On the Design of an Analog-Dyadic Converter CRN

Mathieu Hemery

arxiv: 2605.23745 · v1 · pith:BXJHKABRnew · submitted 2026-05-22 · 🧬 q-bio.QM

On the Design of an Analog-Dyadic Converter CRN

Mathieu Hemery This is my paper

Pith reviewed 2026-05-25 02:23 UTC · model grok-4.3

classification 🧬 q-bio.QM

keywords chemical reaction networksdyadic representationanalog computationmolecular concentrationerror analysisfinite precisionspike sequence

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The pith

A chemical reaction network converts an input concentration into on and off spikes that approximate its dyadic binary bits.

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

The paper designs a chemical reaction network that takes an arbitrary molecular concentration in the unit interval and outputs a timed sequence of on and off spikes meant to track the bits of its dyadic expansion. Standard CRN computation applies only to computable functions with exact output guarantees, so this construction addresses the distinct task of approximating an unknown input concentration to finite precision. The work examines how rate constants shape the resulting errors in the spike sequence and sketches a reader module that could deliver a dyadic encoding to a chosen accuracy. If the construction holds, it would let molecular systems extract numerical values from continuous concentrations in a controlled way.

Core claim

We present an analog-dyadic converter CRN which takes as input one molecular concentration (in [0, 1] but not necessarily computable), and produces as output a sequence of on and off spikes corresponding to some extent to the sequence of bits in the dyadic representation of the input concentration. We provide a detailed analysis of the source of errors and their behavior when varying the reactions rate constants. We conclude by sketching a possible design for a reader module that takes as input an arbitrary concentration and a desired precision and outputs a dyadic encoding approximating the value of the concentration with the desired precision.

What carries the argument

The analog-dyadic converter CRN, a set of elementary reactions with mass-action kinetics that generate timed on and off spikes from a single input concentration.

If this is right

Errors in the produced spike sequence remain bounded and can be reduced by adjusting reaction rate constants.
The sketched reader module can generate a dyadic encoding that approximates any input concentration to a user-specified precision.
The converter works for concentrations that need not be computable functions.
The construction supplies an explicit mechanism for finite-precision readout inside a CRN framework.

Where Pith is reading between the lines

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

The converter could be combined with other CRN modules to read out intermediate results during a larger computation.
Similar reaction designs might produce spikes in different bases or encodings by changing the underlying reactions.
The error analysis supplies a template for designing CRNs that perform measurement rather than pure function computation.

Load-bearing premise

A custom set of reactions with tunable rate constants can produce spike sequences that track the dyadic bits of an arbitrary input concentration within controllable error.

What would settle it

Simulate or realize the proposed reactions with a concrete input concentration such as 0.75, compute its exact dyadic bit sequence, and check whether the observed spike timings and states stay inside the error bounds predicted from the chosen rate constants.

Figures

Figures reproduced from arXiv: 2605.23745 by Mathieu Hemery.

**Figure 1.** Figure 1: Time course of the dyadic converter through 8 complete cycles of the clock. During the kth cycle, the Output species (in bold red) spikes or not according to the comparison of the Input species (in bold orange) to a threshold species. Here we can see that the output is 11100101. Note that this is different from the expected result as, Input(t = 0) = 0.9 = 0.11100110 . . .2. The 6 first bits are correct, bu… view at source ↗

**Figure 2.** Figure 2: A. Schematic representation of a six-step clock. Each species activates the next one and participates in a bidegradation (here depicted as a two crossed connector) with its second neighbor. B. The time course of the CRN presented in panel A. with corresponding colors. Activations have rates kclock = 1 while degradations have rates kf = 1000. At t = 0 one of the species (here light blue) has an initial conc… view at source ↗

**Figure 2.** Figure 2: Even the overlap between Ti and Ti+2 may be too important when strong guarantees are needed: typically when the two steps are likely to create a positive feedback loop that may lead to an exponential divergence as is the case for the two reactions of the copy mechanism: Input → Input + T empo and T empo → Copy + Input. To avoid that, we can always let more steps of the clock pass between such reactions. We… view at source ↗

