{
  "packet_id": "full_manuscript",
  "scope_summary": "The main conclusion is a single deterministic exact algorithm for every Fourier length n, costing O(n(log n)^theta(log log n)^(4-theta)), hence O(n(log n)^(1-10^-13)), with unrestricted complex coefficients, charged scalar preparation and indexing, and one specified root of order below 1024n^3. The audit covers all 22 numbered formal statements and 45 numbered equations in the supplied source, including the optional synthesis appendix. The companion papers' nonuniform and general finite-win results are distinguished from the locally proved uniform construction; their unseen proofs are not claimed audited.",
  "audited_steps": [
    {
      "anchor": "Theorem 1.1 (thm:main)",
      "result": "checked",
      "evidence": "The internally checked network, exact-width compiler, sector traversal, prime-length construction and chirp identity compose without an external mathematical prerequisite. With L=Theta(n) and ell=Theta(log n/log log n), the charged cost n ell^theta(log ell)^4 gives the stated bound. Length 1 is handled directly."
    },
    {
      "anchor": "Corollary 1.2 (cor:decimal)",
      "result": "checked",
      "evidence": "The exact epsilon=Delta/(mW_*) exceeds 28/10^13. Since log m<14, theta<1-2*10^-13. Absorbing (log log n)^(4-theta) into (log n)^(10^-13) proves the corollary asymptotically."
    },
    {
      "anchor": "Corollary 1.3 (cor:polynomial-multiplication)",
      "result": "checked",
      "evidence": "At M=2n-1, support sums lie in [0,M-1], so cyclic convolution equals linear convolution. Two forward transforms, M general pointwise products, and M^-1 J F_M give the result. The additional permission for input-dependent multiplication is stated explicitly."
    },
    {
      "anchor": "Equation (1.1) (eq:main-constants)",
      "result": "checked",
      "evidence": "Delta=6871402692000000 and W_*=2^71 give epsilon=1717850673/590295810358705651712. Thus 1<lambda=m(1-epsilon)<m and 0<theta<1. The proof uses this exact definition, not the displayed decimal approximation."
    },
    {
      "anchor": "Equation (1.2) (eq:main-bound)",
      "result": "checked",
      "evidence": "Substitution of ell=Theta(log n/log log n) into n ell^theta(log ell)^4 yields n(log n)^theta(log log n)^(4-theta). Polynomial logarithmic preparation costs are asymptotically absorbed."
    },
    {
      "anchor": "Lemma 2.1 (net:scalar)",
      "result": "checked",
      "evidence": "RG has coefficient (|S intersect T|-1)/2. JV cancels it for intersections 0 and 2; intersection 1 contributes zero and S=T contributes one. Hence RG+JV=I. The three bank shears produce (-Y,X), and each invocation restores its arbitrary auxiliaries."
    },
    {
      "anchor": "Lemma 2.2 (net:edge)",
      "result": "checked",
      "evidence": "Orthogonality makes support intersections even, so weights add modulo four. An orthonormal residual basis gives phase factors i^(\u00b1[z dot x]), whose Walsh conjugates are C_z or C_z^-1. Since C_z^2=R_z, the inverse is R_z C_z."
    },
    {
      "anchor": "Lemma 2.3 (net:residuals)",
      "result": "checked",
      "evidence": "The displayed residuals follow from E=B perpendicular-sum P and orthogonal neighboring indicators. Only central edges 3-to-4 decrease, by 100 dimensions each. Every nonzero residual has a norm-one vector. The displayed three-vector replacement converts a unit line plus an alternating plane into three orthogonal unit lines."
    },
    {
      "anchor": "Proposition 2.4 (net:finite-interface)",
      "result": "checked",
      "evidence": "Common frames commute with pointwise scalar gates. The exceptional X-to-Y ratio is R_u C^(tensor m), since wt(u)=27. Correcting translations and the signed exchange transforms every role, not merely the banks. A binary basis change makes each directional product C^(tensor f) on fibers."
