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Fourier resolution of the arithmetic boundary layer

2026-09-30. RH OPEN. Existing arithmetic form, periodic Fourier projector and ordinary log-coordinate L² norm are unchanged. The estimates below concern a fixed finite derivative space, not the ground state of the whole finite matrix.

F1. Conventions and the factor of two

Write a=log λ, Y=πe^(2a), β=2Y, and R_a=1_[-a,a]. The boundary coordinate in the previous proof is v=β(t−a). In that coordinate the leading tail is e^(-v). In the coordinate v=Y(t−a) suggested in the request it is e^(-2v).

The Fourier grid is ω_n=πn/a. Set

\Omega_N=\pi N/a,\qquad h_N=\Omega_N/\beta, \qquad c_N=N/(aY),\quad h_N=\pi c_N/2.

Replacing N by N+1 changes h_N by π/(aβ), which tends to zero. Geometric boundary resolution is h_N of order one; asymptotic recovery of the unfiltered boundary requires h_N→∞. This is a different condition from N/a→∞.

The actual conventions for the periodic projector and arithmetic quadratic form are those of CCM, §3.1, (3.19)–(3.21), fixed in the previous boundary calculation. No continuous Fourier cutoff is substituted for that projector.

F2. Exact periodization identity

For an even rapidly decreasing f, define on I_a=[−a,a]

G_af(t)=\sum_{\ell\in\mathbb Z}f(t+2a\ell),\qquad H_af(t)=G_af(t)-f(t)=\sum_{\ell\ne0}f(t+2a\ell).

G_a is an actual periodization, not a new arithmetic relation. H_a is first defined on I_a and then periodically extended for applying P_N; the expression G_a(t)−f(t) on the whole real line is not itself periodic. Their derivatives have different matching behavior at the seam. Let p=P_NR_af, extended by zero. Then δ=p−f satisfies the exact identities

\delta|_{I_a}=(I-P_N)H_af-(I-P_N)G_af, \qquad \delta|_{I_a^c}=-f. \tag{F1}

The coefficients of G_a are the samples of the full transform: (2a)^(-1)\widehat f(ω_n). For r=(I−R_a)f, the coefficients of H_a are (2a)^(-1)\widehat r(ω_n). Thus F1 keeps both the true Fourier/Gamma tail and the periodically folded support tail. Omitting either term is generally wrong.

F3. Uniform jet coordinates, not columnwise cancellation

Use the exact Taylor-jet basis from the support hierarchy. Its members e_{r,Y}, 0≤r≤m, are actual linear combinations of k,k'',…,k^(2m). Put

B_Y=e^{a/2-Y},\quad \kappa_Y=2B_Y^2/\beta=\pi^{-1/2}Y^{-1/2}e^{-2Y}.

For a jet vector z∈C^(m+1), set e_{z,Y}=Σz_r e_{r,Y}. Its normalized right tail is

F_{z,Y}(v)=B_Y^{-1}e_{z,Y}(a+v/\beta) \longrightarrow F_z(v)=e^{-v}\sum_{r=0}^m z_r v^r, \qquad v\ge0. \tag{F2}

Convergence holds uniformly for |z|≤1 with every fixed finite collection of polynomially weighted L¹/L² derivative norms. The proof is the exact polynomial jet construction, followed by Taylor expansion of Ye^(v/Y), splitting at a small positive power of Y, and the exponential theta bound on the complement. The n≥2 theta terms remain exponentially small after the polynomially bounded change of basis. This uniform statement is essential: entrywise asymptotics in the original ill-scaled derivative basis cannot be used after cancellation.

Write φ_{z,Y}(ξ)=∫0^∞F(v)e^(-iξv)dv. Evenness gives the exact grid identity

\widehat r(\omega_n) =\frac{2B_Y}{\beta}(-1)^n\Re\varphi_{z,Y}(\omega_n/\beta) \tag{F3}

for real z. Complex vectors follow by sesquilinear polarization, not by taking the real part of a complex linear combination. Equivalently use the cosine transform Ψ_{z,Y}=∫F_{z,Y}cos(ξv)dv, which is linear in complex z.

