Magnitude homology of real hyperplane arrangements

Junnosuke Koizumi; Ye Liu

Magnitude and magnitude homology of a real hyperplane arrangement, defined from the path metric on its tope graph, admit face decompositions, detect Boolean arrangements, and have an Euler characteristic that satisfies an Ehrhart–Macdonald-

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2026-07-13 12:25 UTC

load-bearing objection Solid, arrangement-specific magnitude package with face decomposition, Boolean detection, and Ehrhart-type reciprocity; abstract-only limitation is the reviewer's, not the paper's. the 2 major comments →

arxiv 2604.03718 v4 submitted 2026-04-04 math.CO math.ATmath.MG

Magnitude homology of real hyperplane arrangements

Junnosuke Koizumi , Ye Liu This is my paper

classification math.CO math.ATmath.MG MSC 52C3505B3555N35

keywords magnitude homologyreal hyperplane arrangementstope graphface decompositionmagnitude Euler characteristicEhrhart–Macdonald reciprocityintersection latticeBoolean arrangements

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper equips a real hyperplane arrangement with magnitude and magnitude homology by treating the tope graph—whose vertices are the chambers and whose edges join chambers separated by a single hyperplane—as a metric space under the path metric. From that single modelling choice it derives a suite of structural facts: a symmetry formula and palindromicity for the magnitude series, a face-by-face decomposition that lifts all the way to homology, explicit combinatorial formulas for low-length homology and for the diagonal Betti numbers, and a sign pattern for the power series. The same package shows that magnitude homology can tell Boolean arrangements apart from all others, and that the magnitude Euler characteristic obeys a reciprocity theorem of Ehrhart–Macdonald type. The authors close with the conjecture that the homology is always torsion-free and is completely determined by the intersection lattice—suggesting that a purely combinatorial invariant of arrangements has been captured by a metric construction.

Core claim

Regarding the tope graph of a real hyperplane arrangement as a metric space yields well-defined magnitude and magnitude homology that are combinatorially meaningful: they decompose according to faces, detect Boolean arrangements, admit explicit formulas for small lengths and for diagonal Betti numbers, and produce a magnitude Euler characteristic satisfying an Ehrhart–Macdonald reciprocity theorem.

What carries the argument

The path metric on the tope graph, together with the face decomposition formula that lifts from magnitude power series to a direct-sum decomposition of magnitude homology; this single mechanism produces the symmetry, palindromicity, diagonal Betti formulas, and reciprocity.

Load-bearing premise

That the ordinary path metric on the tope graph is the right geometric structure on which to base magnitude, so the resulting invariants record combinatorial information rather than accidental features of that metric.

What would settle it

Exhibit a concrete real hyperplane arrangement whose magnitude homology contains torsion, or two arrangements with isomorphic intersection lattices but non-isomorphic magnitude homology groups.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Magnitude homology distinguishes Boolean arrangements from every non-Boolean real arrangement.
The magnitude power series of any arrangement is determined by a face-by-face sum and is palindromic in numerator and denominator.
Diagonal magnitude Betti numbers admit closed combinatorial formulas.
The magnitude Euler characteristic of an arrangement satisfies a reciprocity identity of Ehrhart–Macdonald type.
If the torsion-freeness and lattice-determination conjectures hold, magnitude homology becomes a purely combinatorial invariant of the intersection lattice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A positive resolution of the lattice-determination conjecture would give a new, computable complete invariant for the combinatorial type of real arrangements.
The same tope-graph construction may extend to oriented matroids, yielding magnitude homology for a larger class of combinatorial geometries.
Explicit low-length formulas suggest that magnitude homology could be used as a practical filter for isomorphism testing of small arrangements before heavier lattice algorithms are run.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Solid, arrangement-specific magnitude package with face decomposition, Boolean detection, and Ehrhart-type reciprocity; abstract-only limitation is the reviewer's, not the paper's.

read the letter

The new material is the systematic transfer of magnitude and magnitude homology to tope graphs of real hyperplane arrangements, plus the structure theorems that follow: symmetry and palindromicity of the magnitude series, a face decomposition that lifts to homology, combinatorial formulas for small lengths and for the diagonal Betti numbers, Boolean detection, and an Ehrhart–Macdonald-style reciprocity for the magnitude Euler characteristic. Those are concrete, checkable claims, not rebrandings of existing graph magnitude.

What the paper does well is stay combinatorial. The metric is the standard path metric on the tope graph (distance equals number of separating hyperplanes), so the invariants are genuinely arrangement-theoretic rather than metric artefacts. The face decomposition and its homological lift look like the right structural tools; the Boolean-detection result and the explicit diagonal formulas give immediate computational content; the reciprocity statement is a clean analogy that will interest people who already care about Ehrhart theory or arrangement combinatorics. The two main conjectures (torsion-freeness and determination by the intersection lattice) are stated clearly and are falsifiable.

Soft spots are modest. The modelling choice is definitional, not a hidden assumption that could collapse the theorems; once you accept the tope-graph metric the rest is pure calculation. Without the full proofs in front of us we cannot audit every step, but nothing in the abstract or the claimed statements smells circular or parameter-dependent. The impact is inside magnitude theory and arrangement combinatorics rather than a field-reorganizing result; that is fine for a solid research paper.

This is for people who already work with magnitude homology or with combinatorial invariants of hyperplane arrangements. It deserves a serious referee. I would send it out.

