{
  "artifact": "cheap-gradient-subspace-scoring-decision-memo",
  "as_of": "2026-07-16",
  "claim_boundary": "A literature synthesis, arithmetic cost model, and untested experimental program. Nothing in this record is a new machine measurement or a ViT, LLM, or production-training result.",
  "evidence_tier": "literature_synthesis_and_arithmetic_extrapolation",
  "novelty_search_boundary": "As of July 2026, no published study we found values pretraining examples by alignment with a cross-model weight basis.",
  "common_cost_model": {
    "sequence_tokens": 2048,
    "forward_formula": "F ≈ 2PT",
    "backward_multiple_of_forward": 2.0,
    "ordinary_step_multiple_of_forward": 3.0,
    "offline_total_formula": "cF + 3pF",
    "offline_break_even": "p < 1 - c/3",
    "online_last_block_total_formula": "1.13F + 1.9pF",
    "scales": {
      "100M": { "forward_tflops": 0.410, "ordinary_step_tflops": 1.23 },
      "1B": { "forward_tflops": 4.10, "ordinary_step_tflops": 12.3 },
      "7B": { "forward_tflops": 28.7, "ordinary_step_tflops": 86.0 }
    },
    "worked_cases": [
      { "mode": "online", "scoring_cost_f": 1.13, "backward_cost_per_kept_f": 1.9, "max_keep_fraction": 0.9842105263157894 },
      { "mode": "offline", "scoring_cost_f": 0.043, "backward_cost_per_kept_f": 3.0, "max_keep_fraction": 0.9856666666666667 },
      { "mode": "offline", "scoring_cost_f": 3.0, "backward_cost_per_kept_f": 3.0, "max_keep_fraction": 0.0 }
    ],
    "caveat": "Matmul-dominant estimate; attention, optimizer, utilization, and packing effects change wall time."
  },
  "ranked_avenues": [
    {
      "rank": "1",
      "avenue": "Online last-block alignment gate",
      "cost_f": { "100M": 1.13, "1B": 1.13, "7B": 1.13 },
      "max_keep_fraction": { "mode": "online", "100M": 0.984, "1B": 0.984, "7B": 0.984 },
      "verdict": "Strongest direct economics; reuse forward, backpropagate only to a late layer, then skip the remaining backward for rejected examples.",
      "evidence_tier": "arithmetic_extrapolation"
    },
    {
      "rank": "2",
      "avenue": "Full-gradient proxy scoring",
      "cost_f": { "100M": 0.75, "1B": 0.30, "7B": 0.043 },
      "max_keep_fraction": { "mode": "offline", "100M": 0.75, "1B": 0.90, "7B": 0.986 },
      "verdict": "Best offline scaling economics; ranking transfer remains unproved for this score.",
      "evidence_tier": "arithmetic_extrapolation"
    },
    {
      "rank": "3",
      "avenue": "Online amortized scorer",
      "cost_f": { "100M": 1.06, "1B": 1.06, "7B": 1.06 },
      "max_keep_fraction": { "mode": "online", "100M": 0.97, "1B": 0.97, "7B": 0.97 },
      "verdict": "Excellent economics if a tiny predictor survives training drift; refresh on 2% sparse exact labels.",
      "evidence_tier": "untested_proposal"
    },
    {
      "rank": "4",
      "avenue": "Exact one-last-block matrix score",
      "cost_f": { "100M": 1.10, "1B": 1.10, "7B": 1.10 },
      "max_keep_fraction": { "mode": "offline", "100M": 0.63, "1B": 0.63, "7B": 0.63 },
      "verdict": "Viable only if one layer predicts the useful ranking.",
      "evidence_tier": "arithmetic_extrapolation"
    },
    {
      "rank": "5",
      "avenue": "Structured JL sketch of one last-block matrix",
      "cost_f": { "100M": 1.15, "1B": 1.15, "7B": 1.15 },
      "max_keep_fraction": { "mode": "offline", "100M": 0.62, "1B": 0.62, "7B": 0.62 },
      "verdict": "Storage-efficient, but the partial backward dominates.",
