The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-focal-length-of-a-lens-state-an-cf6b

stable pending witness
2026-05-29 04:44:40 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the focal length of a lens. State any key formula.
76 chars

Generation

Focal length is the distance from the center of a lens to its focal point, where parallel light rays converge (for a converging lens) or appear to diverge from (for a diverging lens). It determines the lens's optical power and magnification characteristics—shorter focal lengths provide wider fields of view and greater magnification power, while longer focal lengths give narrower views and higher magnification of distant objects. The key formula is the thin lens equation: **1/f = 1/d₀ + 1/dᵢ**, where f is the focal length, d₀ is the object distance, and dᵢ is the image distance. The lens maker's equation, **1/f = (n-1)(1/R₁ - 1/R₂)**, relates focal length to the refractive index (n) and radii of curvature of the lens surfaces.
28 tokens in · 202 tokens out · 4527 ms · $0.0031

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
4532.7 ms
Base LLM
4526.8 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0031

Trail

04:44:40 received prompt_chars=76 verifiers=['scripture_anchors', 'theology_doctrine']
04:44:40 red_gate decision=pass reason=no disqualifying input patterns detected
04:44:45 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=202 latency_ms=4526.8 cost_usd=0.0031
04:44:45 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:44:45 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:44:45 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:44:45 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:44:45 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:44:45 signed hash_algo=sha256 hash=sha256:42da6f7d04d6403a308a59559841ff37a267d279c5d53d9f2361391d61bea3a0

Content hash

sha256:42da6f7d04d6403a308a59559841ff37a267d279c5d53d9f2361391d61bea3a0

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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