The Mechanism · Gated-generation receipt

gen-explain-the-master-theorem-as-you-would-to-a-capable-student-aa81

stable pending witness
2026-05-29 05:24:12 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain the master theorem as you would to a capable student, in 3-4 sentences.
79 chars

Generation

The Master Theorem is a formula that quickly tells you the time complexity of divide-and-conquer algorithms without having to solve recurrence relations manually. It applies when you split a problem of size n into *a* subproblems, each of size n/b, and then do O(n^d) work to combine the results. The final complexity depends on comparing a and b^d: if the recursive calls dominate (a > b^d), complexity is O(n^(log_b a)); if combining dominates (a < b^d), it's O(n^d); and if they're balanced (a = b^d), you get O(n^d log n). For example, merge sort splits into 2 subproblems of size n/2 with O(n) merging work, giving O(n log n) since 2 = 2^1.
28 tokens in · 199 tokens out · 5017 ms · $0.0031

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
5024.5 ms
Base LLM
5017.4 ms
Verifiers
0.3 ms
Gates
0.0 ms
Cost
$0.0031

Trail

05:24:12 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
05:24:12 red_gate decision=pass reason=no disqualifying input patterns detected
05:24:17 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=199 latency_ms=5017.4 cost_usd=0.0031
05:24:17 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:24:17 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:24:17 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:24:17 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:24:17 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:24:17 signed hash_algo=sha256 hash=sha256:c5c7f0fc6e52cb7df222410aa14a03a622f74b4a954d148e8bc99eae027686a8

Content hash

sha256:c5c7f0fc6e52cb7df222410aa14a03a622f74b4a954d148e8bc99eae027686a8

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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