The Mechanism · Gated-generation receipt

gen-what-is-the-significance-of-proof-by-cases-answer-concisely-9756

stable pending witness
2026-05-29 05:08:21 UTC · anthropic/claude-sonnet-4-5

Prompt

What is the significance of proof by cases? Answer concisely in 2-3 sentences.
78 chars

Generation

Proof by cases is a fundamental technique that allows mathematicians to establish a general result by breaking it into distinct, exhaustive scenarios and proving each separately. It's significant because many propositions naturally divide into different situations that require different arguments, making direct proof impractical or impossible. This method is particularly powerful when dealing with discrete structures, piecewise definitions, or statements involving "or" conditions.
28 tokens in · 84 tokens out · 3137 ms · $0.0013

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
3152.5 ms
Base LLM
3136.7 ms
Verifiers
0.1 ms
Gates
0.1 ms
Cost
$0.0013

Trail

05:08:21 received prompt_chars=78 verifiers=['scripture_anchors', 'theology_doctrine']
05:08:21 red_gate decision=pass reason=no disqualifying input patterns detected
05:08:24 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=84 latency_ms=3136.7 cost_usd=0.0013
05:08:24 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:08:24 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:08:24 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:08:24 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:08:24 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:08:24 signed hash_algo=sha256 hash=sha256:f2fb3d9ca09c42bac1b189cd4844f81ce99c9a02dc7fe01afd09252ba39bb546

Content hash

sha256:f2fb3d9ca09c42bac1b189cd4844f81ce99c9a02dc7fe01afd09252ba39bb546

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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