The Physics of Pears: A Pataphysical Approach to Geometrodynamics

Marcello Poletti

REVIEW 3 major objections 4 minor 20 references

The Physics of Pears: A Pataphysical Approach to Geometrodynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper's central claim is that a finite graph of proximity relations can be approximately categorized as a three-dimensional space containing matter and energy, and that time defined as the measure of change is spontaneously…

desk verdict An honest, playful philosophical essay whose central technical claims are definitional or false; not a research paper, but a fair-minded reader might enjoy it as a provocation. read the letter →

arxiv 2411.11109 v2 pith:LBEFRK54 submitted 2024-11-17 physics.hist-ph

Marcello Poletti This is my paper · ORCID

classification physics.hist-ph

keywords spaceofpearsgeometrodynamicsAristoteliantimefinitegraphsemergentdimensionsLorentzinvariancepataphysicscosmologyasedgeloss

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is a deliberately speculative "pataphysical" toy theory that treats physical space as a finite undirected graph, called a "space of pears," in which the only structure is which points are proximal. Its central claim is that such a graph could appear to an observer inside it as a roughly curved three-dimensional space full of matter and energy, because any graph can be embedded without crossing edges in three-dimensional Euclidean space and the surrounding metric deformed to match the graph's distances. The paper also revives Aristotle's definition of time as the measure of change, sets $\Delta T = \Delta S / c$, and argues that this definition leads directly to the lightlike interval and, once internal motion is added for massive bodies, to Lorentz transformations. On that basis it sketches a cosmology in which the graph begins as a complete graph and evolves by losing edges, an inexorable "evaporation," so the universe expands at an increasing rate and ends as disconnected points. If these identifications are taken seriously, the distinction between continuous spacetime and a discrete relational structure becomes an accident of perspective rather than a fundamental boundary.

What carries the argument

The central object is the "space of pears": a finite, connected undirected graph in which distance means the length of the shortest chain of adjacent points, a structure the paper calls a finite proximity space. This object carries the spatial argument because it has no built-in dimension, angle, surface, or curvature, yet the paper claims that any such graph can be represented in three-dimensional Euclidean space without crossing edges and with the manifold metric deformed to reproduce the graph's distances, making three-dimensionality and continuity emergent rather than primitive. For time, the carrying identity is $\Delta T = \Delta S / c$, which converts spatial change into temporal ticks and yields the Minkowski lightlike metric. Adding internal circular motion for massive bodies yields Lorentz transformations. The graph-evolution rule—removal of an edge is one unit step into the future, addition one unit step into the past—carries the cosmological story.

What would settle it

A decisive test would be to construct a finite connected graph whose shortest-path distances cannot be matched by the geodesic distances of any three-dimensional Riemannian manifold; for instance, a star graph with three unit-length arms requires the center to support three distinct unit geodesic rays with pairwise leaf distances of two, and one can check whether any smooth three-dimensional metric realizes that pattern without introducing a shorter connection. If such a counterexample exists, the spatial claim fails; if none exists, the claim is supported.

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Extended reading notes

Core claim

The paper's central claim, stated on its own terms, is that a "space of pears"—a finite graph whose distances are shortest-path lengths—"can be approximately categorized as a three-dimensional space containing matter and energy." This rests on the assertion that any graph can be represented in three-dimensional Euclidean space without intersecting edges, after which the ambient space can be deformed so its metric coincides with the graph's metric. The paper further claims that time as the measure of change, written $\Delta T = \Delta S / c$, gives the lightlike interval $dx^2 + dy^2 + dz^2 - c^2 dt^2 = 0$, and that a body modeled as carrying internal circular motion obeys Lorentz transformations, making Aristotelian time "spontaneously relativistic." The cosmological corollary is that a pear universe begins as a complete graph, expands with increasing speed as edges are lost, and ultimately evaporates into isolated points.

Load-bearing premise

The load-bearing premise is that every finite graph can be embedded in three-dimensional Euclidean space without edge crossings and with the surrounding metric deformed to reproduce the graph's shortest-path distances, a claim the paper cites to earlier work without proof.

