{
  "packet_id": "full_manuscript",
  "scope_summary": "Main claim is uniform and self-contained: one deterministic exact algorithm computes unnormalized F_n at every length in O(n(log n)^\u03b8(log log n)^{4-\u03b8}), hence O(n(log n)^{1-10^{-13}}), in the stated exact complex model with one specified root of order <1024n^3. The body proves this from the fixed C-network, exact-width compiler, sector packing, CRT/chirp reduction, and exponent ledger; the appendix is an optional certificate/printer route to the weaker o(n log n) bound and is not used for \u03b8. Companion preprints are credited but not proof dependencies of the included arguments.",
  "audited_steps": [
    {
      "anchor": "eq:main-constants and net:padding",
      "result": "checked",
      "evidence": "v=C(100,3)=161700, d_nbr=C(97,3)+3*97=147731, \u0394=2v^2(v-3h(h+1))=6871402692000000 exactly, W=443195 v^3+303 v^2=1873807244643542670000<2^71, so 1<\u03bb<m and 0<\u03b8<1."
    },
    {
      "anchor": "Lemma net:scalar",
      "result": "checked",
      "evidence": "(RG+JV)_{S,T} is 1,0,0,0 for intersections 3,1,0,2 respectively, so RG+JV=I. Stages send (x,y) to (x,y+x), then (-y,y+x), then (-y,x); auxiliaries cancel per invocation."
    },
    {
      "anchor": "Lemma net:edge and \u03a6_D=C^{\u2297m}",
      "result": "checked",
      "evidence": "J_1 diag(1,i) J_1=C; weight is additive mod 4 on orthogonal vectors because intersections are even; odd weights are 1 or 3 mod 4, giving C_z or C_z^{-1}, and C_z^2=R_z so C_z^{-1}=R_z C_z."
    },
    {
      "anchor": "Lemma net:residuals",
      "result": "checked",
      "evidence": "Only decreases are central edges 3\u21924 of dimension h; side inclusions use neighboring orthogonality t_X\u00b7t_Y=0. Signed telescope gives \u0394=2N_0-2I_0(h+1)h=2v^2(v-3h(h+1))>0. Residuals contain norm-one vectors; the displayed char-2 absorption produces an orthonormal basis."
    },
    {
      "anchor": "Proposition net:finite-interface",
      "result": "checked",
      "evidence": "Terminal ratios are C^{\u2297mf} except X_d\u2192Y_d, whose ratio is R_u C^{\u2297m}; sink translations and undoing (X,Y)\u21a6(-Y,X) restore C^{\u2297mf} on every role, including auxiliaries. Residual sum is s directional C^{\u2297f} steps."
    },
    {
      "anchor": "Lemma net:indexing",
      "result": "checked",
      "evidence": "Fixed radix-2^m digit tables and a prefix tree with <2^{k+1} nodes implement column maps, translations, and contiguous fiber fields in O(2^k), constant depending only on fixed m."
    },
    {
      "anchor": "Theorem net:tensor-bound",
      "result": "checked",
      "evidence": "For k=mf+r\u2265m\u00b772, 2^{k-f} is divisible by 2^{71}. Recurrence t(k)\u2264A+\u03bb t(\u230ak/m\u230b) unrolls to O((k+1)^\u03b8) because d\u2264\u2308log_m(k+1)\u2309 implies \u03bb^d\u2264\u03bb(k+1)^\u03b8. Workspace sums a geometric series of array lengths."
    },
    {
      "anchor": "Lemma loc:newton-reduction",
      "result": "checked",
      "evidence": "N_ik=\u220f_{j<k}(u^i-u^j)=(-1)^k u^{k(k-1)/2} H_i/H_{i-k}; H_j\u22600 for j<r. Unique LDU plus F=F^T gives F=N D_N^{-1} N^T. Holds for composite r."
    },
    {
      "anchor": "Lemma loc:replay",
      "result": "checked",
      "evidence": "Forward map z\u21a6Uz+Tx with FT+D=M; second source-suppressed replay cancels FUz. Greedy coloring uses \u22642\u0394-1 colors. Identity in independent variables covers correlated borrowed data, which is restored."
    },
    {
      "anchor": "Lemma loc:toeplitz",
      "result": "checked",
      "evidence": "Displacement E_ij-E_{i-1,j-1}=-h_{i-p}g_{p-j}; three explicit outer products reconstruct each chunk. Largest admissible b is an integer graph test, and b\u2265\u230av/(D log v)\u230b for large v, giving d(r)=O(1+log^4 r). Direct fallback is only for bounded v."
