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Summary. I model an agent's cognitive reach as a Turing ideal and define its effective closure: the limits it can approach using approximations it can generate itself. Iterating that closure from the computable sets gives exactly the finite levels of the arithmetical hierarchy, and it does so only because convergence is uncertified. If a modulus of convergence were required, the closure would be idempotent and there would be only one rung. Three consequences seem relevant here. (1) No observer, of any computational power, can verify in the limit that a process has left the closure of a countable cognitive space, while next-rung content can be verified from two rungs up. (2) An agent can come to believe correctly, in the limit, that a peer lies within its own space, but no agent can ever come to believe correctly that a peer exceeds it. (3) An agent whose strategy draws on a source it does not hold is unpredictable even when its cognition is fully subsumed. The material is four working papers and a three-page synopsis, linked at the end. They were formalised with Claude under my direction, and they keep their retractions in the text. I'd welcome demolition.
The setup, in five sentences
A cognitive space is a Turing ideal: the class of decision problems an agent can settle, closed under computation and join. The agent can also do something it cannot certify. It can generate a sequence of guesses and commit to the limit. The effective closure cl(C) is the set of all limits of sequences uniformly computable in C. (Without effectivity, the closure of any space is trivial, so boundaries between minds are properties of agents, not of the topology.) Iterating gives R₀ = computable, Rₖ₊₁ = cl(Rₖ), and the Rung Theorem says Rₖ = Δ⁰ₖ₊₁, with each inclusion strict.
That much is the limit lemma in new clothes, and I'll say so before anyone else does. Two pieces of the packaging do work, though.
The ladder exists only because convergence is uncertified. If you define a certified closure, where the limit comes with a modulus m ∈ C after which the guesses stop changing, then clᵐᵒᵈ(C) = C. Requiring certificates collapses the ladder to one rung. Every rung above the first is bought with an unverifiable commitment to a limit. In the papers this act is called promotion: it changes nothing about what the agent computes and everything about which sets its beliefs converge to.
It stalls. The union of the rungs (the arithmetical sets) is a fixed point of the closure, and ∅⁽ω⁾ (the ω-th jump) lies outside it. Convergence alone never leaves the arithmetical world. Crossing requires a leap, which is an oracle, not a method. And voids are generic: the closure of any countable space is meagre and null.
The result that bears on oversight
Theorem 3 (Indistinguishability), Paper B. Let C be a countable Turing ideal and V its void (everything outside cl(C)).
Every finite observation is consistent both with a limit in cl(C) and with a limit in V.
No observer, of any computational power, verifies V in the limit. In particular, none decides cl(C) in the limit.
For C = Rₖ, an observer computable from ∅⁽ᵏ⁺²⁾ verifies Rₖ₊₁ in the limit.
Read with an AI-oversight lens: whether a process's output has left the overseer's effective reach is not a question the overseer can answer from below, however much compute it has, and not even in the limit. The positive half (3) is that one rung of excess can be verified, from two rungs up. So the verification gap is exactly one rung wide on the positive side and unbounded on the negative side.
Theorem 6, Paper B. An observer computable from x′ verifies, in the limit, that a peer lies within x's space. No observer of any power verifies that a peer lies outside it. Subsumption is limit-verifiable. Strict superiority never is.
The general shape is that evidence of being below can accumulate and evidence of being above cannot. I think this is the cleanest formal version I know of the intuition that a sufficiently capable system can't be certified as more capable by its overseer, only as not less. It's also why "the evaluator saw nothing" is weak evidence: the absence of a superior is exactly what no detector can certify (Paper C makes this precise: no sound detector is complete on the non-subsumed points).
"Isn't this Gold / Putnam / Kelly?"
Partly, and the papers say where. Limiting identification (Gold 1965/67, Putnam 1965) and Kelly's limiting verification are the ancestors, and Theorem 3(2) is in the family of results showing that Σ-type properties are verifiable in the limit and their complements are not. What I think is new:
The closure is applied to the learner's own space and iterated, so the hierarchy appears as a ladder of minds rather than of problems.
The modulus result says the ladder exists because of the missing certificate. Certified closure is idempotent.
Agents are pairs (certified space, committed space). Promotion is a first-class operation, and transmission between agents has exactly three routes: method (one rung up), testimony, or certification from strictly above.
The asymmetry about peers (Theorem 6) is a statement about multi-agent verification, not about learning a fixed target.
If this is already in the literature in that form, I'd genuinely like the reference.
