# NS57: independent review of bounded-perturbation detection Reviewed base: `b55d13ef6975e14a1f5b3cffe82381a68d64d943`. Reviewer: delegated mathematical reviewer, 2026-09-29. Scope: the negative perturbation directions and their transfer to compact smooth tests, specialized to `delta_2=-log(2)/100`. This is a dependency audit, not new RH research, a numerical test, or a new research row. **Verdict:** no defect found in these two links. The construction gives real negative compact smooth tests in each parity, conditional on the explicit archived realization and radical inputs listed below. It does not itself prove the subsequent first-zero theorem, locate a first zero, or establish a sign for the original Weil form. **Wall check: Same open gap** for original KERNEL-API. This audit validates part of NS57's existing control dependency; it provides no new arithmetic estimate and does not strengthen PR #63's scoped bounds-only obstruction. ## 1. Source provenance and boundaries The following bytes were read at the reviewed base and hashed locally: | Source | SHA256 | | --- | --- | | `evidence/v156/proof.tex` | `c6ebe402b4555ec18384457f6a24222170a20599ec889dd9848b9d2019b8a384` | | `evidence/v136/new_section.tex` | `78e5f6cb2fe60bff4b6edd41594e8499d815cf4cd20b9d96eb0dec7b628828a3` | | `evidence/v136/adversarial_review.md` | `2075651e73d84fc3b3c698853e96914ac29a2a301324f750c1aeba40230d2cbd` | | `evidence/v135/new_section.tex` | `95b12e631f5f55c987ca8ddfdb1c8712b2a40bbaa01967ec29157742769b77fa` | Relevant labels are `lem:ns57-bounded-detection`, `lem:ns57-symbol-signs`, `cor:ns57-finite-weight-negative`, `lem:v136-total-radicals`, `prop:v135-growing-radical`, `eq:v135-gaussian-tail`, and `eq:v135-weighted-tail`. The earlier independent v1.36 review explicitly checks arbitrary real translates, ordinary rather than form-norm density, complete residuals, parity, and fixed-vector quantifiers. I reconstructed the needed argument below rather than treating that earlier verdict as a replacement for it. Inherited, not independently reproved here: - The canonical complete local operator `A_a` and closed form `q_a`, their support-consistent compact-test realization, and its full archimedean, prime, and pole formula. The inner product is linear in its first slot; the Fourier transform is unitary. - The original Gaussian source `phi` is real, even, nonzero, smooth and ordinary-L2 normalized, with the derivative tail estimate in v1.35. Its normalization is not used as a sign premise. - Every real translate of this source is a polarized global radical: `W phi(.-t)=0` distributionally against compact logarithmic tests. This arithmetic identity is an input. I have not rederived the Gaussian source construction, Poisson summation, or its underlying radical theorem. No bounded-floor implication, strong-resolvent limit, quantitative v1.53 sensitivity rate, compact-resolvent theorem, or first-eigenvalue continuity is needed for these two links. Their validity is not being re-audited here. ## 2. Explicit negative directions, with real and parity checks Put `d=log(2)/100>0`, `ell=log(2)`, and `delta_2=-d`. With the v1.56 convention `(T_t f)(x)=f(x+t)`, the exact perturbation is `B=d(T_ell+T_-ell)` and `q_delta[f]=q[f]+`. Indeed, `-2 delta_2 Re=2d Re`; no sign has been lost by changing conventions for translation. Since `T_t*=T_-t`, B is bounded self-adjoint, preserves real functions, and commutes with reflection. Its Fourier multiplier is `b(xi)=2d cos(ell xi)` and `||B||=2d`. The modified coefficient remains strictly positive: `w_2+delta_2=log(2)(1/sqrt(2)-1/100)>0`. Choose a real smooth Fourier bump `eta` of L2 norm one, with compact support in `J=(2 pi/(3 ell),4 pi/(3 ell))`. On J and its reflection, `b<=-d`. These intervals are disjoint. Set `v_even_hat(xi)=(eta(xi)+eta(-xi))/sqrt(2)`, `v_odd_hat(xi)=i(eta(xi)-eta(-xi))/sqrt(2)`. Both have norm one. Their Fourier Hermitian symmetry `v_hat(-xi)=conjugate(v_hat(xi))` gives real physical vectors; their reflection parities are respectively even and odd. Plancherel gives `<=-d` in each sector. Thus there is no need here to assume independence of prime logarithms, invoke a mean-value limit, or infer odd negativity from an even test. These are initially ordinary L2 (in fact Schwartz) vectors, not compact physical Weil tests. ## 3. One fixed finite radical combination comes before the window limit The v1.35 tail gives every exponential moment of phi, so its Fourier transform is entire and not identically zero. Its real zeros have measure zero. If a vector is orthogonal to every translate, Fourier uniqueness applied to the L1 product of its transform with the conjugate transform of phi makes that vector