# Peano Hydra: where symbolic search should end and learned search begin Peano Hydra is an experiment, not a new trust assumption. We want to combine a proof-producing arithmetic prover with a small language model and ask a narrow, measurable question: > Under the same inference budget, can the language model help solve more new > problems than the strongest system that does not generate language-model > actions? The adjective *new* matters. The phrase *same budget* matters. Most of all, *solve* means that Peano Lab's independent kernel checks a complete certificate against the original formula. A model, a tactic, and an external prover may suggest a route; none may declare the theorem. The campaign's normative protocol is [the binding design](https://github.com/nasqret/vietnam2026/blob/peano-lab/docs/PEANO_HYDRA_DESIGN.md), and the executable milestone gates are in [the campaign plan](https://github.com/nasqret/vietnam2026/blob/peano-lab/PLAN/11_peano_hydra.md). This chapter explains why those rules exist. ## First correct the logical claim Peano Lab uses an intuitionistic proof calculus for first-order arithmetic. Standard first-order Heyting arithmetic is not a decidable theory: there is no terminating algorithm that correctly labels every sentence theorem or non-theorem. A finite website, a bounded search, or a decidable collection of exercises does not change that fact. There are therefore two honest goals: 1. build a **sound theorem prover** that returns a checked proof or `unknown`; 2. separately identify a **restricted decidable fragment**, state its exact grammar and semantics, and supply evidence for both positive and negative answers. The second claim is stronger. Positive proof certificates are already natural: \[ K(\Gamma, p, A)=\mathsf{accept} \] says that the small kernel checked certificate $p$ as a derivation of $A$ from $\Gamma$. A timeout cannot play the corresponding role for non-theorems. To call a fragment decidable we need a terminating procedure and either independently checkable negative certificates or agreement with a genuinely independent reference decision procedure. Until then Hydra is a sound prover, not a decider. ## One authority, many fallible explorers Hydra deliberately has many ways to be clever and one way to be right. ```text original goal -> deterministic symbolic closure -> certificate -> kernel | | critical frontier v typed macro proposal / | \ native search Qwen Vampire/E/SMT hints \ | / transactional engine | symbolic closure resumes | original-goal kernel replay / \ checked theorem rejection ``` The kernel is the sole positive authority. Everything else is untrusted: - native normalization, rewriting, connection or focused search; - theorem retrieval and clause ranking; - Qwen as a student policy or value model; - Codex as a teacher and data generator; - Vampire, E, or an SMT solver as an external search assistant; and - translators and proof reconstruction code. This is especially important for Vampire. Vampire is a powerful classical first-order prover, while Peano Lab's default logic is intuitionistic. A raw Vampire success is not automatically a proof in Heyting arithmetic. We may use it on a separately justified validity-preserving translation or on a stable arithmetic side goal, and we may mine its derivation for instantiations or lemmas. The final result still has to be reconstructed into ordinary Peano proof terms and replayed. There is no trusted `vampire_proved` rule. ## The critical frontier Calling a transformer for every tiny rewrite would be slow and scientifically uninteresting. Symbolic code is excellent when the next move follows from a dense local calculation. Language models are most plausible where search must make a sparse semantic choice. Hydra therefore runs cheap deterministic closure until it reaches a fixed point. The **critical frontier** is a stalled state at which there is no uniquely justified cheap continuation within the current bounds. Examples are: - choosing an existential witness; - inventing an intermediate lemma; - selecting an induction variable and motive; - selecting a useful case split; - retrieving a small premise bundle from hundreds of theorems; or - deciding which bounded solver should explore which subgoal. Only then is the generative model called. After one valid high-level choice, symbolic closure resumes. This leads to the testable decomposition \[ \text{proof search} = \text{cheap closure}^{*} ;\ \text{sparse macro choice} ;\ \text{cheap closure}^{*}. \] A cheap graph or state ranker may score the high-frequency inner loop. Qwen should earn its more expensive call by resolving ambiguity that the cheaper systems do not. ## A macro protocol, not a second proof language The model does not emit kernel constructors. It proposes a small