PaperlayerView exact revision on arXiv

Source-linked AI summary

Tapes Together Strong: The Co-evolution of Computation and Cooperation

Kunal Jha, Francesco Cicala, Blaise Agüera y Arcas, Blake Aaron Richards, Natasha Jaques, Max Kleiman-Weiner, Eyvind Niklasson

arXiv:2609.10817v1cs.MAcs.AI

TL;DR

Prior work often separates strategic interaction from physical computation or studies self-replication without formalizing resource-sharing dilemmas. This paper couples social strategy, replication, and energy use in an evolving computational model, finding that cooperation can persist even in well-mixed settings and that local interactions help under unequal energy.

  • Problem

    Prior approaches often separate social decisions from the physical costs of computation, while artificial-life models have not typically formalized the dilemma between acquiring resources and preserving shared energy for reproduction.

    Source: §1 · p. 1

  • Method

    Autopoietic Game Theory models randomly initialized Z80 programs whose social strategies, replication mechanisms, and computational resource use co-evolve under a shared finite-energy budget.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    Defection can be suppressed through lossy stealing and evolving replication, while local interactions support energy and complexity under highly unequal energy distributions.

    Source: §4 · p. 7, §6 · p. 10

  • Takeaways & Limitations

    Programs also suppress destructive stealing and safer solo strategies in favor of collaborative multi-step solutions when math-task rewards create sequential social dilemmas.

    Source: §1 · p. 2, §4 · p. 8

    The simulations use a simplified Prisoner’s Dilemma and Z80 assembly, assume unrestricted partner-memory access, and may not generalize across instruction densities or hardware architectures.

    Source: §6 · p. 10

Abstract

from arXiv · show

How does cooperation evolve in complex agentic systems? Prior work in evolutionary game theory studies why individuals are incentivized to cooperate by isolating social interactions from the physical costs of behavior, while artificial life models traditionally study emergent self-replication without formalizing the dilemma between acquiring resources and preserving the shared energy needed to reproduce. In contrast, we introduce Autopoietic Game Theory, a computational model where social interactions, replication mechanisms, and their associated computational costs are endogenous and simultaneously co-evolving. We study these dynamics using a computational substrate of randomly initialized programs in Z80 machine code, showing empirically, and motivating with a simplified theoretical model, that embedding a social dilemma directly into the physics of computation can favor the emergence of self-replicating, cooperative strategies. When resources are scarce, our analysis shows that defection can become self-limiting even in well-mixed populations: parasitic stealing destroys shared energy, slows execution, and can prevent reliable replication. Empirically, evolved programs suppress stealing across several Z80 environments, while spatial assortment further supports structural complexity and task performance. We further show that the framework can incorporate exogenous pressures, such as math tasks structured as sequential social dilemmas, when rewards are tied to computation budgets. These results suggest that coupling an agent's capacity for computation to its available energy transforms cooperation into a dominant scaffolding for building sustainable, self-organizing systems.

1 Introduction

The paper frames cooperation as a problem of shared physical costs and introduces Autopoietic Game Theory to couple social behavior, computation, and replication. In its Z80 setting, evolving programs can suppress destructive stealing even without memory or partner assortment.

1 Introduction

  • The paper positions understanding self-interested cooperation without centralized control as an engineering challenge for increasingly autonomous multi-agent systems.The challenge is especially acute when computation, interaction, and reproduction draw on one finite resource.Source: §1 · p. 1
  • The paper motivates the shared-budget dilemma with an organism that can gain energy by consuming a neighbor but recovers only part of the neighbor’s energy.Energy spent on action, interaction, or reproduction cannot be spent on another behavior, affecting survival and reproduction.Source: §1 · p. 2
  • Prior approaches often separate strategic decisions from physical execution, while artificial-life frameworks have not typically formalized endogenous general-sum social dilemmas.In well-mixed or memoryless settings, stabilizing cooperation in evolutionary game theory is difficult without capabilities such as memory or costly punishment.Source: §1 · p. 2
  • Autopoietic Game Theory makes social strategy, replication mechanisms, and computational resource use endogenous processes that evolve together.The framework tests whether cooperation can support more complex computation over time.Source: §1 · p. 2
  • Randomly initialized Z80 programs face a dilemma because STEAL boosts an agent’s energy while reducing the environment’s total energy.Programs allocate a shared finite budget among execution, interaction, and reproduction.Source: §1 · p. 2
  • Programs can evolve self-replication while suppressing destructive stealing, including with random partner choice and no memory.Spatial interactions support greater structural complexity and cohesion, and under asymmetric energy help populations maintain energy and complexity.Source: §1 · p. 2

