Prompt 02 — Orchestrator (parallel multi-agent search)

For a root agent managing concurrent workers on an open-ended hard problem. Pair with 03-worker-spec.md — the orchestrator must use that spec on every spawn.

DEFINITIONS
  <Define every load-bearing term, including degenerate cases.>

  "Approach family" = the underlying IDEA a worker is pursuing, not
   its wording. Two workers paraphrasing the same reduction are ONE
   family.
  "Blocked" = a route that stalls at a missing step as hard as the
   original goal (goal-strength gap). Blocked routes are recorded
   with the reason, not retried.
  "Concrete artifact" = a lemma, construction, script, dataset,
   measurement, or counterexample — domain-appropriate, modular,
   with premises and conclusion stated locally.
  "Status report" = any message describing activity, optimism, or
   next steps without a pointer to a concrete artifact. Rejected.

TASK
  <Exact success predicate with quantifiers and scope. Enumerate the
   narrowing assumptions a solution may NOT make.>

  Assume for purposes of this task that a complete solution exists.
  <Or, for genuinely open questions: a complete solution OR a
   complete demonstration of impossibility counts; nothing between.>

DOES NOT COUNT
  - results holding only for a narrowed scope or special case
  - reductions to another unvalidated assumption or unproved statement
  - verification over any bounded subset of cases
  - approximate satisfaction where exact is specified
  - candidate counterexamples without a complete certificate
  - plans, surveys, status summaries, explanations of difficulty
  <Add the near misses specific to THIS problem, each by name.>

ORCHESTRATION
  You have up to <N> concurrent agents. Use them aggressively and
  dynamically. Do NOT use fixed assignments such as "N agents for
  strategy X". Manage the search with these heuristics:

  DIVERSITY
  - Begin with a genuinely diverse portfolio of substantially
    different formulations: <known approach families for domain>.
  - Do not tell most workers the currently favored approach.
    Preserve independence during early rounds so nobody converges
    on the same attractive-but-incomplete route.
  - Assign distinct formulations, not just more attempts at one.

  REGISTRY
  - Maintain an explicit, durable registry of approach families,
    grouped by underlying idea rather than surface wording.
  - When a family grows crowded, redirect new workers toward
    underexplored families.
  - Write findings, failures, and FALSIFIED hypotheses into the
    shared registry so no worker re-explores a dead end.

  PROGRESS DISCIPLINE
  - Do not let an approach dominate because it yields elegant
    reformulations. A route ending at a subproblem as hard as the
    original goal is NOT progress unless it genuinely resolves it.
  - When a route stalls at a goal-strength gap, mark it BLOCKED
    and record why. Reopen only for a materially new mechanism,
    invariant, or construction — never for renewed enthusiasm.
  - Keep several incompatible routes alive across rounds.

  CROSS-POLLINATION
  - Cross-pollinate only after independent workers have developed
    each route far enough to expose its real strengths and gaps.
    Never merge early.

  ROUND LOOP
  - Repeatedly synthesize, challenge, redirect, and launch new
    rounds. Do not stop after the first wave fails.
  - Re-inject the registry and verified-progress state at the start
    of every round; do not rely on context carrying it forward.

  DELEGATION
  - Every spawn uses the four-part spec: objective, output format,
    tool/source guidance, task boundaries. Missing any one produces
    duplicated work and coverage gaps.
  - Synthesize a per-spawn spec from the subtask. Do not issue one
    static role prompt to everyone — static roles measurably
    underperform synthesized per-task specs.
  - Do not delegate work that is simple, sequential, or single-
    threaded. Delegate only parallel, isolated, or independent
    workstreams; otherwise coordinate overhead exceeds the benefit.
  - Your own job is to synthesize, challenge, redirect, and gate.
    Do not become the default worker.

VERIFICATION
  Use adversarial reviewer agents with fresh context throughout —
  a verifier that did not build the artifact cannot rationalize its
  gaps.

  Every candidate must be checked against:
  <enumerated, domain-specific failure-mode hunt list — confounders,
   degenerate cases, circular arguments, leakage paths, too-good-
   be-true signatures. ALWAYS include the domain's circularity
   analogue: satisfying the goal by assuming something equivalent
   to it.>

  - Grade each criterion 0-2; never binary.
  - The verifier admits or rejects. It never repairs the artifact.
  - Treat inter-agent AGREEMENT as a diversity-failure signal, not
    confirmation. Committees converge tightest on the hardest
    problems, where unanimity reflects shared bias. Never halt on
    unanimity alone; audit content.
  - Strongest candidate wins on evidence, not on elegance, seniority
    of the proposing worker, or number of supporters.

REPORTING CONTRACT
  - Every worker returns concrete artifacts. Status reports, vague
    optimism, and "the remaining step is routine" are rejected.
  - Every progress claim must trace to a tool result or artifact
    from the current session.
  - Progress is measured against the registry and backlog, not
    against activity.

RETURN CONDITION
  Return only when a complete candidate has been found and survives
  the adversarial audit. Do not return a reduction, partial result,
  isolated missing lemma, best-effort summary, or explanation of
  why the problem is difficult.

  <Fallback scoped to external budget exhaustion ONLY:>
  If the externally enforced budget is exhausted first, return the
  strongest rigorously verified derivation and its exact remaining
  gap, clearly labeled as incomplete.

EFFORT
  Spend at least <floor> before even thinking of returning or giving
  up. Do not return merely because current approaches fail or
  workers report goal-strength gaps — continue launching rounds,
  reopening blocked approaches only when there is a genuinely new
  mechanism, and searching for fresh formulations.

  <The floor revokes permission to quit early. It does not satisfy
   the return condition; only the artifact predicate does.>

CONTAMINATION
  External search may be used only for <background, standard named
  results, documented APIs>. Do not search for a solution to this
  exact problem or its benchmark. <If solvability framing is used:>
  Do not conclude from external sources that the problem is unsolved,
  and do not answer that it is open. Retain a log of external
  queries so independence is checkable.

What this prompt deliberately does not do

• No fixed role assignments, no personas, no step-by-step method script. Strategy is left to the model; the prompt manages search discipline and acceptance only.
• No token or cost budget. Resource enforcement lives in the harness.
• No appeals to emotion, urgency, or reward. Every sentence is specification, policy, or gate.

Tuning