The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-rank-of-a-matrix-state-any-key-7aac

stable pending witness
2026-05-29 04:14:10 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the rank of a matrix. State any key formula.
70 chars

Generation

The rank of a matrix is the dimension of the vector space spanned by its columns (or rows), representing the number of linearly independent columns (or rows). It measures the maximum amount of independent information contained in the matrix. **Key formula:** For an m × n matrix A, rank(A) ≤ min(m, n), and importantly, **rank(A) = rank(A^T)** (the column rank equals the row rank). The rank also satisfies the **rank-nullity theorem**: rank(A) + nullity(A) = n, where nullity is the dimension of the null space.
27 tokens in · 140 tokens out · 3344 ms · $0.0022

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
3350.9 ms
Base LLM
3344.2 ms
Verifiers
0.2 ms
Gates
0.1 ms
Cost
$0.0022

Trail

04:14:10 received prompt_chars=70 verifiers=['scripture_anchors', 'theology_doctrine']
04:14:10 red_gate decision=pass reason=no disqualifying input patterns detected
04:14:14 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=27 tokens_out=140 latency_ms=3344.2 cost_usd=0.0022
04:14:14 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
04:14:14 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
04:14:14 floor_gate decision=pass reason=output above protective minimum; no mismatches
04:14:14 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
04:14:14 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
04:14:14 signed hash_algo=sha256 hash=sha256:138f6fdc0f2c7aa07821da698e1a06b50e6a8df6f395d1955623f96f1d7b5101

Content hash

sha256:138f6fdc0f2c7aa07821da698e1a06b50e6a8df6f395d1955623f96f1d7b5101

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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