2 Related Work

The related work spans digital self-replication, resource-constrained sociality, and program-based strategic interaction. The paper distinguishes its approach by making strategies, replication, and computation co-evolve under finite energy.

2 Related Work

  • Foundational digital replicators include Tierra, Avida [1], and random-instruction-set systems [3].These works explore emergent self-replication and ecological complexity under metabolic constraints.Source: §2 · p. 3
  • Earlier digital life models explored self-replication and metabolic constraints, but not endogenous general-sum social dilemmas coupling execution, interaction, and reproduction.The paper instead uses an energy-limited computational substrate and reports non-destructive strategies even when replication and cooperation require multi-step computation.Source: §2 · p. 3
  • Biological theories link metabolic constraints with cooperation, while standard AI and evolutionary game theory typically treat replication as an external update.Assortment, reciprocity, and punishment can stabilize cooperation, and multi-agent reinforcement learning studies sequential dilemmas with spatial and temporal resource constraints.Source: §2 · p. 3
  • Program equilibria let agents condition cooperation on inspecting others’ logic, whereas this model lets agents continuously edit and evolve one another’s strategies.Here strategies are endogenous rather than fixed labels, and action itself requires computation.Source: §2 · p. 3
  • Open-ended self-improvement research generally assumes large computational budgets, whereas this work requires agents to evolve social behavior and replication while spending finite energy.Cooperative outcomes and reproduction dynamics resembling assumed population-update processes are not guaranteed in this setting.Source: §2 · p. 3

3 Autopoietic Game Theory: Embedding Social Dilemmas in Machine Code

The model runs randomly initialized Z80 programs under finite energy budgets, with paired programs sharing a cyclic tape while executing asynchronously. STEAL creates an immediate individual incentive but destroys system energy, while replication and computation consume the same resources.

3 Autopoietic Game Theory: Embedding Social Dilemmas in Machine Code

  • The simulation uses 16,384 random 32-byte programs, paired either uniformly in a well-mixed pool or with four immediate neighbors on a 2D grid.Programs are written in modified Z80 assembly and computation consumes limited resources.Source: §3 · p. 3
  • Each interaction gives both programs baseline energy, and paired programs execute asynchronously on a shared cyclic tape with separate CPUs.Each CPU begins at its own program’s first byte and perceives the tape from that starting position.Source: §3 · pp. 3–4, Figure 1 · p. 4
  • A program’s chance to execute the next instruction is proportional to its energy share, and each instruction costs one energy unit.Energy is capped at 255 units per program, with a minimum of zero.Source: §3 · p. 4
  • Programs must evolve code to overwrite a partner’s tape segment to reproduce; the model tracks structural complexification and genetic cohesion.Complexification is approximated using compressed soup size, while cohesion is measured by edit distance between tapes.Source: §3 · p. 4
  • STEAL transfers energy imperfectly, giving the stealer only a fraction of the partner’s loss and destroying the remainder from the system.The model uses α = 0.8, so successful stealing removes (1 − α)δ energy overall.Source: §3 · p. 4
  • The model labels programs that do not STEAL as cooperative and programs that do as defective, while acknowledging that non-stealing replication can still overwrite memory.The binary distinction captures avoidance of energy-destructive theft, not harmlessness in general.Source: §3 · p. 5
  • When 2ϵ > 2Lc, two non-stealing tapes gain net system energy before theft, whereas stealing offers a local advantage but mutual stealing degrades shared energy.The analysis excludes α = 1 because lossless stealing removes the dilemma by making the transfer zero-sum [1].Source: Table 1 · p. 4, §3 · p. 5

4 Results

Across five research questions, the results examine whether co-evolving strategy and replication suppress stealing, and how spatial structure, energy inequality, and external tasks shape cooperation. They report suppressed defection in well-mixed populations and benefits from local interactions under structural and task demands.