Editorial extensions

If this is right

If a pear space can be approximated by a curved three-dimensional manifold, then dimensions and continuity are not fundamental features of space but appearances of a discrete proximity structure.
If $\Delta T = \Delta S / c$ reproduces Lorentz transformations, a theory need not start from a spacetime continuum; relativistic time can be read as the measure of spatial change.
In this picture, massive bodies cannot reach speed $c$ because part of their change is internal motion, so mass behaves like internal energy and only massless objects travel at $c$ by definition.
A pear universe that starts as a complete graph and loses edges expands at an increasing rate until evaporation dominates, so the toy model contains a Big Bang and a final state of disconnected points.
Time is local in this framework: a change in one subgraph is a tick for that subgraph but not for another, giving time a thermodynamic and observer-dependent flavor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper, one could quantify how closely a three-dimensional Riemannian metric can approximate the shortest-path metric of a given finite graph, producing a distortion measure that would make the embedding claim testable on random graphs.
Going beyond the paper, the edge-loss dynamics could be simulated on random graphs to look for parameter-free scaling laws in the expansion rate and in the onset of evaporation, turning the cosmological sketch into a concrete statistical model.
Going beyond the paper, the identity $\Delta T = \Delta S / c$ suggests building a clock that measures elapsed time as accumulated spatial displacement rather than through an independent temporal coordinate, which would give the Aristotelian definition a direct operational meaning.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is an honest, playful essay, but the technical core is definitional and one central claim is flat wrong as stated.

What's actually new: the juxtaposition of a finite graph space with edge-removal time is unusual, and the philosophical idea that dimensionality is a constraint on distance relations is a useful intuition pump. The paper is also commendably clear about its own status—it calls itself a haphazard game, flags the cosmology as pseudo-scientific, and credits ChatGPT for the abstract. That transparency is rare.

What it does well: the discussion of metric spaces as a generalization of vector spaces, and the local-time/thermodynamic flavor analogy, are thought-provoking. The reading of Aristotle's time as measure of change is coherent.

Soft spots: the central 'result' is built into a definition. Eq. (4) sets ΔT = ΔS/c, and Eq. (5) is the same relation squared, so the lightlike Minkowski metric is assumed, not derived. The Lorentz invariance claim is delegated to the author's preprint [13] with 'a few steps,' which is not reproduced. That would be fine in a different genre, but here it carries the weight.

The bigger problem is Section II's bridge from arbitrary graph to 3-manifold. The stress test is correct: for K_{1,3}, triangle equality forces the center to lie on a geodesic between every pair of leaves, impossible for three distinct leaves in any geodesic metric space. So the exact 'metric coincides' claim is false for that graph. The paper says 'approximately,' but gives no distortion bound or quantifier. Without that, the 'space of pears can be approximately categorized as 3D' claim loses its only technical support.

The rest is qualitative cosmology and metaphysics: fun to read, but not something that can be checked.

Who this is for: readers interested in speculative philosophy of physics or in how toy models can be presented. It could be a lively reading-group discussion piece, but it is not a research contribution. The citation pattern is mostly fine, though the load-bearing items are self-cites to non-peer-reviewed preprints.

Recommendation: I would not send this to peer review as a physics paper; the technical bridge is false and the derivations are definitional. If a history/philosophy journal wanted to treat it as a philosophical essay, it would need heavy revision and a reworking or removal of the embedding claim. For a scientific venue, desk reject.

Referee Report

3 major / 4 minor

Summary. The paper proposes a speculative toy model, the "Physics of Pears," in which physical space is modeled as a finite undirected graph whose edges are removed over time. It claims that any such graph can be approximated by a three-dimensional curved Riemannian manifold with a coincident metric, so that a pear space would be categorized by an agent as three-dimensional space containing matter and energy; it further claims that defining time as the measure of change via ΔT = ΔS/c makes Aristotelian time "spontaneously relativistic," reproducing lightlike Minkowski intervals and Lorentz transformations. The paper also sketches a graph cosmology (edge removal as expansion, increasing diameter, evaporation) and argues for a metaphysical separation between physics and quantum mechanics. The author repeatedly acknowledges the lack of formal rigor and the playful, pataphysical character of the work.

Significance. If the central technical claims were established, the paper would offer a conceptually interesting illustration of how dimensionality and spacetime could emerge from a finite combinatorial structure. Its strengths are its explicitness about the toy model's basic definitions, its clear statement of the finite graph picture, and its honest caveats about the speculative nature of the cosmological and metaphysical chapters. However, the two load-bearing technical assertions are not demonstrated: the graph-to-manifold metric equivalence is false as stated, and the "spontaneously relativistic" time claim is an artifact of definitions rather than a derived consequence. The paper provides no falsifiable predictions, no numerical checks, and no proof beyond self-citations, so its significance as a physics contribution is currently low; its value is confined to the philosophical speculation it explicitly invites.

major comments (3)

