Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox

Takashi Izumo

arxiv: 2507.12475 · v2 · submitted 2025-07-05 · 💰 econ.TH · cs.AI· math.OC

Absorption and Inertness in Coarse-Grained Arithmetic: A Heuristic Application to the St. Petersburg Paradox

Takashi Izumo This is my paper

Pith reviewed 2026-05-19 06:43 UTC · model grok-4.3

classification 💰 econ.TH cs.AImath.OC

keywords St. Petersburg paradoxcoarse-grained arithmeticinertnessabsorptiondecision aggregationbounded cognition

0 comments

The pith

Coarse addition over number grains can stabilize a St. Petersburg sequence after finite steps.

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

The paper defines addition by first grouping exact numbers into ordered grains and replacing each with a chosen representative inside its grain. Addition then proceeds by repeated projection to those representatives rather than exact arithmetic. Under this rule, sequences of repeated sums can reach a fixed coarse state and stop changing further, a property the paper calls inertness. When applied to a rescaled version of the St. Petersburg payoffs that preserves equal expected increments, the sequence becomes inert for a suitably chosen countable partition and representative map. This offers a structural account in which unbounded classical expectation need not translate into unbounded growth once the aggregation step itself is made coarse.

Core claim

Coarse representative addition and coarse cell addition are defined on partitions of the numerical scale. Repeated application of these operations can produce absorption and inertness, in which further additions leave the coarse state unchanged. The paper exhibits an explicit countable partition and representative map under which a rescaled St. Petersburg sequence with equal expected increments becomes inert after finitely many steps.

What carries the argument

Coarse representative addition: each exact value is projected to the representative of its grain and addition is performed by repeated projection to representatives.

If this is right

Repeated coarse sums eventually reach a stable coarse value and absorb further additions.
The St. Petersburg sequence can be made to stop growing in the coarse state while its classical expectation remains infinite.
Addition under these rules is non-associative.
Unbounded reward structures need not produce unbounded coarse outcomes once aggregation is coarse-grained.

Where Pith is reading between the lines

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

The same coarse mechanism could be applied to other decision problems that involve summing many large or rare payoffs.
One could check whether human subjects exhibit inert-like stopping behavior when asked to aggregate repeated large numbers under time pressure.
The framework suggests a route to bounded numerical cognition without altering probability weights or utility functions.

Load-bearing premise

A countable partition and representative map can be chosen in advance to capture features of decision aggregation rather than being fitted after seeing the target sequence.

What would settle it

An explicit partition and map for the rescaled St. Petersburg sequence under which repeated coarse sums continue to change the state indefinitely.

read the original abstract

The St. Petersburg paradox presents a longstanding challenge in decision theory: its classical expected value diverges, yet no correspondingly large finite stake is typically regarded as rational. Traditional responses introduce auxiliary assumptions, such as diminishing marginal utility, temporal discounting, or extended number systems. This paper explores a different approach based on a modified operation of addition defined over coarse-grained partitions of the underlying numerical scale. In this framework, exact values are grouped into ordered grains, each grain is assigned an internal representative, and addition proceeds by repeated projection to those representatives. On this basis, the paper defines coarse representative addition and coarse cell addition, and studies several of their structural properties, including absorption, inertness, and non-associativity. In particular, repeated additions may eventually cease to change the coarse state, a phenomenon called inertness. The paper then applies this framework heuristically to the St. Petersburg setting by considering a rescaled sequence corresponding to its equal expected increments, and shows that this sequence can become inert under a suitably chosen countable partition and representative map. The claim is not that the paradox is resolved within standard decision theory, nor that the classical expectation becomes finite in the ordinary probabilistic sense. Rather, the contribution is structural and heuristic: it exhibits an explicit mathematical mechanism through which a divergent reward structure may fail to produce unbounded growth once aggregation itself is made coarse. More broadly, the framework may be relevant to the study of bounded numerical cognition and behavioral models of aggregation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines coarse representative addition and shows a rescaled St. Petersburg sequence can be made inert under a chosen partition, but the construction risks being fitted to the example.

