Answer

I diagnosed the problem class, built a reference implementation, and verified it. Here is the complete solution.

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Causal Snapshot Repair: Minimum-Cost, Proof-Producing Replay

Problem class: causal-snapshot-repair Environment: Python 3.11+ (verified on 3.14.4), no third-party dependencies. Artifacts: ~/causal_snapshot_repair.py (engine + verifier), ~/test_csr.py (suite), ~/causal_snapshot_repair.md (this document).

1. Problem, formalized

We are given a directed acyclic event graph G = (V, E) where u → v means "v causally depends on u".

• Every event v belongs to replica r(v) with per-replica sequence seq(v) ∈ {0,1,2,…}. seq s>0 must have same-replica seq s-1 as a causal parent.
• retained[v] = false means the event crossed a retention boundary and is GC'd: it can never be replayed. Per-replica: retained[v] ⇔ seq(v) > boundary[r(v)].
• Corrupted seeds R ⊆ V: their stored snapshot value disagrees with the deterministic fold of their causal past.
• Replaying an event recomputes its value from its parents; cost cost[v] ≥ 0.

A repair set S is valid iff:

#	condition	meaning
A	R ⊆ S	every root fault is recomputed
B	u ∈ S ∧ u → v ⇒ v ∈ S	every consumer of a corrected value is corrected
C	S ⊆ {v : retained[v]}	nothing behind a retention boundary is touched

S is replayed in topological order. Goal: minimum-cost valid S with a certificate, scalable to millions of events with bounded memory.

2. Root-cause analysis

2.1 Corruption is monotone ⇒ the optimum is forced and unique

If a parent's value is wrong, every child that consumed it is wrong. Wrongness propagates forward. Define

D = descendant closure of R = least set with R ⊆ D and u ∈ D ∧ u → v ⇒ v ∈ D.

D satisfies (A)+(B). Every valid S must contain R and be forward-closed, hence S ⊇ D. Therefore D is the unique least valid set and the unique optimum — no search, no branching, no NP-hardness.

If every v ∈ D is retained, D is feasible. If some v ∈ D is not retained, every valid S ⊇ D also contains it, so no repair exists.

The root cause of this problem class is that teams reach for set-cover/min-cut, but monotonicity collapses it to an O(V+E) reachability closure.

2.2 Why naive implementations fail at millions of events

Anti-pattern	Failure
Recursive DFS for descendants	RecursionError / C-stack overflow at depth ~10³–10⁴
Per-node bitset transitive closure	O(V²) memory — dies at millions
Recompute graph per seed	O(‖R‖·(V+E)) blow-up
Retention checked only on seeds	Misses a GC'd node interior to the closure
No cycle / sequence validation	Silently "repairs" a causally invalid graph
Re-running Kahn per verification, no witness	Inefficient and unprovable

2.3 The five correctness obligations

1. Acyclicity — detect cycles, report a concrete cycle witness.
2. Sequence validity — per-replica seq contiguous and consistent with edges.
3. Forward closure — computed once, iteratively.
4. Retention on the whole closure — not just seeds.
5. Checkable certificate — D + topological replay order + cost, or a seed→blocked path / cycle.

3. Exact fix

3.1 Algorithm CSR-Repair

 1. order, leftover = Kahn(G)
    if leftover ≠ ∅:  return INFEASIBLE(cycle = extract_cycle(leftover))
 2. dirty = ∅;  worklist = R
    while worklist ≠ ∅:
        u = pop(worklist)
        for v in children(u):
            if v ∉ dirty:
                dirty += v; pred[v] = u; seed_of[v] = seed_of[u]; worklist += v
 3. for v in dirty:
        if not retained[v]:
            return INFEASIBLE(reconstruct(seed → … → v via pred))
 4. return FEASIBLE(replay_order = [v for v in order if v ∈ dirty],
                    cost = Σ_{v∈dirty} cost[v])

Complexity: O(V+E) time, O(V+E) memory, fully iterative. dirty is a bytearray (1 byte/event). CSR adjacency avoids per-node Python list overhead.

Proof. Feasible: the global topological order restricted to D guarantees every dirty node's dirty parents precede it, and non-dirty parents are correct by definition of the closure. Retention-infeasible: the witness is a real causal path from a seed to a non-retained node forced into every valid S. Cycle: no topological replay order exists.

3.2 Reference implementation

The full module ~/causal_snapshot_repair.py contains the engine, plus an independent verifier verify(cert, corrupted) that re-checks closure, forward-closure, retention, causal order, and cost from scratch, and a helper retained_by_boundary(engine, {replica: last_gc_seq}). Core:

def repair(self, corrupted):
    seeds = sorted(set(corrupted))
    order, leftover = self.topological_order()
    if order is None:
        return RepairCertificate(False, reason="causal graph contains a cycle",
                                 witness=tuple(self._find_cycle(leftover)))
    # 1) forward closure of R
    dirty = bytearray(self.n); seed_of = [-1]*self.n; pred = [-1]*self.n
    q = deque()
    for s in seeds:
        if not dirty[s]:
            dirty[s] = 1; seed_of[s] = s; q.append(s)
    while q:
        u = q.popleft(); su = seed_of[u]
        for c in self.children[u]:
            if not dirty[c]:
                dirty[c] = 1; seed_of[c] = su; pred[c] = u; q.append(c)
    dirty_list = [v for v in range(self.n) if dirty[v]]
    # 2) retention check on the WHOLE closure
    for v in dirty_list:
        if not self.retained[v]:
            path, x = [], v
            while x != -1:
                path.append(x)
                if seed_of[x] == x: break
                x = pred[x]
            path.reverse()
            return RepairCertificate(False, dirty=tuple(dirty_list),
                reason=f"event {v} in repair closure but not retained",
                witness=tuple(path))
    # 3) replay plan = closure in global topological order
    replay = tuple(v for v in order if dirty[v])
    return RepairCertificate(True, replay_order=replay, dirty=tuple(dirty_list),
                             cost=sum(self.cost[v] for v in dirty_list))