    },
    {
      "anchor": "Lemma 2.5 (net:indexing)",
      "result": "checked",
      "evidence": "For fixed m, column lookup tables and displaced contributions require O(f+1) work with an absolute, possibly enormous constant. Prefix traversal has 2^k leaves and fewer than 2^(k+1) nodes. Selected bits become the low address field, making recursive fiber batches contiguous."
    },
    {
      "anchor": "Theorem 2.6 (net:tensor-bound)",
      "result": "checked",
      "evidence": "For k>=72m, f=floor(k/m)>=72 and 2^(k-f) is divisible by 2^71. Every recursive batch is full and may contain arbitrary role values. Normalization gives t(k)<=A+lambda t(floor(k/m)); unrolling yields O((k+1)^theta). Scratch arrays and initialization are charged."
    },
    {
      "anchor": "Remark 2.7 (net:role-remark)",
      "result": "checked",
      "evidence": "The scalar identity holds on independent variables for every physical role. Consequently frame telescoping remains valid when recursive fibers occupy auxiliary roles with nonzero data. A zero-initialized-only auxiliary identity would not suffice."
    },
    {
      "anchor": "Equation (2.1) (net:C)",
      "result": "checked",
      "evidence": "a+b=1, a-b=i, det C=i, and C^2 is the coordinate swap. These identities support the phase and inverse-kernel calculations."
    },
    {
      "anchor": "Equation (2.2) (net:parameters)",
      "result": "checked",
      "evidence": "C(100,3)=161700, N_0=4227952113000000, and 100^3=1000000. These are fixed constants independent of the input length."
    },
    {
      "anchor": "Equation (2.3) (net:schedule)",
      "result": "checked",
      "evidence": "Rows 0 and 1 subtract the old auxiliary contribution; rows 4 and 5 add the updated contribution; rows 6 and 7 restore the auxiliaries. Their difference is (RG+JV)x, including for arbitrary initial auxiliary contents."
    },
    {
      "anchor": "Equation (2.4) (net:wire-count)",
      "result": "checked",
      "evidence": "A triple has C(97,3)+3*97=147731 neighbors. Ordered pairs require v*147731 side wires per invocation. With I_0=3v^2=78440670000, the formula gives W=1873807244643542670000."
    },
    {
      "anchor": "Equation (2.5) (net:frame)",
      "result": "checked",
      "evidence": "Nondegeneracy supplies the projection along U-perp, and diagonal phases are invertible. At U=0 the frame is identity. At U=D the weight phase separates over bits, yielding C^(tensor m). Walsh matrices are used to prove identities, not executed."
    },
    {
      "anchor": "Equation (2.6) (net:telescoping)",
      "result": "checked",
      "evidence": "For a gate with common label U, conjugating its touched arrays by Phi_U leaves the scalar gate unchanged. Adjacent frame factors cancel, leaving precisely sink frames, the full scalar network, and inverse source frames."
    },
    {
      "anchor": "Equation (2.7) (net:labels)",
      "result": "checked",
      "evidence": "The stage-boundary labels match after extracting the next triple from Q. In stage 2, reversing the scalar schedule with source Y and target X gives the stated physical touch sequence. On a side wire, t_X perpendicular t_Y makes the row-2 label a subspace of the row-5 label."
    },
    {
      "anchor": "Equation (2.8) (net:s)",
      "result": "checked",
      "evidence": "Source dimensions total N_0 and sink dimensions Wm-N_0. Absolute changes therefore total Wm-2N_0 plus twice the I_0*101 central losses of dimension 100, giving the displayed s."
    },
    {
      "anchor": "Equation (2.9) (net:delta)",
      "result": "checked",
      "evidence": "Delta=2v^2(v-3h(h+1))=2*26146890000*131400=6871402692000000>0. This is the exact saving used by the recurrence."
