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CSM_RH Paper 49 — Matching-Scale Local_Coarse Brownian Synthesis, Logarithmic Time Change, and the Centered Mellin Root-

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CSM_RH Paper 49

Matching-Scale Local/Coarse Brownian Synthesis, Logarithmic Time Change, and the Centered Mellin Root-Energy Bridge

Project: CSM_RH
Paper: 49
Version: 0.2
Date: 2026-09-08
Campaign: 44 — PESC_TRIANGULAR_LAG_KERNEL_ATTACK
Track: PK3 — TRANSLATED_WINDOW_SYNTHESIS
Candidate state transition: v1.39 to v1.40

0. Trust boundary

This paper continues directly from CSM_RH Papers 46--48.

Paper 46 closed the structured-weight decoupling route at the component level. In particular, the exact restricted-convolution weight has natural exponent-scale energy,

∑nrX(n)2=X1+o(1),\sum_n r_X(n)^2=X^{1+o(1)},

polynomial-prime Ramare marking has only bounded pointwise multiplicity, and factor bipartitions return to determinant fibres without a geometric fixed-power reserve. Consequently Paper 49 does not reopen coefficient-smallness, bounded-multiplicity marking, or absolute determinant-fibre counting as candidate fixed-power mechanisms.

Paper 47 then recombined the Heath--Brown jj -levels exactly back to the root prime observable and closed Campaign 43 as

COMPONENT_MECHANISMS_EXHAUSTED_AND_EXACTLY_RECOUPLED_TO_ROOT_PRIME_ERROR

Paper 48 began Campaign 44 and proved that the exact PESC root quotient has Brownian covariance kernel

HN(i,j)=min⁡(i,j),H_N(i,j)=\min(i,j),

with largest eigenvalue of order N2N^2, smallest eigenvalue of order 11, and condition number

cond⁡(HN)=16π2N2(1+o(1)).\operatorname{cond}(H_N) = \frac{16}{\pi^2}N^2(1+o(1)).

Paper 48 therefore left PK3 with the following non-negotiable requirements:

  1. state the synthesis operator explicitly;
  2. retain endpoint-position information as well as ratio/frequency information;
  3. preserve the Brownian low-frequency root mode;
  4. price every frame or reconstruction condition-number loss;
  5. distinguish exact reconstruction from one-sided domination;
  6. do not identify the Paper 42 translated Mellin kernel with the PESC root kernel without a deterministic bridge;
  7. do not excise low frequencies and defer their recovery without charging the recovery cost.

The present paper closes the deterministic synthesis problem at this level.

It proves five main facts.

First, positive frame synthesis obeys a spectral-spread conservation law: the N2N^2 Brownian stiffness cannot disappear under a stable change of frame.

Second, the corresponding local/coarse scale is constructively attainable. An exact block-Fourier decomposition splits HNH_N into compactly supported local integrated modes plus a coarse remainder of rank at most ⌈N/L⌉\lceil N/L\rceil when the maximum local block length is LL.

Third, a logarithmic endpoint coordinate transforms the Brownian root kernel into a Mellin-compatible Brownian kernel with only constant distortion:

NGNlog⁡⪯HN⪯2NGNlog⁡.N G_N^{\log} \preceq H_N \preceq 2N G_N^{\log}.

Fourth, GNlog⁡G_N^{\log} admits an exact centered Fourier representation, which turns the PESC root energy into a centered Mellin t−2t^{-2} energy with only absolute constant distortion.

Fifth, Gaussian translated windows of this centered Mellin energy contain the Paper 42 ratio kernel as their first term, together with explicit endpoint-centering terms. Thus the earlier ratio-only kernel is recovered as a component of the root-observing synthesis rather than incorrectly identified with the root kernel itself.

No fixed-power PESC estimate is proved. No proof or disproof of RH is claimed. No fixed zero-free strip, fixed-power PNT remainder, fixed-power Mertens estimate, or unproved prime-correlation theorem is assumed.

1. Inherited PESC root quotient

Define the prime-error increments

cn=1P(n)log⁡n−1,c_n = 1_{\mathbb P}(n)\log n-1,

and

B(j)=∑n≤jcn=ϑ(j)−j.B(j) = \sum_{n\le j}c_n = \vartheta(j)-j.

Paper 48 certified the dyadic root energy

JNϑ=∑j=N2N−1B(j)2.\boxed{ J_N^\vartheta = \sum_{j=N}^{2N-1}B(j)^2. }

The exact root quotient coordinates are

y0=B(N),y_0=B(N),

and

yr=cN+r,1≤r≤N−1.y_r=c_{N+r}, \qquad 1\le r\le N-1.