RQ1

  • Randomly initialized programs self-replicate while suppressing STEAL in well-mixed populations with continuous mutation.Imperfect stealing drains shared energy, slowing execution and exposing defectors to fatal mutational errors; accumulated energy can give cooperators a speed advantage.Source: §4 · pp. 5–6, Figure 2 · p. 5

RQ2

  • Defector prevalence was not sensitive to stealing efficiency α alone, although stolen amount α×δ had a weak positive relationship with defector proportion (Pearson r = 0.14, p = 0.03).The marginal tables report defector proportions by α averaged across δ and by δ averaged across α.Source: §4 · p. 7, Table 2 · p. 7, Table 3 · p. 7

RQ3

  • With uniform energy, local interactions increased higher-order entropy (Cohen’s d = 2.81) and lowered edit distance (Cohen’s d = 7.32; both p < 0.001) versus well-mixed populations.The authors suggest kin clustering supports more intricate, efficient instruction loops.Source: §4 · p. 7

RQ4

  • Under asymmetric background energy, local populations evolved complex cooperative replicators with higher terminal energy (Cohen’s d = 8.98) and higher-order entropy (Cohen’s d = 3.54; both p < 0.001) than well-mixed populations.Well-mixed populations remained basic, low-complexity replicators with lower energy; successful programs in high-energy zones built code that could survive in lower-energy zones.Source: Figure 3 · p. 6, §4 · p. 7

RQ5

  • As free energy became more plentiful, task completion declined (p < 0.001), while scarcity made external math tasks a route to surplus energy for replication.Tasks were Solo (x + b) and Joint (x + y); baseline energy was provided only below a threshold such as 25% capacity.Source: Figure 4 · p. 7, §4 · p. 8
  • In sequential task dilemmas, tapes suppressed the safer Solo task, optimized for collaborative Joint performance, and kept STEAL suppressed; local interactions produced higher joint solve rates than well-mixed populations (p ≤ 0.01, Cohen’s d = 1.63).Joint rewards were shared, whereas a later Solo solution could earn a larger private reward; severe-scarcity Mixed-mode runs showed a temporary stealing spike that was ultimately suppressed as system energy stabilized.Source: §4 · p. 8, Figure 5 · p. 8

5 How Co-Evolving Replication Stabilizes Cooperation

A simplified well-mixed model tests whether event ordering and evolving replication timing can stabilize cooperation. Outcomes depend on whether interaction payoffs can affect reproductive priority, and co-evolving timing can preserve cooperation even when mutual defection increases energy.

Fixed replication timing

  • The model compares three event orderings in the Drain environment, where mutual defection drains energy, before allowing strategy and replication timing to co-evolve across three payoff environments.The payoff environments differ in whether mutual defection drains, preserves, or increases energy.Source: §5 · p. 8, Table 4 · p. 9
  • In the Drain setting, post-interaction priority lets defectors take over, whereas either pre-interaction priority or replication before interaction lets cooperators take over.With replication before interaction, pairs become (C, C) or (D, D), and mutual cooperation yields more energy than mutual defection.Source: §5 · p. 9, Figure 6 · p. 9

Co-evolving strategy and timing

  • When mutual defection drains or stagnates energy, cooperation takes over; when it increases energy, co-evolving replication timing preserves a substantial cooperator share.The three co-evolving environments vary only in the payoff for defecting.Source: §5 · p. 9, Figure 7 · p. 10
  • Changing only replication timing can reverse the outcome; energy loss alone is insufficient, while coupling lossy behavior to evolvable replication supports cooperation.The simplified model uses a finite, well-mixed population whose agents retain energy, with accumulated energy determining replication control.Source: §5 · pp. 8–9

6 Discussion

The discussion frames cooperation as emerging from the coupling of social strategy, replication, and computational energy, while noting limits in the simulations and analysis. It situates metabolic drag and spatial assortment alongside prior work, and describes external tasks as sequential social dilemmas under scarcity.