**Figure 3.** Figure 3: Numerical integration of one full cycle of the clock for two different initial conditions. One where the Input species is slightly below (A.) the 1 2 threshold and one where it is slightly above (B.). We just show that our simple CRN is able to display the desired behavior by implementing the pseudocode of table 1 to output at least the first bit of the dyadic encoding of its input. We have also seen that … view at source ↗

**Figure 4.** Figure 4: Analysis of the actual behavior of the converter module for kclock = 1 and kexp = kAM = 10 and 1000 different values of the input between 0 and 1. The system is then numerically integrated, and the spikes are automatically detected to determine the system response over 10 full cycles of the clock. A. Comparison between the theoretical precision of the approximate majority analysis (in black) and the actual… view at source ↗

**Figure 5.** Figure 5: Schematic figure of a complete analog machine to compute the function f. The first module takes as input x and the desired precision ϵ and outputs a (possibly not computable) value ˜f(x) while also providing its current estimation of the error on err. A linker module halts the computation once the precision is reached and activates a reader module that takes both the output of the previous module and the d… view at source ↗

read the original abstract

The Chemical Reaction Networks (CRN) interpreted through the differential semantics, even when restricted to elementary reactions with mass action law kinetics, form a Turing-complete language. This means that any computable real function can thus be programmed, and in fact compiled, in an abstract CRN that will compute it with an arbitrarily high precision. In this computational framework, the information carriers are the molecular concentrations, the required precision is given as input, and the output concentration is guaranteed to satisfy the required precision. On the other hand, one can be interested in estimating the derivative of an unknown input signal or in reading the concentration value of an input molecular species. By nature, such problems can only be approximated with a finite precision. Hence, the computation framework proposed previously cannot be applied and we need to design and analyze custom CRNs to perform these tasks. In this paper, we present an analog-dyadic converter CRN which takes as input one molecular concentration (in [0, 1] but not necessarily computable), and produces as output a sequence of ''on'' and ''off'' spikes corresponding to some extent to the sequence of bits in the dyadic representation of the input concentration. We provide a detailed analysis of the source of errors and their behavior when varying the reactions rate constants. We conclude by sketching a possible design for a reader module that takes as input an arbitrary concentration and a desired precision and outputs a dyadic encoding approximating the value of the concentration with the desired precision. We leave as an open question to prove the correctness of our construction.

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

The paper gives a concrete CRN design for mapping a concentration to dyadic bit spikes plus error analysis, but leaves the main correctness claim as an explicit open question.

read the letter

The main takeaway is that this work introduces a specific analog-dyadic converter CRN and walks through error sources when tuning rates, yet states outright that proving the spikes track the input bits remains open. That matches the abstract and the stress-test note exactly. The construction itself is new and not just a rehash of Turing completeness for CRNs; the authors separate the general framework from this custom network aimed at reading an arbitrary input in [0,1]. They also sketch a reader module that would use the converter to output an approximation at a chosen precision. That part is useful as a starting point for anyone trying to build molecular interfaces between continuous signals and discrete sequences. The error analysis looks honest: they identify sources tied to rate constants and describe how the behavior changes with those parameters. No circularity or hidden assumptions beyond the usual mass-action setup. The clear limitation is the missing proof. Without an invariant, Lyapunov argument, or inductive step showing the output sequence matches the dyadic bits for all inputs, the central functionality is not yet established. The paper does not overclaim; it flags the gap. This is the kind of piece that could interest people already working on CRN-based sensing or computation who want a concrete design to experiment with or try to verify. It is not ready as a standalone result because the key correspondence is unproven. I would bring it to a reading group if the group focuses on molecular computing designs, but only as an idea to discuss rather than a finished piece. I would not cite it yet. It deserves peer review because the design and error breakdown are substantive enough to be worth referee time even if heavy revision is needed to close the proof.