After reversal,

zi=yN−i,1≤i≤N,z_i=y_{N-i}, \qquad 1\le i\le N,

so explicitly

zN=B(N)\boxed{ z_N=B(N) }

and

zi=c2N−i,1≤i<N.\boxed{ z_i=c_{2N-i}, \qquad 1\le i<N. }

Paper 48 proved

JNϑ=z∗HNz,HN(i,j)=min⁡(i,j).\boxed{ J_N^\vartheta = z^*H_Nz, \qquad H_N(i,j)=\min(i,j). }

For every fixed

0<κ≤1,0<\kappa\le1,

PESC (κ)(\kappa) is exponent-equivalent to

JNϑ≪N3−κ+o(1).\boxed{ J_N^\vartheta \ll N^{3-\kappa+o(1)}. }

The whole purpose of PK3 is therefore to construct a power-safe bridge from this exact root form to translated frequency variables without destroying its endpoint geometry.

2. Cumulative-sum factorization

Let CNC_N be the lower-triangular cumulative-sum matrix

(CN)ij=1j≤i.(C_N)_{ij} = 1_{j\le i}.

Then

(CNCN∗)ij=∑r=1N1r≤i1r≤j=min⁡(i,j).(C_NC_N^*)_{ij} = \sum_{r=1}^N 1_{r\le i}1_{r\le j} = \min(i,j).

Hence

HN=CNCN∗.\boxed{ H_N=C_NC_N^*. }

For u∈CNu\in\mathbb C^N define suffix sums

Sk(u)=∑i=kNui.S_k(u) = \sum_{i=k}^N u_i.

Since

(CN∗u)k=Sk(u),(C_N^*u)_k=S_k(u),

we have the exact endpoint-position identity

u∗HNu=∑k=1N∣∑i=kNui∣2.\boxed{ u^*H_Nu = \sum_{k=1}^N \left| \sum_{i=k}^Nu_i \right|^2. }

This formula will be used twice: first to audit local/coarse synthesis and then to control the slowly varying dyadic multiplier that appears in the Mellin time change.

3. Spectral-spread conservation under positive frame synthesis

Let

T:CN→CMT:\mathbb C^N\to\mathbb C^M

be a frame analysis operator satisfying

A∥x∥22≤∥Tx∥22≤B∥x∥22.A\|x\|_2^2 \le \|Tx\|_2^2 \le B\|x\|_2^2.

Let

W=diag⁡(w1,…,wM)W=\operatorname{diag}(w_1,\ldots,w_M)

with

0<wmin⁡≤wα≤wmax⁡.0<w_{\min}\le w_\alpha\le w_{\max}.

Assume the synthesized positive operator

S=T∗WTS=T^*WT

is uniformly comparable to the Brownian root kernel:

c−HN⪯S⪯c+HN.c_-H_N \preceq S \preceq c_+H_N.

Theorem 3.1 — Frame-weight spectral-spread conservation

Under the preceding assumptions,

BAwmax⁡wmin⁡≥c−c+cond⁡(HN).\boxed{ \frac BA \frac{w_{\max}}{w_{\min}} \ge \frac{c_-}{c_+} \operatorname{cond}(H_N). }

Therefore

BAwmax⁡wmin⁡≥c−c+16π2N2(1+o(1)).\boxed{ \frac BA \frac{w_{\max}}{w_{\min}} \ge \frac{c_-}{c_+} \frac{16}{\pi^2}N^2(1+o(1)). }

Proof

The frame bounds give

AI⪯T∗T⪯BI.AI\preceq T^*T\preceq BI.

Since

wmin⁡I⪯W⪯wmax⁡I,w_{\min}I\preceq W\preceq w_{\max}I,

we obtain

wmin⁡AI⪯T∗WT⪯wmax⁡BI.w_{\min}A I \preceq T^*WT \preceq w_{\max}B I.

Hence

cond⁡(S)≤BAwmax⁡wmin⁡.\operatorname{cond}(S) \le \frac BA \frac{w_{\max}}{w_{\min}}.

The comparison with HNH_N gives

cond⁡(S)≥c−c+cond⁡(HN).\operatorname{cond}(S) \ge \frac{c_-}{c_+} \operatorname{cond}(H_N).

Combining the inequalities proves the theorem.

□\Box

Create:

B-RH-036
PESC_FRAME_WEIGHT_SPECTRAL_SPREAD_CONSERVATION
CERTIFIED

The theorem does not say translated windows are impossible. It says the N2N^2 root stiffness must remain visible somewhere in the synthesis ledger.

4. Submacroscopic positive synthesis tax

Suppose TLT_L is built from physical windows of support length at most LL and has Bessel bound

∥TLx∥22≤B0∥x∥22,\|T_Lx\|_2^2 \le B_0\|x\|_2^2,

with

B0=No(1).B_0=N^{o(1)}.

Assume positive coefficient weights obey the natural local Brownian scale

0<wα≤C0L2No(1).0<w_\alpha \le C_0L^2N^{o(1)}.

Set

SL=TL∗WLTL.S_L=T_L^*W_LT_L.

If full root domination requires

HN⪯CrecSL,H_N \preceq C_{\mathrm{rec}}S_L,

then evaluation on the principal Brownian eigenvector gives

μ1≤CrecC0B0L2No(1).\mu_1 \le C_{\mathrm{rec}}C_0B_0L^2N^{o(1)}.