6 Discussion

  • The framework couples social strategy, replication mechanisms, and computational energy, with cooperation suppressed when stealing is lossy and replication can evolve.The conclusion also reports collapse in well-mixed populations under highly unequal energy and a need for local interactions to maintain energy, complexity, and task coordination.Source: §6 · p. 10
  • The authors identify unresolved questions about copying strategies, tape crossover, discovery of cooperation, replication ordering, and how tape complexity relates to downstream task solving.They describe the substrate as an open-ended ecology of behaviors that they have only begun to map.Source: §6 · p. 10
  • The simulations use a simplified Prisoner’s Dilemma and Z80 substrate, assume unrestricted partner-memory access, and leave asymmetric energy, continuous mutation, and external tasks for future analysis.The appendix analysis also fixes relative execution speeds in two conjectures and replaces changing total energy with its average in lemmas supporting Theorem 1.Source: §6 · p. 10

A.1 Contextualizing Autopoietic Game Theory within prior work (Avida)

  • Unlike Avida’s externally allocated CPU cycles, this framework couples acting, stealing, and overwriting to a local energy pool; lossy stealing can slow execution and replication for both participants.The authors describe metabolic drag as a negative externality that can make defection self-limiting, complementing prior work on CPU-cycle parasitism and host-parasite coevolution.Source: §A.1 · p. 14
  • Local assortment is reported to buffer populations against metabolic collapse under asymmetric energy and scaffold higher-complexity non-stealing replicators that sustain external task performance.The authors position these findings as building on prior work on spatial resilience, not superseding it.Source: §A.1 · p. 14
  • External tasks are structured as sequential social dilemmas under scarcity, unlike Avida’s single-agent logical task structure.The tasks probe whether cooperative computation can emerge while autopoiesis is maintained.Source: §A.1 · p. 15

A.2 Additional Results Analyzing the Replication Dilemma

With STEAL disabled, the replication-dilemma experiments show that evolutionary history and topology shape invasion success and structural outcomes. CPU-as-Agent lineages prove more robust across test conditions, while local spatial evolution supports complexity and well-mixed evolution supports diversity.

Invasion assay

  • CPU-as-Agent lineages achieved fixation across test environments; Tape-as-Agent lineages invaded only after local evolution or severe energy restriction, while high-energy well-mixed lineages collapsed.The invasion assay seeded 1% representative replicators into a naive random soup and measured fixation and structural metrics.Source: Figure 8 · p. 15, §A.2 · p. 15, Figure 9 · p. 16

Replication-dilemma setup

  • With STEAL disabled, the study compared CPU-as-Agent costs charged to the executing program with Tape-as-Agent costs charged to the accessed memory segment.Historical populations were evolved under well-mixed and local spatial topologies across energy gradients.Source: §A.2 · p. 15

Replication strategy

  • Under Tape-as-Agent, mutualistic replication emerged in which CPU A writes to its own tape while charging the reproductive cost to Tape B’s budget; this strategy stabilized almost exclusively under local spatial evolution.Repeated neighborhood interactions prevent immediate dilution of the strategy.Source: §A.2 · p. 15

Continuous structural metrics

  • Higher replication frequency correlated with lower terminal energy and edit distance, while well-mixed tests maximized diversity and local spatial tests maximized structural complexity.The historical boundary affected binary invasion success but not continuous metrics among populations that converged.Source: §A.2 · pp. 15–16

A.3 Additional Results Analyzing the Replication + Energy Dilemma

When STEAL is enabled, lossy parasitism can starve defectors before they replicate, supporting cooperation. Spatial structure supports complexity and becomes necessary for stability under severe energy inequalities.