Uniformly in the jet unit ball,

|\Psi_{z,Y}(\xi)|\le C_m(1+|\xi|)^{-2},\qquad \int_{h}^{\infty}|\Psi'_{z,Y}(\xi)|d\xi \le C_m(1+h)^{-1},\quad h\ge h_0>0. \tag{F4}

Two integrations by parts prove the first estimate; another integration and the weighted derivative bounds in F2 prove the second. These estimates control both the Riemann sums and the physical-space leakage of the sharp projector.

F4. When the full Fourier tail is negligible

For each original derivative, Stirling and the elementary unconditional polynomial bound for ζ on Re s=1/2 give

|\widehat{k^{(2j)}}(\omega)| \le C_m(1+|\omega|)^{2m+3}e^{-\pi|\omega|/4}.

The jet-to-original coefficients grow at most polynomially in Y. Consequently, for any fixed η>0, if

\boxed{\quad \Omega_N\ge(4/\pi+\eta)Y\quad} \tag{F5}

then b=(I−P_N)G_a e_{z,Y} is exponentially small relative to B_Y, uniformly in the jet unit ball, even after any fixed polynomial factor in Y or Ω_N required below. Indeed its tail begins at e^(-πΩ_N/4), whereas B_Y contains e^(-Y). The gap in F5 absorbs the polynomial factors, including e^a=√(Y/π). In particular weighted BV bounds applied to the actual explicit formula show Q(b,b) and all mixed terms with the F1 boundary error to be o(aκ_Y)|z|². Zero locations are not used as inputs to the vectors.

F5 is a proved sufficient separation condition, not a proved sharp transition. It says liminf c_N>4/π². At or below that constant this argument does not discard the full Fourier term. The balancing constant from an upper bound is not thereby the necessary selection threshold. Regime I and small critical c remain open here.

F5. Universal boundary form from the actual periodic projector

Let E F(v)=F(|v|) be the even extension to R, and let H_h be the ordinary Fourier high-pass multiplier 1_(|ξ|>h) used only to describe the limit. For h_N→h∈(2/π,∞), the right-edge error, in scaled coordinates, is

\mathcal D_hF(v)= \begin{cases} (H_h E F)(v),&v<0,\\ -F(v),&v>0. \end{cases} \tag{F6}

At the left edge it is reflected. Formula F6 follows from F3 by Riemann sums; the frequency spacing is π/(aβ)→0. Parseval on the original circle gives a particularly direct norm proof without identifying pointwise values at the jump:

\frac{\|\delta\|_2^2}{\kappa_Y} \longrightarrow J_h(F):=\|F\|_{L^2(0,\infty)}^2 +\tfrac12\|H_hEF\|_{L^2(\mathbb R)}^2. \tag{F7}

The first summand is the unchanged exterior error; the second is periodic projection leakage on the inside. Thus

\|F\|_2^2\le J_h(F)\le2\|F\|_2^2, \qquad J_\infty(F)=\|F\|_2^2. \tag{F8}

The same assertions are uniform on the fixed jet unit ball. If h_N→∞, the second term tends to zero using F4, and F7 holds with J_∞. No boundary fixed point has been designed; F6 is forced by F1 and the original projector.

For F=e^(-v), the inside profile is

K_h(v)=\frac2\pi\int_h^\infty\frac{\cos(\xi v)}{1+\xi^2}d\xi, \quad J_h(e^{-v})=1-\frac{\arctan h+h/(1+h^2)}\pi. \tag{F9}

The support-only norm is 1/2. The corresponding score multiplier is 2J_h, strictly greater than 1 for finite h and tending to 1 at infinity. This exhibits a genuine boundary-resolution crossover; it does not identify a sharp threshold for selection, which is a different question.

F6. Transfer from profile norms to the actual arithmetic form

This step must not be replaced by ambient positivity. We estimate separately the Gamma, polar and all prime-power terms of the actual form.

