The Mechanism · Gated-generation receipt

gen-define-tangent-lines-to-a-curve-and-give-one-short-worked-ex-6ca0

stable pending witness
2026-05-29 05:00:54 UTC · anthropic/claude-sonnet-4-5

Prompt

Define tangent lines to a curve and give one short worked example.
66 chars

Generation

# Tangent Lines to a Curve ## Definition A **tangent line** to a curve at a point is a straight line that touches the curve at that point and has the same direction as the curve at that point. The slope of the tangent line equals the derivative of the function at that point. For a function f(x), the tangent line at point (a, f(a)) has: - **Slope**: m = f'(a) - **Equation**: y - f(a) = f'(a)(x - a) ## Worked Example **Find the tangent line to f(x) = x² at the point (2, 4).** **Solution:** 1) First, find the derivative: - f'(x) = 2x 2) Find the slope at x = 2: - f'(2) = 2(2) = 4 3) Use point-slope form with point (2, 4) and slope m = 4: - y - 4 = 4(x - 2) - y - 4 = 4x - 8 - **y = 4x - 4** The tangent line to f(x) = x² at (2, 4) is **y = 4x - 4**.
21 tokens in · 309 tokens out · 4674 ms · $0.0047

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
4681.8 ms
Base LLM
4674.2 ms
Verifiers
0.4 ms
Gates
0.1 ms
Cost
$0.0047

Trail

05:00:54 received prompt_chars=66 verifiers=['scripture_anchors', 'theology_doctrine']
05:00:54 red_gate decision=pass reason=no disqualifying input patterns detected
05:00:59 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=21 tokens_out=309 latency_ms=4674.2 cost_usd=0.0047
05:00:59 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
05:00:59 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
05:00:59 floor_gate decision=pass reason=output above protective minimum; no mismatches
05:00:59 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
05:00:59 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
05:00:59 signed hash_algo=sha256 hash=sha256:d8beae2726ce18e816fba849fb0d38311b761331f51a033711f7782b3e901b67

Content hash

sha256:d8beae2726ce18e816fba849fb0d38311b761331f51a033711f7782b3e901b67

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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