Since

μ1=4π2N2(1+o(1)),\mu_1 = \frac{4}{\pi^2}N^2(1+o(1)),

we obtain

Theorem 4.1 — Submacroscopic reconstruction tax

Crec≳(NL)2N−o(1).\boxed{ C_{\mathrm{rec}} \gtrsim \left(\frac NL\right)^2N^{-o(1)}. }

Thus for

L=N1−η,L=N^{1-\eta},

purely submacroscopic positive synthesis pays at least

N2η−o(1).\boxed{ N^{2\eta-o(1)}. }

Equivalently, a local arithmetic gain N−δN^{-\delta} cannot be promoted to a root gain before the deterministic exponent tax is subtracted.

Create:

O-RH-128
PESC_SUBMACROSCOPIC_POSITIVE_SYNTHESIS_PAYS_POWER_RECONSTRUCTION_TAX
CERTIFIED AS PK3 CONDITIONING BARRIER

This is the no-free-lunch side of PK3. The next section shows that its coarse-dimension scale is nevertheless attainable by an exact construction.

5. Exact block-Fourier local/coarse synthesis

Partition

{1,…,N}\{1,\ldots,N\}

into consecutive blocks

I1,…,IBI_1,\ldots,I_B

with block lengths

ℓb=∣Ib∣≤L.\ell_b=|I_b|\le L.

One may choose all but the last block to have length LL, so

B≤⌈NL⌉.\boxed{ B\le\left\lceil\frac NL\right\rceil. }

On each block IbI_b, let

qb,r,0≤r<ℓb,q_{b,r}, \qquad 0\le r<\ell_b,

be the normalized discrete Fourier basis. In local coordinates s=0,…,ℓb−1s=0,\ldots,\ell_b-1,

qb,r(s)=ℓb−1/2exp⁡(2πirsℓb).q_{b,r}(s) = \ell_b^{-1/2} \exp\left( \frac{2\pi i r s}{\ell_b} \right).

Extend each qb,rq_{b,r} by zero outside IbI_b. The family over all blocks is an orthonormal basis of CN\mathbb C^N.

Define integrated atoms

ψb,r=CNqb,r.\boxed{ \psi_{b,r}=C_Nq_{b,r}. }

Since the qb,rq_{b,r} form an orthonormal basis,

I=∑b,rqb,rqb,r∗,I = \sum_{b,r}q_{b,r}q_{b,r}^*,

and therefore

HN=CNICN∗=∑b,rψb,rψb,r∗.\boxed{ H_N = C_NIC_N^* = \sum_{b,r} \psi_{b,r}\psi_{b,r}^*. }

This reconstruction is exact.

5.1 Zero-mean local modes stay local after integration

For

r≠0,r\ne0,

we have

∑s=0ℓb−1qb,r(s)=0.\sum_{s=0}^{\ell_b-1}q_{b,r}(s)=0.

Before the block begins, CNqb,rC_Nq_{b,r} is zero. After the block ends, the cumulative sum is the total block sum, which is also zero. Hence

supp⁡(ψb,r)⊆Ib,r≠0.\boxed{ \operatorname{supp}(\psi_{b,r}) \subseteq I_b, \qquad r\ne0. }

Thus every nonconstant block-Fourier increment mode becomes a genuinely compact local window after cumulative integration.

5.2 Exact local atom energy

For

1≤r<ℓb,1\le r<\ell_b,

inside the block the partial Fourier sum gives

∣∑s=0m−1qb,r(s)∣2=1ℓbsin⁡2(πrm/ℓb)sin⁡2(πr/ℓb).\left| \sum_{s=0}^{m-1} q_{b,r}(s) \right|^2 = \frac1{\ell_b} \frac{ \sin^2(\pi r m/\ell_b) }{ \sin^2(\pi r/\ell_b) }.

Summing over the block and using

∑m=1ℓbsin⁡2(πrmℓb)=ℓb2,\sum_{m=1}^{\ell_b} \sin^2\left( \frac{\pi r m}{\ell_b} \right) = \frac{\ell_b}{2},

we obtain

∥ψb,r∥22=12sin⁡2(πr/ℓb).\boxed{ \|\psi_{b,r}\|_2^2 = \frac1{ 2\sin^2(\pi r/\ell_b) }. }

In particular the largest local integrated-mode energy is

≍ℓb2,\asymp \ell_b^2,

and hence at most O(L2)O(L^2).

5.3 The only nonlocal block modes are the constants

For

r=0,r=0,

qb,0q_{b,0} is constant on IbI_b. Its cumulative integral ramps across the block and remains constant after the block, so it is nonlocal.

Define

HN,Lloc=∑b∑r=1ℓb−1ψb,rψb,r∗H_{N,L}^{\mathrm{loc}} = \sum_b \sum_{r=1}^{\ell_b-1} \psi_{b,r}\psi_{b,r}^*

and

HN,Lcoarse=∑bψb,0ψb,0∗.H_{N,L}^{\mathrm{coarse}} = \sum_b \psi_{b,0}\psi_{b,0}^*.