F6.1 Physical envelope and the arithmetic terms

Abel summation of the F3 coefficients, using F4 and the Dirichlet-kernel geometric sum, gives for d=(I−P_N)H_a e_{z,Y}, h_N≥h_0>0,

|d(t)|\le \frac{C_m B_Y|z|}{1+\beta\,\operatorname{dist}(t,\{-a,a\})}, \quad |t|\le a. \tag{F10}

Near the seam use the absolute coefficient sum; away from it use the partial sum bound C/|sin(π(t−a)/(2a))| and total variation of the coefficient sequence. Outside the interval, the tail is bounded by C_mB_Y|z|e^(-cβ(|t|−a)), with a fixed c>0 after absorbing its fixed-degree polynomial. These bounds are uniform for h_N≥h_0; the discarded b has the stronger F5 bound.

Convolving the two edge envelopes gives a same-edge bound

C_m\frac{B_Y^2|z|^2}{\beta} \frac{\log(2+\beta d)}{1+\beta d},

and, for 0≤d≤2a, an opposite-edge bound with d replaced by |2a−d|. For d>2a an exterior exponential factor is necessary; the bound becomes C_mB_Y²|z|² β^(-1)log(2+aβ)e^(-c'β(d−2a)).

Set X=e^(2a)=Y/π. Summing at d=log n with the actual weights Λ(n)/√n, using only Λ(n)≤log n, gives

|Q_{\rm prime}(\delta)| \le C_m B_Y^2|z|^2\frac{(\log Y)^3}{Y^{3/2}} +o(a\kappa_Y)|z|^2. \tag{F11}

For completeness, n≤X/2 has β|log(n/X)|≳Y and contributes at most this bound. For X/2X the exterior exponential restricts the sum to O(1) integer distance from X; the far tail is exponentially small. Same-edge terms are bounded by Σ_(n≤X)n^(-1/2)=O(√X) after cancellation of the log n denominator. This covers prime multiplicities and threshold events.

Also ∫e^(|t|/2)|δ(t)|dt≤C_mB_Y|z|e^(a/2)log(2+aβ)/β, hence

|Q_{\rm pole}(\delta)| \le C_m B_Y^2|z|^2(\log Y)^2/Y^{3/2} +o(a\kappa_Y)|z|^2. \tag{F12}

Both are o(aκ_Y)|z|². An oscillatory sign assumption about primes was not used.

F6.2 Archimedean logarithm and low frequencies

Let w(ω)=Re ψ(1/4+iω/2)−log π. The needed statement is

\frac1{2\pi}\int w(\omega)|\widehat\delta(\omega)|^2d\omega =\log(\beta/(2\pi))\,\|\delta\|_2^2 +O_m(B_Y^2/\beta)|z|^2+o(a\kappa_Y)|z|^2. \tag{F13}

Here are bounds handling the potentially problematic sharp cutoffs. For any fixed 0<s<1/2 the scaled edge profiles have uniformly bounded H^s norms. For the high-pass piece this follows from |Ψ(ξ)|≤C/(1+ξ²); cutting its physical half-line adds a Gagliardo cross integral C_s∫|g(v)|²|v|^(-2s)dv. On |v|≤1 use the uniform sup bound and s<1/2, and on its complement use the L² bound. The same proof applies to the finite-circle sums by F4 and Parseval, including the jumps at the two endpoints. Thus the high-frequency logarithmic moment is O_m(B_Y²/β)|z|², since log(2+|ξ|)≤C_s(1+|ξ|^(2s)).

At low scaled frequencies |ξ|≤h_0/2, the transform of the interior high-pass piece is uniformly bounded after division by B_Y/β. Indeed integration of each mode over [−a,a] introduces denominators ω_n−βξ of size at least |ω_n|/2; F3–F4 then bound the sum by C_m∫_(h_0)^∞|Ψ(η)|/η dη. The outside tail is L¹. Hence the integrable factor |log|ξ|| causes no loss near zero. The two-edge phases do not affect these bounds. Finally |w(βξ)−log(β/(2π))|≤C+|log|ξ||. These estimates prove F13.