      "evidence_tier": "arithmetic_extrapolation"
    },
    {
      "rank": "6",
      "avenue": "Fixed 100M embedding encoder",
      "cost_f": { "100M": 1.0, "1B": 0.10, "7B": 0.014 },
      "max_keep_fraction": { "mode": "offline", "100M": 0.67, "1B": 0.967, "7B": 0.995 },
      "verdict": "Very cheap at scale, but no demonstrated bridge from embeddings to alignment.",
      "evidence_tier": "arithmetic_extrapolation"
    },
    {
      "rank": "7",
      "avenue": "Full target per-example gradients",
      "cost_f": { "100M": 3.0, "1B": 3.0, "7B": 3.0 },
      "max_keep_fraction": { "mode": "offline", "100M": 0.0, "1B": 0.0, "7B": 0.0 },
      "verdict": "Economically dead for one-pass offline selection; batching and memory tricks do not make the backward free.",
      "evidence_tier": "arithmetic_extrapolation"
    },
    {
      "rank": "ENABLER",
      "avenue": "Streaming or robust basis extraction",
      "cost_f": { "100M": 0.0, "1B": 0.0, "7B": 0.0 },
      "max_keep_fraction": { "mode": "not_applicable", "100M": null, "1B": null, "7B": null },
      "verdict": "Necessary infrastructure with approximately zero marginal scoring cost; one-layer bases remain the tractable target.",
      "evidence_tier": "untested_proposal"
    }
  ],
  "stability_protocol": {
    "evidence_tier": "untested_proposal",
    "source_counts": [12, 20, 32, 50],
    "loso_angle_ucb_degrees": 30,
    "steps": [
      "Use exact-architecture, common-base groups; prefer same-base LoRA deltas.",
      "Sweep source counts N = 12, 20, 32, and 50.",
      "Report bootstrap and leave-one-source-out angle distributions at every N.",
      "Retain a direction only when its eigengap and 95% bootstrap upper confidence bound on LOSO angle clear the threshold.",
      "Stop adding directions when the confidence interval crosses 30 degrees, even below 90% cumulative variance."
    ],
    "projector_formula": "C = E_b[P_b]",
    "robust_score_formula": "s_robust = g^T C g / ||g||^2",
    "caveat": "Robust PCA handles outliers; it cannot manufacture an eigengap."
  },
  "kill_criteria": [
    { "approach": "Online last-block gate", "criterion": "Kill if last-block score has Spearman rho < 0.50 against the multi-layer gold score, top-20% recall below 40%, or measured scoring plus gating saves no GPU time at a 40% drop rate." },
    { "approach": "Proxy scoring", "criterion": "Kill if median proxy-to-target rho < 0.40 across seeds and checkpoints, top-20% recall below 35%, or proxy selection fails to beat random selection's confidence interval at two keep fractions." },
    { "approach": "Amortized scorer", "criterion": "Kill if rolling rho < 0.60, correlation falls by more than 0.10 between refreshes, or refresh plus inference consumes over 10% of the backward FLOPs it saves." },
    { "approach": "Shared basis itself", "criterion": "Independently kill any direction whose 95% bootstrap upper confidence bound on LOSO angle exceeds 30 degrees, even if it improves in-sample explained variance." }
  ],
  "citations": [
    "2405.13954", "1910.00762", "1906.11829", "2402.04333", "2401.12926", "2303.14186",
    "2303.09540", "2308.12284", "2206.07137", "2107.07075", "2211.05610", "1905.03677",
    "1906.03671", "1912.10985", "2009.03106", "2205.10683", "2202.08587", "2303.14753",
    "2006.14651", "1405.0680", "1702.01005", "1709.05667", "2102.03646", "2504.20437"
  ],
  "record_sha256": "187be4eff9183a0461ea7115e227b5e53db908bf6d2b7937c07caf580b0f9497"
}