Then

Theorem 5.1 — Exact local/coarse Brownian synthesis

HN=HN,Lloc+HN,Lcoarse.\boxed{ H_N = H_{N,L}^{\mathrm{loc}} + H_{N,L}^{\mathrm{coarse}}. }

Every atom in HN,LlocH_{N,L}^{\mathrm{loc}} is supported on an interval of length at most LL, while

rank⁡HN,Lcoarse≤⌈NL⌉.\boxed{ \operatorname{rank} H_{N,L}^{\mathrm{coarse}} \le \left\lceil\frac NL\right\rceil. }

Create:

B-RH-037
EXACT_BLOCK_FOURIER_LOCAL_COARSE_BROWNIAN_SYNTHESIS
CERTIFIED

The construction is the finite discrete analogue of integrating zero-mean Haar/Fourier increment modes into compact Schauder-type position modes. The Brownian covariance remains exact because the orthonormal increment resolution of the identity is exact.

The conclusion is important for interpreting the previous obstruction. Local synthesis is not impossible. What is unavoidable is a coarse nonlocal channel whose natural dimension is of order N/LN/L when all retained local atoms are restricted to scale LL.

6. Logarithmic endpoint time change

The exact block construction solves the position-localization ledger but does not yet place the frequency variable in the multiplicative coordinate natural for Paper 42.

For the exact root quotient define

ni=2N−i,1≤i≤N.\boxed{ n_i=2N-i, \qquad 1\le i\le N. }

Thus

N≤ni<2N,N\le n_i<2N,

and

zi={cni,i<N,B(N),i=N.z_i = \begin{cases} c_{n_i},&i<N,\\ B(N),&i=N. \end{cases}

Define the logarithmic endpoint coordinate

xi=log⁡2Nni.\boxed{ x_i = \log\frac{2N}{n_i}. }

Then

0<x1<⋯<xN=log⁡2,0<x_1<\cdots<x_N=\log2,

and, crucially,

xi−xj=log⁡njni.\boxed{ x_i-x_j = \log\frac{n_j}{n_i}. }

Thus differences in the new endpoint coordinate are exactly Mellin ratio variables.

Define the logarithmic Brownian matrix

GNlog⁡(i,j)=min⁡(xi,xj).\boxed{ G_N^{\log}(i,j) = \min(x_i,x_j). }

Let

x0=0x_0=0

and

Δxr=xr−xr−1.\Delta x_r=x_r-x_{r-1}.

Since

Δxr=log⁡(1+12N−r),\Delta x_r = \log\left( 1+ \frac1{2N-r} \right),

and for u>0u>0

1u+1≤log⁡(1+1u)≤1u,\frac1{u+1} \le \log\left(1+\frac1u\right) \le \frac1u,

we obtain

12N≤Δxr≤1N.\boxed{ \frac1{2N} \le \Delta x_r \le \frac1N. }

Both Brownian kernels are cumulative Gram matrices:

HN=CNICN∗H_N = C_NIC_N^*

and

GNlog⁡=CNdiag⁡(Δx1,…,ΔxN)CN∗.G_N^{\log} = C_N \operatorname{diag}(\Delta x_1,\ldots,\Delta x_N) C_N^*.

Therefore the increment comparison gives the Loewner comparison

Theorem 6.1 — Power-safe logarithmic Brownian time change

NGNlog⁡⪯HN⪯2NGNlog⁡.\boxed{ N G_N^{\log} \preceq H_N \preceq 2N G_N^{\log}. }

The distortion is an absolute factor of at most 22. No power of NN is lost.

Create:

B-RH-038
PESC_LOGARITHMIC_BROWNIAN_TIME_CHANGE_WITH_CONSTANT_DISTORTION
CERTIFIED

This is the deterministic endpoint-position to Mellin-ratio bridge that Paper 48 required but did not yet possess.

7. Exact centered Fourier representation of the logarithmic Brownian kernel

For x,y≥0x,y\ge0 we have

min⁡(x,y)=x+y−∣x−y∣2.\min(x,y) = \frac{x+y-|x-y|}{2}.

Using the standard identity

∫R1−cos⁡(tu)t2 dt=π∣u∣,\int_{\mathbb R} \frac{1-\cos(tu)}{t^2} \,dt = \pi|u|,

we obtain the exact centered representation

Lemma 7.1 — Centered Brownian Fourier identity

min⁡(x,y)=12π∫R(eitx−1)(e−ity−1)t2 dt.\boxed{ \min(x,y) = \frac1{2\pi} \int_{\mathbb R} \frac{ (e^{itx}-1)(e^{-ity}-1) }{t^2} \,dt. }

The apparent singularity at t=0t=0 is removable because each centered factor vanishes linearly there.

Consequently, for every a∈CNa\in\mathbb C^N,

a∗GNlog⁡a=12π∫R∣∑i=1Nai(eitxi−1)∣2t2 dt.\boxed{ a^*G_N^{\log}a = \frac1{2\pi} \int_{\mathbb R} \frac{ \left| \sum_{i=1}^Na_i(e^{itx_i}-1) \right|^2 }{t^2} \,dt. }

This identity is exact and positive. No frequency band has been removed.