[Section II] The central bridge claim, that "any graph can be represented in a three-dimensional Euclidean space without intersection of edges" and then the space deformed so that "the manifold's metric coincides with the metric of the graph," is false for even the four-vertex graph K_{1,3}. In K_{1,3}, with center p and leaves a, b, c, the graph metric gives d(a,b)=d(a,c)=d(b,c)=2 and d(p,a)=d(p,b)=d(p,c)=1. In any geodesic metric space, equality in d(a,b) ≤ d(p,a)+d(p,b) forces p to lie on a shortest geodesic from a to b; the same equality for the other leaf pairs would force three distinct geodesic directions from p to be pairwise opposite, which is impossible. No smooth Riemannian manifold, indeed no geodesic metric space, reproduces this finite metric exactly. Since the paper provides neither a distortion bound nor a precise definition of the quantifier "approximately," the claim that a space of pears "can be approximately categorized as a three-dimensional space containing matter and energy" loses its only technical support.
[Section III, Eqs. (4)-(5)] Equation (4), ΔT = ΔS/c, is a definition of time as spatial path length divided by a constant c, and Eq. (5) is simply that same definition squared for infinitesimal motion. Thus the lightlike Minkowski interval ds^2 = dx^2+dy^2+dz^2−c^2dt^2 = 0 is put in by construction, not derived. The statement that "Aristotelian time is spontaneously relativistic" is therefore circular unless the Lorentz transformations are actually obtained from the internal-motion model; however, the derivation is delegated to "A few steps [13]" and reference [13] is a self-citation whose content is not reproduced in this paper. As it stands, the claim that time as change plus internal circular motion yields Lorentz invariance is unverified.
[Section III, internal motion paragraphs] The model of a material point with internal circular motion is introduced without any equations of motion or a precise relation between internal motion and translation. The assertions that "the body cannot, by definition, reach or exceed velocity c," that "internal motion is slowed in the presence of external motion," and that "mass is a form of motion, or internal energy" are not consequences derived from the graph model or from Eq. (4); they are ad hoc postulates. Because these postulates carry the entire weight of the claimed Lorentz invariance for massive bodies, the central relativistic conclusion is asserted rather than demonstrated.

minor comments (4)

[Section II, paragraph on four points] The text states that for the four-point distance pattern of Eq. (3), "any set of coordinates that satisfies Equation 3 results in a value of d(P1,P4) in (0,√3)" in analytic geometry, but in the plane the admissible values are only 0 and √3; the interval [0,√3] appears only when solid geometry is allowed. This should be rephrased to avoid contradicting the preceding sentence.
[Section III, definition of temporal units] The graph-time definition "two graphs isomorphic except for the addition or removal of an edge are one temporal unit apart" and the additional rule that edge removal is a step into the future are arbitrary assumptions, not consequences of the Aristotelian time definition in Eq. (4); the relationship between these two notions of time is left unclear.
[Section IV, cosmology] The claims that the diameter increases with accelerated motion, that the evaporation rate eventually dominates, and that the universe ends in a cloud of disconnected points are presented qualitatively without a derivation or simulation. If these are intended as more than analogical speculation, the paper should state the precise stochastic or deterministic rule for which edges are removed and show the claimed behavior.
[References] Reference [13] is cited for the "few steps" proving Lorentz invariance, but the proof is not reproduced and the reference is not publicly described in this manuscript; the reader cannot verify the claim without consulting a separate, self-published preprint.

Circularity Check

3 steps flagged · score 9.0 of 10

The paper's time result is definitional (Eq. 4 squared is Eq. 5), its Lorentz invariance is delegated to a self-cited preprint [13], and its 3D-space emergence is imported from another self-cited preprint [9].

self definitional [Section III, Equations (4)-(5)]
"∆ T = ∆ S / c (4) where c is a constant with the dimensions of a velocity that only serves to convert units. In diﬀerential form: dx2 + dy2 + dz2 − c2dt2 = 0 (5) This is the metric for light-like paths in a Minkowski space."

Equation (5) is exactly Equation (4) squared: if ΔT = ΔS/c, then c^2 ΔT^2 = ΔS^2 = dx^2+dy^2+dz^2, so dx^2+dy^2+dz^2 − c^2 dt^2 = 0. The lightlike Minkowski interval is therefore not derived from graph dynamics; it is the definition of the time coordinate, with c serving only as a unit-conversion constant. The text's own statement that c 'only serves to convert units' confirms that the light-cone structure is put in by construction, not obtained as a consequence. The claimed alignment with relativity is a restatement of the initial definition, not an independent prediction.
self citation load bearing [Section III, paragraph following Eq. (5)]
"A few steps [13] are suﬃcient to show that the motion of the body, as deﬁned here, perfectly respects the Lorentz transformations. Aristotelian time is spontaneously relativistic."

