flow

Flow

A deep-thinker that conquers one hard problem by autonomous search, modeled on a CDCL/DPLL SAT solver. Given a goal, it iterates — assume, follow consequences, hit walls, learn from the walls, jump back, and re-aim when truly stuck — until a solution emerges or it converges on an equally-valuable reachable goal.

Scope. Goal-locked, fully autonomous, domain-agnostic. This is not brainstorm-ideas (open-ended, collaborative, research-only). Use flow when you already have a specific hard target and want a relentless solver thrown at it. KB-optional: if <project>/.knowledge/INDEX.md exists, treat its papers as a fact source for propagation; never require it.

The mental model

Solver concept	Here
Variable assignment (Decide)	what-if: assume a new condition
Unit propagation (BCP)	simulate: run consequences forward, reflect
Conflict	a contradiction or dead-end on the current branch
Conflict analysis + learned clause	the note taken after every trial
Non-chronological backtracking	backjump to the real cause, not one step
Restart (clauses kept)	pivot: re-aim the goal; notes survive
Decision heuristic (VSIDS)	pick the lever that shrinks distance most + is easiest; favor conditions seen in recent conflicts

State to track

Maintain these throughout (in the journal file, see below):

• GOAL — restated crisply, with a concrete success test: how will I know it is solved?
• TRAIL — ordered list of decisions + propagated facts, each tagged with a decision level.
• GOAL_STACK — subgoals spawned by what-if recursion (current goal on top).
• NOTES — learned clauses: trigger conditions → outcome → reusable lesson. These are the whole point — they prune future search and guide backjumps.
• distance — a rough estimate of how far the current state is from GOAL.
• no_progress — consecutive trials with no distance drop (drives PIVOT).

Preflight (gate — pass before the loop)

A solver loads every known fact before it searches. Before the first trial, ask yourself two questions and do not proceed until both are honestly answered:

1. Am I clear about the goal? Can I state GOAL in one sentence and write a success test that would unambiguously tell me it is solved? If not — the goal is underspecified. Resolve it: derive the missing constraint from context if you can, otherwise ask the user one sharp clarifying question. A blurry goal makes every later distance estimate noise.

2. Have I gathered every piece of information I already have? Sweep all available sources before assuming anything:

   • the conversation context (constraints, examples, prior attempts the user mentioned),
   • the project knowledge base if present (<project>/.knowledge/INDEX.md + NOTES.md, and relevant rendered papers) — these become facts/unit clauses on the TRAIL for free,
   • the repository / files when the goal is about code or a concrete artifact.

   Anything you assume that was actually knowable up front is a self-inflicted dead-end. List what you found in the journal's "Levers & facts" section.

Only when GOAL is testable and the known facts are loaded do you enter Setup and the loop.

Setup

1. Restate GOAL and write its success test. If vague, sharpen it into something testable.
2. Pick the journal path docs/flow/<goal-slug>.md (relative to the project working directory). Create it from skills/flow/journal-template.md and fill the header (GOAL, success test, date).
3. List the obvious levers (conditions you could assume) and known facts (from reasoning, or from <project>/.knowledge/ if present). Estimate the initial distance.

The loop

Run autonomously, one trial per iteration, until SOLVED / PIVOTED-SOLVED / EXHAUSTED.

1. ASSESS    Estimate distance to GOAL given the current TRAIL.
             Success test appears to pass?  → run the FINAL CHECK (below)
             before declaring SOLVED.
2. SELECT    Choose the move via the heuristic:
               · a promising untried lever exists      → WHAT-IF
               · you just made a decision              → SIMULATE
               · current branch is a dead-end/conflict → ANALYZE → NOTE → BACKJUMP
               · no_progress ≥ 3                       → PIVOT
3. EXECUTE   Carry out the move (below).
4. NOTE      Append a trial entry to the journal — ALWAYS, even on success.
5. UPDATE    no_progress: reset to 0 if distance dropped, else +1.  Loop.

Heuristic for picking the next what-if (VSIDS analogy): prefer conditions that (a) most shrink distance to GOAL if true, (b) are easiest to achieve, and (c) involve variables that showed up in recent conflicts (bump their priority). Break ties toward conditions that, if false, also teach you something.

Move: what-if (= Decide)

Pick a candidate condition C and apply two tests:

1. Relevance — assuming C is true, are we measurably closer to GOAL? If no → discard C and note why it doesn't help (a learned clause too).
2. Tractability — is achieving C easier than GOAL itself?
   • Yes → push C onto GOAL_STACK as a subgoal and recurse: the loop now works to establish C (a new, deeper decision level). When C is established, pop back and resume on the parent goal with C now a fact on the TRAIL.
   • Closer but not easier → C is a useful reframing, not progress. Record the insight, keep C as a tentative assumption at a new decision level, and continue searching.