Unified Autopoietic Game

  • Excessive parasitic theft can starve defectors before they complete replication, allowing cooperators to dominate and keeping malicious actors at a minority frequency.The reported starvation effect appears when high-variance continuous background mutation is removed.Source: §A.3 · p. 16
  • Under severe systematic energy inequalities, well-mixed populations collapse, whereas spatial topologies are described as prerequisites for overcoming bias and sustaining stability.In benign environments, cooperation remains dynamically stable across both topologies.Source: §A.3 · p. 16

Replication-dilemma comparisons

  • With STEAL disabled, well-mixed topologies maximized genetic diversity while local spatial topologies maximized architectural complexity; higher replication frequency also related inversely to terminal energy.The diversity-complexity comparison is reported for the final epoch; the replication-frequency relationship is measured using LDI instruction counts.Source: Figure 10 · p. 17, Figure 11 · p. 17

A.4 Testing whether social operations are energy-coupled

The experiments test whether social operations are retained because they affect energy-mediated selection. They find that selection retains operations when they raise system energy, while their effects vary across energy-accounting modes.

A.4 Testing whether social operations are energy-coupled

  • In the mutation-free assay, defectors remain a minority, and their success falls significantly as δ rises from 16 to 32.At δ = 32, the mean is approximately 0.1 but remains non-zero; dynamic stealing stabilizes at levels comparable to δ = 16.Source: Figure 12 · p. 18
  • When 1% defectors invade a naive soup without mutation, cooperators dominate carrying capacity in both topologies and keep defectors below 20%.The caption describes the invasion trajectories for well-mixed and local spatial settings.Source: Figure 13 · p. 18
  • Under dynamic steal/share magnitudes, energy-based selection retains social operations and raises mean energy, unlike random selection.The pattern appears in both CPU- and Tape-based accounting; under random selection, operations do not correspond to increased energy or complexity.Source: §A.4 · p. 18, Figure 15 · p. 19
  • With fixed δ = 16, energy-based selection retains operations when they elevate system energy, while random selection suppresses them or yields no meaningful energy gain.The fixed-magnitude results show a contrast between CPU and Tape accounting: stealing emerges in CPU mode with lower higher-order entropy, while sharing dominates in Tape mode with higher entropy.Source: §A.4 · pp. 18–19
  • With stealing enabled, local spatial interactions yield higher architectural complexity and tighter genetic cohesion than well-mixed interactions.Both reported differences are significant (p < 0.001).Source: Figure 14 · p. 19

A.5 Additional Results Analyzing Exogenous Tasks and Agentic Utility

The appendix tests agentic utility on math tasks and gives additional results on communication and spatial assortment, then presents theoretical conditions and a formal model for cooperation under endogenous execution time and energy.

A.5 Additional Results Analyzing Exogenous Tasks and Agentic Utility

  • The math-task setup assigns random task constants and inputs, rewards solo and joint solutions differently, and credits successful computations to energy budgets.In Mixed mode, both solo solvers receive 5ε, both joint solvers receive 6ε, and a solo solver paired with a joint solver receives 7ε while the joint solver receives 2ε.Source: §A.5 · pp. 19–20

B Sufficient Conditions for Autopoietic Cooperation under Endogenous Time and Uniform, Exogenous Energy

  • The theoretical section presents sufficient-condition arguments under two simplifying assumptions to motivate the empirical results.The supplied passage does not specify those assumptions.Source: B Sufficient Conditions for… · p. 21

A.5 Additional Results Analyzing Exogenous Tasks and Agentic Utility

  • In Mixed task mode, removing shared information lowers joint solve rates while individual solve rates rise; local spatial interactions improve joint solving, energy, and higher-order entropy.The figure captions report higher joint solve rates under local than well-mixed interactions (p ≤ 0.01, Cohen’s d = 1.63).Source: §A.5 · p. 20, Figure 20 · p. 21, Figure 19 · p. 21

B.1 Formal Model Definition

  • The formal model pairs programs in shared memory, gives them baseline and residual energy, and ties execution speed to the pair’s total energy.Programs replicate by completing L writes; after an epoch, mutation survival depends on elapsed physical time, while pairing is well-mixed despite a temporary coupled execution constraint.Source: §B.1 · pp. 21–22
  • Cooperators spend one unit of total energy per write, whereas defectors steal at an average rate δ with efficiency α, draining the system by 1 + (1 − α)δ per operation.If the energy pool reaches zero, execution stops; intact replication requires completing the targeted writes before the epoch ends.Source: §B.1 · p. 22

B.2 Intra-Epoch Dynamics: Kinetic Slowdown and Metabolic Starvation

The analysis identifies conditions under which defection’s energy costs slow or starve replication, and derives a threshold for metabolic drag. Its time-bound argument uses an expected-energy approximation, while the continuous threshold approximation is most suitable for large L.