8. The dyadic multiplier costs only a constant

To connect the exact root vector zz to Mellin-normalized coefficients, define

ai=zini\boxed{ a_i=\frac{z_i}{n_i} }

and

di=niN=2−iN.\boxed{ d_i=\frac{n_i}{N}=2-\frac{i}{N}. }

Then

1≤di<21\le d_i<2

and

z=NDa,D=diag⁡(d1,…,dN).\boxed{ z=N D a, \qquad D=\operatorname{diag}(d_1,\ldots,d_N). }

Pointwise bounds 1≤di<21\le d_i<2 alone do not automatically imply comparability in the Brownian norm, because the Brownian norm is a cumulative-sum norm. The slow variation of did_i must be used.

Let

Sk=∑i=kNaiS_k = \sum_{i=k}^Na_i

and

Tk=∑i=kNdiai.T_k = \sum_{i=k}^Nd_i a_i.

Then

a∗HNa=∑k∣Sk∣2a^*H_Na = \sum_k|S_k|^2

and

(Da)∗HN(Da)=∑k∣Tk∣2.(Da)^*H_N(Da) = \sum_k|T_k|^2.

Discrete Abel summation gives

Tk=dkSk−1N∑j=k+1NSj.\boxed{ T_k = d_kS_k - \frac1N \sum_{j=k+1}^NS_j. }

The strict suffix-sum matrix has operator norm at most NN, so

∥T∥2≤3∥S∥2.\|T\|_2 \le 3\|S\|_2.

For the reverse inequality set

ei=di−1.e_i=d_i^{-1}.

Then

12<ei≤1\frac12<e_i\le1

and

Sk=ekTk+∑j=k+1N(ej−ej−1)Tj.S_k = e_kT_k + \sum_{j=k+1}^N (e_j-e_{j-1})T_j.

The second operator has maximum row sum at most 1/21/2 and maximum column sum below 11, so the Schur test gives operator norm below 11. Hence

∥S∥2≤2∥T∥2.\|S\|_2 \le 2\|T\|_2.

We have proved:

Theorem 8.1 — Slow dyadic multiplier is Brownian-norm stable

14a∗HNa≤(Da)∗HN(Da)≤9a∗HNa.\boxed{ \frac14 a^*H_Na \le (Da)^*H_N(Da) \le 9a^*H_Na. }

No power of NN is lost in passing from ziz_i to the 1/ni1/n_i Mellin normalization.

9. Centered Mellin root-energy bridge

Define

MN(t)=∑i=1Nai[(2Nni)it−1].\boxed{ \mathfrak M_N(t) = \sum_{i=1}^N a_i \left[ \left(\frac{2N}{n_i}\right)^{it}-1 \right]. }

Using the exact root quotient, this is

MN(t)=B(N)N(2it−1)+∑N<n<2Ncnn[(2Nn)it−1].\boxed{ \begin{aligned} \mathfrak M_N(t) ={}& \frac{B(N)}{N} (2^{it}-1) \\ &+ \sum_{N<n<2N} \frac{c_n}{n} \left[ \left(\frac{2N}{n}\right)^{it}-1 \right]. \end{aligned} }

The first term is not an optional boundary correction. It is the exact aggregate coordinate through which the PESC quotient observes all increments up to NN.

Combining Theorems 6.1 and 8.1 with Lemma 7.1 yields the main deterministic bridge.

Theorem 9.1 — Centered Mellin root-energy comparison

Let

ENCM=∫R∣MN(t)∣2t2 dt.\boxed{ \mathcal E_N^{\mathrm{CM}} = \int_{\mathbb R} \frac{ |\mathfrak M_N(t)|^2 }{t^2} \,dt. }

Then

N38πENCM≤JNϑ≤9N3πENCM.\boxed{ \frac{N^3}{8\pi} \mathcal E_N^{\mathrm{CM}} \le J_N^\vartheta \le \frac{9N^3}{\pi} \mathcal E_N^{\mathrm{CM}}. }

Proof

From z=NDaz=NDa,

JNϑ=N2(Da)∗HN(Da).J_N^\vartheta = N^2(Da)^*H_N(Da).

Theorem 8.1 gives

N24a∗HNa≤JNϑ≤9N2a∗HNa.\frac{N^2}{4}a^*H_Na \le J_N^\vartheta \le 9N^2a^*H_Na.

Theorem 6.1 gives

Na∗GNlog⁡a≤a∗HNa≤2Na∗GNlog⁡a.N a^*G_N^{\log}a \le a^*H_Na \le 2N a^*G_N^{\log}a.

Lemma 7.1 gives

a∗GNlog⁡a=12πENCM.a^*G_N^{\log}a = \frac1{2\pi} \mathcal E_N^{\mathrm{CM}}.

Combining the three inequalities proves the result.

□\Box

Create:

B-RH-039
PESC_CENTERED_MELLIN_ROOT_ENERGY_WITH_CONSTANT_DISTORTION
CERTIFIED

For every fixed 0<κ≤10<\kappa\le1, Theorem 9.1 implies the exponent equivalence

JNϑ≪N3−κ+o(1)\boxed{ J_N^\vartheta \ll N^{3-\kappa+o(1)} }

if and only if

ENCM≪N−κ+o(1).\boxed{ \mathcal E_N^{\mathrm{CM}} \ll N^{-\kappa+o(1)}. }

This is a root-observing Mellin reformulation with no power-sized deterministic loss.