Referee Report

2 major / 1 minor

Summary. The paper claims to design an analog-dyadic converter CRN that maps an arbitrary input molecular concentration in [0,1] to a sequence of on/off spikes approximating the dyadic bits of the input. It supplies an informal network design, an analysis of error sources and their dependence on rate constants, and a sketch of a reader module that approximates the input to a desired precision; the authors explicitly leave a formal proof of correctness as an open question.

Significance. If the construction were proven correct, the work would provide a mechanism for digitizing arbitrary analog concentrations into dyadic expansions using CRNs, extending beyond the computation of computable functions to the reading of non-computable signals. The error analysis offers practical guidance on rate-constant tuning. However, the absence of any invariant, Lyapunov analysis, or inductive argument substantially reduces the immediate significance.

major comments (2)

[Abstract] Abstract: The manuscript states verbatim that 'We leave as an open question to prove the correctness of our construction.' This is load-bearing for the central claim, as no invariant, Lyapunov function, or inductive argument is supplied to establish that the continuous trajectories produce the required dyadic bit sequence for arbitrary inputs in [0,1].
[CRN design and error analysis sections] CRN design and error analysis sections: The informal design and error-source discussion assert that error is controllable by tuning rate constants, yet supply no formal demonstration that the output spikes track the dyadic bits; without this, the claimed functionality remains unverified.

minor comments (1)

[Reader module sketch] The description of the reader module sketch would benefit from an explicit diagram or pseudocode showing how the desired precision is used to terminate the spike sequence.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed review. We acknowledge that the absence of a formal correctness proof is a substantive limitation, as the manuscript itself states. We respond to each major comment below, indicating where revisions are feasible.

read point-by-point responses

Referee: [Abstract] Abstract: The manuscript states verbatim that 'We leave as an open question to prove the correctness of our construction.' This is load-bearing for the central claim, as no invariant, Lyapunov function, or inductive argument is supplied to establish that the continuous trajectories produce the required dyadic bit sequence for arbitrary inputs in [0,1].

Authors: We agree that the formal proof is load-bearing and currently absent. The manuscript's contribution is the informal CRN design together with error analysis; the proof is explicitly identified as open. We will revise the abstract to state more explicitly that the work proposes a construction and provides error analysis but does not include a formal verification, thereby aligning the abstract with the paper's actual scope. revision: yes
Referee: [CRN design and error analysis sections] CRN design and error analysis sections: The informal design and error-source discussion assert that error is controllable by tuning rate constants, yet supply no formal demonstration that the output spikes track the dyadic bits; without this, the claimed functionality remains unverified.

Authors: The design and error sections supply an informal network and analyze dependence of errors on rate constants. We concur that these do not amount to a formal demonstration that spikes track dyadic bits. Because the manuscript leaves formal correctness open, we cannot add such a demonstration. We will partially revise by adding explicit discussion of this limitation and its implications in the error-analysis and conclusion sections. revision: partial

standing simulated objections not resolved

Formal proof of correctness (via invariant, Lyapunov analysis, or induction) that the proposed CRN produces the required dyadic bit sequence for arbitrary inputs in [0,1].

Circularity Check

0 steps flagged

No circularity; paper explicitly leaves correctness unproven

full rationale

The manuscript presents an informal CRN design for analog-to-dyadic conversion and analyzes error sources but states verbatim that 'We leave as an open question to prove the correctness of our construction.' No derivation chain, invariant, or formal argument is supplied that could reduce to its own inputs. The work invokes the established Turing-completeness of CRNs (an external result) without self-citation load-bearing or any fitted-parameter-as-prediction pattern. The central claim is therefore not asserted as proven, eliminating any possibility of circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The claim rests on the prior result that CRNs are Turing-complete plus the introduction of an unproven custom network whose rate constants act as tunable parameters.

free parameters (1)

reaction rate constants
The paper states that error behavior is analyzed when varying these constants, making them design parameters chosen by the engineer.

axioms (1)

standard math CRNs interpreted through differential semantics with elementary reactions and mass-action kinetics are Turing-complete
Invoked in the first sentence of the abstract as established background.

invented entities (1)

analog-dyadic converter CRN no independent evidence
purpose: Maps input concentration to sequence of on/off spikes approximating dyadic bits
Newly proposed construction whose correctness is left unproven.

pith-pipeline@v0.9.0 · 5806 in / 1163 out tokens · 32829 ms · 2026-05-25T02:23:39.981032+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/Foundation/Breath1024.lean period8, flipAt512 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.

