{
  "packet_id": "full_manuscript",
  "scope_summary": "Main claim (Thm 1.1, Cor 1.2): one deterministic algorithm computes F_n x exactly for every n in O(n(log n)^\u03b8(log log n)^{4-\u03b8}) = O(n(log n)^{1-1e-13}). The model is exact complex arithmetic with unit-cost field operations, unrestricted coefficients, one supplied root \u03b6_{D*} with D* < 1024n^3, and O(log n)-bit address words. Preparation and indexing are charged. I audited the whole manuscript: \u00a72 network (Lemmas 2.1, 2.2, 2.3, Prop 2.4, Lemma 2.5, Thm 2.6), \u00a73 local compiler (Prop 3.1, Lemmas 3.2\u20133.5), \u00a74 synchronization (Lemma 4.1, Prop 4.2), \u00a75 working lengths, Bluestein reduction, root ledger and exponent arithmetic, Cor 1.3, and Appendix A (Lemmas A.1, A.2, printer amplification, Thm A.4). Appendix A and the companion papers [exactfourier], [motif] are not needed for the main exponent. The main proof is self-contained as written.",
  "audited_steps": [
    {
      "anchor": "Lemma 2.1 (net:scalar), schedule (2.3)",
      "result": "checked",
      "evidence": "(RGx)_S = \u03a3_T x_T(|S\u2229T|-1)/2, giving 1 for S=T, 0 for |\u2229|=1, and \u22131/2 for |\u2229|=0,2. JV cancels the even-intersection (neighbor) terms, so RG+JV = I. Rows 0\u20131 subtract JA+Rc, rows 4\u20135 add J(A+Vx)+R(c+Gx), rows 6\u20137 restore, so y\u2192y+x for arbitrary A,c. The three stages give (x,y+x) \u2192 (-y,y+x) \u2192 (-y,x)."
    },
    {
      "anchor": "(2.4)\u2013(2.5), (2.11)\u2013(2.12), (2.13)",
      "result": "checked",
      "evidence": "d_nbr = C(97,3)+3\u00b797 = 147731. v^2 = 26146890000, so I0 = 78440670000 and N0 = 4227952113000000. W = 2N0 + I0(v\u00b7147731+101) = 1873807244643542670000, with 2^70 < W < 2^71. \u0394 = 2N0 \u2212 2I0\u00b710100 = 6871402692000000 = 2v^2(v\u221230300) > 0. s = Wm \u2212 \u0394 matches. \u03b5 = 1717850673/2^69 \u2248 2.910e-12 > 2.8e-12. \u03b8 \u2248 1 \u2212 2.106e-13, matching the stated decimal."
    },
    {
      "anchor": "Lemma 2.2 (net:edge), (2.6)",
      "result": "checked",
      "evidence": "Weight is additive mod 4 on orthogonal vectors. P_V x = P_U x + \u03a3 z_j[z_j\u00b7x], with each wt z_j odd. i^{[z\u00b7x]} = a + b(\u22121)^{z\u00b7x}. Walsh conjugation gives C_z. C_z^2 = (a^2+b^2)I + 2ab R_z = R_z, hence C_z^{-1} = R_z C_z."
    },
    {
      "anchor": "Frame telescoping (2.7) and labels (2.8)",
      "result": "checked",
      "evidence": "Write actual contents as \u03a6_label\u00b7s. Edge maps change the label only. Gates with a common label commute with \u03a6_U \u2297 I, so s evolves by the scalar network; the output is \u03a6_sink S \u03a6_source^{-1}. I checked every gate's wires share one label. Rows 0, 2, 5 group side wires A_{ST} with Y_S or X_T using t_S/t_T consistently. The stage-2 touch order Y-side, Y-center, X-side, X-center, Y-center, Y-side, X-center, X-side matches the reversed schedule. Stage outputs: X ends at E\u2297D\u2297Q, and Y ends at (P\u2297t)^\u22a5\u2297Q = B'\u2297\u27e8t'\u27e9\u2297Q'. Terminals are U_d/0 at sources and \ud835\udc9f/U_d^\u22a5 at sinks."
    },
    {
      "anchor": "Lemma 2.3 (net:residuals)",
      "result": "checked",
      "evidence": "I recomputed every residual in the table. Nesting at side edge 2\u21925 uses t_X \u22a5 t_Y. The only decrease is center 3\u21924, of dimension h. Signed sum: sinks total Wm \u2212 N0, sources total N0. Absolute sum s = Wm \u2212 2N0 + 2\u00b7I0\u00b7101\u00b7100. The orthonormal-basis argument (unit vector plus absorbing alternating planes via w+p, w+q, w+p+q) checks: each has norm 1 and the three are pairwise orthogonal."
