{
  "id": "scalar_kalman",
  "title": "Scalar Kalman filter \u2014 proven optimal, on silicon at 100 MHz",
  "blurb": "The flagship. A scalar Kalman filter carrying state across measurements (P \u2190 P\u00b7r/(P+r) \u2014 a division in the feedback path), proven MMSE-optimal and fixed-point-error-bounded, running bit-exact and tracking at 100 MHz on a real Arty A7-100T.",
  "emit_targets": 32,
  "proofs": [
    {
      "theorem": "kalman_recursive_mmse",
      "claim": "each recursive step's posterior mean is MMSE-optimal (conjugacy self-composition)",
      "clean": true,
      "trail": "kalman_recursive_mmse.axioms.txt"
    },
    {
      "theorem": "kalman_variance_recursion_fixed_point",
      "claim": "the Q16.16 variance recursion stays within a proven bound of the exact recursion over N steps",
      "clean": true,
      "trail": "kalman_variance_recursion_fixed_point.axioms.txt"
    },
    {
      "theorem": "kalman_estimate_recursion_coupled",
      "claim": "the estimate recursion's fixed-point error is bounded \u2014 the gain error derived from the variance error",
      "clean": true,
      "trail": "kalman_estimate_recursion_coupled.axioms.txt"
    }
  ],
  "checks": [
    {
      "quantity": "filter RMS vs measurement RMS",
      "value": "0.071 / 0.325",
      "relation": "ratio",
      "bound": "0.22 < 1",
      "holds": true
    },
    {
      "quantity": "estimate converges to truth",
      "value": "|2.4812 \u2212 2.5| = 0.0188",
      "relation": "<",
      "bound": "0.10",
      "holds": true
    },
    {
      "quantity": "posterior variance shrinks",
      "value": "1.000 \u2192 0.01538",
      "relation": "<",
      "bound": "0.05",
      "holds": true
    }
  ],
  "silicon": [
    "kalman_filter_step",
    "kalman_update_1d",
    "eml_reciprocal"
  ],
  "silicon_hw": {
    "board": "Arty A7-100T",
    "clock_mhz": 100,
    "anchors": [
      [
        "eml_reciprocal",
        "+1.102"
      ],
      [
        "kalman_update_1d",
        "+0.587"
      ],
      [
        "kalman_filter_step",
        "+0.866"
      ]
    ],
    "summary": "all three anchors close 100 MHz WHOLE-DESIGN (0 failing endpoints, DRC clean); 170 passes bit-exact to the golden recursion, the estimate tracks and P shrinks 1.0 \u2192 0.015",
    "evidence_ref": "monogate-research/electronics_intake/KALMAN_SILICON_FLAGSHIP.md"
  }
}