    },
    {
      "anchor": "Equation (2.10) (net:padding)",
      "result": "checked",
      "evidence": "2^70<W<2^71. Adding W_*-W roles at m directions each gives S=W_*m-Delta and S/W_*=lambda. Padding preserves the absolute saving and enables exact power-of-two batches."
    },
    {
      "anchor": "Proposition 3.1 (loc:compiler)",
      "result": "checked",
      "evidence": "Newton reduction, exact-width Toeplitz replay and fixed six-C shear replacement give the claimed slot count. Topology and workspace decisions use integers only. Root inverses reproduce coefficient conjugates, and all divisions have explicit nonzero witnesses."
    },
    {
      "anchor": "Lemma 3.2 (loc:newton-reduction)",
      "result": "checked",
      "evidence": "The monic Newton coefficient matrix is unit upper triangular and F_r P=N is lower triangular with nonzero diagonal. Symmetry and uniqueness of unit-lower/diagonal/unit-upper factorization give F_r=N D_N^-1 N^T. For r=1 all factors are identity; for r=2, N=[[1,0],[1,-2]]."
    },
    {
      "anchor": "Lemma 3.3 (loc:replay)",
      "result": "checked",
      "evidence": "The first compute/read/uncompute adds Mx+FUz to y and restores z. The source-suppressed replay subtracts FUz and restores z again. Bounded-degree edge coloring schedules each depth level; four sweeps and two readouts use at most 8g+2a shears."
    },
    {
      "anchor": "Lemma 3.4 (loc:toeplitz)",
      "result": "checked",
      "evidence": "The cross block has displacement rank at most three, with explicit convolution formulas. The integer measured-fit test reserves every gate register inside the parent width. Serial chunk updates restore all borrowed coordinates. O(log^3 v) cross depth along a balanced recursion gives O(log^4 r)."
    },
    {
      "anchor": "Lemma 3.5 (loc:fourier)",
      "result": "checked",
      "evidence": "Compose the exact-width words for N^T, D_N^-1 and N. Transposition reverses factor order and preserves shear and monomial layers. Only nonzero Newton products, Toeplitz diagonals and Fourier normalization constants are divided by."
    },
    {
      "anchor": "Equation (3.1) (loc:newton)",
      "result": "checked",
      "evidence": "Factoring u^i-u^j=-u^j(1-u^(i-j)) gives (-1)^k u^(k(k-1)/2) H_i/H_(i-k). Every H_j with j<r is nonzero for a primitive rth root."
    },
    {
      "anchor": "Equation (3.2) (loc:fourier-factorization)",
      "result": "checked",
      "evidence": "From F=L D_N P^-1, symmetry supplies another unit-lower/diagonal/unit-upper factorization. Uniqueness yields P^-1=L^T and hence F=N D_N^-1 N^T."
    },
    {
      "anchor": "Equation (3.3) (loc:convolution-bounds)",
      "result": "checked",
      "evidence": "A power-of-two FFT, fixed diagonal multiplication and inverse FFT have O(v log(2+v)) scalar gates and O(log(2+v)) depth. Butterfly fan-out is bounded, including output reads. Fixed-vector transforms are coefficient preparation, not uncharged variable operations."
    },
    {
      "anchor": "Equation (3.4) (loc:reciprocal)",
      "result": "checked",
      "evidence": "The recurrence makes the convolution of h and g equal (1,0,...), so it inverts every relevant triangular Toeplitz block. Its sole parameter divisor is h_0, nonzero by hypothesis."
    },
    {
      "anchor": "Equation (3.5) (loc:cross)",
      "result": "checked",
      "evidence": "Multiplying B by the lower triangular inverse T_p^-1 gives exactly the sum over nu=j,...,p-1. Thus the diagonal block transforms followed by the cross shear compute T_v."