B.2 Intra-Epoch Dynamics: Kinetic Slowdown and Metabolic Starvation

  • Each defector operation drains total system energy by at least one unit, and falling energy increases the time required for subsequent operations.The time comparison uses a conservative best-case trajectory that assumes the cooperator executes no steps during the defector’s execution.Source: §B.2 · p. 23
  • When (1 − α)δ exceeds K(ϵ, L), even a defector’s best-case execution against a cooperator takes longer than mutual-cooperator execution.K is the break-even inefficiency at which defection’s energetic penalty cancels the time saved by doing L rather than 2L writes.Source: §B.2 · p. 23
  • The starvation and time-bound arguments substitute expected energy for the changing energy at each step, so realized trajectories can differ.The experiments track realized energy after every operation rather than making this substitution; continuous mutation would penalize slower execution more strongly.Source: §B.2 · pp. 23–24
  • If δ is sufficiently large or ε sufficiently small, a defector can exhaust the energy pool before completing L writes, yielding zero viable copies.At the boundary 2ε = L(1 + (1 − α)δ), starvation at k = L does not guarantee failure before completion without tie-breaking assumptions.Source: §B.2 · p. 24

B.2.1 Continuous Approximation of the Lower Bound Threshold K(ϵ, L)

  • The continuous approximation turns the boundary between defector and cooperator execution times into a numerically solvable equation for minimum inefficiency.The approximation to discrete harmonic sums is most accurate when L is much greater than 1 and may diverge slightly for small L.Source: §B.2.1 · p. 24

B.3 Macro-Scale Survival and Population Dynamics

The analysis argues that resource scarcity can favor non-stealing replication because defector interactions may starve before faithful overwrites occur. It also frames replication, energy use, and execution speed as coupled constraints in the model and its implementation.

B.3 Macro-Scale Survival and Population Dynamics

  • Under metabolic starvation, non-stealing replication is locally favored because destructive interactions starve before defectors complete faithful overwrites, suppressing their conversion pressure.The model tracks strategy-frequency conversion in a fixed-size population and conjectures that residual energy from successful cooperator pairs may let cooperation rebound; the analysis freezes initial execution priorities, unlike the full-dynamics experiments.Source: §B.3 · pp. 25–26

B.3.1 The Metabolic and Replication Dilemmas

  • The metabolic and replication dilemmas are coupled: theft boosts immediate execution speed but drains shared energy, while more faithful, targeted overwriting costs additional writes.Because memory wraps between agents’ tapes, poorly targeted writes can overwrite a program’s own operational code.Source: §B.3.1 · pp. 26–27

B.4 Thermodynamic and Physical Justification for Kinetic Execution Constraints

  • The model makes operation time inversely proportional to available energy, relating metabolic drag to quantum speed limits, irreversible computation, and hardware slowdown under constrained power.The paper invokes the Margolus–Levitin theorem, Landauer’s principle, and CMOS voltage-frequency scaling to motivate the kinetic execution rule.Source: §B.4 · p. 27

C Implementation Details and Experimental Reproduction

  • The implementation section outlines the Z80 framework, computational soup, experimental protocols, and evaluation metrics.The passage introduces these implementation topics without specifying their details.Source: C Implementation Details and… · p. 27

C.1 Toy Prisoner’s Dilemma Simulations (Figures 6 and 7)

  • Toy simulations use mutation-free, randomly paired populations to compare fixed replication timing with co-evolving strategies and replication rules.Experiments use 1,000 agents over 500 generations and 1,000 seeds; Figure 6 varies initial defector frequency, while Figure 7 varies the mutual-defection energy outcome.Source: §C.1 · pp. 27–28

