Open vs. Closed Recursion in AI: A Four-Literature Synthesis

TL;DR

Most production AI is closed-loop, goal-seeking recursion. A recursion becomes a mirror when it loses coupling to an external source of variety.

On the central open question — the minimum perturbation needed before a recursion stops being a mirror — there is no single universal number, but three sharp, partly-reconcilable answers: (1) empirically, model-collapse work shows that accumulating/retaining real data rather than replacing it bounds error, and even ~10% real-data preservation reduces degradation to "minor," while the pessimistic "strong collapse" result finds even 1% synthetic data can cap gains; (2) formally, Ashby's Law of Requisite Variety gives an information-theoretic lower bound — the regulator's variety must match the disturbance variety; (3) dynamically, von Foerster's eigenform theory implies convergence to a fixed point is the default attractor of any closed recursion, so "becoming a mirror" is the null outcome you must actively perturb your way out of.
Goal-seeking is not inherently incompatible with open recursion, but pure objective-maximization against a stationary self-generated distribution is. The open-endedness literature (Stanley, Clune, Hughes et al.) shows objectives become "deceptive" straitjackets at long horizons, while RL practice quietly concedes the same point by injecting entropy/noise to prevent premature convergence. Open recursion requires an external, non-stationary source of novelty (real data, a co-evolving adversary, environmental perturbation, or an observer who keeps redefining "interesting").

Key Findings

Model collapse is real, quantified, and reversible by anchoring to external real data — not by abstinence from synthetic data. Shumailov et al. (Nature, 2024) show recursive self-training causes early collapse (tails vanish) then late collapse (variance shrinks to a point). Gerstgrasser et al. (2024) prove accumulating rather than replacing data bounds test error by a constant; Bertrand et al. (2023) prove a fixed-point stability condition on the real-data fraction; Dohmatob et al. (2024–25) show even tiny synthetic fractions can break scaling. The minimum-signal question therefore has a partial quantitative answer that depends on the data regime.
Autopoiesis contradicts the user's framing at the level of the word "closed" but agrees at the level of mechanism. Living systems are operationally closed (self-producing) yet structurally coupled (open to perturbation). What the user calls a "dead mirror loop" is what Maturana–Varela would call a system that has lost structural coupling — not one that is closed.
Open-endedness research gives a formal, observer-relative definition (novelty + learnability) that is essentially a restatement of the mirror question, and its founders argue objectives are the enemy of sustained novelty.
Cybernetics supplies the two missing formal tools: Ashby's requisite variety (how much variety a regulator needs) and von Foerster's eigenforms (why recursion converges to fixed points — i.e., why the mirror is the default).

Details

Thread 1 — Model Collapse / Autophagy: the empirical, quantitative answer

Shumailov, Shumaylov, Zhao, Papernot, Anderson & Gal, "AI models collapse when trained on recursively generated data," Nature 631:755–759 (2024) (extending their 2023 arXiv preprint "The Curse of Recursion: Training on Generated Data Makes Models Forget"; Author Correction Nature 640:E6, 2025). The canonical result. Training a generative model (they test GMMs, VAEs, and the OPT-125m language model on wikitext2) on the previous generation's output causes model collapse:

Early collapse: the model loses the tails — rare/low-probability events disappear first.
Late collapse: the model converges to a low-variance distribution bearing little resemblance to the original — a near-delta "mirror" of itself.
Three compounding error sources: statistical approximation error (finite sampling drops rare events), functional expressivity error (model class can't represent the true distribution), functional approximation error (learning-procedure bias).

Concrete numbers, directly on the user's question: in the "no original data retained" regime, task performance degraded "from 20 to 28 perplexity points" across generations; in the "Ten epochs, 10% of original training data preserved" regime (a random 10% of the original data resampled each generation), the authors report that "preservation of the original data allows for better model fine-tuning and leads to only minor degradation of performance." So ~10% fresh real signal per generation substantially arrests collapse in their setup, without eliminating it.