typed action: ```text Use(name, specializations*) Cut(kind = have | suffices, name, formula) Witness(term) Induct(variable, motive) Rewrite(source, direction, location) Split(kind) Dispatch(solver, premises, bounds) ``` Each action has one canonical serialization and compiles deterministically to existing Peano Lab commands. For example, `Witness(t)` becomes the public `exists t` action; a `Cut` becomes ordinary `have` or `suffices`. `Dispatch` starts a bounded untrusted search and may return hints or a reconstructable derivation, but never closes a goal merely because a solver printed “theorem.” Compilation and execution are transactional. If parsing, specialization, rewriting, or reconstruction fails, the proof state and undo history remain exactly unchanged. The trace retains the raw model text, parsed action, compiled commands, intermediate states, solver transcript, resource use, and kernel result. This makes it possible to distinguish model failure, interface failure, search failure, and certificate failure. The first executable plumbing test uses a deliberately smaller compatibility format: one complete public Peano line with a structural head. It accepts such actions as `have`, `induction`, `exists`, `cases`, and `rewrite`, while rejecting `simp`, `ring`, `compact_arith`, tactical wrappers, session commands, and multiline scripts. This is enough to test the trust boundary and the symbolic/model hand-off. It is not yet the structured version-1 action schema above, and it is not training evidence. Recorded teacher routes require a complete kernel-checked QED by default; an open trace can enter only through an explicit partial-evidence option and never becomes a positive proof label. ## Why the library must have epochs The source-bound model-v3 training epoch contains 247 independently checked runtime theorems, including a constructive, conservatively encoded Fundamental Theorem of Arithmetic. It is a powerful training source. It is not a fair test of whether a model can prove the same theorems after reading their proofs. At the start of a campaign we freeze an ordered epoch $L_0$. Its content root commits to each theorem's canonical statement, dependencies, source and script, certificate, proof size and depth, and declaration order. A model can train on eligible $L_0$ material. The final benchmark is lineage-disjoint from it. If the mathematical library later grows, those theorems enter $L_1$. They cannot silently appear in retrieval or prompts for the $L_0$ experiment. A new epoch needs a newly sealed benchmark. This rule avoids a surprisingly easy mistake: improving the prover by adding the answer to its library and then reporting the improvement as search intelligence. ## Leakage follows mathematical ancestry Randomly splitting tactic rows is not enough. One authored proof can produce hundreds of state/action rows, and two differently worded statements can be the same mathematical problem. Hydra partitions *before* row expansion. Every target and artifact receives a lineage identity. The separation graph contains at least: - proof dependencies and reverse dependencies; - equivalent or stronger reformulations; - shared problem-family and generator ancestry; - generator seed and template relationships; and - authored, symbolic, teacher, and student provenance. For a sealed theorem $T$, training and run-time retrieval mask $T$, equivalent forms, its family, proof and trace, generator seed, stronger capstones, descendants that reveal it, and any theorem whose certificate depends on a masked node. The connected components are split first; only then are prompts, actions, negatives, or paraphrases produced. This is stricter than matching statement strings. It should be. A theorem prover that retrieves a disguised copy of the target is demonstrating lookup, not discovery. ## Quadratic reciprocity as a sealed experiment Quadratic reciprocity is a demanding stress test because its formal development requires a long dependency chain and several useful choices of representation. The current 384-theorem library contains 137 checked reciprocity-infrastructure certificates, but the reciprocity law itself is not admitted. For a reciprocity endpoint $Q$ to serve as a test, we must deposit its statement before its proof enters the library. We then mask the whole $Q$ lineage: definitions introduced only for the route, residue-theory lemmas, generated variants, equivalent formulations, authored scripts, teacher sketches, stronger consequences, and retrieval records whose proofs use them. The split must use lineage IDs and the dependency graph, not theorem names. If instead we first publish the complete development, it becomes excellent training data for the *next* questions, but it can no longer be clean headline evidence for proving quadratic