[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"YbERIRlJVH":3},"# climber\n\nA reasonable-reflection artifact in the Beklemishev shape: a system\nwhose **logical strength climbs under proposer/gate control**, with\neach rung kernel-blessed by an independent metalanguage soundness\ncertificate.\n\n> **LCF checks theorem construction; climber checks theory\n> construction.**\n\nThe proposer (an LLM, in the full version) does not propose theorems\nor tactics. It proposes **new sound axiom-schema extensions** —\nschemas with soundness certificates — and the kernel admits or\nrefuses based on whether the certificate type-checks against the\nmetalanguage interp. Admitted extensions enter the theory; new\ntheorems become derivable; the system has climbed.\n\nThis is the LCF discipline, applied not to proof construction but\nto **proof-system construction**. In the keynote portfolio:\n\n| artifact | what the gate governs |\n|---|---|\n| lean-grey | evaluator modification |\n| lean-green | causal `set!` on a heap cell |\n| LeanDisco | discovery of theorems and heuristics |\n| sc-mini | program-transformation rewrites |\n| **climber** | **the right to extend the proof system itself** |\n\n## What lives here\n\n- **`Climber/Object.lean`** — the object theory T₀.\n  - `Formula`: ⊥, atom, imp.\n  - `interp`: standard truth-functional interpretation in Lean's\n    metalanguage.\n  - `Derivable₀`: minimal Hilbert system — K, S, ⊥-elim, MP.\n    The implicational fragment of intuitionistic propositional\n    logic plus ex falso quodlibet.\n  - `soundness₀`: every Derivable₀ formula is metalanguage-true.\n- **`Climber/Climb.lean`** — the climb infrastructure.\n  - `SoundExtension`: a schema (predicate on formulas) + a soundness\n    certificate (a Lean term of the soundness type). The certificate\n    is what the kernel checks at admission.\n  - `Theory`: list of admitted extensions on top of T₀.\n  - `derives T`: closure of T₀ + admitted schemas under MP.\n  - `climb_sound`: **headline metatheorem.** Every theory built by\n    climbing preserves metalanguage truth. By induction over the\n    derivation, with each admitted-extension case discharged by the\n    extension's own `sound` field. Fully proved, no sorry.\n  - `climb_consistent`: corollary — no climbed theory derives ⊥.\n- **`Climber/Demo.lean`** — one rung, end-to-end.\n  - `peirceFormula`: Peirce's law `((p → q) → p) → p`.\n  - `peirceSound`: classical truth of Peirce by `Classical.em`. This\n    is the certificate the kernel checks.\n  - `peirceExtension`: the proposed `SoundExtension`.\n  - `T₁ := T₀.extend peirceExtension`.\n  - `T₁_derives_peirce`: T₁ derives any instance of Peirce's law.\n  - `T₁_sound`: corollary of `climb_sound`.\n- **`Climber/Counter.lean`** — countermodel showing T₀ does not\n  derive Peirce.\n  - `H3`: 3-element linear Heyting algebra `{bot \u003C mid \u003C top}`.\n  - `Formula.heyting`: interpretation into H3, parameterized over\n    a `provVal` assignment for `prov φ` atoms.\n  - `Derivable₀.h3Valid`: every T₀-derivable formula evaluates to\n    top under any H3 environment and any `provVal`. K, S, ⊥-elim\n    valid; MP preserves.\n  - `counterEnv`, `counterProvVal`: the assignment used for Peirce.\n  - `peirce_h3_value`: Peirce evaluates to `mid`.\n  - `peirce_not_derivable_in_T₀`: corollary — `¬ Derivable₀ peirce`.\n- **`Climber/Reflection.lean`** — internal RFN(T₀) and Con(T₀).\n  - `rfn0Schema`: the Beklemishev-shape schema admitting any\n    formula of the form `prov φ → φ`.\n  - `rfn0Extension`: the SoundExtension whose certificate is\n    `soundness₀` itself — RFN(T₀) is sound iff T₀ is sound.\n  - `T₁_rfn`: T₀ extended with RFN(T₀).\n  - `consistencyFormula`: `prov ⊥ → ⊥`, i.e., Con(T₀).