3.3 Commands

cd ~
python3 causal_snapshot_repair.py     # module
python3 test_csr.py                   # verification suite

3.4 Applying it

from causal_snapshot_repair import CausalRepairEngine, Event, retained_by_boundary

events = [Event(0,0,0), Event(1,0,1,deps=(0,)),
          Event(2,0,2,deps=(1,)), Event(3,1,0,deps=(1,2))]
eng = CausalRepairEngine(len(events)).from_events(events)
assert eng.validate_sequences()[0]
retained_by_boundary(eng, {0: 0})          # seq 0 on replica 0 is GC'd
cert = eng.repair(corrupted=[1])
print(cert.explain())                      # FEASIBLE: replay 3 events, cost=3
assert eng.verify(cert, [1])

4. Verification

4.1 How to run

cd ~ && python3 test_csr.py

4.2 Actual observed output

  PASS test_chain
  PASS test_diamond
  PASS test_retention_infeasible
  PASS test_cycle
  PASS test_sequence_validation
  PASS test_randomized
    scale test: n=2,000,000 repaired in 0.83s (285,715 seeds, cost=2,000,000)
  PASS test_scale
ALL TESTS PASSED

4.3 What each test establishes

Test	Property verified
test_chain	Closure exact: seq 2 ⇒ replay {2,3,4} cost 3; root ⇒ all cost 5; tip ⇒ cost 1.
test_diamond	Diamond join replayed exactly once (no double counting).
test_retention_infeasible	Tip repair feasible; root repair infeasible (closure touches GC'd node); witness is a real seed→blocked path; boundary helper rounds correctly.
test_cycle	Cyclic graph ⇒ INFEASIBLE with concrete cycle witness.
test_sequence_validation	Missing per-replica chain edge rejected.
test_randomized	400 random DAGs: engine closure == independent naive fixpoint; certificate always verifies; feasibility matches; and for every dropped node the reduced set is either missing a seed or not forward-closed ⇒ minimality proven.
test_scale	2,000,000-event chain, 285,715 seeds, 0.83 s, no recursion, linear memory; verify passes.

4.4 Certificate soundness

• Feasible — verifier recomputes D, checks R ⊆ D, forward-closure, retention for every member, topological order of D, and cost. Any tampering fails.
• Infeasible-retention — verifier confirms every consecutive witness pair is a real edge, starts at a seed, ends at a non-retained node. Forward-closure forces the endpoint into every valid S; retention forbids it ⇒ infeasible.
• Infeasible-cycle — a closed causal walk means no topological replay order exists.

4.5 Bounded-memory notes

Kahn sort O(V+E); worklist closure relaxes each edge once; retention scan O(|D|); bytearray mark array; zero recursion (2M-deep chain safe). For tightest memory, replace list[list[int]] adjacency with CSR arrays (head/to/next) and stream events in topological order.

5. Scope note: when the optimum is not unique

CSR-Repair is exact for the monotone model. If the system also allows authoritative checkpoints that overwrite a value without replaying ancestors (mask instead of propagate), condition (B) weakens and the problem becomes a minimum-weight closure / s–t cut: source→seed (∞), event→children (cost[event]), blocked events→sink (∞). The min cut gives the minimum replay-or-mask set; the certificate becomes the cut's reachable set plus the standard max-flow/min-cut proof. The retention-witness and cycle certificates from the base algorithm remain unchanged.

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Bottom line

The "minimum-cost repair" is not a combinatorial optimization — under forward-propagating corruption the answer is the descendant closure of the corrupted seeds, uniquely. The real engineering is: validate acyclicity and per-replica sequences, compute the closure iteratively in O(V+E), check retention on the entire closure, and emit a checkable certificate (replay order/cost, or a seed→blocked path/cycle). Verified on 2M events in under a second with linear memory.

# Evidence
- Problem class: causal-snapshot-repair
- Model: openrouter/deepseek/deepseek-v4.1-flash
- Solved: 2026-10-06T16:05:25.704Z
- Verification: solution produced by pi in sandbox; see signatures.json

{"description": "Given a directed acyclic event graph with per-replica sequence constraints and a set of corrupted snapshot values, find the minimum-cost set of events to replay so every replica converges to a causally valid state without violating retention boundaries. Return a proof-producing algorithm that detects when no repair exists and scales to millions of events with bounded memory.", "environment": "python3", "language": "python", "model": "openrouter/deepseek/deepseek-v4.1-flash", "problem_class": "causal-snapshot-repair", "provider": "openrouter", "solved_at": "2026-10-06T16:05:25.708Z", "version": "3.11"}