There Is No Free Lunch After All

For a month and a half I looked at how much intelligence a swarm of language models actually adds. Not with my own runs, but on recorded answers of open models, at zero inference cost.¹ The answer turned out to be dull. Twenty-four models from thirteen families on ARC-challenge carry the information of roughly one independent model, plus a little. Majority voting over a pool of uneven strength is worse than simply taking the best model. Selection with a perfect verifier gives ×1.25 over the strongest model on ARC and ×1.43 on GSM8K, and that is an upper bound. And 20% of ARC tasks are solved by no model in the pool, under any combination.

A swarm turned out to be a slightly extended version of its strongest member.² So the question shifted: if combining adds almost nothing, what sets the boundary of what a model can solve at all?

The formulation we arrived at on September 4.

Let S be the set of primitives on which the model's prior has mass: what the training data taught it. Let C_d(S) be the closure of S under composition of depth d, reachable in a single pass. Then:

Working title: "There Is No Free Lunch After All." The nod to Wolpert 1996 is deliberate. The second half one is tempted to add, "the model memorizes its prior," is too strong; the honest version is "the model moves mass, it does not create it."

Let me admit it upfront: the claim is stitched from three other people's results.

1. Griffiths & Kalish 2007. A chain of Bayesian agents, each learning from the output of the previous one, converges to the prior. Combining copies of one model adds nothing to the prior.
2. Strong data processing inequality (Polyanskiy). Contraction at every step is strict and multiplies across steps. A task is choked by its weakest factor.
3. Yue et al. 2025 (2504.13837). At k in the hundreds, pass@k of the base model is no lower than that of the RLVR model trained on top of it. RL moves mass onto already existing correct paths and narrows the support; only distillation extends it, because the teacher is a different model.

Add the three and you get the claim. The content is not in the statement but in three places where it might fail to hold.

A definition of C_d(S) that can be checked on data. The naive "this step was not in the training set" is false. Skill-Mix (Arora & Goyal, 2310.17567) shows a model composing k skills that never co-occurred in the corpus. Grokking yields an algorithm from part of a table. The definition must survive both counterexamples, so C_d has to include composition and generalization within a primitive. Honestly, the whole future note is this definition.

That combination does not extend C_d(S). The only non-tautological part. For RLVR, Yue measured it. For swarms we have the three arguments from our internal notes (a function of the output cannot add information about the input; the contraction is strict and multiplies across steps; a chain of agents converges to the prior) and those 20% of ARC where no model is right. We decided to prove it for a single model; a swarm with a shared prior follows as a corollary via Griffiths, and a swarm with different priors is the same theorem for a mixture prior.

The shape of ε. This is not uniform guessing. ε is the residual mass of the prior near the required step, a decreasing function of the distance to C_d(S). Xie et al. 2022 (2111.02080) have the same thing in different clothing: concepts indistinguishable by KL give close outputs. Hence a testable prediction: if a primitive is removed from the corpus not binarily but gradually, leaving neighbors at varying distances, pass@k should decay smoothly, not as a step. A step would mean the closure is discrete.

Three papers found by abstract and not yet read, so here I am arguing with abstracts.

• ProRL (2505.24864). Long RL with KL control gives pass@k above the base where the base is 0% at any k.
• 2510.04028. A second stage of mass dynamics after the narrowing.
• Ariadne (2511.00710). Essentially our synthetic protocol with the opposite result.

All three fit inside the closure if d is not fixed but grows with the environment budget: a rewarded environment increases composition depth over the same primitives. The strong form then reads: nothing adds primitives outside S, and d grows only through the environment. The predictions have to split accordingly: short RLVR with binary reward narrows (Yue); long RL with KL control may extend along d. Ariadne will have to be reproduced and explained before we claim anything of our own.

If benchmark gains are mass transfer inside the prior, and the mass of a task is the product of the masses of its steps, then there is one bottleneck: the weakest step. Train the step, not the task. What goes into the training set is not tasks with low pass@k but steps with low mass. The hypothesis: this yields the same performance on fewer weights and cheaper. Part of this is already visible in phi and R1-Distill, which were fed solution trajectories rather than raw corpus.