    },
    {
      "anchor": "Prop 2.4 (finite interface)",
      "result": "checked",
      "evidence": "For the exceptional route, q_{U\u22a5} \u2212 q_U \u2261 wt x + 2[u\u00b7x] mod 4, so the ratio is R_u C^{\u2297m} per column. A directional step C_z^{\u2297f}, after a column basis change mapping z to e_1, is C^{\u2297f} on f selected bits per fiber. The step count equals s."
    },
    {
      "anchor": "Lemma 2.5 (indexing), Thm 2.6 (tensor bound)",
      "result": "checked",
      "evidence": "The prefix tree has internal degree \u2265 2, giving < 2^{k+1} nodes. For k \u2265 K = 72m: f \u2265 72 and k \u2212 f \u2265 71, so batches are exact. Normalized cost: S\u00b72^{k\u2212f}/W*\u00b7T(f) gives \u03bb t(\u230ak/m\u230b). The recurrence t \u2264 A + \u03bbt(k/m) with 1 < \u03bb < m gives O((k+1)^\u03b8). Correctness uses restoration of arbitrary auxiliaries. The radix-2^{10^6} tables are absolute constants and are used only when k \u2265 K, i.e. log n \u226b 10^6."
    },
    {
      "anchor": "Lemma 3.2 (Newton reduction)",
      "result": "checked",
      "evidence": "N_ik = \u03a0_{j<k}(u^i \u2212 u^j) = (\u22121)^k u^{k(k\u22121)/2} H_i/H_{i\u2212k}, and the H_j are nonzero for j < r. LDU uniqueness plus F = F^T gives P^{-1} = L^T, hence F = N D_N^{-1} N^T. T is unit lower Toeplitz."
    },
    {
      "anchor": "Lemma 3.3 (replay)",
      "result": "checked",
      "evidence": "Sweep: z \u2192 Uz + Tx. Readout and unsweep add Mx + FUz. The source-free sweep subtracts FUz. The result is an identity in independent x, y, z. Shears: 4 sweeps \u00d7 \u2264 2g plus 2a. Per-level greedy edge coloring needs \u2264 2max(2,\u03b4) \u2212 1 colors."
    },
    {
      "anchor": "Lemma 3.4 (Toeplitz) (3.4)\u2013(3.11)",
      "result": "checked",
      "evidence": "E_ij \u2212 E_{i\u22121,j\u22121} = \u2212h_{i\u2212p} g_{p\u2212j} by index shift. The displacement has three outer products, and nilpotent reconstruction gives two convolutions. The fit test g + a + e \u2264 v is passed by b0 = \u230av/(D log v)\u230b for large v, so O(log v) chunks and O(log^2 v) serial pairs at O(log v) layers each, giving depth O(log^4 r). Preparation is poly(r); the roots needed have power-of-two order < 8r."
    },
    {
      "anchor": "Lemma 3.5 / Prop 3.1 kernel (3.12)\u2013(3.14)",
      "result": "checked",
      "evidence": "SCS = [[a,a],[a,\u2212a]] = aH'. H'diag(2,1)H'diag(\u22123,1)H' = [[\u22128,\u221210],[0,\u22126]]. diag(\u22121/8,\u22121/6)K = [[1,5/4],[0,1]], and conjugation by diag(4t/5,1) gives a shear with coefficient t. In the split, |\u03bc \u2212 \u03ba| \u2265 1 + |\u03bc|^2 \u2212 |\u03bc| > 0. Conjugates come from evaluating the DAG at inverse roots, which is valid because the DAG has only rational literals and roots."
    },
    {
      "anchor": "Lemma 4.1 (sector address), Prop 4.2",
      "result": "checked",
      "evidence": "S formula: classify sectors by the first differing axis, with \u03c4_j = \u03a0_{i>j} r_i. Prefix nodes < 2R. A tensor pair slot is a direct sum of C^{\u2297k}, k \u2264 \u2113, of total width R. Multiplicativity (4.4) makes local-word borrowing irrelevant across axes. Total cost is R(\u2113+1)^\u03b8 per slot times O(log^4 r_max) slots."
    },
    {
      "anchor": "Lemma 5.1, (5.3)\u2013(5.4)",
      "result": "checked",
      "evidence": "The central-binomial bound gives r_j = O(j^2). R_\u2113 \u2265 (\u2113+2)!/2, so \u2113 = \u0398(log n/log log n). Since R_\u2113 is odd and \u2264 2n, e \u2265 1 and 2n \u2264 L < 4n. Also 2^{e\u22121} < r_{\u2113+1}."
