Exploring the unknown (`recurve explore`)¶

The gate has exactly one currency: GREEN. That is what makes closure trustworthy — but it also means an agent driving recurve will always route around a genuinely-open problem. It decomposes down to the already-provable, closes those, correctly labels the hard remainder "not known," and parks. That is not timidity; GREEN is the only thing it is rewarded for, and the unknown space contains none of it.

recurve explore adds the second gradient — one that rewards walking into the unknown — without ever loosening the honesty the gate exists to enforce. It does this by inverting the trap.

The inversion¶

A trap is a known-bad a probe must reject: seeing it go RED proves the probe can fail, and that is what licenses trusting its GREEN. A trap is a bullet the probe must dodge — it guards a claim you believe true.

A falsifier is the mirror image, fired at a conjecture you do not yet know: a genuine attempt to construct a counterexample.

If it lands — finds a counterexample — the conjecture is FALSIFIED and pruned. A dead lead is real information.
If a battery of potent falsifiers all miss, the conjecture SURVIVES — a trustworthy lead, because it withstood real attempts to destroy it, not a model's optimism.

A survival is a lead, never a proof

Explore mode scores whether an open lead is alive, not whether it is true. Only a kernel/probe GREEN mints proof. A SURVIVING conjecture stays RED — it is a graded, honestly-labeled hypothesis worth a proof attempt.

The one rule that keeps it honest¶

The same discipline as the gate, pointed the other way:

No survival counts until the falsifier has demonstrably killed a decoy.

A conjecture that "survives" only because nobody tried hard is worthless — the exact failure mode recurve exists to kill. So every falsifier ships a calibration decoy: a known-false variant it must refute. A probe is only trusted once its trap has been seen RED; a survival is only trusted once its falsifier has been seen to KILL. A battery with no calibrated falsifier is BROKEN, never SURVIVING.

Arming a conjecture¶

A claim becomes a conjecture by carrying a falsifiers/ directory beside its probe — the structural inverse of a probe's .trap/ dir. No new claim class:

probes/
  my-claim.sh                 # the probe — is it PROVEN? (RED until it is)
  my-claim.trap/…             # (optional) the closure-side trap
  my-claim.falsifiers/        # ← this makes it a conjecture
    ode-search/
      run.sh                  # exit 0 KILLED · 1 SURVIVED · other BROKEN
      kind                    # numeric | fuzz | symbolic | adversary | partial-proof
      decoy/                  # the known-false variant run.sh MUST kill

run.sh reads RECURVE_FALSIFY_TARGET: when set (to the decoy/ path) it must attack the decoy and KILL it; when unset it attacks the real conjecture.

#!/usr/bin/env bash
# exit 0 = KILLED (counterexample found) · 1 = SURVIVED · other = BROKEN
exec python3 "$(dirname "${BASH_SOURCE[0]}")/search.py"   # your kill-attempt

Running it¶

recurve explore

For every conjecture it reports the survival gradient:

PROMOTED — proven (probe kernel-clean GREEN); hand to the closure loop:
  ★ FOO-1              probe GREEN — no longer a conjecture

SURVIVING — live leads, ranked by strength (the exploration frontier):
  ○ R8-RATIO           survived 1 calibrated · strength 1 · 1×numeric (unsound-proxy only)

FALSIFIED — dead leads (a counterexample was found); prune:
  ✗ BAR-2              KILLED — R ran away from R0=0.05 …

BROKEN — batteries with no demonstrated teeth; fix before trusting a survival:
  ▲ BAZ-3              failed calibration: did not KILL its decoy

frontier: surviving 1 · falsified 0 · broken 1 · promoted 1

PROMOTED — the probe is GREEN: the lead is proven. It leaves the survival axis and becomes an ordinary closed claim for the closure loop (the jackpot).
SURVIVING — a live lead, ranked by frontier_rank (strength, then count). That ranking is the reward gradient.
FALSIFIED — a calibrated falsifier found a counterexample. Prune it.
BROKEN — the battery has no demonstrated teeth. --strict exits nonzero on any of these, so CI can gate on it — a lead nobody can trust is a defect, exactly as a probe without a trap is.

The falsifiers/ directory is invisible to the closure gate — matrix, gate, and validate ignore it. A conjecture is scored on two independent axes: proven? (its probe) and alive? (its battery).

Graded survival¶

Survival is evidence, never a single "safe" bit — its strength is graded by how sound the oracle underneath the falsifier is (the same spectrum as the rest of recurve):

`kind`	what it is	a survival is…
`numeric`	simulate and measure	weak — unsound proxy, a hint
`fuzz`	random-instance counterexample search	medium
`symbolic`	symbolic sign / monotonicity check	medium
`adversary`	an agent constructing a counterexample	medium–strong
`partial-proof`	a kernel-checked restricted case	strong — bottoms out in the oracle

A conjecture reports its whole profile (count · strength · by-kind), and numeric survivals are marked unsound-proxy wherever they appear. To lift a lead above a hint, arm it with more, and stronger, independent falsifiers.

The actor stays yours¶

As with the closure loop, recurve is BYO-agent: the actor that proposes conjectures and authors falsifiers is any agent you plug in. recurve explore is the deterministic spine that scores it — it never proposes and never trusts the proposer. The trust comes from the calibrated battery, never from the agent.

Exploring the unknown (recurve explore)¶