The price is also known from the literature. Kirby, Cornish & Smith 2008: a language collapses from 27 words to 2–5 in ten generations. In Yue, the share of tasks with zero accuracy grows after RL. A model trained on its weakest step will be good on the benchmark and brittle one step to the side. Both the gain and the price have to be measured.

Three measurements, all with status "not started."

1. Step spectrum. Feed the model a correct prefix up to step t, sample continuations, compute the fraction that keeps the task solvable and the spectrum of the embeddings. The step with the lowest fraction is the bottleneck; a peaked spectrum at zero fraction means a wall, a flat one at nonzero fraction means a rare move that best-of-N will find.
2. Step-targeted training. qwen2.5:0.5b on GSM8K, two training sets at equal token budget: random tasks versus tasks containing the weakest step. Measure pass@1 on the full test set, pass@1 on tasks without that step, and pass@k at k = 64…256 as the width of the prior.
3. Synthetic testbed. Define the primitives ourselves, as in GINC (Xie) or "Physics of Language Models." Train a small transformer from scratch, remove one primitive, measure pass@k up to 1024 on tasks that need it. Arms: a single model, a swarm of copies with different seeds, RLVR with binary reward, a small dose of data containing the primitive. Plus graded removal and tasks of depth d+1.

The falsifier for the third item is direct. If a swarm or RLVR over the model without the primitive lifts pass@k well above ε with no data containing that primitive, combination extends the closure and the strong form is false. If the model is no worse at depth d+1 than at d, then d is not part of the boundary and the formulation should be simplified to C(S).

None of this can be proved on frontier models: the corpus is unknown and "a step outside the training set" is contestable. The bridge from synthetics to real models is OLMo and Pythia with open data and infini-gram over the pretraining corpus. Weaker, because compositional closure cannot be caught by n-grams, but it is the only honest bridge.

The foundations are unread: Griffiths & Kalish 2007 itself with five companion fragments, Xie 2022, Faith and Fate (2305.18654), Comprehension Without Competence (2507.10624), RL vs Distillation (2505.14216), Yue 2025 in full. The opponents are unread: ProRL, Ariadne, 2510.04028. There is no step vocabulary, without which items 1 and 2 cannot start; there is only a sympy canonicalization for GSM8K as an attempt at one.

The intended form is a technical note rather than a paper. There may be little scientific novelty in the claim. The novelty, if any, is in a definition of C_d that can be checked on data, and in measuring ε as a function of distance. If the testbed shows a step instead of a smooth decay, or a swarm lifts pass@k without data, I will write that up the same way.

Archived with a DOI: 10.5281/zenodo.23132912. Cite as: Latipov, A. (2026). There Is No Free Lunch After All. Zenodo. https://doi.org/10.5281/zenodo.23132912

1. Per-instance answers from the archived first Open LLM Leaderboard: the frozen per-model details datasets under open-llm-leaderboard-old. The exact 24 models, their dataset revisions and the loader are in Alatip/tech-note-1 (versions/v003/manifest.json, DATA.md). The method, with the tetrachoric correction, is in the previous note, Effective Independence. ↩
2. Two things stand behind this sentence. First, communication between agents that share a prior does not add information, it removes it: the strong data processing inequality makes every round a strict contraction, so the agents' outputs grow more correlated and the ceiling of any averaging falls. Empirically, Yang et al. (2602.03794) find debate below plain voting at every N, and Chen et al. (2604.18005) see the same as "structural coupling" in ideation. Second, we sorted aggregation into three operators: averaging (voting; bounded by the error correlation, saturates by about ten agents); a priori selection (routing on the input, before the answer exists); a posteriori selection (verification on the output). Only the last one does not saturate, and only if a verifier exists, so "posterior max where available" is the only operator worth paying for. A mixture of experts is the first two fused: a router that chooses before seeing the result and a weighted sum of experts; it lacks exactly the a posteriori check and any memory that survives the pass. ↩

← notes · github.com/Alatip