Alemohammad, Casco-Rodriguez, Luzi, Humayun, Babaei, LeJeune, Siahkoohi & Baraniuk, "Self-Consuming Generative Models Go MAD," ICLR 2024 (arXiv:2307.01850). Coins Model Autophagy Disorder (MAD) by analogy to mad cow disease. Studies three autophagous-loop families: fully synthetic; synthetic augmented with a fixed real set; and synthetic mixed with fresh real data each generation. Verbatim primary conclusion: "Our primary conclusion across all scenarios is that without enough fresh real data in each generation of an autophagous loop, future generative models are doomed to have their quality (precision) or diversity (recall) progressively decrease," and "appreciable MADness arises in just a few generations." Only loops with sufficient fresh (not merely fixed) real data fully avoid both precision and recall loss.

Gerstgrasser et al., "Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data" (arXiv:2404.01413, 2024). The most important reframing. Prior work assumed data is replaced each generation; they argue accumulation is the realistic web dynamic. Result: replacing → collapse; accumulating → collapse avoided. In their analytically tractable linear model, where replace causes test error to grow linearly with iteration count (extending Dohmatob et al.), accumulation makes test error bounded by a finite constant independent of iteration count. Validated on GPT-2/Llama-2 (TinyStories), GeoDiff diffusion (molecules), and VAEs (images). This is a provably different regime, not merely better practice: the rule is "accumulate, don't replace."

Bertrand, Bose, Duplessis, Jiralerspong & Gidel, "On the Stability of Iterative Retraining of Generative Models on their own Data" (arXiv:2310.00429, ICLR 2024). Proves a fixed-point stability theorem (Theorem 1): iterative retraining is stable iff (1) the initial model approximates the real distribution well enough AND (2) the proportion of real ("clean") data in each retraining set is high enough. Below that threshold, collapse. This is the formal statement that a critical real-data fraction exists — a minimum perturbation — though the paper does not give a universal numerical value, and its condition is sufficient, not necessary.

Dohmatob, Feng, Yang, Charton & Kempe, "A Tale of Tails: Model Collapse as a Change of Scaling Laws" (ICML 2024, arXiv:2402.07043), plus "Model Collapse Demystified: The Case of Regression" and Dohmatob, Feng, Subramonian & Kempe, "Strong Model Collapse" (ICLR 2025, arXiv:2410.04840). These sharpen and worsen the picture. "A Tale of Tails" recasts collapse as a modification of neural scaling laws — loss of scaling, shifted scaling, skill "un-learning," and grokking when human and synthetic data are mixed. "Strong Model Collapse" proves the pessimistic result verbatim: "even the smallest fraction of synthetic data (e.g., as little as 1% of the total training dataset) can still lead to model collapse: larger and larger training sets do not enhance performance" (the OpenReview spotlight puts the threshold as low as "1 per 1000"). It further shows that "in a simplified regime where neural networks are approximated via random projections of tunable size … larger models can amplify model collapse" (below the interpolation threshold, by overfitting to synthetic artifacts), while they may mitigate it near/above the threshold.

The critique / "overstated" camp. (a) Gerstgrasser et al. themselves are the strongest counter: under realistic accumulation, "the curse of recursion may not be as dire as portrayed." (b) The a16z-style objection quoted in that literature: mass deletion of real data "is not what is happening on the internet — we won't replace the Mona Lisa." (c) Curated synthetic data demonstrably helps: Gunasekar et al., "Textbooks Are All You Need" (arXiv:2306.11644, 2023) trained phi-1, a 1.3B-parameter Transformer, "for 4 days on 8 A100s, using a selection of 'textbook quality' data from the web (6B tokens) and synthetically generated textbooks and exercises with GPT-3.5 (1B tokens)," attaining "pass@1 accuracy 50.6% on HumanEval and 55.5% on MBPP" — performance "comparable to models 10× larger and trained on 100× more data." Synthetic data with a verification/curation step (cf. Feng et al., "Beyond Model Collapse: Scaling Up with Synthesized Data Requires Verification") behaves fundamentally differently from indiscriminate autophagy. (d) The "Position: Model Collapse Does Not Mean What You Think" paper argues many collapse results rest on strong, often unrealistic assumptions (e.g., Bertrand's condition is sufficient, not necessary).