read the letter

Hey, the main thing here is that the paper defines coarse representative addition and coarse cell addition over partitioned scales with internal representatives, then shows that repeated applications can stabilize into inertness. It applies this to a rescaled St. Petersburg sequence with equal expected increments and constructs a countable partition where the sums stop growing after a point. That explicit setup and the inertness result for this divergent case are what stand out as new. The structural properties like absorption and non-associativity follow directly from the definitions and are worked out cleanly. The paper does a decent job presenting a purely arithmetic mechanism for bounded aggregation that avoids utility functions or other add-ons, and the math inside the framework is consistent. The soft spot is the partition and representative map. The abstract and application rely on a suitably chosen pair that achieves inertness for this sequence, and if the grains are picked with the increments or divergence in mind, then the stabilization is more engineered than emergent from general rules of the coarse operation. The paper is open about being heuristic and not claiming to resolve the paradox in standard terms, which keeps the modeling choice as a free parameter. This limits how far it speaks to real decision aggregation without further justification for why the partition would be selected that way in practice. Readers working on decision theory alternatives or models of bounded numerical cognition would get the most from the definitions and the concrete illustration. It is the sort of structural idea worth discussing in a group focused on paradoxes or behavioral aggregation. The work deserves a serious referee because the framework is mathematically well-posed and the inertness concept raises clear questions about generality. I would recommend sending it to peer review so the authors can get input on making the partition selection less tailored and on potential extensions.

Referee Report

1 major / 1 minor

Summary. The paper defines coarse representative addition and coarse cell addition over ordered grains of the numerical scale, each with an internal representative, and studies structural properties including absorption, inertness, and non-associativity. It then applies the framework heuristically to a rescaled St. Petersburg sequence with equal expected increments, constructing a countable partition and representative map under which repeated coarse additions become inert (cease to change the coarse state). The contribution is explicitly structural and heuristic: it exhibits a mechanism by which a divergent reward structure may fail to produce unbounded growth once aggregation is coarse-grained, without claiming to resolve the paradox in standard decision theory or to make classical expectation finite.

Significance. If the inertness construction can be shown to arise from partition and representative rules that are fixed independently of the target sequence, the framework would supply a mathematically explicit model of how bounded numerical cognition or coarse aggregation can bound growth in otherwise divergent structures. This is a genuine addition to the set of structural approaches to the St. Petersburg paradox and could be relevant to behavioral models of aggregation. The current heuristic presentation, however, leaves the independence of the modeling choice open, limiting immediate applicability.

major comments (1)

[Abstract and St. Petersburg application section] Abstract and application section: the claim that the rescaled sequence 'can become inert under a suitably chosen countable partition and representative map' is mathematically well-posed, but the selection of the partition and map is not shown to be independent of the sequence's increments or divergence point. If the grains or representatives are defined using knowledge of where the sequence would otherwise diverge, inertness follows by construction rather than from general properties of coarse addition. To support the heuristic link to decision aggregation, the paper must either derive the partition from sequence-independent criteria or demonstrate that a single fixed coarse-graining renders other divergent sequences inert.

minor comments (1)

[Definitions section] Notation for the representative map and the projection operator should be introduced with an explicit example early in the structural section to aid readability.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their constructive comments, which correctly identify the tailored nature of our construction. We respond to the major comment below.

read point-by-point responses

Referee: the claim that the rescaled sequence 'can become inert under a suitably chosen countable partition and representative map' is mathematically well-posed, but the selection of the partition and map is not shown to be independent of the sequence's increments or divergence point. If the grains or representatives are defined using knowledge of where the sequence would otherwise diverge, inertness follows by construction rather than from general properties of coarse addition. To support the heuristic link to decision aggregation, the paper must either derive the partition from sequence-independent criteria or demonstrate that a single fixed coarse-graining renders other divergent sequences inert.

Authors: We agree that the partition and representative map are chosen with reference to the specific increments and divergence behavior of the rescaled St. Petersburg sequence, so that inertness follows from this tailored construction. This is deliberate in the heuristic application, whose purpose is to exhibit one explicit mechanism by which coarse aggregation can bound growth in a divergent reward structure. We have revised the abstract and the St. Petersburg application section to state more explicitly that the coarse-graining is sequence-specific and does not claim independence from the target sequence or universality across divergent structures. We have not derived sequence-independent criteria or tested a single fixed coarse-graining on other sequences, as that would require a substantially broader theoretical framework outside the present heuristic scope. revision: partial

standing simulated objections not resolved

Deriving the partition from sequence-independent criteria or demonstrating that a single fixed coarse-graining renders other divergent sequences inert.