\n  - `T₁_rfn_derives_con`: T₁_rfn derives Con(T₀) via the RFN\n    instance at φ = ⊥.\n  - `T₁_rfn_sound`: corollary of `climb_sound`.\n  - `con_not_derivable_in_T₀`: H3 countermodel shows T₀ provably\n    cannot reach Con(T₀). The Beklemishev rung made formal.\n- **`Climber/Bedrock.lean`** — AWS Bedrock invoke wrapper for the\n  LLM proposer. Defaults to `claude-sonnet-4-6` in `us-east-1`.\n- **`Climber/Elab.lean`** — splices an LLM `SoundExtension` term\n  into a wrapper file. Outcomes: `ADMITTED` (the kernel verified\n  the `sound` field type-checks against the metalanguage interp;\n  the schema enters the climbed theory) or `ELAB-ERROR`\n  (compilation failed; the diagnostic is fed back).\n- **`Climber/Runner.lean`** — orchestrates the cascade. Each\n  round prompts Claude for an EXTENSION, classifies, retries on\n  elab errors. After each non-error round the runner regenerates\n  `Climbed.lean` containing the accumulated extensions and a\n  composite theory `T_climbed`, then verifies that `Climbed.lean`\n  elaborates. Unreachable-line claims are *not* part of the\n  cascade verdict — they live in the static artifact\n  (`peirce_not_derivable_in_T₀`, `con_not_derivable_in_T₀`).\n- **`Smoke.lean`** — `lake exe smoke`. Reports the load-bearing\n  facts at runtime; the kernel did the verification at compile\n  time.\n- **`BedrockSmoke.lean`** — `lake exe bedrock-smoke`. One-shot\n  Bedrock connectivity check.\n- **`RunnerMain.lean`** — `lake exe runner [N]`. Runs `N` rounds\n  of the LLM/Lean cascade (default 3). Each admitted extension\n  enters the accumulated `T_climbed` as a kernel-blessed sound\n  axiom-schema; after each round, `Climbed.lean` is regenerated\n  and verified.\n- **`Climbed.lean`** — *generated*. The accumulated kernel-blessed\n  composite theory written by the runner across rounds. A\n  committed snapshot serves as evidence of a particular run; any\n  fresh run will overwrite it.\n\n## Status\n\n- **Builds clean** on `leanprover/lean4:v4.29.1`. `lake build`\n  finishes in seconds.\n- **Zero sorries.** All headline results — `climb_sound`,\n  `climb_consistent`, `T₁_derives_peirce`, `T₁_sound`, `peirceSound`,\n  `soundness₀`, `Derivable₀.h3Valid`, `peirce_not_derivable_in_T₀`,\n  `T₁_rfn_derives_con`, `T₁_rfn_sound`, `con_not_derivable_in_T₀` —\n  are fully proved.\n- **The climb crosses an unreachable line, twice.** The Peirce\n  rung shows the architecture works for sound axiom-schema\n  extension. The RFN rung shows it works for *internal reflection*:\n  T₁_rfn derives Con(T₀), T₀ provably cannot.\n- **The cascade is interactive.** `lake exe runner` runs the\n  proposer/gate loop end-to-end against AWS Bedrock; the LLM\n  proposes `SoundExtension` terms; the kernel checks the\n  soundness certificate; admitted extensions accumulate into\n  `Climbed.lean` (regenerated and re-verified after every round).\n  The unreachable-line claims live in the static artifact, not in\n  the cascade verdict.\n- **Honest scope.** This is the *propositional, one-rung skeleton*\n  of a reflection progression — not a formalization of Beklemishev's\n  full hierarchy. The architecture is right; the engineering to\n  iterate (level-indexed `prov`, stratified Π_n reflection,\n  ordinal analysis) is sketched in the *Path to iterated\n  reflection-principle climbing* section below.\n\n## What this demonstrates\n\n**The climb crosses a provably unreachable line, and stays sound.**\nT₀ is shown by the H3 countermodel to be incapable of deriving\nPeirce's law (`peirce_not_derivable_in_T₀`). The proposer offers\nPeirce as a `SoundExtension` with a kernel-checked classical truth\ncertificate; T₁ now derives it (`T₁_derives_peirce`); the headline\n`climb_sound` carries T₀'s metalanguage truth across to T₁\n(`T₁_sound`). Three theorems, one rung — the system's logical\nstrength has provably grown without leaving the metalanguage's\ntruth.