Fix the recipient specification, task family, reveal schedule, interaction permissions and resource accounting. Abstract a contributor \(v\) by a causal interaction law, written as budget-indexed kernels

Here \(H\) is the history available to the contributor, \(b\) is its remaining contribution budget, and \(h\) includes the next message or failure/stop output with its joint resource record. The kernels arise from one consistent causal process under stopping: different budgets cannot be assigned unrelated ideal answers. No access to unrevealed test instances or their answers is permitted.

Two internal realizations are equivalent for this abstraction when they induce the same joint laws of transcripts, costs and stopping for every admitted recipient protocol and budget. Mathematically equivalent message contents need not be equivalent interactions: the recipient may find one encoding much easier to interpret or verify. A useful contribution may develop through several exchanges. Producing, interpreting, rejecting, correcting and learning from suggestions all consume resources.

Conditional theorems may assume properties of this interface without simulating a brain or reconstructing a civilization. An unconditional existence result additionally requires a realizable member of the assumed class, for example an executable source with disclosed initial state and costs. A controlled human protocol instead supplies empirical estimates. An arbitrary answer kernel or assumed success probability proves neither.

The target statements, truth contract, checker and reveal schedule stay fixed. Contributors may have different broad backgrounds or search biases; these endowment differences are disclosed. Relevant prior results must be justified under the target checker. A hidden target parameter or precomputed answer table is a different mechanism from conceptual insight. Equalizing all background material is an optional stronger comparison. Giving the recipient textual access to the relevant material can test whether acquiring and using the connection remains costly. Historical preparation follows the symmetric accounting convention in Definition [ftip-00MK].

Fix a development law and recipient interaction protocol \(\Pi \) before final evaluation. For development input \(D\) and resulting contribution transcript \(H_v\), choose an independently specified structural predicate \(S(D,H_v)\). Define

A possible property is a sound reduction to a declared tractable class with bounded translation and certificate-expansion costs. A reusable invariant with a proved consequence or a curriculum satisfying a separate learning condition are other possibilities. A fallible analogy may require charged recipient completion before the property holds. The predicate is not simply that the final score improved; soundness, scope and checking costs require their own arguments. Quality is relative to this task, protocol and budget, not a universal ranking of minds.

For \(0<q_0\leq 1\), the restricted class

is legitimate. An existence theorem about a member of this class need not help every contributor outside it. Nonemptiness remains a separate obligation; choosing sources on final test outcomes does not establish pre-evaluation availability. Recruitment and screening costs are necessary when claiming an affordable procedure to find a good contributor, but mere existence does not require a population recruitment theorem.

To discuss a median, additionally declare a population law \(\mu \) on admissible contributors with measurable quality \(v\mapsto q_v\). Let \(q_{50}\) satisfy \(\mu (q_v\leq q_{50})\geq \frac 12\) and \(\mu (q_v\geq q_{50})\geq \frac 12\). Then define

If \(q_0\leq q_{50}\), this class has at least half the population mass. If \(q_0>q_{50}\), even nonemptiness needs evidence. The median can be zero, so belonging to the upper half alone does not guarantee useful assistance. There is no default uniform law over all possible contributors, and an unattained quality supremum need not have a best contributor.

Fix one contributor and one recipient protocol with almost-sure total resource caps, including failed consultations. Let \(Z\in [0,1]\) be the realized fresh-task score of its frozen artifact with contributor access removed. Suppose, for the same joint process and remaining resources,

Then \(Q_{\mathrm {acq}}=\mathbb E Z\geq q_0\beta \).

No independence assumption is used. A uniform recipient guarantee for qualified transcripts and compatible development histories can establish the transfer premise, including interpretation, verification and acquisition costs. A guarantee for a different, easier message distribution cannot.

Together with a universal closed bound \(Q_{\mathrm {acq}}(P)\leq \tau ^-\) and an admissible member of \(\mathcal G(q_0)\), the strict inequality \(q_0\beta \geq \tau ^+>\tau ^-\) yields the assisted crossing in § [ftip-00MN]. The substantive obligations are affordable source quality, recipient transfer and the bound over every admitted unaided alternative. The elementary expectation inequality alone establishes none of these.

Fix a scalar accounting model with the other constraints declared. Suppose an admitted assisted campaign \((R,v)\) at total budget \(B\) has an admitted closed simulator at budget \(g(B)\). Require that the simulator can reproduce the contributor's endowment, observations and causal interaction, together with the recipient's allowed final artifact and evaluation law. All simulation work and initialization count. Then, for \(j\in \{\mathrm {disc},\mathrm {acq}\}\),

If the simulation is uniform across instance sizes with \(g(B)\leq cB\) for a fixed constant \(c\), an actual assisted procedure reaching \(\tau \) at work \(U(n)\) implies \(B_{\mathrm {closed}}(n,\tau )\leq cU(n)\). This excludes a lower bound \(L(n)\) with \(U(n)/L(n)\to 0\). Unavailable background state, observations, hardware or large simulation overhead can invalidate the premise; different origin alone does not. The argument does not assume that a language model already has an affordable simulation of a human contributor.

Admission includes discovery of any required reconstruction program from the declared initial state. A simulator written with advance knowledge of the useful representation does not satisfy this premise merely because its later execution is cheap. If the program is generated during the campaign, its generation law, failed attempts and selection costs belong in the simulation. This conditional proposition supplies no lower bound on that discovery cost; § [ftip-00N6] distinguishes program existence from its availability to the lineage.

In this exchange, the human is the contributor and the agent is the recipient: the system receiving and processing the contribution. The agent's proposals arose within an already human-directed conversation; they were not outputs of a separate unaided experiment.

1. The agent proposed a resource-bounded formulation and a finite missing-evidence construction. The human rejected missing facts as the intended mechanism and described conceptual leaps across model generations: new perspectives, paradigms and connections between domains. The agent revised the target to conceptual discovery and separated prohibitive finite work from permanent unreachability using the enumeration countercheck.
2. The agent specified an executable source with bounded observations. The human challenged an oracle-like abstraction, pointed to diverse cultural backgrounds and developed representations, and asked for an abstract account of what the source contributes. The agent formulated the causal contribution interface, with background differences disclosed rather than a mandatory model of a brain or lifetime.
3. The human emphasized that existence of good contributors need not imply usefulness of all contributors, and suggested an upper-median restriction. The agent separated independent structural quality, class nonemptiness and recipient transfer, while showing why a median requires a population and need not have positive quality.
4. The human suggested studying recursive self-improvement papers for the gap between the problems they diagnose and the portions their methods repair. The agent examined contemporary work and checked a revised primary source that narrowed stronger self-imitation claims. The resulting thesis distinguishes affordable quality, recipient acquisition and a universal unaided bound, with proof, refutation and precise obstacles all possible.

In Definition 1, \(H\) corresponds to the exchange so far and \(h\) to the human's next criticism, analogy or research direction. The agent's retained conversation and revised artifacts form its evolving state; reasoning, literature search and checking are its processing actions. The human did not supply a completed theorem: the assistance changed the problem representation and choice of inquiry. The observed outcome was a revised conjecture, before any fixed-target acquisition experiment. No structural-quality probability, cost advantage, held-out acquisition score or universal unaided bound was measured, so it does not establish the separation conjecture.