For any n ≥ 4, the CRN: Ti → Ti + Ti+1, Ti + Ti+2 → ∅ ... gives us an n-clock ... we try to minimize ... n=8
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow, LogicNat recovery 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.

the output is 11100101 ... Input(t=0)=0.9=0.11100110...2 ... precision is 7

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

17 extracted references · 17 canonical work pages

[1]

Campagnolo, Daniel S

Olivier Bournez, Manuel L. Campagnolo, Daniel S. Graça, and Emmanuel Hainry. Polynomial differential equations compute all real computable functions on com- putable compact intervals.Journal of Complexity, 23(3):317–335, 2007

work page 2007
[2]

Graça, and Amaury Pouly

Olivier Bournez, Daniel S. Graça, and Amaury Pouly. Polynomial Time corre- sponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. In 43rd Int. Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 ofLIPIcs, pages 109:1– 109:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Infor...

work page 2016
[3]

Graça, and Amaury Pouly

Olivier Bournez, Daniel S. Graça, and Amaury Pouly. On the functions generated by the general purpose analog computer.Information and Computation, (accepted under minor revision), 2017

work page 2017
[4]

Thecellcycleswitchcomputesapproximate majority

LucaCardelliandAttilaCsikász-Nagy. Thecellcycleswitchcomputesapproximate majority. Scientific Reports, 2, 09 2012

work page 2012
[5]

On biocham symbolic com- putation pipeline for compiling mathematical functions into biochemistry.ACM Commun

François Fages, Mathieu Hemery, and Sylvain Soliman. On biocham symbolic com- putation pipeline for compiling mathematical functions into biochemistry.ACM Commun. Comput. Algebra, 58(2):15–22, January 2025

work page 2025
[6]

Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs

FrançoisFages,GuillaumeLeGuludec,OlivierBournez,andAmauryPouly. Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs. InCMSB’17: Proc. of the fiveteen international Conf. on Computational Methods in Systems Biology,volume10545of LNCS,pages 108–127. Springer-Verlag, September 2017

work page 2017
[7]

Graça and J.F

D.S. Graça and J.F. Costa. Analog computers and recursive functions over the reals. Journal of Complexity, 19(5):644–664, 2003

work page 2003
[8]

On the computational complexity of al- gorithms

Juris Hartmanis and Richard E Stearns. On the computational complexity of al- gorithms. Transactions of the American Mathematical Society, 117:285–306, 1965

work page 1965
[9]

On the complexity of quadratization for polynomial differential equations

Mathieu Hemery, François Fages, and Sylvain Soliman. On the complexity of quadratization for polynomial differential equations. In CMSB’20: Proc. of the 18 eighteenth international Conf. on Computational Methods in Systems Biology , LNCS. Springer-Verlag, September 2020

work page 2020
[10]

Compiling elementary mathematical functions into finite chemical reaction networks via a polynomializa- tion algorithm for ODEs

Mathieu Hemery, François Fages, and Sylvain Soliman. Compiling elementary mathematical functions into finite chemical reaction networks via a polynomializa- tion algorithm for ODEs. InCMSB’21: Proc. of the nineteenth international Conf. on Computational Methods in Systems Biology, volume 12881 ofLNCS. Springer- Verlag, September 2021

work page 2021
[11]

A polynomialization al- gorithm for elementary functions and odes, and their compilation into chemical reaction networks