The paper's main advertised result for time, 'Aristotelian time is spontaneously relativistic,' is not demonstrated in the paper. The reader is referred to reference [13], M. Poletti, 'Time as change' (arXiv:2201.01944), which is prior work by the same author. The in-paper derivation ends at Eq. (4), which defines time as spatial change divided by c, and the Lorentz-invariance conclusion is imported from this self-citation. It is not machine-checked or independently reproduced here, so the central relativistic claim rests on a self-citation chain rather than on the graph model presented in this paper.

1 more flagged steps

self citation load bearing [Section II, paragraph after planar graph discussion]
"similarly, any graph can be represented in a three-dimensional Euclidean space without intersection of edges. Once such a representation is produced, the Euclidean space can be deformed so that the manifold's metric coincides with the metric of the graph. [9]"

The geometrodynamic hypothesis 'A space of pears can be approximately categorized as a three-dimensional space containing matter and energy' is the central spatial claim. Its only technical support is this exact assertion, cited to the author's own prior work [9] (Poletti, 'Space as relation', arXiv:2202.02985). The metric-coincidence statement is precisely the nontrivial content that needs proof, and no proof or quantitative approximation bound is given in this paper. The conclusion that a pear space can be seen as a curved 3D manifold is therefore imported from a self-citation rather than derived in the present argument, making the central 'emergence' claim dependent on a citation to the same author.

full rationale

The time section is the clearest circular step. Equation (4) defines ΔT as ΔS/c; Equation (5), presented as 'the metric for light-like paths in a Minkowski space,' is algebraically identical to that definition squared. Hence the lightlike structure is not a prediction of the pear model but a restatement of the chosen time variable. The further claim of Lorentz invariance is outsourced to the author's own [13], so the central relativistic conclusion is either definitional or self-cited, not derived. Similarly, the spatial emergence claim depends on the assertion that any graph can be embedded so that the manifold metric 'coincides' with the graph metric, cited to the author's own [9] with no proof. That is not an externally verified or machine-checked theorem in this paper, so the central 'approximately three-dimensional' conclusion is load-bearing on a self-citation. Because both advertised results (space as 3D, time as relativistic) reduce to definition or self-citation, the paper's circularity score is high. The philosophical and literary content is independent, but the derivation chain itself is the citation/definition.

Assumptions & free parameters 2 free parameters · 5 assumptions · 3 invented entities

The central model rests on five unproved assumptions and two arbitrary conventions. The main mathematical bridge to 3D space is the graph-to-manifold embedding claim, cited to the author's previous work rather than demonstrated. The relativistic result is deliberately constructed: time is defined as distance divided by c, so the lightlike metric follows by definition. The paper itself disclaims formal rigor, calling the work a 'haphazard game' and its cosmology chapter 'pseudo-scientific'.

free parameters (2)

temporal unit per edge removal = 1 (arbitrary unit)
Section III states: 'Given two graphs, isomorphic except for the addition or removal of an edge, they will be said to be one temporal unit apart.' This fixes a unit of time to a single edge flip without calibration to any physical measurement.
direction of time = removal = future, addition = past
Section III makes an 'arbitrary assumption': removal of an edge is one unit step into the future and addition is one unit step into the past. This convention sets the arrow of time in the model.

assumptions (5)

domain assumption Physical space is a finite undirected graph whose distance is shortest-path length.
Section II defines 'space of pears' this way and gives no derivation; this is the basic modeling choice.
domain assumption Time is the measure of change, so a change in the graph is time.
Section III adopts Aristotle's definition: 'the measure of this change is called time.'
domain assumption Any graph can be isometrically represented by a curved three-dimensional manifold.
Section II states this and cites only the author's own preprint [9]; no proof is included.
ad hoc to paper A massive body at rest has internal circular motion, and translation slows that motion.
Section III introduces this to explain why non-lightlike bodies obey Lorentz transformations.
ad hoc to paper The initial pear universe is a complete graph of diameter 1.
Chapter 4 postulates the Big Bang state as a complete, symmetric graph without independent evidence.

invented entities (3)

Space of pears (finite proximity graph)
purpose: Replacement for continuous manifold as model of physical space
No observable signature is specified beyond the unfalsifiable claim that it looks like a 3D curved space to an agent.
Internal motion of a material point
purpose: Gives massive bodies time dilation by diverting displacement
The internal motion is not independently measured; it is a picture invoked to make relativity emerge from the definition of time.
Evaporation (loss of connected subgraphs)
purpose: Accounts for cosmic dissolution into isolated points
Defined as an effect of edge removal; no quantitative prediction allows a test.