    },
    {
      "anchor": "Equation (3.6) (loc:displacement-entry)",
      "result": "checked",
      "evidence": "Reindexing E_(i-1,j-1) with mu=nu+1 changes its summation range to j,...,p. Subtracting E_ij leaves -h_(i-p)g_(p-j), valid at interior chunk entries."
    },
    {
      "anchor": "Equation (3.7) (loc:displacement)",
      "result": "checked",
      "evidence": "After removing the explicit interior outer product, the residual is supported in the first row and first column. The two stated boundary outer products reconstruct it without a rank or zero test."
    },
    {
      "anchor": "Equation (3.8) (loc:reconstruction)",
      "result": "checked",
      "evidence": "Iterating M=D+J_a M J_e^T telescopes. Nilpotence of the shifts terminates the sum, and each outer-product contribution has entries sum_l v_(i-l)w_(j-l)."
    },
    {
      "anchor": "Equation (3.9) (loc:two-convolutions)",
      "result": "checked",
      "evidence": "Applying the reconstructed outer-product contribution gives b_l=sum_(j>=l)w_(j-l)x_j and c_i=sum_l v_(i-l)b_l. Reversing x identifies b_l with convolution coefficient e-1-l."
    },
    {
      "anchor": "Equation (3.10) (loc:chunk-bounds)",
      "result": "checked",
      "evidence": "Three branches, each containing two FFT convolutions, followed by a fixed-size output sum retain bounded fan-out and logarithmic depth. Gate count is O((a+e)log(2+a+e)), with absolute constants."
    },
    {
      "anchor": "Equation (3.11) (loc:fit)",
      "result": "checked",
      "evidence": "Source and target chunks are disjoint; g+a+e<=v leaves g distinct coordinates for gate storage. For sufficiently large v, b_0=floor(v/(D log_2 v)) passes the test, proving only finitely many widths use direct fallback."
    },
    {
      "anchor": "Equation (3.12) (loc:three-kernel)",
      "result": "checked",
      "evidence": "H' diag(2,1) H'=[[3,1],[1,3]]. Multiplication by diag(-3,1) and H' gives [[-8,-10],[0,-6]]. Also a^-1 S C S=H'."
    },
    {
      "anchor": "Equation (3.13) (loc:nonzero-shear)",
      "result": "checked",
      "evidence": "diag(-1/8,-1/6)K=[[1,5/4],[0,1]]. Conjugating by diag(4t/5,1) changes its upper-right entry to t. The diagonal factors are invertible for t nonzero."
    },
    {
      "anchor": "Equation (3.14) (loc:nonzero-split)",
      "result": "checked",
      "evidence": "kappa=1+|mu|^2>0 and kappa-|mu|=(|mu|-1/2)^2+3/4>0. Therefore both kappa and mu-kappa are nonzero, including mu=0. Evaluating the coefficient DAG at inverse roots supplies its conjugate without a complex zero test."
    },
    {
      "anchor": "Lemma 4.1 (lem:sector-address)",
      "result": "checked",
      "evidence": "Earlier sectors partition by their first differing block, giving the start formula. Digit-prefix and block-prefix traversal both cost O(R). For two width-3 axes split into a pair then singleton, starts are 0,4,6,8 and widths 4,2,2,1."
    },
    {
      "anchor": "Proposition 4.2 (prop:tensor-fourier)",
      "result": "checked",
      "evidence": "Common slot patterns give tensor monomials or sectors carrying C^(tensor k). Sector widths sum to R and k<=ell, yielding O(R(ell+1)^theta) per pair slot. Local compiler preparation is polynomial in ell, r_max and log R, separate from full-array scalar products."
    },
    {
      "anchor": "Equation (4.1) (eq:prefix-nodes)",
      "result": "checked",
      "evidence": "Since every radix is at least two, each prefix product is at most R*2^(j-ell). Summing the geometric series gives fewer than 2R nodes. Depth-first traversal updates states rather than copying entire prefixes."