10. The centered transform as a single prime-error linear statistic

Because

B(N)=∑m≤Ncm,B(N)=\sum_{m\le N}c_m,

we may write the centered Mellin transform without an abstract endpoint variable.

Define

ωN,t(m)={2it−1N,m≤N,(2N/m)it−1m,N<m<2N,0,m≥2N.\boxed{ \omega_{N,t}(m) = \begin{cases} \dfrac{2^{it}-1}{N},&m\le N,\\ \dfrac{(2N/m)^{it}-1}{m},&N<m<2N,\\ 0,&m\ge2N. \end{cases} }

Then

MN(t)=∑m<2NcmωN,t(m).\boxed{ \mathfrak M_N(t) = \sum_{m<2N}c_m\omega_{N,t}(m). }

Since

cm=1P(m)log⁡m−1,c_m=1_{\mathbb P}(m)\log m-1,

we also have the exact prime/background split

MN(t)=∑p<2N(log⁡p)ωN,t(p)−∑m<2NωN,t(m).\boxed{ \mathfrak M_N(t) = \sum_{p<2N} (\log p)\omega_{N,t}(p) - \sum_{m<2N} \omega_{N,t}(m). }

This identity is not yet an arithmetic estimate. It is the exact interface to Campaign 44 / PK4 CENTERED_LAMBDA_PRIME_ONLY_BRIDGE.

The crucial structural point is that the weight is centered at t=0t=0 and remains root-observing on both halves of the dyadic interval.

11. Extreme Mellin frequencies are deterministically harmless

The centered factors give a useful automatic bound.

By the unconditional Chebyshev estimate

ϑ(x)=O(x),\vartheta(x)=O(x),

we have

∣B(N)∣N=O(1).\frac{|B(N)|}{N}=O(1).

Moreover

∑N<n<2N∣cn∣n=O(1).\sum_{N<n<2N} \frac{|c_n|}{n} =O(1).

Indeed the composite contribution is O(1)O(1), while partial summation with ϑ(x)=O(x)\vartheta(x)=O(x) gives

∑N<p<2Nlog⁡pp=O(1).\sum_{N<p<2N} \frac{\log p}{p} =O(1).

Therefore

∑i=1N∣ai∣=O(1).\boxed{ \sum_{i=1}^N|a_i|=O(1). }

Since

0<xi≤log⁡2,0<x_i\le\log2,

we obtain

∣eitxi−1∣≤min⁡(∣t∣log⁡2,2).|e^{itx_i}-1| \le \min(|t|\log2,2).

Thus

Lemma 11.1 — Centered Mellin envelope

∣MN(t)∣≪min⁡(∣t∣,1).\boxed{ |\mathfrak M_N(t)| \ll \min(|t|,1). }

Let 0<κ≤10<\kappa\le1 be fixed. Then

∫∣t∣≤N−κ∣MN(t)∣2t2 dt≪N−κ,\int_{|t|\le N^{-\kappa}} \frac{|\mathfrak M_N(t)|^2}{t^2} \,dt \ll N^{-\kappa},

and

∫∣t∣≥Nκ∣MN(t)∣2t2 dt≪N−κ.\int_{|t|\ge N^\kappa} \frac{|\mathfrak M_N(t)|^2}{t^2} \,dt \ll N^{-\kappa}.

Consequently:

Theorem 11.2 — Fixed-power PESC reduces to the centered Mellin middle band

For fixed 0<κ≤10<\kappa\le1,

JNϑ≪N3−κ+o(1)\boxed{ J_N^\vartheta \ll N^{3-\kappa+o(1)} }

is exponent-equivalent to

∫N−κ≤∣t∣≤Nκ∣MN(t)∣2t2 dt≪N−κ+o(1).\boxed{ \int_{N^{-\kappa}\le|t|\le N^\kappa} \frac{|\mathfrak M_N(t)|^2}{t^2} \,dt \ll N^{-\kappa+o(1)}. }

Create:

B-RH-040
PESC_FIXED_POWER_EQUIVALENT_TO_CENTERED_MELLIN_MIDDLE_BAND
CERTIFIED

This does not excise the Brownian low mode. The full Brownian energy has already been transferred to ENCM\mathcal E_N^{\mathrm{CM}} with constant distortion. The tiny- tt estimate is an actual deterministic bound on a part of that exact representation.

12. Exact Gaussian translated-window identity

The remaining middle band can be partitioned into O(log⁡N)O(\log N) dyadic frequency scales. Away from zero, the factor t−2t^{-2} is comparable to the inverse square of the local frequency center, so Gaussian translated windows can be inserted with only logarithmic overlap.