Two further consequences
The Gambler (Paper C). A strategic agent is (C, P, r), where r is a source its strategy may depend on. A Gambler has r ∉ P: its policy has taken an oracle leap that its cognition hasn't. If the strategy actually computes r (for example, by consuming its bits), then an agent that holds everything the Gambler can settle, and can prove it, still cannot compute the Gambler's strategy unless it computes r. Cognitive subsumption does not give strategic predictability. With a random source, the Gambler's moves can be audited in hindsight from three rungs above its content, with a delay bound the auditor knows in advance, but never predicted. A delay-coded source built from a generic construction can be made unreadable at any prescribed finite rung. Demonstrating superiority spends concealment. The source buys unreadability but not height.
Five filters (Paper D, the Fermi application). Between a physical population and an observed one there are five filters: existence, persistence, causal reach, channel access, cognitive access. For a marked Poisson population they compose into an accessible intensity Λ, with P(silence) = exp(−Λ). Each kind of darkness is the failure of one filter. Liu's dark forest is one of four: temporal, strategic, archaeological, cognitive. The modelling result that surprised me: a lineage that withholds testimony still leaves an observable record unless it can also close the passive channel, so "choosing silence" doesn't buy concealment in general. That distinction only appeared after the model caught its own conflation of being above a peer with having access to it.
Method, including what went wrong
I'm a former lattice-QCD physicist now working in finance. The ideas and questions are mine. The formalisation, proofs, simulations and drafting were done with Claude under my direction, and the drafts were read adversarially by a separate model acting as reviewer. Three errors were caught that way, and they are kept in the papers as findings:
A lemma claiming that "excess" is decided at the first jump was false; it used an inequality backwards. Its replacement, a sandwich bound, led directly to the hidden-information theorem in Paper A.
A phase diagram in Paper D showed a "silence floor of one half". It turned out to be the fraction of lineages not yet born, sitting in the denominator.
Paper D first claimed that switching off testimony changed nothing. That was because the model treated height as access. Separating them produced the active/passive channel result and the fifth filter.
None of these was caught by the model that made it. The pattern I'd draw for anyone doing theory this way: author for ideas and judgement, model for formalisation and execution, a separate adversarial reader for certification. Remove the third and I would expect undetected errors. Fittingly, that is Theorem 3 again: the system producing the convergent approximation is not the one that can supply the modulus.
What I'm unsure of
Paper A (the geometry of the rung distance) has one result proved only in sketch, and its strongest result is conditional on an embedding hypothesis for jump partial orders that I believe is open. If you know otherwise, please say.
Some of Paper C's strategic results rest on two postulates (elimination requires a model of the target; persistence has lexical priority). These are choices, and the conclusions are only as good as they are.
Paper D is a model paper. Its inputs are scenarios and its outputs are sensitivities. The robust claims are qualitative.
Everything is also in one place at github.com/n-goodman/effective-horizons. arXiv versions to follow. The most useful comments are counterexamples, prior art, and "this theorem doesn't say what the prose says it says."
Summary. I model an agent's cognitive reach as a Turing ideal and define its effective closure: the limits it can approach using approximations it can generate itself. Iterating that closure from the computable sets gives exactly the finite levels of the arithmetical hierarchy, and it does so only because convergence is uncertified. If a modulus of convergence were required, the closure would be idempotent and there would be only one rung. Three consequences seem relevant here. (1) No observer, of any computational power, can verify in the limit that a process has left the closure of a countable cognitive space, while next-rung content can be verified from two rungs up. (2) An agent can come to believe correctly, in the limit, that a peer lies within its own space, but no agent can ever come to believe correctly that a peer exceeds it. (3) An agent whose strategy draws on a source it does not hold is unpredictable even when its cognition is fully subsumed. The material is four working papers and a three-page synopsis, linked at the end. They were formalised with Claude under my direction, and they keep their retractions in the text. I'd welcome demolition.
The setup, in five sentences
A cognitive space is a Turing ideal: the class of decision problems an agent can settle, closed under computation and join. The agent can also do something it cannot certify. It can generate a sequence of guesses and commit to the limit. The effective closure cl(C) is the set of all limits of sequences uniformly computable in C. (Without effectivity, the closure of any space is trivial, so boundaries between minds are properties of agents, not of the topology.) Iterating gives R₀ = computable, Rₖ₊₁ = cl(Rₖ), and the Rung Theorem says Rₖ = Δ⁰ₖ₊₁, with each inclusion strict.