zero. Consequently the finite complex translate span is dense in ordinary L2. This is not a density statement in any global Weil form norm. For either of the fixed real vectors above, take a finite translate approximation and then apply its bounded real-part and parity projections. Because phi is real and even, the resulting `h_sigma` is still a finite linear combination of real translates, is real, has the requested parity, and can satisfy `||h_sigma-v_sigma||<1/16`. Boundedness now provides a numerical margin without any unbounded-form continuity assumption: `|-| <= ||B|| ||h-v|| (||h||+||v||)` `< (2d)(1/16)(33/16)=33d/128`. Therefore `<-95d/128<-d/2`. Fix this finite combination, including every coefficient and translate center, at this point. It will not depend on the later growing window. The proof needs no uniform bound on those coefficients as an approximation tolerance changes. ## 4. Complete cutoff transfer, including the exterior rows Use the original even physical cutoff `chi_a` and set `g_a=chi_a h_sigma`. It is real, has the same parity, and is smooth with support strictly inside `I_a=(-a,a)`. It converges to `h_sigma` in ordinary L2. The smooth compact test is in the actual operator domain: the complete fixed-window archimedean action maps it to L2, and the remaining fixed-window prime and pole contributions are bounded operators. Boundedly perturbing this operator leaves its domain unchanged. Here is the part of the complete residual estimate checked from v1.35. For any one of the finitely many fixed translate centers t, eventually `|t|<=a/2`. Write `r_(a,t)=(1-chi_a)phi(.-t)` on the whole line. The cutoff and Gaussian derivative bound give `||r_(a,t)||_(H1)+sup_x e^(2|x|)(|r_(a,t)(x)|+|r'_(a,t)(x)|)` `<= C exp(-pi e^a/(4e^2)) =: delta_a`. The exterior-tail derivation uses only `|t|<=a/2`, not membership in the spaced lattice used for the separate growing-Gram theorem. On the tail, `X=e^(2|x-t|)>=e^a/e^2` and `e^(2|x|)<=e^a X<=e^2 X^2`; the Gaussian absorbs these polynomial factors. Thus no lattice Gram estimate or increasing-rank argument is imported into this fixed-combination proof. The inherited global radical identity yields `A_a(chi_a phi(.-t))=-W r_(a,t)` on I_a. Every term of the right side is retained: - The archimedean multiplier is bounded by `C(1+|xi|)`, giving L2 norm at most `C delta_a` from the H1 bound. - Every prime row, including `n>e^(2a)`, is covered by `sum_(n>=2) Lambda(n)n^(-1/2)|r_(a,t)(x +/- log n)|` `<=delta_a e^(2|x|) sum_(n>=2) Lambda(n)n^(-5/2)`. The latter sum converges absolutely, including prime powers. Thus its L2 norm on I_a is at most `C sqrt(a)e^(2a)delta_a`. - Both pole kernels are kept. Absolute values bound each integral by `C delta_a e^(|x|/2)`, since `e^(+/-y/2)e^(-2|y|)` is integrable. In particular the negative odd pole term has not been silently deleted or made positive; only a residual norm is being bounded. These estimates give a complete operator residual tending to zero, with the weaker convenient bound `||A_a(chi_a phi(.-t))||<=C sqrt(a)exp(-e^a/100)`. The finite fixed coefficient sum then gives `||A_a g_a||->0`, with a constant allowed to depend on the already fixed h_sigma. There is no finite-head approximation or interchange with a divergent prime sum. ## 5. Compact negative tests and exact quantifier order As a consequence of the preceding domain and residual facts, `|q_a[g_a]|=||<=||A_a g_a|| ||g_a|| -> 0`. Ordinary convergence and boundedness of B also give ` -> <-d/2`. Choose a sufficiently large finite a so that the first absolute value is less than d/8 and the second expression differs from its limit by less than d/8. Then `q_(delta,a)[g_a]<-d/4<0`. The negative value ensures that g_a is nonzero. It is the required real compact smooth negative test in that parity; support consistency makes it a test of the perturbed global compact-test form as well. The entire construction is repeated in the other parity, or the larger of the two finite window thresholds is chosen. The order is: fix delta; choose a negative ordinary vector; choose and fix a finite radical approximation with a strict bounded-B margin; only then increase a until cutoff and complete residual errors are small. No limit of coefficients depending on a, no graph-norm density, no uniform positive spectral gap, and no assumption of RH enters this step. The conclusion is exactly the negative-test input to NS57. Turning it into a finite nonnegative first-zero window requires the separately reviewed small-window positivity, continuity and compact-resolvent links. Those links and the original source/radical construction remain explicit dependencies of any use of the complete control theorem.