\n\nThe architecture: **proposer/gate-controlled extension of a formal\nsystem's derivation calculus, with metalanguage soundness preserved\nacross the climb.**\n\n- Smith's reflective tower could mutate the evaluator but had no\n  way to verify the mutation. The climber's analogue is `derives`,\n  and the gate is the kernel's check on each `SoundExtension`'s\n  certificate.\n- LCF tactics produce theorems gated by a kernel; the climber\n  produces *theories* gated by the same kernel.\n- Beklemishev's hierarchy is a *theorem* about which extensions of\n  PA are conservative / sound. The climber is an *artifact* that\n  navigates such hierarchies — with a proposer choosing which\n  extension to add and a kernel checking the navigation step.\n\n### A note on the H3 separating model\n\n`Counter.lean`'s 3-element Heyting algebra is used in two ways\nthat are worth distinguishing.\n\nThe `prov` constructor has an **intended interpretation** — under\n`interp`, `prov φ` reduces to the metalanguage proposition\n`Derivable₀ φ`. This is the interpretation under which `soundness₀`\nholds and under which RFN(T₀) is sound.\n\nThe H3 model uses an **arbitrary `provVal` assignment**. This is\n*not* an intended interpretation; it is a *separating model* for\nT₀'s syntactic proof calculus. Since T₀ has no rule introducing\n`prov`, every T₀-derivable formula is H3-valid for *any* `provVal`,\nwhich is what makes the model a separator: any single `provVal`\nthat makes a target formula H3-invalid is enough to rule out\nT₀-derivability.\n\nThe asymmetry is not a defect. It is the architecture: the gate\nconnects the inert object-level `prov` to its intended metalanguage\nmeaning (for soundness), while the H3 separator certifies syntactic\nnon-derivability without committing to any single interpretation\n(for the unreachable-line claims). Different jobs, different\nmodels, both kernel-checked.\n\n## Path to iterated reflection-principle climbing\n\nThe present artifact has the Beklemishev rung at level 1: T₁_rfn\nderives Con(T₀) via an internal RFN(T₀) schema with soundness₀ as\nthe certificate. The path forward is *iteration* — admitting\nRFN(T₁_rfn) at level 2, RFN(T₂) at level 3, and so on, climbing\nthrough the Beklemishev hierarchy.\n\nThe remaining gaps:\n\n1. **Internal provability predicate per rung.** At level 1, the\n   `prov` constructor in `Formula` is hard-coded to T₀ via\n   `interp env (.prov φ) := Derivable₀ φ`. To climb past T₁_rfn we\n   need `prov` to be indexed by a level — `prov (n : Nat) (φ : Formula)`\n   — and the interpretation parameterized over a sequence of theories.\n   ~50 LOC of refactoring.\n\n2. **Soundness lemma per rung.** Each rung n+1's RFN extension\n   needs the soundness theorem of T_n. For T₀ this is `soundness₀`.\n   For T₁_rfn it's `T₁_rfn_sound` (already proved as a corollary of\n   `climb_sound`). General: each rung's soundness comes free from\n   `climb_sound` once `prov` and `interp` are level-indexed.\n\n3. **Stratification (Π_n reflection).** The full Beklemishev\n   hierarchy stratifies reflection by formula complexity, giving\n   finite presentations at each level. Required for transfinite\n   ordinals; not required for finite-rung demos. Engineering, not\n   research.\n\nEach of these is finite work. The architectural claim — that a\nkernel-checked metalanguage soundness certificate is the right gate\nfor theory extension, including for internal reflection principles —\nis already demonstrated by `climb_sound` together with\n`T₁_rfn_derives_con` and `con_not_derivable_in_T₀`.