The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-dispersion-in-a-prism-state-any-key-58f0

stable pending witness
2026-05-29 06:22:54 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain dispersion in a prism. State any key formula.
71 chars

Generation

When white light enters a prism, different wavelengths (colors) refract by different amounts because the refractive index of the material varies with wavelength—a phenomenon called **dispersion**. Shorter wavelengths (blue/violet light) bend more than longer wavelengths (red light), causing the light to spread into a spectrum. The key formula is **Cauchy's equation**: n(λ) = A + B/λ² + C/λ⁴ + ... where n is the refractive index and λ is the wavelength. This relationship, combined with Snell's law (n₁sinθ₁ = n₂sinθ₂), determines the angle of refraction for each color.
28 tokens in · 170 tokens out · 4590 ms · $0.0026

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
4597.0 ms
Base LLM
4590.0 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0026

Trail

06:22:54 received prompt_chars=71 verifiers=['scripture_anchors', 'theology_doctrine']
06:22:54 red_gate decision=pass reason=no disqualifying input patterns detected
06:22:58 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=28 tokens_out=170 latency_ms=4590.0 cost_usd=0.0026
06:22:58 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
06:22:58 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
06:22:58 floor_gate decision=pass reason=output above protective minimum; no mismatches
06:22:58 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
06:22:58 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
06:22:58 signed hash_algo=sha256 hash=sha256:ddaf7d9b21e25030b15bc40dec87e5b010d8012b2da82755a77b112940d436ff

Content hash

sha256:ddaf7d9b21e25030b15bc40dec87e5b010d8012b2da82755a77b112940d436ff

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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