The Mechanism · Gated-generation receipt

gen-verify-whether-the-derivative-of-x-2-is-2x-7872

stable pending witness
2026-05-29 02:10:43 UTC · anthropic/claude-sonnet-4-5

Prompt

Verify whether the derivative of x^2 is 2x.
43 chars

Generation

# Verifying that the derivative of x² is 2x I'll verify this using the limit definition of the derivative. ## Using the Limit Definition The derivative of f(x) = x² is: f'(x) = lim[h→0] [f(x+h) - f(x)]/h **Step 1:** Find f(x+h) - f(x+h) = (x+h)² - f(x+h) = x² + 2xh + h² **Step 2:** Substitute into the limit formula f'(x) = lim[h→0] [(x² + 2xh + h²) - x²]/h **Step 3:** Simplify f'(x) = lim[h→0] [2xh + h²]/h **Step 4:** Factor out h f'(x) = lim[h→0] h(2x + h)/h **Step 5:** Cancel h (valid since h ≠ 0 in the limit) f'(x) = lim[h→0] (2x + h) **Step 6:** Evaluate the limit f'(x) = 2x + 0 = **2x** ## ✓ Verified Yes, the derivative of x² is indeed **2x**. This can also be verified using the power rule: d/dx(xⁿ) = nxⁿ⁻¹, which gives d/dx(x²) = 2x²⁻¹ = 2x.
22 tokens in · 384 tokens out · 5728 ms · $0.0058

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
5741.0 ms
Base LLM
5728.4 ms
Verifiers
0.4 ms
Gates
0.1 ms
Cost
$0.0058

Trail

02:10:43 received prompt_chars=43 verifiers=['scripture_anchors', 'theology_doctrine']
02:10:43 red_gate decision=pass reason=no disqualifying input patterns detected
02:10:49 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=22 tokens_out=384 latency_ms=5728.4 cost_usd=0.0058
02:10:49 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:10:49 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:10:49 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:10:49 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:10:49 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:10:49 signed hash_algo=sha256 hash=sha256:2cb33ab17899992eedb53dc626dd2260de47700e001ac518aa52bc185135fa39

Content hash

sha256:2cb33ab17899992eedb53dc626dd2260de47700e001ac518aa52bc185135fa39

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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