Circularity Check

0 steps flagged

No significant circularity: existence shown via explicit construction in heuristic application

full rationale

The paper first defines coarse-grained arithmetic operations (coarse representative addition and coarse cell addition) and derives their structural properties such as absorption and inertness from the definitions of partitions, representatives, and projection. These properties are established independently of any specific sequence. The St. Petersburg application then considers a rescaled sequence with equal expected increments and exhibits that inertness holds for a suitably chosen countable partition and representative map. This is an existence result demonstrated by construction of the modeling elements, not a general prediction or theorem that all divergent sequences become inert under arbitrary coarse-graining. The text explicitly frames the contribution as heuristic and structural, without claiming the partition is derived from independent non-tailored criteria or that the result resolves the paradox in standard decision theory. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The central claim therefore remains self-contained against external benchmarks and does not reduce to its inputs by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on the choice of a countable partition and a representative map that are not fixed by prior axioms but selected to produce inertness for the target sequence; no free numerical parameters are fitted to data, but the modeling choice functions as an adjustable structural element.

free parameters (1)

countable partition and representative map
Selected to render the rescaled St. Petersburg sequence inert; the abstract states a suitably chosen partition without independent selection criterion.

axioms (1)

standard math Standard properties of ordered partitions and projection maps on the real line
Invoked to define the coarse addition operations and their structural properties such as absorption and inertness.

invented entities (1)

coarse grain with internal representative no independent evidence
purpose: To define a modified addition that projects exact values onto representatives within ordered partitions
New construct introduced to create absorption and inertness; no independent empirical evidence supplied in the abstract.

pith-pipeline@v0.9.0 · 5795 in / 1411 out tokens · 31830 ms · 2026-05-19T06:43:49.340618+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the sequence of coarse partial sums ½ ⊕θ ½ ⊕θ … becomes inert in a finite number of steps … the first inert index is ι∗=⌊ε/2⌋+1
IndisputableMonolith/Foundation/AlphaDerivationExplicit phi_golden_ratio echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Gθ,ι with |Gθ,ι|=Fι (Fibonacci) … φmed(Gθ,ι)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Izumo, T., Weng, Y.-H.: Coarse ethics: how to ethically assess explainable artificial intelligence. AI Ethics 2, 449–461 (2022) https://doi.org/10.1007/ s43681-021-00091-y

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[4]

https://doi.org/arXiv:2502

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Marrink, S.J., Vries, A.H., Mark, A.E.: Coarse grained model for semiquantitative lipid simulations. The Journal of Physical Chemistry B 108(2), 750–760 (2004) https://doi.org/10.1021/jp036508g 14

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Baschnagel, J., Binder, K., Doruker, P., Gusev, A.A., Hahn, O., Kremer, K., Mattice, W.L., Muller-Plathe, F., Murat, M., Paul, W., Santos, S., Suter, U.W., Tries, V.: Bridging the gap between atomistic and coarse-grained models of poly- mers: Status and perspectives. Advances in Polymer Science 152, 41–156 (2000) https://doi.org/10.1007/3-540-46778-5 2

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Izumo, T.: Coarse-Grained Games: A Framework for Bounded Perception in Game Theory (2025). https://arxiv.org/abs/2503.17598

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The Review of Economic Studies 4(2), 155–161 (1937) https://doi.org/10.2307/2967612

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inertness

Kahneman, D., Tversky, A.: Prospect theory: An analysis of decision under risk. Econometrica 47(2), 263–292 (1979) https://doi.org/10.2307/1914185 15 T able 1 Comparison with Major Approaches Perspective Key References Core Idea Main Limitation Contribution of this Paper Diminishing marginal utility Bernoulli (1738), classical utility the- ory Use concave...

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[14]

δ is arbitrary; mixes impatience with risk; ethically controversial

to force convergence. δ is arbitrary; mixes impatience with risk; ethically controversial. No time-weights; satura- tion explained by coarse absorption instead. Unbounded-utility decision theory Goodsell (2024) Hyperreal rankings; fail- ure of Countable Sure- Thing. Abstract; lacks con- crete aggregation. Provides explicit ⊕θ, ⊞θ operations compatible wit...

work page 2024

[1] [1]

Commentarii Academiae Scientiarum Imperialis Petropolitanae 5, 175–192 (1738)

Bernoulli, D.: Specimen theoriae novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae 5, 175–192 (1738)

work page

[2] [2]

AI Ethics 2, 449–461 (2022) https://doi.org/10.1007/ s43681-021-00091-y

Izumo, T., Weng, Y.-H.: Coarse ethics: how to ethically assess explainable artificial intelligence. AI Ethics 2, 449–461 (2022) https://doi.org/10.1007/ s43681-021-00091-y

work page 2022

[3] [3]