reciprocity itself. Both choices are useful; they answer different questions. ## The teacher experiment is only an interface test Before spending GPU time, a strong teacher such as Codex may attempt the symbolic system's unsolved DEV frontier using only the frozen macro interface. This answers: > Does the interface expose actions that could bridge the symbolic gaps? If the teacher cannot solve even 10% of those cases, the likely bottleneck is the action space, observation, or symbolic backend—not the size of Qwen. If it solves at least 20%, we have useful headroom for distillation. But the teacher's score is not a student score. Its outputs may seed tagged training examples after kernel replay, provided their lineages do not intersect the final set. It must never see the final benchmark. A spectacular teacher pilot still does not show that Qwen learned the behavior or that Hydra beats a symbolic baseline. The earlier model-v3 four-goal run illustrates the same restraint. The trained adapter solved three shallow goals at $k=1$, while a revision/configuration- pinned pretrained comparison solved none; the induction-heavy goal remained unsolved. Those three scripts kernel-check. Four goals, however, are a launch smoke, not a statistically defensible capability result and not evidence for Hydra's new architecture. ### The first functional plumbing test The repository now contains a provider-neutral bootstrap in `training/peano_hydra/` and a runnable pilot: ```console python3 scripts/eval_peano_hydra.py --include-trace ``` It uses the checked consecutive-product script `forall n. exists x. n * (n + 1) = 2 * x`. Both lanes receive the same fixed, state-independent symbolic candidates, `compact_arith` and `compact_arith [IH_witness]`, and the same three-slot, depth-13, beam-1 search budget. The control's third slot is an identified null head. The hybrid's third slot supplies the script's ten structural actions only at their ten exact canonical states. The control exhausts at the root. The hybrid reproduces the 13-command route, then a fresh retained-trace replay checks its 180-node certificate against the original formula. A related mutated statement activates none of the recorded macro states and remains `unknown`; that is transcript non-reuse, not a non-theorem certificate. The complete deterministic evidence is committed as `artifacts/peano-hydra/teacher-oracle-pilot-v1.json`. This result is useful and deliberately modest. It proves that portfolio quotas, exact-state gating, public tactics, proposal provenance, independent replay, and the kernel compose. The structural route came from the answer, and the contextual symbolic candidate was human-selected for this example. No Qwen or Codex was called, so the pilot measures neither model capability nor a hybrid advantage. The report also marks every lane comparison-ineligible. At this bootstrap stage the ledger retains extracted tactic lines but not raw decoder text, token and latency measurements, or a campaign provider attestation, and the critical-state allowlist is copied from the teacher route rather than detected as a symbolic fixed point. Clean execution therefore means “the plumbing ran as specified,” not “this row may enter a model comparison.” ## Build the strongest baseline first The LLM should not receive credit for work that a good algorithm already does. Before model training, Hydra freezes a symbolic portfolio containing as much of the following as the fragment permits: - canonical normalization and equality rewriting; - arithmetic closure and bounded witness enumeration; - focused intuitionistic or connection/tableau search; - induction-candidate enumeration; - deterministic theorem retrieval; and - reconstructed hints from external first-order or SMT solvers. Each component is measured alone and in portfolio. Development-only scheduling chooses the strongest solved-versus-resource envelope. That frozen system is $S$, the real baseline. Then we add increasingly expensive learned components: \[ S \subset S+\mathrm{BM25} \subset S+R \subset S+C \subset S+P \subset S+P+V, \] where $R$ is learned retrieval, $C$ a cheap clause/state ranker, $P$ the macro policy, and $V$ a value model for best-first or PUCT search. Shuffled scores, random valid actions, no-retrieval, no-value, no-symbolic, and LLM-only runs show which component caused an improvement. If cheap retrieval or ranking captures the gain, increasing the transformer is the wrong engineering move. ## Training data must end in QED A positive policy row is admissible only when it lies on a complete trajectory whose final certificate checks against the original goal. Partial progress, an attractive lemma, a solver assertion, and a syntactically valid tactic are useful diagnostic or negative data but not positive proof