The Mechanism · Gated-generation receipt

gen-state-the-master-theorem-precisely-and-explain-why-it-matter-54ad

stable pending witness
2026-05-29 05:22:14 UTC · anthropic/claude-sonnet-4-5

Prompt

State the master theorem precisely and explain why it matters in 2-3 sentences.
79 chars

Generation

# Master Theorem For recurrence relations of the form **T(n) = aT(n/b) + f(n)** where a ≥ 1, b > 1: 1. If f(n) = O(n^(log_b(a) - ε)) for some 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)) for some ε > 0, and af(n/b) ≤ cf(n) for some c < 1 and sufficiently large n, then **T(n) = Θ(f(n))** ## Why It Matters The Master Theorem provides a cookbook method for analyzing the time complexity of divide-and-conquer algorithms without solving recurrences manually. It directly applies to fundamental algorithms like merge sort, binary search, and Strassen's matrix multiplication, making it an essential tool for algorithm analysis.
25 tokens in · 290 tokens out · 5439 ms · $0.0044

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
5447.2 ms
Base LLM
5439.3 ms
Verifiers
0.3 ms
Gates
0.1 ms
Cost
$0.0044

Trail

05:22:14 received prompt_chars=79 verifiers=['scripture_anchors', 'theology_doctrine']
05:22:14 red_gate decision=pass reason=no disqualifying input patterns detected
05:22:20 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=290 latency_ms=5439.3 cost_usd=0.0044
05:22:20 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:22:20 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:22:20 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:22:20 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:22:20 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:22:20 signed hash_algo=sha256 hash=sha256:f4748ee4c84f8a936e782be52114459903dfb89a8dc216d6a695262cd5b1aa16

Content hash

sha256:f4748ee4c84f8a936e782be52114459903dfb89a8dc216d6a695262cd5b1aa16

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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