Verdict: collapse is a genuine attractor of the pure closed loop, but the "minimum signal to escape" depends entirely on whether data accumulates, whether synthetic data is curated/verified, and the mixing ratio — it is not a fixed universal constant.

Thread 2 — Autopoiesis and Operational Closure: the apparent contradiction

Maturana & Varela, Autopoiesis and Cognition (1980) and The Tree of Knowledge (1987/1992). An autopoietic system is "a network of processes of production of components which … continuously regenerate the network of processes that produced them" and constitute the system as a unity. The load-bearing distinction:

Operational (organizational) closure: the system's organization is closed — its processes produce only more of its own processes; it maintains identity "in the face of perturbations."
Structural coupling: though organizationally closed, the system is in constant interaction with an environment that acts as a source of perturbation — perturbation triggers (but does not specify or instruct) internal state changes. Maturana explicitly rejected the input-output/information-processing model: environmental interactions are "not information transfers but structural perturbations."

So the autopoietic slogan — closed to information (meaning/instruction), open to perturbation (and energy/matter) — is the direct antithesis of the user's "closed = incestuous mirror." In autopoietic terms, the user's healthy "open recursion" is a structurally coupled operationally-closed system, and the user's "dead mirror" is a system that has lost structural coupling (or, in Maturana's terms, suffered a destructive interaction and died). The word "closed" is doing two incompatible jobs.

Luhmann extended operational closure to social systems: function systems (law, science, economy) are operationally closed (law produces only law via the legal/illegal code) yet structurally coupled to one another and to psychic systems; communication, not information transfer, is the operation. This demonstrates operational closure scaling to non-biological systems.

Enactivism / Varela's later work. Varela's move from autopoiesis to enaction ("Laying Down a Path in Walking," 1987; Varela, Thompson & Rosch, The Embodied Mind, 1991) reframes cognition as sense-making: the organism "brings forth a world" through structural coupling, and meaning is a "surplus of significance" the organism enacts, not a signal it receives. The signature metaphor — "there is no path, you lay down a path in walking" (Machado, via Varela) — is itself a description of open recursion: the loop's output reshapes the very landscape that perturbs its next iteration.

Application to LLMs — Zönnchen, Dzhimova & Socher, "From intelligence to autopoiesis: rethinking artificial intelligence through systems theory," Frontiers in Communication 10:1585321 (2025). A peer-reviewed Luhmannian conceptual analysis directly on point, with a deliberately nuanced conclusion:

LLMs are NOT fully autopoietic sense-making systems: "while language models exhibit self-referential and recursive properties, they do not engage in their own sense-making, as they do not produce or reproduce their own system/environment distinction. Their outputs are generated through probabilistic distributions rather than reflexive attributions of sense."
Yet they DO grant LLMs operational closure as computation: minds think, society communicates, and "language models … compute — asserting themselves by their operational closure." LLMs occupy "an ambiguous position: they are neither mere tools nor autopoietic sense-making cognitive agents."
On the self-output loop vs. human coupling: an LLM revisiting its own text (they cite chain-of-thought) shows only "processual self-referentiality" / "an implicit feedback loop," but "this self-reference is not intrinsic but must be explicitly reintroduced through input." Decisively: "a system that references itself too rigidly risks either infinite recursion or collapse, while one that ignores self-reference entirely remains externally determined. The key challenge, then, is achieving a balance — an oscillation between self-reference and external reference." This is, in Luhmannian language, exactly the user's "how much outside signal" question, and the answer is qualitative: neither pure self-reference (collapse) nor pure external determination, but an oscillation.
Human–LLM interaction is a "structural coupling" producing "virtual double contingency" and "artificial communication" (Esposito). The mirror metaphor appears explicitly in the primary literature: "Much like a mirror reflects an individual's image from an external viewpoint, language models reflect psychological and societal contingency in a machine-processed form."