These auxiliary Sobolev estimates are estimates on the existing vectors, not a replacement metric or topology for selecting a state.

F6.3 Full effective matrix

All e_{z,Y} lie in the global radical, so Q(p)=Q(δ). Since log(β/(2π))=2a, F7 and F11–F13 prove

\boxed{ \frac{Q_W(P_NR_a e_{z,Y})}{2a\kappa_Y} \longrightarrow J_h(F_z) } \tag{F14}

uniformly for |z|≤1, when h_N→h>2/π or h_N→∞. By polarization this is matrix convergence. More generally, if liminf h_N>2/π, whether or not h_N converges, there are fixed positive constants b_-,b_+ such that, eventually,

b_-\,2a\kappa_Y|z|^2 \le Q_W(P_NR_a e_{z,Y}) \le b_+\,2a\kappa_Y|z|^2. \tag{F15}

This is uniform coercivity only on the a-dependent finite jet span. It is not Weil positivity on all test functions, nor positivity of the whole finite matrix.

F7. What this implies for the restricted selector

The jet selection lemma in the hierarchy note uses precisely F15, together with convergence of the original derivative Gram and the fixed physical columns P_NR_af_j→f_j in L²(R). Both forms of convergence hold whenever N/a→∞, in particular in F5, by SH8 of the support proof. Gram convergence alone would not rule out a common isometric rotation of all columns. With the actual fixed-column convergence it yields, for every fixed m≥1,

\operatorname{span}u_{a,N,m}\to\operatorname{span}k \quad\text{in the original }L^2(\mathbb R), \tag{F16}

for the lowest direction restricted to P_NR_a R_m, along every path satisfying liminf N/(aY)>4/π². It is simple eventually. Thus the resolved regime N/(aY)→∞ is included, and so are sufficiently large finite critical ratios. The full matrix ground is not being identified.

For resolved paths F14 reproduces the support-only effective jet matrix. For finite critical ratios it gives a different positive jet matrix J_h, while the same physical direction k is selected. Therefore “full selection requires infinite boundary resolution” is stronger than what the analysis supports.

The regimes N/(aY)→0, 0<c≤4/π², and the exact boundary value c=4/π² are not settled by this proof. No sharp necessary transition constant is claimed.

F8. Scope and remaining gap

This is a fixed-m asymptotic statement with original inner product and actual prime/Gamma/pole formula. No m→∞ limit, full-ground gap, ES condition, normalized Fourier comparison G*, or RH conclusion follows. The unknown full-ground coupling from the prior audits remains. A high-resolution restricted result does not retrospectively turn the earlier under-resolved computations into counterexamples or confirmations of the new theorem.

F9. Limited diagnostics, separately from the proof

Scaled-profile experiment checks 12 small J_h variational problems, the scalar F9 identity and 20 sampled Fourier coefficients. The theta-tail diagnostic uses the first theta term and finite quadrature, and is labeled accordingly; it is not an exact interval enclosure. For example 2J_h(e^(-v)) is about 1.181690 at h=1, 1.040519 at h=2 and 1.000419 at h=10.

Actual-cutoff experiment uses the unchanged arithmetic matrix builder at λ=3 and N=4,8,12,20,40. It diagonalizes only the restricted derivative spans, not the full ground problem. The coefficient of k'' in the m=1 restricted minimum, normalized to coefficient 1 on k, changes as follows:

N	h=Ω/(2Y)	coefficient of k''
4	0.202275	+0.00444311
8	0.404551	+0.00153959
12	0.606826	−0.000347017
20	1.01138	−0.000382056
40	2.02275	−0.000378821

The 192→256-node column-quadrature difference is about 1.57·10^(-95) at 100 working digits. The Q assembly uses 448-bit Arb midpoints; subsequent arithmetic is not interval-certified. This table diagnoses a substantial change with resolution at one endpoint. It proves neither a critical constant nor an asymptotic selector. The preceding theorem is based on F1–F15, not a fit to this table.

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