That much is the limit lemma in new clothes, and I'll say so before anyone else does. Two pieces of the packaging do work, though.
The ladder exists only because convergence is uncertified. If you define a certified closure, where the limit comes with a modulus m ∈ C after which the guesses stop changing, then clᵐᵒᵈ(C) = C. Requiring certificates collapses the ladder to one rung. Every rung above the first is bought with an unverifiable commitment to a limit. In the papers this act is called promotion: it changes nothing about what the agent computes and everything about which sets its beliefs converge to.
It stalls. The union of the rungs (the arithmetical sets) is a fixed point of the closure, and ∅⁽ω⁾ (the ω-th jump) lies outside it. Convergence alone never leaves the arithmetical world. Crossing requires a leap, which is an oracle, not a method. And voids are generic: the closure of any countable space is meagre and null.
The result that bears on oversight
Theorem 3 (Indistinguishability), Paper B. Let C be a countable Turing ideal and V its void (everything outside cl(C)).
Read with an AI-oversight lens: whether a process's output has left the overseer's effective reach is not a question the overseer can answer from below, however much compute it has, and not even in the limit. The positive half (3) is that one rung of excess can be verified, from two rungs up. So the verification gap is exactly one rung wide on the positive side and unbounded on the negative side.
Theorem 6, Paper B. An observer computable from x′ verifies, in the limit, that a peer lies within x's space. No observer of any power verifies that a peer lies outside it. Subsumption is limit-verifiable. Strict superiority never is.
The general shape is that evidence of being below can accumulate and evidence of being above cannot. I think this is the cleanest formal version I know of the intuition that a sufficiently capable system can't be certified as more capable by its overseer, only as not less. It's also why "the evaluator saw nothing" is weak evidence: the absence of a superior is exactly what no detector can certify (Paper C makes this precise: no sound detector is complete on the non-subsumed points).
"Isn't this Gold / Putnam / Kelly?"
Partly, and the papers say where. Limiting identification (Gold 1965/67, Putnam 1965) and Kelly's limiting verification are the ancestors, and Theorem 3(2) is in the family of results showing that Σ-type properties are verifiable in the limit and their complements are not. What I think is new:
If this is already in the literature in that form, I'd genuinely like the reference.
Two further consequences
The Gambler (Paper C). A strategic agent is (C, P, r), where r is a source its strategy may depend on. A Gambler has r ∉ P: its policy has taken an oracle leap that its cognition hasn't. If the strategy actually computes r (for example, by consuming its bits), then an agent that holds everything the Gambler can settle, and can prove it, still cannot compute the Gambler's strategy unless it computes r. Cognitive subsumption does not give strategic predictability. With a random source, the Gambler's moves can be audited in hindsight from three rungs above its content, with a delay bound the auditor knows in advance, but never predicted. A delay-coded source built from a generic construction can be made unreadable at any prescribed finite rung. Demonstrating superiority spends concealment. The source buys unreadability but not height.
Five filters (Paper D, the Fermi application). Between a physical population and an observed one there are five filters: existence, persistence, causal reach, channel access, cognitive access. For a marked Poisson population they compose into an accessible intensity Λ, with P(silence) = exp(−Λ). Each kind of darkness is the failure of one filter. Liu's dark forest is one of four: temporal, strategic, archaeological, cognitive. The modelling result that surprised me: a lineage that withholds testimony still leaves an observable record unless it can also close the passive channel, so "choosing silence" doesn't buy concealment in general. That distinction only appeared after the model caught its own conflation of being above a peer with having access to it.
Method, including what went wrong
I'm a former lattice-QCD physicist now working in finance. The ideas and questions are mine. The formalisation, proofs, simulations and drafting were done with Claude under my direction, and the drafts were read adversarially by a separate model acting as reviewer. Three errors were caught that way, and they are kept in the papers as findings:
None of these was caught by the model that made it. The pattern I'd draw for anyone doing theory this way: author for ideas and judgement, model for formalisation and execution, a separate adversarial reader for certification. Remove the third and I would expect undetected errors. Fittingly, that is Theorem 3 again: the system producing the convergent approximation is not the one that can supply the modulus.
What I'm unsure of
Links
Everything is also in one place at github.com/n-goodman/effective-horizons. arXiv versions to follow. The most useful comments are counterexamples, prior art, and "this theorem doesn't say what the prose says it says."