Thread 3 — Open-Endedness: novelty, learnability, and the enmity of objectives

Stanley & Lehman, Why Greatness Cannot Be Planned: The Myth of the Objective (2015). The foundational anti-objective argument, grounded in Lehman & Stanley's novelty search (2011): rewarding behavioral novelty alone often solves deceptive problems better than rewarding the objective, because "the precursors to a great achievement often do not resemble that achievement." In their words, objectives "limit our freedom and become straightjackets around our desire to explore," and the paradoxical thesis is that greatness is most likely when it is not set as an objective — objectives become obstacles once the goal is more than "one stepping-stone away." Novelty search is framed as an information accumulator: "you have to acquire some kind of knowledge to continue to produce novelty." This is the purest statement of "open recursion": a loop driven by divergence from its own past rather than convergence to a fixed target.

Clune, "AI-GAs: AI-Generating Algorithms" (2019), and POET — Wang, Lehman, Clune & Stanley, "Paired Open-Ended Trailblazer" (arXiv:1901.01753, 2019) + Enhanced POET (arXiv:2003.08536, 2020). POET co-evolves environments and their solutions simultaneously (originally on a 2-D bipedal-walker obstacle course): it generates its own curriculum of challenges (held in a "minimal-criterion" band — neither too easy nor too hard) and transfers stepping-stone solutions between niches. The architectural insight for the debate: open-endedness is engineered by making the environment itself a non-stationary, co-evolving source of perturbation — the loop is prevented from becoming a mirror by continuously manufacturing new outside.

Hughes, Dennis, Parker-Holder, Behbahani, Mavalankar, Shi, Schaul & Rocktäschel (Google DeepMind), "Position: Open-Endedness is Essential for Artificial Superhuman Intelligence" (ICML 2024, arXiv:2406.04268). Provides the field's cleanest formal, observer-relative definition:

A system S producing artifacts X_t is open-ended iff, from the perspective of an observer O, the sequence is both novel and learnable. Novelty: artifacts become increasingly unpredictable w.r.t. O's fixed-time model — ∀t, ∀T>t, ∃T′>T : E[ℓ(t,T′)] > E[ℓ(t,T)] ("there is always a less predictable artifact coming further in the future"). Learnability: conditioning on a longer history makes artifacts more predictable — E[ℓ(t′,T)] < E[ℓ(t,T)] for t′>t.

The noisy-TV thought experiment is a direct answer to the mirror question: a TV of uniform random noise is learnable but not novel (once O's model converges to uniform, novelty dies — aleatoric noise is "collapsed by the expectation"); a TV randomly switched to arbitrary channels is novel but not learnable (history doesn't help). Pure randomness is not open-endedness. Critically, the paper concedes present foundation models are not open-ended: "If the distribution of this data is learnable, which it must be, for the foundation model learned it in the first place, then it cannot be endlessly novel, because eventually the observer will have modelled the epistemic uncertainty." Static-dataset training is closed by construction.

Self-referential self-improvement in LLM systems. Promptbreeder (Fernando et al., ICML 2024, "self-referential self-improvement via prompt evolution"); Gödel Agent (Yin et al., ACL 2025, arXiv:2410.04444); the Darwin Gödel Machine (Zhang et al., 2025); SICA (Robeyns et al., 2025). These replace Schmidhuber's intractable provable self-modification (the Gödel machine, 2003/2007) with empirical acceptance: a modification is kept only if it improves a held-out benchmark score. This is the operational crux: the benchmark is the external anchor. A self-improving loop with no external evaluator is a pure mirror; these systems stay productive precisely by coupling self-modification to an outside signal. Reported behaviors include continuous improvement but also "rare catastrophic failures" mitigated by rollback (and cost-efficient convergence, e.g. ~$15 for 30 Gödel-Agent iterations) — the loop degrades without the anchor.

Thread 4 — Cybernetics: first- vs second-order, requisite variety, eigenforms

First-order cybernetics (Wiener). Negative feedback, homeostasis, goal-seeking control — the thermostat, with the observer outside the system. This is exactly the user's "closed, goal-seeking loop," and the user is right that it describes most production AI: a controller minimizing an error signal against a fixed setpoint/objective.