Add C to the TRAIL at a fresh decision level either way.

Move: simulate (= Propagate)

From the current assumptions, follow the forced or strongly-implied consequences forward several steps (if C then likely D, then E…). At each step, feel and reflect: note surprises, tensions, symmetries, emerging structure — insight often arrives here, not at the decision.

• Consequences trend toward GOAL → record the gained insight, decrease distance, keep going.
• Consequences hit a contradiction or clearly trend away → declare a conflict and switch to ANALYZE.

Parallel paths (optional, for wide forks). When a fork has 2–3 genuinely competing continuations and choosing blindly would waste effort, dispatch one subagent per path to simulate it independently, then compare their reflections and adopt the most promising (and note the others' dead-ends so they are never re-tried). Use this only when the fork is wide enough to justify the token cost; otherwise simulate one path at a time.

Move: analyze → note → backjump (= clause learning)

On a conflict:

1. Analyze — find the minimal subset of assumptions on the TRAIL responsible for the dead-end.
2. Note — write a reusable constraint: {X, Y} ⇒ dead-end because Z. This is the learned clause; it must be general enough to fire again later.
3. Backjump — undo assumptions back to the deepest decision level still consistent with the new note (not merely one step). Resume the loop from there. The note guarantees you won't re-enter the same trap.

Move: pivot (= meta-restart, NOTES kept)

Trigger when no_progress ≥ 3 across the whole search, or conflicts stop teaching anything new. Step out of the search and ask, in order:

1. Feasibility — is GOAL actually achievable with the available tools, facts, and time?
2. Re-aim — is GOAL really what we want, or is there an equally valuable goal that is easier?
3. Relaxation — can GOAL be weakened, split, or specialized into a version reachable now?

Auto-select the most promising re-aimed/relaxed goal, keep all NOTES (they carry over — that is what makes this CDCL, not a fresh start), reset the TRAIL, update GOAL/GOAL_STACK, log the pivot rationale, and continue the loop.

Final check (verify the model before declaring SOLVED)

A solver that claims SAT verifies the assignment satisfies every clause before stopping. Do the same: the last round is always a simulate pass over the candidate solution, not a decision. Take the proposed solution as the starting statement and run it forward, asking:

"Is this solution clear, actionable, and does it actually achieve the GOAL?"

• Clear — could a competent reader follow it without filling gaps? No hand-waving, no "and then it works."
• Actionable — are the concrete steps / construction / proof spelled out, not just gestured at?
• Achieves the GOAL — simulate it against the success test and against every NOTE (learned clause): does any prior dead-end still apply? Does it meet the original goal, not a drifted one?

If any answer is no, the check is a conflict: ANALYZE → NOTE the gap → BACKJUMP and keep searching (do not declare SOLVED). Only when all three hold do you settle. Record this final verification as the last trial in the journal.

Termination

• SOLVED — success test passes and the final check confirms it. Write the solution and a clean summary of the reasoning trail.
• PIVOTED-SOLVED — a re-aimed goal was solved. Report what was achieved versus the original, and what gap remains to the original GOAL.
• EXHAUSTED — after at most 3 pivots still stuck. Stop (do not loop forever). Report: the best partial result, the map of dead-ends (the NOTES), the current best lever, and what new fact or tool would unblock it.

Always end by ensuring the journal's final section is filled in.

Journal

Append a trial entry after every trial (this is the user's standing rule and the solver's clause store). Format in skills/flow/journal-template.md. Keep entries terse — one block per trial — so the file stays scannable and survives context compaction (the journal is the recoverable state of the run).

Worked example (abbreviated)

GOAL: prove a given graph property P holds for all planar graphs. Success test: a complete argument with no unjustified step.

• Trial 1 — what-if: assume the graph is a maximal planar triangulation. Relevance: yes (general case reduces to it). Tractability: easier (extra edges only help). → subgoal pushed. distance ↓.
• Trial 2 — simulate: propagate Euler's formula on triangulations → degree-sum bound emerges. Insight noted. distance ↓.
• Trial 3 — simulate: push the bound toward P → contradiction with a degree-2 vertex case. conflict.
• Trial 4 — analyze/note/backjump: note {triangulation assumption} ⇒ misses low-degree vertices. Backjump to Trial 1's level; refine assumption to "triangulation after removing degree-≤2 vertices."
• Trial 5 — simulate: re-propagate → P follows by induction on vertex removal. Success test appears to pass.
• Trial 6 — final-check: simulate the full argument forward — clear? actionable? meets the original GOAL with no NOTE still firing? Yes on all three. SOLVED.