The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-master-theorem-state-any-key-fo-c80b

stable pending witness
2026-05-29 04:23:51 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the master theorem. State any key formula.
68 chars

Generation

The Master Theorem provides a method for solving recurrence relations of the form **T(n) = aT(n/b) + f(n)**, where a ≥ 1 and b > 1. It compares f(n) with n^(log_b(a)) to determine which dominates the running time. The three cases are: 1. If f(n) = O(n^(log_b(a) - ε)) for ε > 0, then **T(n) = Θ(n^(log_b(a)))** 2. If f(n) = Θ(n^(log_b(a))), then **T(n) = Θ(n^(log_b(a)) log n)** 3. If f(n) = Ω(n^(log_b(a) + ε)) and af(n/b) ≤ cf(n) for c < 1, then **T(n) = Θ(f(n))**
25 tokens in · 246 tokens out · 4182 ms · $0.0038

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
4189.5 ms
Base LLM
4182.0 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0038

Trail

04:23:51 received prompt_chars=68 verifiers=['scripture_anchors', 'theology_doctrine']
04:23:51 red_gate decision=pass reason=no disqualifying input patterns detected
04:23:55 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=246 latency_ms=4182.0 cost_usd=0.0038
04:23:55 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:23:55 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:23:55 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:23:55 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:23:55 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:23:55 signed hash_algo=sha256 hash=sha256:a645151381d730ca219f38663256688d5eb1f75014efc8f074e882837b309453

Content hash

sha256:a645151381d730ca219f38663256688d5eb1f75014efc8f074e882837b309453

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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