Mathieu Hemery, François Fages, and Sylvain Soliman. A polynomialization al- gorithm for elementary functions and odes, and their compilation into chemical reaction networks. InCASC’21: Computer Algebra in Scientific Computing, pro- ceedings of short papers, September 2021

work page 2021
[12]

C.E. Shannon. Mathematical theory of the differential analyser.Journal of Math- ematics and Physics, 20:337–354, 1941

work page 1941
[13]

On an integrating machine having a new kinematic principle

James Thomson. On an integrating machine having a new kinematic principle. Van Nostrand’s Eclectic Engineering Magazine (1869-1879), 15(94):301, 1876

work page
[14]

On computable numbers, with an application to the entscheidungsproblem

Alan Mathison Turing et al. On computable numbers, with an application to the entscheidungsproblem. J. of Math, 58(345-363):5, 1936

work page 1936
[15]

150 years of the mass action law.PLoS computational biology, 11(1):e1004012, 2015

Eberhard O Voit, Harald A Martens, and Stig W Omholt. 150 years of the mass action law.PLoS computational biology, 11(1):e1004012, 2015. 19 A Appendix A.1 Precision computation Most differential equations involved in the dyadic converter harbor converging exponential as solutions, that is, functions of the form: f (t) = f∞ + (f0 − f∞) e−kexpt, where f0 is...

work page 2015
[16]

The functionfi is symetrical under the relation˜a → 1 − ˜a and describes the evolution of the system over time

log(√s − 2 √ 2˜a + 2˜a − 1) −( √ 2 − 1) log(√s + 2(1 + √s)˜a − 1) = fi(˜a) (22) where for readability, we denote: p 4˜a(1 − ˜a) + 1 = √s and c1 is an integration constant that should be computed from the initial condition. The functionfi is symetrical under the relation˜a → 1 − ˜a and describes the evolution of the system over time. That is, if we want a ...

work page
[17]

(24) All this derivation has been done under the assumption thatk1 ≫ k2 which is the most natural one

log δ log δ = − log 4r2 1 + √ 2 − k2τ 1 + √ 2 (23) From this, we can estimate the scaling of the critical threshold, below which the approximate majority is not able to discern the majority species as a function of k2 and τ: δ⋆ ∝ exp − k2τ 1 + √ 2 . (24) All this derivation has been done under the assumption thatk1 ≫ k2 which is the most natural one. To b...

work page

[1] [1]

Campagnolo, Daniel S

Olivier Bournez, Manuel L. Campagnolo, Daniel S. Graça, and Emmanuel Hainry. Polynomial differential equations compute all real computable functions on com- putable compact intervals.Journal of Complexity, 23(3):317–335, 2007

work page 2007

[2] [2]

Graça, and Amaury Pouly

Olivier Bournez, Daniel S. Graça, and Amaury Pouly. Polynomial Time corre- sponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length. In 43rd Int. Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 ofLIPIcs, pages 109:1– 109:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Infor...

work page 2016

[3] [3]

Graça, and Amaury Pouly

Olivier Bournez, Daniel S. Graça, and Amaury Pouly. On the functions generated by the general purpose analog computer.Information and Computation, (accepted under minor revision), 2017

work page 2017

[4] [4]

Thecellcycleswitchcomputesapproximate majority

LucaCardelliandAttilaCsikász-Nagy. Thecellcycleswitchcomputesapproximate majority. Scientific Reports, 2, 09 2012

work page 2012

[5] [5]

On biocham symbolic com- putation pipeline for compiling mathematical functions into biochemistry.ACM Commun

François Fages, Mathieu Hemery, and Sylvain Soliman. On biocham symbolic com- putation pipeline for compiling mathematical functions into biochemistry.ACM Commun. Comput. Algebra, 58(2):15–22, January 2025

work page 2025

[6] [6]

Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs

FrançoisFages,GuillaumeLeGuludec,OlivierBournez,andAmauryPouly. Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs. InCMSB’17: Proc. of the fiveteen international Conf. on Computational Methods in Systems Biology,volume10545of LNCS,pages 108–127. Springer-Verlag, September 2017

work page 2017

[7] [7]

Graça and J.F

D.S. Graça and J.F. Costa. Analog computers and recursive functions over the reals. Journal of Complexity, 19(5):644–664, 2003

work page 2003

[8] [8]

On the computational complexity of al- gorithms

Juris Hartmanis and Richard E Stearns. On the computational complexity of al- gorithms. Transactions of the American Mathematical Society, 117:285–306, 1965

work page 1965

[9] [9]

On the complexity of quadratization for polynomial differential equations

Mathieu Hemery, François Fages, and Sylvain Soliman. On the complexity of quadratization for polynomial differential equations. In CMSB’20: Proc. of the 18 eighteenth international Conf. on Computational Methods in Systems Biology , LNCS. Springer-Verlag, September 2020

work page 2020

[10] [10]

Compiling elementary mathematical functions into finite chemical reaction networks via a polynomializa- tion algorithm for ODEs

Mathieu Hemery, François Fages, and Sylvain Soliman. Compiling elementary mathematical functions into finite chemical reaction networks via a polynomializa- tion algorithm for ODEs. InCMSB’21: Proc. of the nineteenth international Conf. on Computational Methods in Systems Biology, volume 12881 ofLNCS. Springer- Verlag, September 2021

work page 2021

[11] [11]

A polynomialization al- gorithm for elementary functions and odes, and their compilation into chemical reaction networks

Mathieu Hemery, François Fages, and Sylvain Soliman. A polynomialization al- gorithm for elementary functions and odes, and their compilation into chemical reaction networks. InCASC’21: Computer Algebra in Scientific Computing, pro- ceedings of short papers, September 2021

work page 2021

[12] [12]

C.E. Shannon. Mathematical theory of the differential analyser.Journal of Math- ematics and Physics, 20:337–354, 1941

work page 1941

[13] [13]

On an integrating machine having a new kinematic principle

James Thomson. On an integrating machine having a new kinematic principle. Van Nostrand’s Eclectic Engineering Magazine (1869-1879), 15(94):301, 1876

work page

[14] [14]

On computable numbers, with an application to the entscheidungsproblem

Alan Mathison Turing et al. On computable numbers, with an application to the entscheidungsproblem. J. of Math, 58(345-363):5, 1936

work page 1936

[15] [15]

150 years of the mass action law.PLoS computational biology, 11(1):e1004012, 2015

Eberhard O Voit, Harald A Martens, and Stig W Omholt. 150 years of the mass action law.PLoS computational biology, 11(1):e1004012, 2015. 19 A Appendix A.1 Precision computation Most differential equations involved in the dyadic converter harbor converging exponential as solutions, that is, functions of the form: f (t) = f∞ + (f0 − f∞) e−kexpt, where f0 is...

work page 2015

[16] [16]

The functionfi is symetrical under the relation˜a → 1 − ˜a and describes the evolution of the system over time

log(√s − 2 √ 2˜a + 2˜a − 1) −( √ 2 − 1) log(√s + 2(1 + √s)˜a − 1) = fi(˜a) (22) where for readability, we denote: p 4˜a(1 − ˜a) + 1 = √s and c1 is an integration constant that should be computed from the initial condition. The functionfi is symetrical under the relation˜a → 1 − ˜a and describes the evolution of the system over time. That is, if we want a ...

work page

[17] [17]

(24) All this derivation has been done under the assumption thatk1 ≫ k2 which is the most natural one

log δ log δ = − log 4r2 1 + √ 2 − k2τ 1 + √ 2 (23) From this, we can estimate the scaling of the critical threshold, below which the approximate majority is not able to discern the majority species as a function of k2 and τ: δ⋆ ∝ exp − k2τ 1 + √ 2 . (24) All this derivation has been done under the assumption thatk1 ≫ k2 which is the most natural one. To b...

work page