Ashby's Law of Requisite Variety (An Introduction to Cybernetics, 1956): "only variety can destroy [absorb] variety." The formal answer to "how much outside signal." A regulator R can hold an essential variable within bounds against disturbances D only if the regulator's variety is at least as large as the disturbance variety it must compensate. The Conant–Ashby theorem adds that "every good regulator of a system must be a model of that system." Heylighen & Joslyn's information-theoretic formulation (following Aulin) shows disturbance variety in the essential variables can be reduced by (1) buffering, (2) increasing regulator action-variety H(R), or (3) reducing uncertainty about which action to select, H(R|D) — i.e., knowledge. This is the formal minimum-perturbation bound: an adaptive loop must internally command at least as much variety as its environment throws at it, or "the environment will dominate and ultimately destroy that system."

Beer's Viable System Model (VSM). Applies requisite variety to organizations: a viable system recursively balances variety between operational units and management via variety engineering (attenuating environmental variety, amplifying regulatory variety), with algedonic alerts bypassing hierarchy in emergencies. Relevance: viability is recursive (each subsystem is itself a viable system) and is maintained by matching internal to external variety — a systems-engineering instantiation of open recursion.

Second-order cybernetics (von Foerster): the cybernetics of observing systems. "First-order = cybernetics of observed systems; second-order = cybernetics of observing systems." The observer is inside the loop. This maps precisely onto Hughes et al.'s observer-relative open-endedness and onto the autopoietic claim that "everything said is said by an observer" (Maturana).

Von Foerster's eigenforms / eigenbehaviors ("Objects: Tokens for (Eigen-)Behaviours," 1976; formalized in Kauffman, "EigenForm," 2005). The deepest result for the mirror question. When an operator (a recursion) is iterated, it converges to fixed points — eigenforms — invariant under further application: "an eigenform is a fixed point of an operator, seen as the emergence of an 'object' … in the course of a recursion." What we experience as stable "objects" are the eigen-behaviors of our recursive sensorimotor/cognitive loops. Kauffman's crucial observation: "at the abstract level every recursion has a fixed point." The consequence for AI is stark: convergence to a stable, self-reproducing form is the default attractor of any closed recursive process — "becoming a mirror" is not a failure mode, it is the generic terminal state. To avoid the eigenform (to stay open-ended) you must continuously inject something that resets or perturbs the operator — precisely the outside signal the user is asking about.

RL exploration, entropy, and noise — the same question in disguise. Maximum-entropy RL (Ziebart 2008/2010; Haarnoja et al., Soft Actor-Critic 2018) and entropy regularization in policy-gradient methods (Williams & Peng 1991; A3C, Mnih et al. 2016; PPO, Schulman et al. 2017) add an entropy bonus so the policy does not "prematurely converge to a small set of high-reward actions." Without it, the policy collapses to a deterministic eigenform (a mirror of its early lucky rollouts). Parameter-space noise (Plappert et al.) and count/curiosity-based intrinsic rewards do the same. This is production AI quietly admitting Ashby's law: the learning loop must be dosed with a minimum entropy/noise to keep exploring — and the entropy coefficient is typically annealed down over training precisely because you want eventual convergence to an eigenform once the environment has been sufficiently sampled. The "how much noise" question is literally the entropy-coefficient hyperparameter — with the documented failure mode that "uniform entropy regularization can mislead optimization in control tasks requiring precise, low-entropy policies" (entropy traps).

Recursive self-improvement in safety discourse. I. J. Good's 1965 "intelligence explosion" ("the first ultraintelligent machine is the last invention man need ever make"); Yudkowsky's "seed AI"; Bostrom's Superintelligence. The hard-takeoff argument assumes the self-improvement loop is productively open (each gain enables further gains); Yudkowsky argues it "should either flatline or blow up." Chollet's "The Implausibility of Intelligence Explosion" (2017) is the key counter: intelligence is embedded in context; there is "no such thing as general intelligence independent of environment"; recursive self-improvement faces diminishing returns, not exponential blow-up — because a loop cut off from an ever-richer external environment exhausts its own variety. This is the model-collapse debate one abstraction level up: is a self-referential improvement loop open (structurally coupled to a rich world) or closed (feeding on its own outputs)? Empirically, current self-improving agents stay open only via external benchmarks/verifiers.