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Cite this review

Pith. "Pith review of The Physics of Pears: A Pataphysical Approach to Geometrodynamics." pith.science (2026). https://pith.science/paper/LBEFRK54

@misc{pith2026241111109,
  author       = {Pith},
  title        = {Pith review of: The Physics of Pears: A Pataphysical Approach to Geometrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBEFRK54}},
  note         = {Machine review of arXiv:2411.11109}
}

read the original abstract

This work explores a pataphysical approach to the concept of space and time, inspired by Aristotelian philosophy and the insights of John Wheeler. We propose the "Physics of Pears", a theory that conceives space as a graph and time as a measure of change, within a geometrodynamical context. The goal is to stimulate reflections on the nature of physical laws and the boundaries between physics and metaphysics.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages

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[1]

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F or instance, in spherical geometry, it is allowed that P1P4 = 2

This ﬂexibility corresponds to possible non-Euclidean distances. F or instance, in spherical geometry, it is allowed that P1P4 = 2. The addition of dimensions and non-Euclidean geometries work toget her to make the concept of distance more ﬂexible. A Riemannian manifold of appropriate dimension allows access to distan ces between points that are solely co...

work page
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This constraint, unlike Equation 2, is speciﬁc to planar geometry; in s olid geometry, it loosens, allowing P1P4 to take any value within the interval [0 , √ 3]

In analytic geometry, this corresponds to the fact that any set of coordinates that satisﬁes Equation 3 results in a value of d(P1, P4) ∈ (0, √ 3). This constraint, unlike Equation 2, is speciﬁc to planar geometry; in s olid geometry, it loosens, allowing P1P4 to take any value within the interval [0 , √ 3]. The number of dimensions, in this sense, operat...

work page
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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [13]

Smolin, Time Reborn: From the Crisis in Physics to the Future of the Un iverse (Houghton Miﬄin Harcourt, 2013)

L. Smolin, Time Reborn: From the Crisis in Physics to the Future of the Un iverse (Houghton Miﬄin Harcourt, 2013)

work page 2013

[2] [1]

J. A. Wheeler, Geometrodynamics (Academic Press, New York, 1962)

work page 1962

[3] [2]

F or instance, in spherical geometry, it is allowed that P1P4 = 2

This ﬂexibility corresponds to possible non-Euclidean distances. F or instance, in spherical geometry, it is allowed that P1P4 = 2. The addition of dimensions and non-Euclidean geometries work toget her to make the concept of distance more ﬂexible. A Riemannian manifold of appropriate dimension allows access to distan ces between points that are solely co...

work page

[4] [3]

This constraint, unlike Equation 2, is speciﬁc to planar geometry; in s olid geometry, it loosens, allowing P1P4 to take any value within the interval [0 , √ 3]

In analytic geometry, this corresponds to the fact that any set of coordinates that satisﬁes Equation 3 results in a value of d(P1, P4) ∈ (0, √ 3). This constraint, unlike Equation 2, is speciﬁc to planar geometry; in s olid geometry, it loosens, allowing P1P4 to take any value within the interval [0 , √ 3]. The number of dimensions, in this sense, operat...

work page

[5] [4]

Einstein, Die feldgleichungen der gravitation, Sitz ungsberichte der K¨ oniglich Preußischen Akademie der Wissenschaften (Berlin) , 844 (1915)

A. Einstein, Die feldgleichungen der gravitation, Sitz ungsberichte der K¨ oniglich Preußischen Akademie der Wissenschaften (Berlin) , 844 (1915)

work page 1915

[6] [5]

Kaluza, Zum unit¨ atsproblem in der physik, Sitzungsb erichte der K¨ oniglich Preußischen Akademie der Wissensch aften (Berlin) , 966 (1921)

T. Kaluza, Zum unit¨ atsproblem in der physik, Sitzungsb erichte der K¨ oniglich Preußischen Akademie der Wissensch aften (Berlin) , 966 (1921)

work page 1921

[7] [6]

Klein, Quantum theory and ﬁve-dimensional theory of r elativity, Zeitschrift f¨ ur Physik37, 895 (1926)

O. Klein, Quantum theory and ﬁve-dimensional theory of r elativity, Zeitschrift f¨ ur Physik37, 895 (1926)

work page 1926

[8] [7]

Vi` ete, In Artem Analyticam Isagoge (Typographia Plantiniana, 1591) geometria, quae prius sol ebat verborum interpre- tatione, nunc verorum numerorum calculatione tractanda es t

F. Vi` ete, In Artem Analyticam Isagoge (Typographia Plantiniana, 1591) geometria, quae prius sol ebat verborum interpre- tatione, nunc verorum numerorum calculatione tractanda es t

work page