The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-lagrangian-duality-answer-concis-1961

stable pending witness
2026-05-29 05:40:25 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of Lagrangian duality? Answer concisely in 2-3 sentences.
82 chars

Generation

Lagrangian duality provides a systematic way to find lower bounds on optimization problems by converting constrained problems into unconstrained ones through dual variables. It enables the development of efficient algorithms and provides optimality conditions (like KKT conditions) that characterize when a solution is optimal. For convex problems, strong duality often holds, meaning the dual problem yields the exact optimal value of the original (primal) problem.
31 tokens in · 91 tokens out · 2746 ms · $0.0015

The four gates

RED
pass
no disqualifying input patterns detected
FLOOR
pass
output above protective minimum; no mismatches
WAY
pass
no declared way_path (Way check NA); no coercion language
EXECUTION
deferred
no witnesses available (Witness Roll not yet constituted)

Verifier results

scripture_anchors NOT_APPLICABLE
0 Scripture citation(s) extracted
theology_doctrine NOT_APPLICABLE
no doctrine keyword patterns detected

Metrics

Total latency
2762.8 ms
Base LLM
2746.4 ms
Verifiers
0.1 ms
Gates
0.0 ms
Cost
$0.0015

Trail

05:40:25 received prompt_chars=82 verifiers=['scripture_anchors', 'theology_doctrine']
05:40:25 red_gate decision=pass reason=no disqualifying input patterns detected
05:40:28 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=31 tokens_out=91 latency_ms=2746.4 cost_usd=0.0015
05:40:28 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:40:28 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:40:28 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:40:28 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:40:28 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:40:28 signed hash_algo=sha256 hash=sha256:826917d081a48a4bbca846096531c9277ca3e7777d9e9a3f0f47372b0b55f0b3

Content hash

sha256:826917d081a48a4bbca846096531c9277ca3e7777d9e9a3f0f47372b0b55f0b3

SHA256 over the canonical JSON (excluding this field). Tamper detection. Ed25519 signing planned for v2 once the operator’s signing key is provisioned on disk.

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