The Mechanism · Gated-generation receipt

gen-define-logical-equivalence-and-give-one-short-worked-example-5bc0

stable pending witness
2026-05-29 04:29:06 UTC · anthropic/claude-sonnet-4-5

Prompt

Define logical equivalence and give one short worked example.
61 chars

Generation

# Logical Equivalence **Definition:** Two logical statements P and Q are **logically equivalent** (written P ≡ Q or P ⇔ Q) if they have the same truth value in all possible cases. That is, they are true under exactly the same conditions. ## Worked Example **Show that ¬(p ∧ q) ≡ ¬p ∨ ¬q** (De Morgan's Law) **Proof by truth table:** | p | q | p ∧ q | ¬(p ∧ q) | ¬p | ¬q | ¬p ∨ ¬q | |---|---|-------|----------|----|----|---------| | T | T | T | F | F | F | F | | T | F | F | T | F | T | T | | F | T | F | T | T | F | T | | F | F | F | T | T | T | T | Since the columns for ¬(p ∧ q) and ¬p ∨ ¬q are identical, the statements are **logically equivalent**. ✓
18 tokens in · 318 tokens out · 6542 ms · $0.0048

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
6552.7 ms
Base LLM
6541.6 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0048

Trail

04:29:06 received prompt_chars=61 verifiers=['scripture_anchors', 'theology_doctrine']
04:29:06 red_gate decision=pass reason=no disqualifying input patterns detected
04:29:12 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=18 tokens_out=318 latency_ms=6541.6 cost_usd=0.0048
04:29:12 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:29:12 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:29:12 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:29:12 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:29:12 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:29:12 signed hash_algo=sha256 hash=sha256:d3e548b2f7c5799bac711419fec723663d79ff3683d6666da92a9c3affab265d

Content hash

sha256:d3e548b2f7c5799bac711419fec723663d79ff3683d6666da92a9c3affab265d

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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