{
  "packet_id": "fourth_moment_section18",
  "scope_summary": "Audited the internal \u00a718 fourth-moment argument for Lemma plain, including mask erasure, reflection length bounds, prime-slot exponent \u03baz, induction order, both Poisson transforms, shared-prime and artificial off-coprime extensions, zero-extended principal powers, Gauss-row amplification, normalization and subunit clipping, exceptional-row forcing, centered cancellation, and the finite loss quantifiers. Fully included prerequisites (coefficient conventions, Lemmas 4.4\u20134.5, 4.9\u20134.10 with the logarithmic-control proof, and \u00a713 correlation lemmas) were checked. Unseen citations were not treated as verified.",
  "audited_steps": [
    {
      "anchor": "lem:prime-power-fourier eq:gauss-local",
      "result": "checked",
      "evidence": "Splitting x=x0+py gives the y-sum P^{a-1} iff p^{a-1}|k. For 6|a the unit sum is P*1_{p|k0}-1, reproducing P^{a/2}1_{p^a|k}-P^{a/2-1}1_{p^{a-1}|k}. For 6\u2224a the nontrivial Gauss sum has size \u221aP only at v_p(k)=a-1."
    },
    {
      "anchor": "lem:full-correlation eq:correlation-local",
      "result": "checked",
      "evidence": "Field substitution y=(k/n1)t, x=(k/n2)(1+t) yields A(n1/n2)\u2211_{z\u22600,1}A(z), hence -1 or P-2. The exactly-one-residual case forces x=k/n2 and vanishes unless A is principal and p\u2224k. Absolute bounds used later are at least these values."
    },
    {
      "anchor": "lem:complete-support-correlation and amplifier factorization",
      "result": "checked",
      "evidence": "CRT x=uy+mz separates G(p^i u,h)=\u03c7_{p^i}(u)\u03c7_u(p^i)G(p^i,h)G(u,h). Reciprocity \u03c7_u(p)=R(p,u)\u03c7_p(u) gives exactly R(p,u)^i \u03c7_p(u)^{2i}. For i=6 this is the zero-extended principal factor, matching e_6=0."
    },
    {
      "anchor": "old-eq:2.6",
      "result": "checked",
      "evidence": "With r=1_{6\u2224(i-j)}, (3i-5j-r)_+/6 \u2264 i+j-2-r. The right side is nonnegative because r=1 forces i\u2265j+1. If the left side is positive, 6 times the gap is 3i+11j-12-5r\u22650 (equality at i=2,j=1,r=1). Global positive parts do not exceed the sum of local ones."
    },
    {
      "anchor": "old-eq:2.13 and \u039b_c ledger",
      "result": "checked",
      "evidence": "Substituting a0=A-c-w and K0=2A-c-d+R+E-m gives M'=M+J-g-g2+t2\u2264M-\u03c3 since g-J\u2265\u03c3 and g2\u2265t2. Subtracting the second-transform exponent sum from \u039b_c returns exactly M'+\u0394_child with \u0394_child=b2-p2+w+B_c+\u2113."
    },
    {
      "anchor": "old-eq:2.15 through old-eq:2.19",
      "result": "checked",
      "evidence": "Full expansion of (m'-f)/6+a0-b2-(M'+\u0394_child) equals A-5M/6-F1-F2. Local F2 contributions are 7i/6, (7i-5)/6+(1/3)1_{i=1}, 7i/6-1, and i+j0/6-1, each at least (2/3) of the local b2. The centered deficit A-5M/6-(2/3)v-(L-v)_+ has maximum A-M at v=L=M/4, hence \u2264(A-M)_+."
    },
    {
      "anchor": "eq:centered-clipped-rectangle and eq:centered-divisor-boundary",
      "result": "checked",
      "evidence": "(x)_+-(x-a)_+=a-[(-y)_+-(-x)_+] in all sign cases, so r_clip=r\u0303-(\u03c0-\u03c00). Then t_--(r_clip,1+r_clip,2)/2+(e+\u03c00)/2\u22642\u03b8_N because r\u0303_j+\u03c9_j\u2265t_- and |e_j+\u03c9_j|\u2264\u03b8_N on disjoint slot subsets. Nonempty shells force a0\u2265-\u03b8_N and the added zero-frequency sixth-power contribution fits \u039b_c after the 2\u03b8_N allowance."
    },
    {
      "anchor": "lem:logarithmic-control proof and induction quantifiers old-eq:2.1i",
      "result": "checked",
      "evidence": "Disk edges are Re=a+je; r0=49/100 lies in Re\u2265151/100 and r0<R6<R4. For b\u2265\u03b2_*, e<v/8 puts the R2-disk in Re>\u03b2_* and the R8-disk over Re\u2265b+v. Separately, \u03ba\u22653/4 gives 6\u03ba-1\u22657/2 and z<M/21; T_term+C_*D(\u03b7+\u03be+\u03b50)<\u03b5 under the stated order, with C_*\u03be<\u03c3/4 so each child drops by more than \u03c3/2."
    }
  ],
  "objections": [],
  "unverified_obligations": [
    {
      "anchor": "lem:hecke-strip-growth and reflection scale C_k=3Q_k/(2\u03c0)^2",
      "obligation": "The strip-growth proof and the primitive functional equation it records are not in the packet; the reflection profile and conductor scale are conditional on that unseen argument."
    },
    {
      "anchor": "lem:fixed-numerator-ray and eq:row-fixed-ray-reduction",
      "obligation": "Cited to bound unit and S-supported row factors by a fixed ray family; the statement is not included, so the finite exceptional choices at S were not rederived."
    },
    {
      "anchor": "lem:kernel-seminorms",
      "obligation": "Arbitrary radial decay of \u03a6\u0302 and its Euler derivatives, uniformly for every R_sc>0, is cited but not proved in the packet."
    },
    {
      "anchor": "lem:ray-prime-normalizer and the amplifier pool",
      "obligation": "Both asymptotic prime counts are the fixed-extension case of Chebotarev in the unseen reference TZ; no conductor-uniform error is claimed or checked."
    },
    {
      "anchor": "eq:reciprocity-four-class",
      "obligation": "The bicharacter table defining R on noncoprime primary pairs is external; internal checks used only the stated identity \u03c7_b(a)=R(a,b)\u03c7_a(b) for coprime primary arguments outside S."
    }
  ],
  "assessment": "no_demonstrated_error_found"
}