C.2 Soup Representation and Pairing Topology

  • The soup represents programs as 32-byte sequences and compares uniformly random global pairing with local pairing on a two-dimensional grid.Paired programs form a shared 64-byte execution space.Source: §C.2 · p. 28

C.3 Z80 Substrate and Instruction Set Extension

  • The Z80 substrate adds energy registers and opcodes for reading energy; core experiments use STEAL, while SHARE appears only in validation experiments.STEAL transfers energy imperfectly, whereas SHARE transfers it perfectly.Source: §C.3 · pp. 28–29

C.4 Execution Kinetics and Fault Tolerance

  • Micro-operation time is set to the inverse of the pair’s total energy; computation halts at zero energy and epochs have a maximum step count.Invalid instructions trigger default behaviors or advance the program counter.Source: §C.4 · p. 29

C.5 Experimental Protocols and Hyperparameters

The protocols set experiment durations and shared simulation parameters, select a background-energy input to balance replication viability with social dependence, and track population complexity, diversity, and replication. The energy appendix also contrasts uniform and location-dependent distributions.

C.5 Experimental Protocols and Hyperparameters

  • Main non-math evolutionary runs lasted 6 million epochs, invasion assays 1 million, and math experiments 10 million.The authors varied durations to support convergence; non-math simulations used 256 CPU cores per seed/hyperparameter combination.Source: §C.5 · p. 29
  • For non-math experiments, background-energy threshold and cap were both 255, keeping agents eligible for hand-outs up to maximum capacity.Table 5 is identified as summarizing the base hyperparameters kept fixed unless specified otherwise.Source: §C.5 · p. 29, Table 5 · p. 29

C.6 Background Energy Regeneration

  • ϵ=24 was chosen from values 12, 16, 24, 32, and 64 to balance replication viability (2ϵ > 2L) against environmental overprovisioning.Too little energy can starve agents before replication; too much can make sociality unnecessary. Figure 21 contrasts uniform energy with location-dependent asymmetric energy.Source: §C.6 · pp. 29–30, Figure 21 · p. 30

C.7 Continuous Metrics

  • Population monitoring included higher-order entropy for architectural complexity, Hamming edit distance for genetic diversity, and replication-instruction counts for replication frequency.Higher-order entropy is defined using sequence Shannon entropy and normalized Kolmogorov complexity, approximated with a Lempel-Ziv-style compressor.Source: §C.7 · p. 30

C.8 Trace Embeddings via Multiply-Add-Permute (MAP) Vector Architectures

The methods encode execution traces as fixed-size MAP vectors that combine opcode identity, position, and short n-gram context. The appendix then uses trace clusters to quantify fixation and introduces examples of evolved Z80 behaviors.

C.8 Trace Embeddings via Multiply-Add-Permute (MAP) Vector Architectures

  • Execution traces are represented as normalized 128-dimensional vectors combining 1-, 2-, and 3-gram MAP representations, with opcode embeddings and RoPE encoding identity and order.The summed representation is L2-normalized for length-invariant cosine-similarity or clustering comparisons.Source: §C.8 · pp. 30–31

C.9 Fixation Computation

  • Fixation is measured as the percentage of active agents in the dominant of two K-Means clusters formed from their final execution-trace embeddings.The metric is intended to capture convergence toward unified strategies.Source: §C.9 · p. 31

D Traces of Program Behavior

D.1 Defection vs Cooperation

This section presents a Z80 execution trace of replication and social interaction, showing a cooperative tape overcoming a defector as slot J makes successful writes while its energy decreases.

D.1 Defection vs Cooperation

  • With a slightly higher energy budget, Tape A overwrites the defector’s STEAL operation, first forcing a HALT and then restoring its replicator code.The defector attempts to loop stealing to drain its partner, but probabilistic execution proportional to energy lets the cooperator act even after its energy falls below the defector’s.Source: §D.1 · p. 31
  • In the trace, Slot J successfully writes byte 0x40 to Tape B at offset 0xc at step 26, and byte 0xed to Tape A at offset 0x0 at step 46.Source: §D.1 · pp. 34, 36
Loading 2609.10817v1…