labels. The first curriculum target is deliberately large and balanced: at least 100,000 unique macro transitions from at least 20,000 checked QED roots, every macro head represented, and at least 2,000 examples for each open-ended frontier choice. Clean generation must be byte-for-byte reproducible. The tokenizer must reject examples that do not fit; silent truncation changes the task and can remove the answer. The initial student stays modest—roughly 1.7–3 billion Qwen parameters—until the data and search design pass causal gates. Supervised training must beat the identical pretrained model on DEV, solve a meaningful number of registered frontier cases, and have a positive paired confidence bound. Value search and expert iteration have their own incremental gates. Only newly discovered, independently checked QEDs enter expert iteration. ## What “matched compute” means We compare three frozen systems on the same sealed targets: - $S$: strongest purely symbolic portfolio; - $S+R$: strongest non-generative learned system; and - $H$: full Hydra with the generative model at critical frontiers. Each is measured at 1, 10, 60, and 300 seconds per problem on the same hardware class. Wall time alone can hide very different work, so the report also gives CPU instructions or symbolic activations where possible, GPU/CPU energy, peak memory, and cost. Training cost is reported separately and as an amortized break-even curve. The main curve is independently checked solved fraction versus resource. We also report time-to-proof, PAR-2, proof size, invalid actions, calls, and the asymmetric sets \[ H\setminus(S\cup(S+R)) \quad\text{and}\quad (S\cup(S+R))\setminus H. \] The first shows genuinely hybrid-only solves; the second prevents an average score from hiding regressions. The preregistered headline gate is intentionally hard. At two adjacent time budgets, Hydra must beat the better baseline by at least three percentage points, the lower paired stratified 95% interval must remain above zero, and a corrected exact paired test must reject equality. Every counted proof must replay, with no negative-decision regression. Otherwise the result is simply: > No demonstrated LLM advantage under these budgets. That is a useful result. We do not rescue a miss by reopening the benchmark, tuning after inspection, changing to `pass@k`, or asking the teacher. ## The seven gates The campaign proceeds in order: | Gate | Question | Required evidence | |---|---|---| | H0 | Is the logic and fragment exact? | conformance, reference agreement, mutation rejection | | H1 | Is there clean headroom? | frozen $L_0$, sealed lineage split, symbolic and teacher DEV probes | | H2 | Is the non-LLM baseline strong? | proof-producing portfolio and replayed resource curves | | H3 | Is the curriculum real proof data? | deterministic corpus, complete QED roots, zero leakage | | H4 | Which learned component helps? | model ladder and matched causal ablations | | H5 | Does the LLM win once, fairly? | one-shot sealed matched-compute comparison | | H6 | Can another group reproduce it? | source, environments, raw traces, certificates, tables, review | H0–H4 should take roughly eight to ten weeks for a serious prototype. The full campaign, including benchmark authorship, independent evaluation, replication, and release, is more realistically four to six months. GPU training is not the first step; it is one guarded step after semantics, leakage control, and a strong baseline exist. ## What would be novel The novelty would not be “an LLM printed a Peano proof.” The stronger result would be a clean demonstration that sparse learned semantic decisions improve a sound, proof-producing intuitionistic arithmetic prover over strong symbolic and cheap learned baselines under equal resources. A negative result with the same controls would also teach us where language models are unnecessary. That is why the Hydra metaphor fits. The system has many exploratory heads, but every path returns to one small kernel. More heads may find more routes; none gets a vote on truth. ## Further reading - [Peano Lab binding design](https://github.com/nasqret/vietnam2026/blob/peano-lab/docs/PEANO_LAB_DESIGN.md) - [Peano Hydra binding design](https://github.com/nasqret/vietnam2026/blob/peano-lab/docs/PEANO_HYDRA_DESIGN.md) - [Training a Peano policy](training-a-peano-policy.md) - [AlphaGeometry](https://www.nature.com/articles/s41586-023-06747-5) - [AlphaProof](https://www.nature.com/articles/s41586-025-09833-y) - [Thor](https://proceedings.neurips.cc/paper_files/paper/2022/hash/377c25312668e48f2e531e2f2c422483-Abstract-Conference.html) - [Efficient Neural Clause-Selection Reinforcement](https://arxiv.org/abs/2503.07792) (evaluated inside Vampire) - [Intuitionistic Logic Theorem Proving library](https://www.iltp.de/)