    },
    {
      "anchor": "Equation (4.2) (eq:sector-start)",
      "result": "checked",
      "evidence": "The first differing axis j contributes preceding-prefix width times preceding-block total width p_j times full suffix width tau_j. These classes are disjoint and exhaust earlier sectors."
    },
    {
      "anchor": "Equation (4.3) (eq:sector-updates)",
      "result": "checked",
      "evidence": "The S update uses the old prefix width Q. The offset recurrence is mixed-radix indexing within the sector, and I is original mixed-radix indexing. At a leaf, S+o is therefore the packed address, even for nonconsecutive pair coordinates."
    },
    {
      "anchor": "Equation (4.4) (eq:tensor-fourier)",
      "result": "checked",
      "evidence": "There are O((1+log r_max)^4) common slots. Multiplying the checked per-slot array cost by this count gives the first term; printing schedules, preparing coefficients and constructing local tables gives the separate polynomial term."
    },
    {
      "anchor": "Equation (4.5) (eq:tensor-multiply)",
      "result": "checked",
      "evidence": "Kronecker multiplication gives the identity directly. Consequently simultaneously executing another axis does not invalidate a local word that temporarily modifies and eventually restores borrowed coordinates."
    },
    {
      "anchor": "Lemma 5.1 (lem:prime-lengths)",
      "result": "checked",
      "evidence": "The central-binomial estimate gives r_j=O((j+1)^2); r_j>=j+2 gives the matching product-log lower bound. Maximality bounds the unused-prime logarithm by O(log ell). Trial division over polynomially bounded candidates and product maintenance use polynomial logarithmic time."
    },
    {
      "anchor": "Equation (5.1) (eq:axis-count)",
      "result": "checked",
      "evidence": "Maximality and log R_ell=Theta(ell log(ell+2)) imply log n=Theta(ell log ell). Inverting this relation gives ell=Theta(log n/log log n) and log ell=Theta(log log n)."
    },
    {
      "anchor": "Equation (5.2) (eq:binomial-primes)",
      "result": "checked",
      "evidence": "The largest binomial coefficient is at least 4^u/(2u+1). Each prime valuation is a sum of zero-or-one terms over powers at most 2u, so its prime-power contribution is at most 2u. Multiplication gives the upper bound."
    },
    {
      "anchor": "Equation (5.3) (eq:working-length)",
      "result": "checked",
      "evidence": "Successive doubling of R_ell determines the least e with R_ell 2^e>=2n. The chosen factors are pairwise coprime, and all relevant integer sizes are O(log(n+2)) bits up to absolute constant factors."
    },
    {
      "anchor": "Equation (5.4) (eq:working-bounds)",
      "result": "checked",
      "evidence": "Minimal doubling gives L<4n. Maximality gives 2n/R_ell<r_(ell+1), hence 2^e<2r_(ell+1). The prime bound then gives e=O(log(ell+2)). At n=1, R_0=1 and L=2 satisfy the bounds."
    },
    {
      "anchor": "Equation (5.5) (eq:crt-fourier)",
      "result": "checked",
      "evidence": "E_q E_s is zero modulo L for distinct factors and E_q^2=E_q modulo L. Hence jk=sum_q E_q j_q k_q modulo L, and E_q/L=d_q/q gives the stated product. Multiplication of an output digit by d_q is a permutation."
    },
    {
      "anchor": "Equation (5.6) (eq:working-transform)",
      "result": "checked",
      "evidence": "The odd-prime tensor transform is repeated on 2^e fibers, sharing local preparation. The remaining radix-two transforms cost O(L(1+e)). CRT permutations use prefix traversal, and F_L^-1=L^-1 J F_L adds only linear work."
    },
    {
      "anchor": "Equation (5.7) (eq:chirp)",
      "result": "checked",
      "evidence": "k^2-(k-j)^2+j^2=2kj and eta^2=zeta_n. The fixed kernel's 2n-1 supported residues are distinct for L>=2n, so convolution outputs k<n contain exactly the required Fourier sums."