\n\n## Relationship to the rest of the keynote portfolio\n\n- **lean-grey** governs reflective modifications via a parametric\n  policy table; `installPolicy` is itself reflective. Climber is\n  in the same architectural family — both are proposer/gate\n  systems with kernel-checked admission — but the modified object\n  is different. lean-grey modifies the apply rule of an evaluator;\n  climber modifies the rule set of a derivation calculus.\n- **lean-green** realizes Smith's \"meta-level is data\" via real\n  heap mutation. Climber's analogue is the `Theory.extras` list,\n  which grows by `extend`; the `SoundExtension` admitted at each\n  step is the `set!`-equivalent.\n- **LeanDisco** discovers theorems and heuristics with kernel\n  verification. Climber discovers *derivation rules* — a deeper\n  reflective layer.\n- **sc-mini** is the proposer/gate pattern in a non-reflective\n  substrate. Climber is the pattern at the deepest reflective\n  layer — modifying the system's idea of what counts as a proof.\n\nThe climber is the artifact that puts *self-governance with\nmonotone soundness* on the table as a working system, not a\nresearch wish.\n\n## Build\n\n```bash\nlake build       # builds the library\nlake build smoke # builds the smoke executable\nlake exe smoke   # runs it\n```\n\n## Running the cascade\n\nThe LLM/Lean cascade requires the `aws` CLI on PATH with\nBedrock-enabled credentials. With those in place:\n\n```bash\nlake exe bedrock-smoke    # one-shot connectivity check\nlake exe runner           # 3 rounds (default)\nlake exe runner 10        # 10 rounds\n```\n\nEach round, Claude proposes an **EXTENSION** — a `SoundExtension`\nterm (schema + soundness certificate). One gate runs: the\nextension must elaborate. On success, `ADMITTED` — the kernel\nverified the soundness certificate, the schema enters the\naccumulated climbed theory. On failure, `ELAB-ERROR` with Lean's\ndiagnostic fed back for one retry.\n\nAfter every non-error round, the runner regenerates `Climbed.lean`\ncontaining all admitted extensions as `round_0`, `round_1`, … and\na composite theory\n`T_climbed := Theory.base.extend round_0.extend round_1…`, plus\n`T_climbed_sound` as a corollary of `climb_sound`. The runner\nverifies `Climbed.lean` elaborates after writing it. The file is\n**replayable evidence** that the cascade-built theory is sound:\nthe accumulated `SoundExtension`s and the soundness corollary\nsurvive `lake env lean Climbed.lean`.\n\nA typical run admits classical schemas like Peirce's law\n`((φ → ψ) → φ) → φ`, double-negation elimination\n`((φ → ⊥) → ⊥) → φ`, or consequentia mirabilis\n`((φ → ⊥) → φ) → φ`.\n\n**Where the unreachable-line claims live.** The cascade does not\nverdict on T₀-non-derivability. Those claims are in the static\nartifact: `peirce_not_derivable_in_T₀` (in `Counter.lean`) and\n`con_not_derivable_in_T₀` (in `Reflection.lean`) prove the climb\ncrossed unreachable lines via H3 separating models — kernel-\nchecked, independent of any cascade run. The cascade is the same\nproposer/gate soundness loop made interactive; the headline\n\"unreachable line crossed\" sentences in the talk are anchored by\nthe static theorems, not by cascade verdicts.\n\n## References\n\n- Feferman 1962, *Transfinite Recursive Progressions of Axiomatic\n  Theories.* The original construction of climbing theories by\n  iterated reflection.\n- Beklemishev 2005, *Provability Algebras and Proof-Theoretic\n  Ordinals.* The modern proof-theoretic account of the hierarchy\n  the climber walks.\n- Smith 1984, *Reflection and Semantics in LISP.* The reflective\n  architecture climber's `Theory` extends.\n- Milner 1972 / LCF lineage. The kernel discipline climber's\n  admission gate instantiates.\n- Heyting 1930. The 3-element linear Heyting algebra used in\n  `Counter.lean` to separate intuitionistic from classical\n  implicational logic.\n",1780846776757]