The Mechanism · Gated-generation receipt

gen-state-the-inscribed-angle-theorem-precisely-and-explain-why-332e

stable pending witness
2026-05-29 05:56:16 UTC · anthropic/claude-sonnet-4-5

Prompt

State the inscribed angle theorem precisely and explain why it matters in 2-3 sentences.
88 chars

Generation

## Inscribed Angle Theorem **Precise Statement:** An inscribed angle (formed by two chords sharing an endpoint on a circle) measures half the central angle that subtends the same arc. Equivalently, the inscribed angle equals half the measure of its intercepted arc. **Why It Matters:** This theorem is fundamental for solving countless geometric problems involving circles, as it provides a consistent relationship between angles and arcs. It also leads to important corollaries, such as the fact that all inscribed angles subtending the same arc are equal, and that any angle inscribed in a semicircle is a right angle—results widely used in construction, navigation, and geometric proofs.
27 tokens in · 150 tokens out · 3689 ms · $0.0023

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
3698.9 ms
Base LLM
3688.9 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0023

Trail

05:56:16 received prompt_chars=88 verifiers=['scripture_anchors', 'theology_doctrine']
05:56:16 red_gate decision=pass reason=no disqualifying input patterns detected
05:56:20 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=150 latency_ms=3688.9 cost_usd=0.0023
05:56:20 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:56:20 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:56:20 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:56:20 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:56:20 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:56:20 signed hash_algo=sha256 hash=sha256:73d74d527991ca8d41575747fd218ecb60e4c56b3e76ee5d8a2c2b58b4dc3dbc

Content hash

sha256:73d74d527991ca8d41575747fd218ecb60e4c56b3e76ee5d8a2c2b58b4dc3dbc

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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