    },
    {
      "anchor": "CRT (5.5), Bluestein (5.7), roots (5.8)\u2013(5.9)",
      "result": "checked",
      "evidence": "jk \u2261 \u03a3 E_q j_q k_q mod L, so \u03b6_L^{E_q j_q k_q} = \u03b6_q^{d_q j_q k_q}. L \u2265 2n \u2212 1 prevents aliasing. Every required root order (2n, divisors of L, local 2-power orders \u2264 8r \u2264 16L) divides D* = 2nL\u00b72^{\u2308log2 16L\u2309} < 64nL^2 < 1024n^3."
    },
    {
      "anchor": "Proof of Thm 1.1 / Cor 1.2",
      "result": "checked",
      "evidence": "T = O(n\u2113^\u03b8 log^4 \u2113) = O(n(log n)^\u03b8(log log n)^{4\u2212\u03b8}). The exponent inequality uses 1 \u2212 \u03b5 < 1 \u2212 \u03b70 log m \u2264 exp(\u2212\u03b70 log m)."
    },
    {
      "anchor": "Cor 1.3",
      "result": "checked",
      "evidence": "M = 2n \u2212 1 with no wraparound. F_M^2 = MJ. Three length-M transforms with one reused root give D* < 8192n^3."
    },
    {
      "anchor": "Lemma A.1",
      "result": "checked",
      "evidence": "Call count g \u2264 (Sf + 71W* + 2H)2^{mf\u22121} = b2^{b\u22121} \u2212 (\u0394f \u2212 2H)2^{mf\u22121}. det C = i."
    },
    {
      "anchor": "Lemma A.2, idempotent resolution, Remark A.3",
      "result": "checked",
      "evidence": "Noether normalization with weights (d+1)^i; the Nakayama/adjugate contradiction; termination after \u2264 a decreases. Projection by p is a unital homomorphism onto pE."
    },
    {
      "anchor": "Printer amplification (A.4)\u2013(A.5), Thm A.4",
      "result": "conditional",
      "evidence": "The binomial sector recurrence and the concavity bound are plausible and consistent. Thm A.4 checks: D = 2nPR < 8n^3, and (1+p)/log n + 1/log m + s/(m log m) \u2192 0. This appendix is optional and not needed for the main exponent; I did not fully re-derive the inductive constant."
    }
  ],
  "objections": [
    {
      "anchor": "\u00a75.2 eq. (5.6) and \u00a75.3 convolution step",
      "classification": "unverified_step",
      "claim": "Preparation of L^{-1} for F_L^{-1} = L^{-1}JF_L is not explicitly described in the main-theorem section.",
      "evidence": "The proof of Cor 1.3 gives the method (binary doubling, then one nonzero division). Section 5 relies on it implicitly.",
      "derivation": "Forming L from 1 costs O(log L) field operations; L \u2260 0 justifies the division.",
      "hypotheses_checked": [
        "model permits division in preparation with nonzero denominator"
      ],
      "possible_repair": "Cite the Cor 1.3 construction in \u00a75.2.",
      "dependency_impact": "None on the bound; cost O(log n)."
    },
    {
      "anchor": "\u00a75.2 'each digit of the power-of-two axis fixed in turn'",
      "classification": "unverified_step",
      "claim": "The array layout for fibers of the 2^e axis versus Prop 4.2's contiguous mixed-radix input is not spelled out.",
      "evidence": "CRT ordering interleaves the 2^e digit with the odd digits.",
      "derivation": "One O(L) permutation by prefix traversal (4.1) makes the fibers contiguous, so the cost stays linear.",
      "hypotheses_checked": [
        "(4.1) prefix bound",
        "L = \u0398(n)"
      ],
      "possible_repair": "Add an explicit O(L) reordering.",
      "dependency_impact": "None on the bound."
    }
  ],
  "unverified_obligations": [
    {
      "anchor": "(A.5) amplification",
      "obligation": "Fully verify the induction f(k) = O((k+1)^\u03b1), including the leftover-axis and monomial terms. This is optional and does not affect Thm 1.1."
    },
    {
      "anchor": "External citations [motif], [exactfourier]",
      "obligation": "Not audited. The main proof reproduces the needed network and compiler, so the main theorem does not depend on them."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "claude-opus-5-5",
    "model_key": "claude-opus-5.5",
    "provider": "anthropic",
    "effort": "high",
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    "started_at": "2026-10-13T05:26:26.370225+00:00",
    "finished_at": "2026-10-13T05:32:26.872092+00:00",
    "elapsed_seconds": 360.503,
    "selected_attempt": "recovery",
    "source_only_independent_first_pass": true
  }
}