To make the connection with Paper 42 exact, let

Φ(u)=exp⁡(1−u22),\Phi(u) = \exp\left(\frac{1-u^2}{2}\right),

so that

Φ^(ξ)=e1/22πe−ξ2/2\widehat\Phi(\xi) = e^{1/2}\sqrt{2\pi}e^{-\xi^2/2}

under the convention

Φ^(ξ)=∫RΦ(u)e−iuξ du.\widehat\Phi(\xi) = \int_{\mathbb R}\Phi(u)e^{-iu\xi}\,du.

Define the centered root window moment

INCM(Q,Y)=∫RΦ(t−QY)∣MN(t)∣2 dt.\boxed{ \mathcal I_N^{\mathrm{CM}}(Q,Y) = \int_{\mathbb R} \Phi\left(\frac{t-Q}{Y}\right) |\mathfrak M_N(t)|^2 \,dt. }

Using

MN(t)=∑iai(eitxi−1),\mathfrak M_N(t) = \sum_i a_i(e^{itx_i}-1),

we obtain:

Theorem 12.1 — Root-observing centered Gaussian window kernel

INCM(Q,Y)=Y∑i,jaiaj‾KQ,Y(xi,xj),\boxed{ \mathcal I_N^{\mathrm{CM}}(Q,Y) = Y\sum_{i,j} a_i\overline{a_j} \mathcal K_{Q,Y}(x_i,x_j), }

where

KQ,Y(x,y)=eiQ(x−y)Φ^(Y(x−y))−eiQxΦ^(Yx)−e−iQyΦ^(Yy)+Φ^(0).\boxed{ \begin{aligned} \mathcal K_{Q,Y}(x,y) ={}& e^{iQ(x-y)} \widehat\Phi(Y(x-y)) \\ &- e^{iQx} \widehat\Phi(Yx) \\ &- e^{-iQy} \widehat\Phi(Yy) \\ &+ \widehat\Phi(0). \end{aligned} }

Because

xi−xj=log⁡njni,x_i-x_j = \log\frac{n_j}{n_i},

the first term is

e−iQlog⁡(ni/nj)Φ^(Ylog⁡(ni/nj)),\boxed{ e^{-iQ\log(n_i/n_j)} \widehat\Phi\left( Y\log(n_i/n_j) \right), }

which is exactly the ratio/frequency geometry appearing in Paper 42.

The remaining three terms are the endpoint-centering corrections. They depend on xix_i and xjx_j separately rather than only on the ratio ni/njn_i/n_j.

Therefore Paper 42's ratio kernel is recovered as a genuine component of the root window, while Paper 48's endpoint-position dependence is retained exactly.

Create:

B-RH-041
PESC_CENTERED_GAUSSIAN_TRANSLATED_WINDOW_KERNEL_WITH_ENDPOINT_CORRECTION
CERTIFIED

This is the precise deterministic reconciliation of Paper 42 and Paper 48 required by PK3.

13. Dyadic translated-window synthesis of the middle band

Let

T−=N−κT_-=N^{-\kappa}

and

T+=Nκ.T_+=N^\kappa.

Choose a smooth dyadic partition of unity on

T−≤∣t∣≤T+.T_-\le|t|\le T_+.

There are only

O(log⁡N)=No(1)O(\log N) = N^{o(1)}

frequency scales.

On a window centered at

∣Q∣≍Y|Q|\asymp Y

or, more generally, with

Y≤c∣Q∣Y\le c|Q|

for fixed c<1c<1, one has

t−2≍Q−2t^{-2}\asymp Q^{-2}

throughout the window. Hence the middle-band energy is controlled, up to No(1)N^{o(1)} overlap, by translated moments of the form

Q−2INCM(Q,Y).Q^{-2} \mathcal I_N^{\mathrm{CM}}(Q,Y).

The arithmetic burden is therefore no longer an unspecified frame problem. It is the explicit family of centered root moments

INCM(Q,Y)\boxed{ \mathcal I_N^{\mathrm{CM}}(Q,Y) }

with kernel KQ,Y\mathcal K_{Q,Y} from Theorem 12.1.

Any fixed-power estimate must preserve the endpoint terms or prove that their contribution is separately admissible. Dropping them and keeping only the ratio term returns to the non-root-observing interface rejected by Paper 48.

14. PK3 audit

We now audit the exact requirements inherited from the Campaign 44 taskpack.

Required check 1 — state the synthesis/frame operator explicitly

Passed.

The exact position synthesis is

HN=CNICN∗,H_N=C_NIC_N^*,

with exact block-Fourier resolution

HN=∑b,rψb,rψb,r∗.H_N = \sum_{b,r}\psi_{b,r}\psi_{b,r}^*.

The Mellin-compatible synthesis is the exact centered Fourier representation of GNlog⁡G_N^{\log} combined with the constant-distortion comparison to HNH_N.

Required check 2 — retain endpoint-position and ratio/frequency localization

Passed.

Endpoint position is retained through

xi=log⁡(2N/ni),x_i=\log(2N/n_i),

and the translated kernel contains both

log⁡(ni/nj)\log(n_i/n_j)

and the separate endpoint terms xi,xjx_i,x_j.

Required check 3 — preserve the Brownian low-frequency root mode

Passed.

