{
  "assumption_pins": {
    "P1_terminal_feedback": "terminal binary ACK/NACK feedback: before termination the feedback history is all-NACK, so the j-round transmit prefix is a FIXED (non-adaptive) length-jn code and the no-feedback converse applies; with richer feedback the bound is FALSE (Schalkwijk-Kailath)",
    "P2_abstention_to_guess": "'no ACK yet' is an erasure; guessing on abstain only increases P(correct), so P(ACK by j AND correct) <= 1 - eps_conv(jn); zero false-ACKs gives P(tau<=j) = P(ACK by j AND correct)",
    "P3_never_abandon_truncation": "restricted to schemes that never abandon within the budget B=4 (the v48 guard and the fixed baseline are both in this class); E[tau] truncates to the first B tail terms",
    "P4_equal_energy_on_sphere": "BPSK symbols have constant unit energy \u2014 exactly on-sphere \u2014 so the Shannon-1959 equal-energy sphere-packing converse applies verbatim to every constant-envelope scheme"
  },
  "bi_awgn_constants": {
    "c_bits_per_symbol": 0.563597799171,
    "method": "deterministic trapezoid quadrature, 200001 points, 12 sigma span",
    "v_bits2_per_symbol": 0.648126823942
  },
  "bound": "E[tau] = sum_{j=0}^{B-1} P(tau>j) >= 1 + sum_{j=1}^{B-1} eps_conv(j*n, k, snr); P(tau>j) >= eps_conv(j*n) because decoding correctly within j rounds cannot beat the sphere-packing converse (P1/P2), and truncation is valid in the never-abandon class (P3)",
  "bpsk_na_note": "NORMAL APPROXIMATION (with the 1/2*log2 N term) of the BPSK-input meta-converse \u2014 an approximation, NOT an exact converse; the exact floors are the sphere-packing rows above",
  "captured_share_of_feasible_savings": {
    "bpsk_na": 0.845906952532,
    "complex_unconstrained": 0.840000503469,
    "real_signaling": 0.841497535834
  },
  "chase_class_note": "class-specific converse for repetition/chase combining ONLY; its j=1 term is identical to the real-signaling term, so it does NOT lift the floor \u2014 reported as a labeled secondary column, never the headline",
  "eps_terms": {
    "bpsk_na": [
      0.020948944259,
      0.0,
      0.0
    ],
    "chase_class_SECONDARY": [
      0.00533882431,
      0.0,
      0.0
    ],
    "complex_unconstrained": [
      1.798102e-06,
      0.0,
      0.0
    ],
    "real_signaling": [
      0.00533882431,
      0.0,
      0.0
    ]
  },
  "executed": true,
  "explicit_non_claim": "Does NOT claim the measured 1.48 approaches the channel floor (~1.0); the optimality claim is over STOPPING POLICIES for the given code+combiner, and the floor rows quantify exactly how much a better CODE could still recover. No dB/byte magnitude moves; the v48 measurement is quoted, not re-run.",
  "floors_expected_rounds": {
    "bpsk_na": 1.020948944259,
    "complex_unconstrained": 1.000001798102,
    "real_signaling": 1.00533882431
  },
  "headline": "physics floor 1.0000018 rounds (unconstrained) / 1.0209 (BPSK class, NA): the guard's measured 1.48 is EXACTLY optimal among zero-false-ACK stopping rules and captures 84.0% of the entire converse-feasible savings range; physics does not forbid one-shot decoding here \u2014 the residual 0.48 rounds is the real code's finite-length/BPSK penalty, not a channel limit",
  "honest_scope": "floor for zero-false-ACK HARQ with terminal binary ACK/NACK feedback, equal per-round energy, budget 4, over the MODELED AWGN channel at the v48 operating point; the unconstrained-input converse is conservative (a BPSK-constrained exact converse would sit near the labeled NA row); NOT a claim that 1.48 is near a physical limit",
  "operating_point": {
    "budget_rounds": 4,
    "esn0_db": -2.0,
    "fixed_avg_rounds": 4.0,
    "guard_avg_rounds": 1.48,
    "k_bits": 256,
    "n_complex_uses_per_round": 256,
    "n_real_dims_per_round": 512
  },
  "policy_optimality": {
    "guard_attains": true,
    "guard_measured_avg_rounds": 1.48,
    "proof": "path-by-path: a zero-false-ACK rule may not ACK before the first successful decode (that would be a false ACK on some path), and never-abandon forces it to keep transmitting until then, so tau_any >= tau_first_decode on EVERY sample path; continuing past the first decode only adds rounds. Hence E[tau_any] >= E[tau_first_decode], with equality iff the rule stops at first decode.",
    "statement": "among zero-false-ACK stopping rules for the GIVEN code+combiner that never abandon within the budget, stop-at-first-successful-decode minimizes E[rounds] EXACTLY (path-by-path), and the v48 guard IS that rule"
  },
  "schema": "rounds-converse-v54",
  "skipped": false
}