Proof Dead Ends: Guardrails and Fix Pattern

When a reasoning chain tries to prove a claim but never closes the loop.
You see endless subgoals, repeated restatements, or a jump to a conclusion without obligations being discharged.
This page localizes proof stalls and gives a minimal, testable repair plan using ΔS, λ_observe, and E_resonance.

Open these first

Visual map and recovery
→ rag-architecture-and-recovery.md
Logic and symbols
→ logic-collapse.md
→ symbolic-collapse.md
Infinite recursion and overload
→ recursive-loop.md
→ entropy-overload.md
If the chain shifts into meta or paradox
→ philosophical-recursion.md
Traceability and contracts
→ retrieval-traceability.md
→ data-contracts.md

Symptoms

Symptom	What you see
Goal never shrinks	Subgoals keep being restated with new labels but same content
Lemma spiral	New lemmas depend on each other in a cycle without a base
Premise drop	A needed premise disappears after paraphrase or tool call
Unproven jump	Model asserts Q since P but never shows P or its bridge
Invariant drift	Claimed invariant changes wording or unit across steps
Citation vacuum	Steps claim support yet no snippet or rule is referenced

Why dead ends happen

No proof obligations. Steps do not state what must be shown to advance.
Missing bridge. The link from premise to subgoal is never written.
Drifting invariants. The property that should stay fixed keeps mutating.
Symbol table absent. Names rebind silently so obligations mismatch.
Plan length without checkpoints. Long chains lose ΔS and λ stability.
Retrieval anchor unstable. The cited snippet shuffles, the goal resets.

Acceptance targets

ΔS(question, retrieved) ≤ 0.45
Coverage ≥ 0.70 to the target section
λ remains convergent across 3 paraphrases and 2 seeds
E_resonance flat across step joins
All obligations discharged or explicitly marked unsatisfied with a cited reason

Fix in 60 seconds

Declare the goal and obligations
Write goal, known, need_to_show, and the current invariants.
If any field is empty, stop. Fetch or restate before continuing.
Add a BBCR bridge
Produce a short cited bridge from current premises to the next subgoal.
Reject continuation if the bridge lacks citation or rule tag.
Clamp variance with BBAM
If λ flips on paraphrase, freeze the symbol table and the invariant set.
Re-run the step with the same bindings.
Apply BBPF when stuck
Branch to two small proof tactics, attempt each for N steps, then backtrack.
Keep the best branch where ΔS drops and obligations shrink.

Minimal proof contract

Require every step to carry this schema. Refuse steps that miss fields.

{
  "step_id": "S7",
  "goal": "Prove claim C",
  "known": ["P1", "P2"],
  "need_to_show": ["L1 -> C"],
  "bridge": {
    "rule": "modus_ponens | contradiction | induction | algebra",
    "citations": ["S12#CH2.3", "S09#APP.A"],
    "ΔS_bridge": 0.33
  },
  "invariants": [
    {"name": "unit_price_nonnegative", "scope": "calc", "status": "holds"}
  ],
  "symbols": [
    {"name": "x", "kind": "var", "namespace": "calc", "unit": "USD/kg"}
  ],
  "λ_state": "convergent"
}

Structural repairs

If anchors or citations are missing → retrieval-traceability.md and → data-contracts.md
If long chains decay → entropy-overload.md and → recursive-loop.md
If symbol or unit meanings drift → symbolic-collapse.md
If logic itself collapses → logic-collapse.md
If meaning vs similarity conflict → embedding-vs-semantic.md

Verification

Run three paraphrases and two seeds. The set {goal, invariants, symbols} must remain identical.
All need_to_show items become empty or a single clearly blocked item with a cited missing premise.
ΔS(question, retrieved) ≤ 0.45 and coverage ≥ 0.70 in every run.
Logs show at least one BBCR bridge with citations and a monotone decrease in open obligations.

Copy-paste prompt

You have TXT OS and the WFGY Problem Map loaded.

We suspect a proof dead end.

Inputs:
- question: "{q}"
- snippets: [{snippet_id, section_id, source_url}]
- current plan trace: [{step_id, text}]
- last symbol table and invariants (if any)

Do:
1) Build the proof contract with {goal, known, need_to_show, invariants, symbols}.
2) Create a BBCR bridge with a named rule and citations. If missing, refuse and return the exact missing item.
3) If λ flips across paraphrase, apply BBAM and retry with the frozen table.
4) If still stuck, branch two BBPF tactics for at most N=3 steps each, keep the branch with lower ΔS and fewer obligations.
5) Return JSON:
   {
     "plan": [...steps...],
     "open_obligations": [],
     "ΔS": 0.xx,
     "λ_state": "convergent",
     "verdict": "proved | blocked_missing_premise | not_provable_from_given"
   }
Refuse to assert the final claim unless all obligations are discharged or the verdict is explicitly "not_provable_from_given" with a cited reason.

Common gotchas

Rule without scope. A named rule is used but no citation or domain. Add both or reject.
Induction without base. The inductive step is written yet the base case is missing.
Contradiction shortcut. The chain says “assume not C” then jumps to C with no witness.
Bridge drift. The bridge cites new snippets each run. Lock the query and add a reranker if anchors shuffle.

When to escalate

The same wrong assertion returns after a correction → pattern_hallucination_reentry.md
Multi-agent handoff reopens closed obligations → Multi-Agent_Problems.md and → multi-agent-chaos/role-drift.md

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