The Mechanism · Gated-generation receipt

gen-explain-proof-by-cases-as-you-would-to-a-capable-student-in-b980

stable pending witness
2026-05-29 07:36:39 UTC · anthropic/claude-sonnet-4-5

Prompt

Explain proof by cases as you would to a capable student, in 3-4 sentences.
75 chars

Generation

Proof by cases (also called proof by exhaustion) is a technique where you break down a problem into several distinct scenarios that cover all possibilities, then prove your statement is true in each scenario separately. For example, to prove "every integer squared is non-negative," you could split into two cases: when the integer is positive or zero (clearly gives non-negative result), and when it's negative (multiplying two negatives gives a positive). Since these cases cover all integers and the statement holds in each case, you've proven it for all integers. The key is ensuring your cases are exhaustive—they must cover every possible situation with no gaps.
28 tokens in · 137 tokens out · 5251 ms · $0.0021

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
5261.7 ms
Base LLM
5251.4 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0021

Trail

07:36:39 received prompt_chars=75 verifiers=['scripture_anchors', 'theology_doctrine']
07:36:39 red_gate decision=pass reason=no disqualifying input patterns detected
07:36:45 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=137 latency_ms=5251.4 cost_usd=0.0021
07:36:45 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
07:36:45 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
07:36:45 floor_gate decision=pass reason=output above protective minimum; no mismatches
07:36:45 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
07:36:45 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
07:36:45 signed hash_algo=sha256 hash=sha256:efc22300a0839ef545557aeeeabe85d2de59c19d1b6cdd89f0da21bf54cb30de

Content hash

sha256:efc22300a0839ef545557aeeeabe85d2de59c19d1b6cdd89f0da21bf54cb30de

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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