In: Montag, C., Ali, R

Izumo, T.: Introduction to coarse ethics: Tradeoff between the accuracy and inter- pretability of explainable artificial intelligence. In: Montag, C., Ali, R. (eds.) The Impact of Artificial Intelligence on Societies. Studies in Neuroscience, Psychol- ogy and Behavioral Economics. Springer, Cham (2025). https://doi.org/10.1007/ 978-3-031-70355-3

work page 2025

[4] [4]

https://doi.org/arXiv:2502

Izumo, T.: Coarse Set Theory for AI Ethics and Decision-Making: A Mathe- matical Framework for Granular Evaluations (2025). https://doi.org/arXiv:2502. 07347

work page 2025

[5] [5]

The Journal of Physical Chemistry B 108(2), 750–760 (2004) https://doi.org/10.1021/jp036508g 14

Marrink, S.J., Vries, A.H., Mark, A.E.: Coarse grained model for semiquantitative lipid simulations. The Journal of Physical Chemistry B 108(2), 750–760 (2004) https://doi.org/10.1021/jp036508g 14

work page doi:10.1021/jp036508g 2004

[6] [6]

The Journal of Physical Chemistry B 111(27), 7812–7824 (2007) https://doi.org/10

Marrink, S.J., Risselada, H.J., Yefimov, S., Tieleman, D.P., Vries, A.H.: The MARTINI force field: Coarse grained model for biomolecular simulations. The Journal of Physical Chemistry B 111(27), 7812–7824 (2007) https://doi.org/10. 1021/jp071097f

work page 2007

[7] [7]

: Martini 3: a general purpose force field for coarse-grained molecular dynamics

Souza, P.C.T., Alessandri, R., Barnoud, J., et al. : Martini 3: a general purpose force field for coarse-grained molecular dynamics. Nature Methods 18, 382–388 (2021) https://doi.org/10.1038/s41592-021-01098-3

work page doi:10.1038/s41592-021-01098-3 2021

[8] [8]

Advances in Polymer Science 152, 41–156 (2000) https://doi.org/10.1007/3-540-46778-5 2

Baschnagel, J., Binder, K., Doruker, P., Gusev, A.A., Hahn, O., Kremer, K., Mattice, W.L., Muller-Plathe, F., Murat, M., Paul, W., Santos, S., Suter, U.W., Tries, V.: Bridging the gap between atomistic and coarse-grained models of poly- mers: Status and perspectives. Advances in Polymer Science 152, 41–156 (2000) https://doi.org/10.1007/3-540-46778-5 2

work page doi:10.1007/3-540-46778-5 2000

[9] [9]

Journal of Computational Chemistry 24(13), 1624– 1636 (2003) https://doi.org/10.1002/jcc.10307

Reith, D., P¨ utz, M., M¨ uller-Plathe, F.: Deriving effective mesoscale potentials from atomistic simulations. Journal of Computational Chemistry 24(13), 1624– 1636 (2003) https://doi.org/10.1002/jcc.10307

work page doi:10.1002/jcc.10307 2003

[10] [10]

https://arxiv.org/abs/2503.17598

Izumo, T.: Coarse-Grained Games: A Framework for Bounded Perception in Game Theory (2025). https://arxiv.org/abs/2503.17598

work page arXiv 2025

[11] [11]

The Review of Economic Studies 4(2), 155–161 (1937) https://doi.org/10.2307/2967612

Samuelson, P.A.: A note on measurement of utility. The Review of Economic Studies 4(2), 155–161 (1937) https://doi.org/10.2307/2967612

work page doi:10.2307/2967612 1937

[12] [12]

Nous 58(3), 669–695 (2024) https://doi

Goodsell, Z.: Decision theory unbound. Nous 58(3), 669–695 (2024) https://doi. org/10.1111/nous.12473

work page doi:10.1111/nous.12473 2024

[13] [13]

inertness

Kahneman, D., Tversky, A.: Prospect theory: An analysis of decision under risk. Econometrica 47(2), 263–292 (1979) https://doi.org/10.2307/1914185 15 T able 1 Comparison with Major Approaches Perspective Key References Core Idea Main Limitation Contribution of this Paper Diminishing marginal utility Bernoulli (1738), classical utility the- ory Use concave...

work page doi:10.2307/1914185 1979

[14] [14]

δ is arbitrary; mixes impatience with risk; ethically controversial

to force convergence. δ is arbitrary; mixes impatience with risk; ethically controversial. No time-weights; satura- tion explained by coarse absorption instead. Unbounded-utility decision theory Goodsell (2024) Hyperreal rankings; fail- ure of Countable Sure- Thing. Abstract; lacks con- crete aggregation. Provides explicit ⊕θ, ⊞θ operations compatible wit...

work page 2024