The Contradiction Map (the core deliverable)

(A) "Closed = life" (autopoiesis) vs. "closed = death" (user). This is ~80% terminological, ~20% substantive. Terminologically, Maturana–Varela "operational closure" refers to closure of the organization (the system produces its own components) and is explicitly paired with structural openness to perturbation; the user's "closed" refers to closure of the information/perturbation channel (nothing new gets in). Different axes. Substantively, the two frameworks agree: both hold that a recursion sustains interesting behavior only while coupled to an external source of perturbation, and both predict degeneration when that coupling is lost. The genuine residue is about description: autopoiesis insists the organization must be closed (self-producing) for there to be a "self" at all — a pure open pipe with no closure is not an entity, just a channel. Sharpest synthesis: a living/open recursion is operationally closed and structurally coupled; a dead/mirror recursion is operationally closed and structurally decoupled. The user should say "structurally coupled vs. decoupled," not "open vs. closed," to avoid talking past a DeepMind scientist who will hear "closed" in the Maturana/Luhmann sense.

(B) Objectives vs. open-endedness — false dichotomy with a real core. Stanley's strong claim ("objectives are the enemy of novelty") and RL's objective-maximization look contradictory but are reconciled by the stepping-stone distance and non-stationarity qualifiers. Objectives are fine when local (one stepping stone away — the thermostat, most deployed systems); they are deceptive at long horizons. POET, novelty search, and MaxEnt RL all resolve this the same way: keep the loop open by making either the objective non-stationary (co-evolving environments/adversaries) or the reward partly novelty/entropy-based. Goal-seeking per se is not the enemy; a fixed objective optimized against a stationary distribution of your own outputs is. Note that Hughes et al. — themselves DeepMind RL researchers — argue open-endedness is essential and not at odds with foundation-model/RL methods; their paper is a reconciliation, not a rejection. So the opponent's likely position (that objective-driven RL is compatible with open recursion) is defensible; the user's stronger claim needs the qualifier "against a stationary self-generated distribution."

(C) Do the model-collapse ratio and Ashby's requisite variety agree? Partially — they are the same law at different resolutions. Ashby is a necessary information-theoretic condition: regulator variety ≥ disturbance variety. Model collapse is the empirical realization in generative training: as the fresh-real-data fraction (the external variety being injected) drops toward zero, the model's effective variety decays and it can no longer match the real distribution's variety — and the tails (rare, high-information events) vanish first, exactly as Ashby predicts (low-probability states are the ones a low-variety regulator cannot cover). They agree in direction and mechanism but are not numerically unifiable into one constant, because (i) Ashby's bound is on variety/entropy while collapse thresholds are stated in data-mixing fractions that depend on model class, accumulation vs. replacement, and curation; and (ii) Bertrand's "sufficiently high real fraction," Shumailov's "~10% preservation," and Dohmatob's "even 1% synthetic hurts" measure different things (a sufficient anchoring fraction vs. a destructive contamination fraction). Honest synthesis: the minimum perturbation is not a universal constant but a ratio between injected external variety and the intrinsic variety-loss rate of the specific loop. Ashby says the ratio must be ≥1 in variety terms; the collapse literature says that in practice, for LLM pretraining under accumulation, a persistent nonzero real-data anchor suffices, whereas under replacement no fraction is safe indefinitely.

(D) Is "becoming a mirror" the default attractor? Yes — this is the strongest cross-cutting conclusion. Von Foerster's eigenform theorem says every recursion has fixed points and iterating drives you toward them. Late model collapse (convergence to a low-variance fixed distribution), RL policy collapse without entropy, and MAD are all eigenform phenomena. Therefore the mirror is not a pathology requiring explanation; it is the null hypothesis, and open-endedness is the thing that requires explanation and active engineering**.** Every "open" system across these literatures maintains openness by the same trick: continuous coupling to an external, non-stationary, higher-variety source — real data (collapse), perturbation/energy (autopoiesis), a co-evolving environment or an observer who keeps moving the "interesting" goalposts (open-endedness), or disturbance variety plus entropy injection (cybernetics/RL). The eigenform is what you get when that coupling stops.

