The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-proof-by-cases-state-any-key-formul-c4ae

stable pending witness
2026-05-29 02:45:59 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain proof by cases. State any key formula.
64 chars

Generation

# Proof by Cases Proof by cases is a logical technique where you prove a statement by breaking it into exhaustive, mutually exclusive scenarios and proving each one separately. The key idea is: if you want to prove P is true, you identify cases C₁, C₂, ..., Cₙ that cover all possibilities, then show that P holds under each case. The logical formula is: **(C₁ ∨ C₂ ∨ ... ∨ Cₙ) ∧ (C₁ → P) ∧ (C₂ → P) ∧ ... ∧ (Cₙ → P) ⊢ P** For example, to prove "all integers n satisfy n² ≥ n," you might split into cases: n < 0, n = 0, n = 1, and n > 1, proving the inequality holds in each case.
25 tokens in · 223 tokens out · 5222 ms · $0.0034

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
5229.2 ms
Base LLM
5221.9 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0034

Trail

02:45:59 received prompt_chars=64 verifiers=['scripture_anchors', 'theology_doctrine']
02:45:59 red_gate decision=pass reason=no disqualifying input patterns detected
02:46:04 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=25 tokens_out=223 latency_ms=5221.9 cost_usd=0.0034
02:46:04 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:46:04 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:46:04 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:46:04 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:46:04 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:46:04 signed hash_algo=sha256 hash=sha256:3a95fcb760266211203e526883494f18281d36a7770cd7ab9132c60441f6280e

Content hash

sha256:3a95fcb760266211203e526883494f18281d36a7770cd7ab9132c60441f6280e

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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