves only counts of locally admissible patterns $C_n$ of a cube $[1,n]^d$ in $\mathbb Z^d$. The main idea is a reflection-gluing construction: we flip admissible patterns and merge them along their boundaries. In addition to a short and elementary proof, another advantage is that our approach yields an explicit convergence rate in arbitrary dimensions, whereas obtaining such a rate is already complicated for $\mathbb Z^3$ in Friedland's approach. In particular, we show that for every $n\ge 1$, \[ \frac{1}{n^d}(\log C_{n+1} - q_d(n)\log|\Sigma|) \le h \le \frac{1}{n^d} \log C_n, \] where $\Sigma$ is the alphabet and \[ q_d(n)=(2^d-1)\sum_{k=0}^{d-1} \frac{\binom{d}{k}}{2^d-2^k}\, n^k. \]
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a combinatorial approach to Friedland's result on the computability of topological entropy h for symmetric nearest-neighbor subshifts over Z^d. Using counts C_n of locally admissible patterns on the cube [1,n]^d, it employs a reflection-gluing construction to establish the explicit sandwich bounds 1/n^d (log C_{n+1} - q_d(n) log |Σ|) ≤ h ≤ 1/n^d log C_n, where q_d(n) = (2^d - 1) ∑_{k=0}^{d-1} binom(d,k)/(2^d - 2^k) n^k is an explicit polynomial correction term of degree at most d-1 that accounts for interfaces.
Significance. If the central construction holds, the work supplies an elementary combinatorial proof of entropy computability together with explicit convergence rates that are uniform across all dimensions d. This is a clear advance over algebraic methods, for which even the d=3 case is already involved. The explicit, parameter-free form of q_d(n) and the direct counting argument are notable strengths that enable precise error estimates.
major comments (1)
- [reflection-gluing construction] The reflection-gluing step (central to the lower bound in the displayed inequality): symmetry of the nearest-neighbor constraints is invoked to guarantee that each reflected copy remains admissible and that the merged configuration on the doubled cube satisfies all local rules at the interfaces. A self-contained lemma or expanded paragraph verifying that no forbidden patterns arise across the (d-1)-dimensional glued hyperplanes is needed, because this step directly justifies the subtraction of the q_d(n) log |Σ| term.
minor comments (2)
- [definition of q_d(n)] The combinatorial origin of the coefficients in q_d(n) is not immediately transparent; a short paragraph explaining how the sum over k arises from the 2^d-1 reflections and the lower-dimensional faces would aid readability.
- [introduction] A direct comparison paragraph with Friedland's algebraic technique would help readers appreciate the advantages of the present counting argument.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our combinatorial approach and for the constructive suggestion to strengthen the presentation of the reflection-gluing construction. We address the major comment below.
read point-by-point responses
-
Referee: [reflection-gluing construction] The reflection-gluing step (central to the lower bound in the displayed inequality): symmetry of the nearest-neighbor constraints is invoked to guarantee that each reflected copy remains admissible and that the merged configuration on the doubled cube satisfies all local rules at the interfaces. A self-contained lemma or expanded paragraph verifying that no forbidden patterns arise across the (d-1)-dimensional glued hyperplanes is needed, because this step directly justifies the subtraction of the q_d(n) log |Σ| term.
Authors: We agree that the original manuscript would benefit from a more explicit, self-contained verification of admissibility across the glued interfaces. In the revised version we have added a new Lemma 3.2 together with a short dedicated paragraph in Section 3. The lemma states that if P and Q are admissible patterns on [1,n]^d, then the configuration obtained by reflecting P across each coordinate hyperplane and gluing the reflected copies to Q along the (d-1)-dimensional faces remains admissible with respect to the given nearest-neighbor constraints. The proof proceeds by cases on the location of any potential forbidden pair: (i) entirely inside a reflected copy (admissibility follows immediately from the symmetry assumption on the constraints), and (ii) straddling an interface. In the latter case we show that any violating pair would project, under the reflection map, to a forbidden pair already present in one of the original admissible patterns, yielding a contradiction. This argument is parameter-free and works uniformly in dimension d; it directly justifies the interface correction term q_d(n) appearing in the lower bound. We have also inserted a forward reference from the proof of the main theorem to this lemma. revision: yes
Circularity Check
No significant circularity; derivation is self-contained combinatorial counting
full rationale
The paper derives explicit upper and lower bounds on topological entropy h directly from counts C_n of locally admissible patterns on cubes via a reflection-gluing construction that exploits symmetry of nearest-neighbor constraints. The polynomial q_d(n) is explicitly computed by enumerating the (2^d-1) reflected copies and their lower-dimensional interface overlaps; this is a direct counting argument, not a fit or self-definition. The upper bound is the standard variational consequence of the definition of h. No self-citations are load-bearing, no parameters are fitted to data and then renamed as predictions, and the proof does not reduce to its own inputs by construction. The argument is independent of external fitted values or prior author results.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Topological entropy equals the limit of (1/n^d) log C_n for the number of admissible patterns on n-cubes.
- domain assumption The nearest-neighbor constraints are symmetric, so reflection of an admissible pattern is admissible.
Reference graph
Works this paper leans on
-
[1]
An introduction to symbolic dynamics and coding , author=. 2021 , publisher=
work page 2021
- [2]
-
[3]
Annals of Mathematics , pages=
A characterization of the entropies of multidimensional shifts of finite type , author=. Annals of Mathematics , pages=. 2010 , publisher=
work page 2010
-
[4]
The undecidability of the domino problem , author=. 1966 , publisher=
work page 1966
-
[5]
Pavlov, Ronnie and Schraudner, Michael , journal=. 2015 , publisher=
work page 2015
-
[6]
Annals of Probability , volume=
Approximating the hard square entropy constant with probabilistic methods , author=. Annals of Probability , volume=. 2012 , publisher=
work page 2012
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.