    },
    {
      "anchor": "Lemma loc:fourier and Proposition loc:compiler",
      "result": "checked",
      "evidence": "Shear factorization K=[[-8,-10],[0,-6]] and D_t diag(-1/8,-1/6)K D_t^{-1} is an exact matrix identity. \u03ba=1+|\u03bc|^2 and t_2=\u03bc-\u03ba are nonzero; conjugates are inverse-root evaluations of the same DAG. Six-C pattern synchronizes in O(1+log^4 R) slots."
    },
    {
      "anchor": "Lemma lem:sector-address and Proposition prop:tensor-fourier",
      "result": "checked",
      "evidence": "Sector start sums disjoint first-difference classes. Prefix traversal is O(R). Sector widths sum to R and k\u2264\u2113, so C-slots cost O(R(\u2113+1)^\u03b8); tensor multiplication preserves local words."
    },
    {
      "anchor": "Lemma lem:prime-lengths and working length",
      "result": "checked",
      "evidence": "4^u/(2u+1)\u2264C(2u,u)\u2264(2u)^{\u03c0(2u)} gives r_j=O((j+1)^2) and \u2113=\u0398(log n/log log n). Minimal e yields 2n\u2264L<4n and e=O(log \u2113)."
    },
    {
      "anchor": "CRT, chirp, and root ledger in sec:all-lengths",
      "result": "checked",
      "evidence": "E_q^2\u2261E_q mod L and cross terms vanish, so F_L is the permuted tensor of local F_q. Chirp \u03b7^{2kj}=\u03b6_n^{kj} with 2n-1 distinct kernel residues. D_*=(2n)L 2^{\u2308log_2(16L)\u2309}<64n L^2<1024n^3; all compiler and chirp roots are powers of \u03b6_{D_*}."
    },
    {
      "anchor": "Theorem thm:main and Corollary cor:decimal",
      "result": "checked",
      "evidence": "Three charged transforms give T(n)=O(n \u2113^\u03b8(log \u2113)^4). Substitution of \u2113 produces the stated (log log n)^{4-\u03b8} factor. Exact fraction \u03b5>28/10^{13}, ln(10^6)<14, and exp(-t)\u22651-t give \u03b8<1-2\u00b710^{-13}, hence the decimal corollary."
    },
    {
      "anchor": "Corollary cor:polynomial-multiplication",
      "result": "checked",
      "evidence": "Length M=2n-1 cyclic convolution has no wraparound. F_M^2=MJ gives the inverse after one extra forward transform; M^{-1} is a nonzero scalar division. Model extension to one variable-variable product per coordinate is explicit."
    },
    {
      "anchor": "Lemma syn:local-certificate",
      "result": "checked",
      "evidence": "Separate application of S directional steps costs Sf 2^{mf-1} calls; adding 71 role bits preserves g<b2^{b-1} for large f. Used only to terminate the appendix search."
    },
    {
      "anchor": "Lemma syn:extraction and Theorem syn:printer",
      "result": "checked",
      "evidence": "Included Noether/Nakayama search is effective and terminates. Printer budget B=\u230alog_2 n\u230b forces selected m\u2192\u221e, and the mixed-radix/chirp reduction is o(n log n). This appendix bound is weaker than thm:main and is not an input to it."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "citations exactfourier and motif",
      "obligation": "Not used as proof inputs to the included network, compiler, or all-length argument; their external theorems were not audited and are not required for thm:main."
    },
    {
      "anchor": "decorative decimal after eq:main-constants",
      "obligation": "Non-normative expansion of \u03b8 was checked only to magnitude; the exact definition \u03b8=log_m \u03bb and the corollary inequality do not depend on those extra digits."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "grok-4.7",
    "model_key": "grok-4.7",
    "provider": "xai",
    "effort": "high",
    "packet_sha256": "e9dc36257106c27125cfeb73d82a6af0b4caac4f129012d449b3a097251df921",
    "started_at": "2026-10-13T05:17:09.466610+00:00",
    "finished_at": "2026-10-13T05:28:43.510116+00:00",
    "elapsed_seconds": 694.044,
    "selected_attempt": "primary",
    "source_only_independent_first_pass": true
  }
}