    },
    {
      "anchor": "Equation (5.8) (eq:master-root)",
      "result": "checked",
      "evidence": "D_* is divisible by 2n, L and a power of two exceeding 16L. It therefore supplies the chirp, CRT, FFT, local convolution and i roots by integer powers. Extracting polynomially many local roots costs polynomial logarithmic work."
    },
    {
      "anchor": "Equation (5.9) (eq:root-size)",
      "result": "checked",
      "evidence": "2^ceil(log_2(16L))<32L, so D_*<64nL^2. Since L<4n, D_*<1024n^3. Root orders and powering exponents therefore fit the stated integer-word model."
    },
    {
      "anchor": "Section 5.3, charged preparation of the fixed chirp operand",
      "result": "checked",
      "evidence": "Both chirp tables can be generated through successive geometric ratios in O(n) field operations. The fixed operand's transform is charged as one full transform; its later products with variable data use prepared scalars only. No uncharged nonlinear input operation is introduced."
    },
    {
      "anchor": "Section 5.4, final integer and workspace ledger",
      "result": "checked",
      "evidence": "Working arrays have linear input-dependent lengths with absolute constant multipliers. Recursive network lengths shrink geometrically and scratch arrays are reusable. Local tables are polynomially logarithmic. Each address, stride, root order and counter uses a fixed number of logarithmic-size words."
    },
    {
      "anchor": "Lemma A.1 (syn:local-certificate)",
      "result": "checked",
      "evidence": "Directional steps cost S f 2^(mf-1) forward C calls; fixed pointwise shears cost at most H2^mf. Adding 71 role axes gives b=mf+71 and a saving (Delta f-2H)2^(mf-1), positive for f>2H/Delta. All roles contain arbitrary data and the width is exactly 2^b."
    },
    {
      "anchor": "Lemma A.2 (syn:extraction)",
      "result": "checked",
      "evidence": "Triangular monic relations make the quotient finite over the remaining polynomial ring. The fiber is computed by finite rational linear algebra. A zero fiber with positive m forces a nonzero elimination relation by the supplied Nakayama argument. Weighted triangular substitution makes it monic and decreases m, proving termination."
    },
    {
      "anchor": "Remark A.3 (syn:simulation)",
      "result": "checked",
      "evidence": "A nonzero unital Q-algebra has an injective scalar embedding. In a basis beginning with 1_E, multiplication by an algebra element is a fixed d-by-d complex matrix. Inputs and scalar Fourier outputs in the 1_E direction are recovered from their first coordinates."
    },
    {
      "anchor": "Theorem A.4 (syn:printer)",
      "result": "conditional",
      "evidence": "The deduction is checked under its explicit complete-record printer and s_t=o(m_t log m_t) assumptions. A log_2 n-bit printing budget bounds literal preparation and root orders; the selected radix tends to infinity. Mixed-radix recursion and chirp convolution give the stated o(n log n) cost. The appendix supplies a separate local printer construction."
    },
    {
      "anchor": "Equation (A.1) (syn:win)",
      "result": "conditional",
      "evidence": "This is the assumed general certificate saving, not a theorem for every matrix. Lemma A.1 independently supplies an actual witness for C, so the locally used certificate prerequisite is established."
    },
    {
      "anchor": "Equation (A.2) (syn:certificate)",
      "result": "checked",
      "evidence": "Ordered embedded calls are permutation conjugates of A direct-sum identity, so their placements can be absorbed into monomials. Matrix and diagonal inverse witnesses enforce units. Idempotent permutation entries can be projected successively to 0 or 1 in a nonzero algebra while preserving identities and units."
    },
    {
      "anchor": "Equation (A.3) (syn:amplification)",
      "result": "conditional",
      "evidence": "Given the certificate, sector-width weighting is Binomial(floor(k/b),q/q^b). Since gq/(bq^b)<1, choose alpha<1 with g(q/(bq^b))^alpha<1. Jensen's inequality and induction absorb the additive constant and prove f(k)=O((k+1)^alpha)."
