The Mechanism · Gated-generation receipt

gen-in-3-4-sentences-explain-the-hamiltonian-formulation-state-a-8fab

stable pending witness
2026-05-29 08:16:07 UTC · anthropic/claude-sonnet-4-5

Prompt

In 3-4 sentences, explain the Hamiltonian formulation. State any key formula.
77 chars

Generation

The Hamiltonian formulation is a reformulation of classical mechanics where the system's state is described by generalized coordinates **q** and their conjugate momenta **p**, rather than coordinates and velocities. The central quantity is the Hamiltonian function **H(q,p,t)**, typically representing the total energy of the system, given by **H = Σpᵢq̇ᵢ - L**, where **L** is the Lagrangian. The equations of motion are Hamilton's equations: **∂H/∂p = q̇** and **∂H/∂q = -ṗ**, which form a set of first-order differential equations. This formulation is particularly powerful for analyzing conservation laws, performing canonical transformations, and forms the foundation for quantum mechanics.
29 tokens in · 182 tokens out · 4807 ms · $0.0028

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
4813.9 ms
Base LLM
4807.2 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.0028

Trail

08:16:07 received prompt_chars=77 verifiers=['scripture_anchors', 'theology_doctrine']
08:16:07 red_gate decision=pass reason=no disqualifying input patterns detected
08:16:12 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=29 tokens_out=182 latency_ms=4807.2 cost_usd=0.0028
08:16:12 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
08:16:12 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
08:16:12 floor_gate decision=pass reason=output above protective minimum; no mismatches
08:16:12 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
08:16:12 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
08:16:12 signed hash_algo=sha256 hash=sha256:fa544e50478b5a67b9195437d0362f24e54dace4d4056f5d188408ea70ce3514

Content hash

sha256:fa544e50478b5a67b9195437d0362f24e54dace4d4056f5d188408ea70ce3514

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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