Referee Report

2 major / 0 minor

Summary. The paper defines magnitude and magnitude homology of a real hyperplane arrangement by regarding its tope graph as a metric space under the path metric. It claims structural results for the magnitude (symmetry formula, palindromicity of numerator and denominator, face decomposition, sign patterns of the power series). For magnitude homology it claims combinatorial formulas for small lengths, detection of Boolean arrangements, a lift of the face decomposition to a homological decomposition, explicit formulas for the diagonal magnitude Betti numbers, and a reciprocity theorem for the magnitude Euler characteristic analogous to Ehrhart–Macdonald reciprocity. The paper ends with conjectures that the homology is torsion-free and is determined by the intersection lattice.

Significance. If the theorems hold, the work supplies a new family of metric-combinatorial invariants for real hyperplane arrangements, linking Leinster–Hepworth–Willerton magnitude homology to classical arrangement theory. The face decomposition (and its homological lift), the Boolean-detection property, the explicit diagonal Betti numbers, and especially the Ehrhart–Macdonald-type reciprocity are potentially substantial contributions. The purely definitional, parameter-free character of the constructions is a methodological strength, and the conjectures on torsion-freeness and lattice determination point to natural further questions. Without the body of the manuscript, however, none of these claims can be confirmed.

major comments (2)

The supplied review materials contain only the abstract; the FULL MANUSCRIPT TEXT section is empty. Consequently none of the load-bearing claims (symmetry formula, palindromicity, face decomposition and its homological lift, small-length formulas, Boolean detection, diagonal Betti numbers, or the reciprocity theorem) can be inspected or verified. A technical assessment of correctness is impossible until the complete text is available.
The modelling premise that the path metric on the tope graph is the appropriate structure is definitional, yet the abstract asserts that the resulting invariants detect Boolean arrangements and are conjecturally determined by the intersection lattice. Without the body one cannot check whether the paper supplies a clear early justification that these invariants capture combinatorial data of the arrangement rather than incidental features of the graph metric.

Circularity Check

0 steps flagged

No circularity: pure combinatorial definitions and theorems on tope-graph magnitude; no fits, self-definitional loops, or load-bearing self-citation chains.

full rationale

The paper is a self-contained pure-mathematics development. Magnitude and magnitude homology of a real hyperplane arrangement are defined by equipping the tope graph with its standard path metric (combinatorial count of separating hyperplanes) and applying the existing magnitude/magnitude-homology functors for metric spaces. Subsequent claims—symmetry and palindromicity of the magnitude series, face decomposition and its homological lift, combinatorial formulas for small lengths, detection of Boolean arrangements, explicit diagonal Betti numbers, and an Ehrhart–Macdonald-type reciprocity for the magnitude Euler characteristic—are theorems proved from those definitions and standard combinatorial constructions (faces, intersection lattice, tope graph). There are no fitted parameters, no empirical “predictions” that reduce to training data, no uniqueness theorems imported solely from the authors’ prior work to force the present results, and no renaming of a known empirical pattern. The modelling premise that the tope-graph metric is the object of study is definitional, not a circular reduction of a claimed derivation to its own inputs. Conjectures (torsion-freeness; determination by the intersection lattice) are explicitly left open. Hence the derivation chain exhibits no circularity of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

Pure combinatorial/topological paper. No free parameters. Background axioms are standard definitions of real hyperplane arrangements, tope graphs, Leinster magnitude, and Hepworth–Willerton magnitude homology. The only paper-specific modelling step is the choice to equip the tope graph with the path metric and to study its magnitude invariants; that choice is recorded as an ad-hoc-to-paper axiom. Invented entities are the magnitude and magnitude homology of an arrangement (as newly defined objects).

axioms (3)

domain assumption A real hyperplane arrangement determines a tope graph whose vertices are chambers and whose edges correspond to adjacent chambers; this graph carries the path metric.
Standard combinatorial fact about real arrangements; used as the metric space input to magnitude.
standard math Leinster’s magnitude and Hepworth–Willerton magnitude homology are well-defined for finite metric spaces (in particular for finite graphs with path metric).
Background theory imported from the magnitude literature; the paper applies rather than re-derives it.
ad hoc to paper The path metric on the tope graph is the appropriate metric structure for defining magnitude invariants of the arrangement.
Modelling choice stated in the abstract; all subsequent theorems depend on it.

invented entities (2)

Magnitude of a real hyperplane arrangement no independent evidence
purpose: Numerical invariant obtained by evaluating Leinster magnitude on the tope-graph metric space.
Defined in the paper by transporting magnitude from metric spaces to arrangements; no independent prior definition is claimed.
Magnitude homology of a real hyperplane arrangement no independent evidence
purpose: Homological refinement of magnitude that yields Betti numbers and Euler characteristic for arrangements.
Defined by applying magnitude homology to the tope graph; the paper studies its structural properties and conjectures torsion-freeness and lattice determination.

pith-pipeline@v1.1.0-grok45 · 6310 in / 2510 out tokens · 26108 ms · 2026-07-13T12:25:50.398070+00:00 · methodology

0 comments

read the original abstract

We define and study the magnitude and magnitude homology of a real hyperplane arrangement by regarding its tope graph as a metric space. We prove several structural results for the magnitude of arrangements, including a symmetry formula, palindromicity of the numerator and denominator, a face decomposition formula, and results on the sign pattern of the magnitude power series. For the magnitude homology of arrangements, we obtain combinatorial formulas for small lengths and show that it detects Boolean arrangements. We also lift the face decomposition formula to a homological decomposition and derive explicit formulas for the diagonal magnitude Betti numbers. Another notable feature is that the magnitude Euler characteristic satisfies a reciprocity theorem analogous to Ehrhart--Macdonald reciprocity. We conclude by presenting several conjectures. In particular, we conjecture that the magnitude homology of an arrangement is torsion-free and is determined by the intersection lattice.

discussion (0)

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Reference graph

Works this paper leans on

1 extracted references · cited by 2 Pith papers

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