    },
    {
      "anchor": "Equation (A.4) (syn:radix-size)",
      "result": "checked",
      "evidence": "The locally obtained certificate supports O(log^4 r_t) synchronized slots, each costing O(m_t(t+1)^alpha). Since r_t=O_q((t+1)^2), log m_t>=t log 2 and alpha<1, the displayed size is o(m_t log m_t). Symbolic coefficient printing may take arbitrary finite time."
    },
    {
      "anchor": "Equation (A.5) (syn:uniform-cost)",
      "result": "conditional",
      "evidence": "With a selected complete radix record, each of H radix-m levels costs O(P(1+s/m)); the terminal direct radix-p level costs O(Pp). Root tables and reindexing cost linear work. The preparation term is polynomial in the printing budget B."
    },
    {
      "anchor": "Appendix A.2, cyclotomic preparation and substitution of shear words",
      "result": "checked",
      "evidence": "Local denominators are nonzero at every primitive specialization and hence units modulo the cyclotomic polynomial. Only finitely many positive integers can equal a specialization of c, so testing d until c-d is a unit terminates. Disjoint q-tuples fit because q floor(r_i/q)>=floor(r_i/2) for r_i>2q."
    },
    {
      "anchor": "Appendix A.3, bounded-printer root and exponent ledger",
      "result": "conditional",
      "evidence": "For an explicit record of at most B bits, the product R of literal root orders satisfies R<=2^B<=n. With 2n-1<=P<4n, D=2nPR<8n^3 supplies every required root. Since p=O(sqrt(log n)), the normalized costs (1+p)/log n, 1/log m and s/(m log m) all tend to zero."
    },
    {
      "anchor": "Introduction and Appendix A: exactfourier companion claims",
      "result": "unverified",
      "evidence": "The cited general finite-win, positive-generation and nonuniform subsequence results are not supplied. They are not needed to validate the main quantitative theorem: its network and compiler are reproduced, and Lemma A.1 supplies the appendix's search-termination witness locally."
    },
    {
      "anchor": "Introduction: Alman-Rao and Morgenstern/Ailon comparisons",
      "result": "unverified",
      "evidence": "The quoted leading constant and restricted-model lower bounds require inspection of their cited external sources. None is used in the proof or as evidence for or against the manuscript's unrestricted exact-arithmetic conclusion."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "exactfourier citation: introductory companion claims; Appendix A references to Theorem 8.3 and Lemmas 2.2-2.3",
      "obligation": "Inspect the unseen companion proofs to verify their general finite-win, positive-generation and nonuniform conclusions. The present audit checks the local replacement certificate and displayed amplification argument, not those external proofs."
    },
    {
      "anchor": "motif citation: Proposition 8 and Section 3.5",
      "obligation": "Inspect the unseen article to verify the attribution and its own statement and arithmetic model. The reproduced scalar network and its use here were checked independently; no later tape or precision result was imported."
    },
    {
      "anchor": "Introduction: almanrao, morgenstern, ailon and ailonConditioned citations",
      "obligation": "Verify the quoted constants and exact hypotheses against the external papers. These contextual assertions have no dependency impact on the internally audited main construction."
    }
  ],
  "assessment": "no_demonstrated_error_found",
  "_meta": {
    "model_id": "gpt-6.1-sol",
    "model_key": "sol",
    "provider": "openai-responses",
    "effort": "high",
    "packet_sha256": "e9dc36257106c27125cfeb73d82a6af0b4caac4f129012d449b3a097251df921",
    "started_at": "2026-10-13T05:16:53.843193+00:00",
    "finished_at": "2026-10-13T05:28:07.718560+00:00",
    "elapsed_seconds": 673.878,
    "selected_attempt": "primary",
    "source_only_independent_first_pass": true
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}