Quantitative linear logic semantics, when used as the differentiable constraint layer in neuro-symbolic loop invariant synthesis, will reduce proof-search token cost on Lean-QuantumAlg-Bench by enforcing resource-bounded inference that prevents the combinatorial branching responsible for the scalability bottleneck observed in cloud-API-dependent verification tools.

Replacing (or augmenting) a standard classical/intuitionistic differentiable constraint layer with a quantitative linear logic (QLL) resource-semantics layer in a neuro-symbolic loop-invariant synthesizer will reduce the mean and tail (p90) proof-search token cost by ≥30% on Lean-QuantumAlg-Bench (a benchmark of quantum-algorithm loop invariants formalized in Lean), at equal or better proof success rate (≥ baseline success rate − 2 percentage points), when measured against a cloud-API-dependent (GPT-4/Claude-class) verification baseline of matched model class and prompting budget. The effect is attributed specifically to QLL's resource-bounded inference (bang/!-modality-controlled reuse, additive vs. multiplicative branching discipline) suppressing combinatorial search-tree branching, not to unrelated confounds (prompt engineering, model choice, retrieval).

Minimum viable test (MVT): 2-arm controlled comparison on a stratified 150-problem subset of Lean-QuantumAlg-Bench (or a constructed equivalent if the named benchmark does not yet exist publicly — see Dependencies/Abort checkpoints), comparing: (A) baseline neuro-symbolic loop invariant synthesizer using a standard classical-logic differentiable constraint layer + cloud-API LLM backend, vs. (B) identical architecture with QLL constraint layer substituted in, same LLM backend, same decoding budget. Both measured for: total tokens consumed per solved problem, proof search tree size (nodes expanded), wall-clock time, and success/failure. Repeat across 3 random seeds. Use paired statistical tests (Wilcoxon signed-rank on per-problem token cost) plus bootstrap CIs.

# Pseudocode: QLL-constrained neuro-symbolic invariant synthesis loop

for problem in LeanQuantumAlgBench:
    context = build_prompt(problem.loop_code, problem.spec)
    candidates = []
    search_tree = InitTree(root=context)

    while not search_tree.solved and token_budget_remaining():
        node = search_tree.select_frontier_node()
        # LLM proposes candidate invariant continuations
        proposals = LLM.sample(node.context, n=k, temperature=T)

        for p in proposals:
            # Classical baseline: soft logical consistency score
            score_classical = classical_constraint_layer(p, node.state)

            # QLL variant: resource-bounded relaxation
            #   - bang(!) modality: penalize unbounded reuse of hypotheses
            #   - multiplicative (⊗): enforce joint resource consumption
            #   - additive (&): allow branching only within resource budget
            score_qll = qll_constraint_layer(
                candidate=p,
                resource_budget=node.remaining_budget,
                reuse_penalty=lambda_bang,
                branch_discipline='additive_bounded'
            )

            node.children.append(
                ScoredCandidate(p, score_qll or score_classical)
            )

        search_tree.prune(keep_top_m_by_score)
        search_tree.expand(node)

        if lean_kernel.check(node.candidate_proof):
            search_tree.solved = True
            log(tokens_used, nodes_expanded, success=True)
            break

    if not search_tree.solved:
        log(tokens_used, nodes_expanded, success=False)

# Metrics: tokens_used, nodes_expanded (branching factor proxy),
#          success rate, wall_clock -- compared across classical vs QLL arms