Recommendations (debate strategy and thresholds)

Staged positioning for next week:

Concede and reframe the terminology first. Open with: "By 'closed' I mean structurally decoupled — informationally sealed — not operationally closed in the Maturana sense. With that fixed, autopoiesis is on my side: a living loop is operationally closed and structurally coupled; a dead loop has lost the coupling." This preempts the single strongest rebuttal a DeepMind scientist can make.
Anchor the empirical claim on Gerstgrasser + Bertrand + Dohmatob, not Shumailov alone. The Nature paper is famous but has been partly relativized by the accumulate-vs-replace distinction; leading with it invites the "collapse is overstated" counter. The defensible strong claim: under replacement, no synthetic fraction is safe (error grows without bound); under accumulation with a persistent real anchor, error is bounded. That is a proven dichotomy.
Make the eigenform point the intellectual centerpiece. "The mirror is the default attractor of any recursion (von Foerster); everything we call 'open' is an active fight against convergence to an eigenform." This is the most powerful and least obvious claim, and it unifies all four threads.
On the minimum-perturbation question, state the tri-partite answer explicitly (Ashby ratio ≥ 1; empirical anchoring fraction under accumulation; entropy coefficient in RL) and concede there is no universal constant — the answer is a ratio of injected variety to intrinsic variety-decay, which is exactly why it feels unanswerable as a single number.

Benchmarks/thresholds that would change these positions:

If someone demonstrates a genuinely closed self-improvement loop (no external evaluator, no fresh data) that sustains novelty+learnability over many iterations without converging, the eigenform-default claim (D) is falsified. To date none exists — Gödel-agent variants all rely on external benchmarks.
If accumulation is shown to fail at web scale (compute can't keep pace, forcing downsampling → effective replacement — the "Collapse or Thrive?" regime), the empirical optimism in (C) weakens.
If a formal bridge is built expressing collapse mixing-ratios directly in Ashby variety/entropy units, the "not numerically unifiable" caveat in (C) is resolved — this is a genuinely open research problem worth proposing.

The sharpest unresolved questions to debate:

Is there any recursion that stays open without an external variety source — or is "openness" always parasitic on an outside? (All four literatures currently say: always parasitic.)
Is the minimum perturbation a ratio (injected variety ÷ intrinsic decay rate) rather than an absolute threshold — and can it be expressed in one currency (bits of entropy) across data-mixing, structural coupling, and RL exploration?
Does the tails-first signature of collapse (rare events die before the mode) prove that the mirror consumes information/variety in exactly Ashby's sense — making model collapse the empirical proof of the requisite-variety law?
If the eigenform is the default, is the right design goal not "avoid convergence" but "control which eigenform, and oscillate deliberately between self-reference and external reference" (Zönnchen et al.'s framing)?

Caveats

Terminological hazard is severe. "Closed," "open," "recursion," and "novelty" each mean different things across the four literatures; several apparent contradictions dissolve on disambiguation and are flagged as such above. Do not present terminological differences as substantive ones.
Model-collapse numbers are setting-specific. The ~10% figure is Shumailov's OPT/wikitext2 fine-tuning setup; the 1% (≈1-per-1000) figure is Dohmatob's regression/scaling-law setting; Bertrand's "high enough" is unquantified. None is a universal threshold, and combining them would be an error.
Autopoiesis-to-AI is contested and largely philosophical. Maturana himself resisted extending autopoiesis beyond biology; Luhmann's social extension and the LLM extension (Zönnchen et al.) are analogical/conceptual, not empirical. Zönnchen et al. explicitly label the "LLM recognizes its own system/environment distinction" scenario as purely speculative.
Open-endedness formalism is a 2024 position paper, not settled theory; some researchers (Stepney & Hickinbotham) doubt open-endedness can be quantified at all.
Speculation flags. The intelligence-explosion / hard-takeoff literature is heavily forecast-driven ("could," "would"); Chollet's diminishing-returns critique and the empirical self-improvement results are the more grounded anchors. Treat takeoff claims as contested predictions, not findings.
The central question is genuinely open. There is no derived closed-form for "minimum perturbation to escape mirror-hood" that spans domains; the strongest honest statement is the variety-ratio framing — a structure for the answer, not a number.