ROOT PUBLISHER INDEPENDENT READING AND COMPARISON
October18,2026 project date. Exact sealed TeX SHA-256:
63f1a1a2400046d0083e8c654a5730a6239a713f09560fce83bcea06153b05b1.

Root read the complete nine-page manuscript source and visually inspected all
nine rendered PDF pages independently of the separate proof/code/literature
reviewers. Prior qutrit approval was not treated as proof of general p.

For the local estimate, direct expansion gives leading coefficient
 p/(p-1) - (p-1)/(4p) = (3p-1)(p+1)/(4p(p-1)).
Using P<=min(1,(p+1)f-1) and the computational collision lower bound gives exactly
the two displayed factorizations. The first additional root is >1 and the second
<=2/(p+1), so both intervals have the claimed sign, including all endpoints.
At p3 these reduce to the previously derived mixed-qutrit factors.

Nonzero scalar weights with sum of squares0 give an ordinary Weyl tensor
representation. Character projectors have rankD^(k-2) and form a full PVM.
The scaled law A_t(x)=A(t^-1 x) has transform|chi(tu)|^2; hence the difference
law is the positive convolution of the scaled A factors, with no assumed A
sampler. Each nonzero scalar preserves the linear hyperplane and its complement
inside S. Two or three disjoint one-escape-factor events yield the stated F4
or F6 lower bound. All-n reduction and n1 endpoint are valid.

For p=3mod4, square-set intersection supplies a²+b²=-1 and neither scalar can
vanish. For p=1mod4, sqrt(-1) supplies the two-register branch. The p5 zero-
weight pitfall is avoided explicitly. Finding weights by finite search is
polynomial in p, compatible with the stated model, not polynomial in logp.

Root independently checked the threshold logic:
 Delta=2sin²(pi/p)>=8/p²;
 1-c>=Delta(1-sum_j z_j²);
 c>1-3/(4p²) forces a spectral outcome>29/32>7/8.
Noncommuting Weyl eigenspaces have product norm1/sqrt(p), so two such high
outcomes contradict ||E+G||<=1+1/sqrt(p)<7/4. For low c, reverse triangle
bounds each outcome by (1+sqrt(1-eta))/2<=1-eta/4=q. These are conservative
p-dependent thresholds; no qutrit-only uncertainty estimate was reused blindly.

The finite-precision estimate can use rational root-of-unity enclosures at
O(logp) fractional precision: empirical outcome weights sum to one, so averaging
does not accumulate a p-sized precision loss. Counter bit lengths account for h.
The uniform subspace union bound handles the data-dependent H without enumerating
subspaces in the algorithm. Code-support forces every retained nonzero correlation
to commute with the existing code, allowing isotropic completion.

Conditioning fixes the same original optimizer and makes its conditional optimum
F/w. Only the favorable witness, killed on first unfavorable jump and absorbed
at first recorded optimizer, needs preparation acceptance>=F>=tau. Candidates
survive later physical aborts. All wrong paths still have a deterministic cap;
there is no assumed zero-weight oracle. Candidate validation uses original rho.

Root obtains J=O(p²L), kappa^-1<=2p², r<=6, and
 log(1/b0)=O(L+logp). Thus log(1/s0)=Jlog(1/b0)=O(p²L(L+logp))
with absolute constants. Other factors in RQK+RJ+Nv have only the displayed
polynomial/absorbed dependence. The exact prime-dimensional operation model is
important: no fixed finite universal gate-set approximation theorem is included.

Primary sections read directly by root in this continuation include Allcock
2510.06848v3 Cor44/46 and testing scope, and Hinsche2505.15770v3 introduction's
agnostic-extension expectation and mixed-input SectionII.4. The former already
contains the residue arithmetic and pure weighted marginal; the latter explicitly
proposes a qudit extension but does not prove the arbitrary-mixed optimizer bound.
Earlier source-bound qutrit review already read Chen2408.06967v3, Qi2607.15559v1,
Noller2510.08070v2, Iyer-Liang2411.08765v2, Holt2609.38085v2 and the MPS/TTN
alternative. Separate new originality review refreshes the all-prime claim and
recent sources rather than extrapolating the qutrit search.

This is one strengthened publication candidate within the preserved qutrit
research family. All-prime scope is mathematically stronger than p3 alone;
neither version should be counted as a separate parallel original paper. The
main value remains removal of the pure-input restriction using the mixed
estimate and complete resource analysis, not a new sampler or new bootstrap.
