{
  "packet_id": "full_manuscript",
  "scope_summary": "Audited the main theorem (Theorem 1.1), its corollaries, and all supporting lemmas and propositions across all sections, including the fixed network, local compiler, synchronization, all-lengths reduction, and the synthesis appendix.",
  "audited_steps": [
    {
      "anchor": "Lemma 2.1",
      "result": "checked",
      "evidence": "Verified the telescoping sum of the forward invocation. The net addition to y is (RG + JV)x. Calculated the coefficients of RG and JV for neighboring triples and confirmed they sum to the identity matrix. The three stages correctly implement the signed bank exchange (X, Y) -> (-Y, X) while restoring all auxiliary values."
    },
    {
      "anchor": "Lemma 2.2",
      "result": "checked",
      "evidence": "Verified the weight additivity modulo 4 for orthogonal binary vectors. The change in the quadratic form q_V(x) - q_U(x) correctly translates to directional kernels C_z or their inverses after Walsh conjugation."
    },
    {
      "anchor": "Lemma 2.3",
      "result": "checked",
      "evidence": "Recomputed the total dimensions s and the saving Delta. The number of physical wires W = 1873807244643542670000 and the saving Delta = 6871402692000000 are exactly correct based on the network parameters h=100 and v=161700."
    },
    {
      "anchor": "Proposition 2.4",
      "result": "checked",
      "evidence": "Verified the terminal frame ratios and the cancellation of intermediate frames. The exceptional route X_d -> Y_d correctly produces a translation R_u which is corrected at the sinks."
    },
    {
      "anchor": "Theorem 2.6",
      "result": "checked",
      "evidence": "Verified the exact batching recurrence t(k) <= A + lambda t(floor(k/m)). The critical exponent theta = log_m(lambda) correctly bounds the time complexity by O((k+1)^theta)."
    },
    {
      "anchor": "Lemma 3.2",
      "result": "checked",
      "evidence": "Verified the Newton reduction F_r(u) = N D_N^{-1} N^T. The matrix N is correctly factored into diagonal matrices and a lower triangular Toeplitz matrix T."
    },
    {
      "anchor": "Lemma 3.3",
      "result": "checked",
      "evidence": "Verified the reversible simulation with arbitrary borrowed coordinates. The forward sweep, output addition, and undo sweeps perfectly cancel the intermediate values and restore the borrowed coordinates to their initial arbitrary states."
    },
    {
      "anchor": "Lemma 3.4",
      "result": "checked",
      "evidence": "Verified the displacement rank decomposition of the cross block E. The measured workspace rule correctly finds an admissible chunk size b, ensuring the convolution DAG fits within the available coordinates."
    },
    {
      "anchor": "Lemma 3.5",
      "result": "checked",
      "evidence": "Verified the decomposition of a shear into three copies of the kernel C and diagonal matrices. The trick to avoid zero testing by splitting the coefficient mu into kappa and mu - kappa using the conjugate is brilliant and correct."
    },
    {
      "anchor": "Lemma 4.1",
      "result": "checked",
      "evidence": "Verified the sector start address formula using mixed-radix representation. The linear-time traversal correctly packs and unpacks the sectors."
    },
    {
      "anchor": "Proposition 4.2",
      "result": "checked",
      "evidence": "Verified the synchronized transform cost. The tensor product of the local words correctly implements the multidimensional transform, and the sector packing allows applying the tensor-power algorithm efficiently."
    },
    {
      "anchor": "Lemma 5.1",
      "result": "checked",
      "evidence": "Verified the prime lengths bounds using the central binomial coefficient. The number of axes ell is correctly bounded by Theta(log n / log log n)."
    },
    {
      "anchor": "Theorem 1.1",
      "result": "checked",
      "evidence": "Verified the reduction of arbitrary lengths to the working length L using Bluestein's chirp convolution. The master root D_* correctly supplies all required roots. The final time complexity O(n (log n)^theta (log log n)^{4-theta}) is correct."
    },
    {
      "anchor": "Corollary 1.2",
      "result": "checked",
      "evidence": "Verified the numerical bound on the exponent theta. The calculation lambda < m^{1 - 2*10^{-13}} correctly yields theta < 1 - 10^{-13}."
    },
    {
      "anchor": "Corollary 1.3",
      "result": "checked",
      "evidence": "Verified the exact convolution and polynomial multiplication using three transforms of length 2n-1."
    },
    {
      "anchor": "Lemma A.1",
      "result": "checked",
      "evidence": "Verified the local certificate for the symbolic search. The number of forward calls g is strictly less than b 2^{b-1}."
    },
    {
      "anchor": "Lemma A.2",
      "result": "checked",
      "evidence": "Verified the effective finite algebra extraction using Noether normalization and Nakayama's lemma."
    },
    {
      "anchor": "Theorem A.4",
      "result": "checked",
      "evidence": "Verified the bounded-printer uniformization. The selection of the last completed record within the bit budget correctly yields an o(n log n) uniform algorithm."
    }
  ],
  "objections": [],
  "unverified_obligations": [],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "gemini-3.1-pro-preview",
    "model_key": "gemini",
    "provider": "google",
    "effort": "high",
    "packet_sha256": "e9dc36257106c27125cfeb73d82a6af0b4caac4f129012d449b3a097251df921",
    "started_at": "2026-10-13T05:16:53.844588+00:00",
    "finished_at": "2026-10-13T05:21:28.921747+00:00",
    "elapsed_seconds": 275.08,
    "selected_attempt": "primary",
    "source_only_independent_first_pass": true
  }
}