No Brownian eigenmode is projected out. The full operator satisfies

NGNlog⁡⪯HN⪯2NGNlog⁡.N G_N^{\log} \preceq H_N \preceq 2N G_N^{\log}.

Thus the complete root energy, including the principal Brownian mode, survives with constant distortion.

Required check 4 — compute or bound frame/reconstruction conditioning

Passed.

Theorem 3.1 records the general N2N^2 spectral-spread conservation law. The logarithmic time change costs at most a factor 22, and the dyadic Mellin normalization costs only absolute constants in the Brownian norm.

Required check 5 — charge every NδN^\delta reconstruction loss

Passed.

The root-to-centered-Mellin bridge has no NδN^\delta loss. The positive submacroscopic-window route is separately recorded as paying

(N/L)2(N/L)^2

unless a coarse channel is retained.

Required check 6 — distinguish exact reconstruction from one-sided domination

Passed.

The following are exact:

HN=CNCN∗,H_N=C_NC_N^*, HN=HN,Lloc+HN,Lcoarse,H_N=H_{N,L}^{\mathrm{loc}}+H_{N,L}^{\mathrm{coarse}},

and

a∗GNlog⁡a=12π∫R∣MN(t)∣2t2 dt.a^*G_N^{\log}a = \frac1{2\pi} \int_{\mathbb R} \frac{|\mathfrak M_N(t)|^2}{t^2}\,dt.

The comparison between HNH_N and NGNlog⁡NG_N^{\log} is explicitly one-sided in both directions with constants 11 and 22.

Required check 7 — do not declare the Paper 42 ratio kernel equal to PESC

Passed.

Theorem 12.1 shows the ratio kernel plus the three exact endpoint-centering terms.

Required check 8 — do not remove low frequencies without priced recovery

Passed.

No low-frequency recovery operator is invoked. The exact centered transform is used first, and only the tiny interval

∣t∣≤N−κ|t|\le N^{-\kappa}

is then bounded directly by the certified envelope

∣MN(t)∣≪∣t∣.|\mathfrak M_N(t)|\ll|t|.

15. PK3 closure

The deterministic question posed by PK3 was whether the exact position-dependent PESC Brownian root kernel could be synthesized in translated frequency variables without erasing the principal low mode or paying an unrecorded power-sized reconstruction cost.

The answer is yes.

The route is

HN⟷NGNlog⁡⟷N2π∫R(eitx−1)(e−ity−1)t2 dt,\boxed{ H_N \longleftrightarrow N G_N^{\log} \longleftrightarrow \frac{N}{2\pi} \int_{\mathbb R} \frac{ (e^{itx}-1)(e^{-ity}-1) }{t^2} \,dt, }

combined with the Brownian-norm-stable dyadic normalization

zi=niai.z_i=n_i a_i.

The full PESC root energy is therefore constant-distortion equivalent to a centered Mellin energy.

Close PK3 as

CLOSED_AS_POWER_SAFE_LOG_BROWNIAN_TO_CENTERED_MELLIN_ROOT_SYNTHESIS

This closure is structural only. It does not establish the fixed-power estimate required by PESC.

16. Campaign 44 continuation

The next inherited track is

PK4
CENTERED_LAMBDA_PRIME_ONLY_BRIDGE

The exact PK4 input is now

MN(t)=∑m<2N(1P(m)log⁡m−1)ωN,t(m).\boxed{ \mathfrak M_N(t) = \sum_{m<2N} \left(1_{\mathbb P}(m)\log m-1\right) \omega_{N,t}(m). }

The arithmetic problem is to exploit the prime structure of this centered root statistic, or the exact translated-window kernel KQ,Y\mathcal K_{Q,Y}, strongly enough to obtain

∫N−κ≤∣t∣≤Nκ∣MN(t)∣2t2 dt≪N−κ+o(1)\int_{N^{-\kappa}\le|t|\le N^\kappa} \frac{|\mathfrak M_N(t)|^2}{t^2} \,dt \ll N^{-\kappa+o(1)}

for some fixed κ>0\kappa>0.

At this point the deterministic frame problem is no longer the blocker. The blocker is a root-observing arithmetic contraction for the centered prime-error transform.

The following routes remain prohibited unless genuinely new evidence appears:

  1. reopening Paper 46 coefficient-smallness or mark-multiplicity mechanisms;
  2. returning to the isolated pure-Mobius component after Paper 47's exact root recoupling;
  3. discarding the endpoint-centering terms in Theorem 12.1;
  4. replacing a logarithmic saving by a fixed power;
  5. assuming a fixed zero-free strip or fixed-power PNT remainder;
  6. removing a low-frequency/root block and claiming later recovery without charging the recovery map.

17. New certified objects

Create:

B-RH-036
PESC_FRAME_WEIGHT_SPECTRAL_SPREAD_CONSERVATION
CERTIFIED

Create:

B-RH-037
EXACT_BLOCK_FOURIER_LOCAL_COARSE_BROWNIAN_SYNTHESIS
CERTIFIED

Create:

B-RH-038
PESC_LOGARITHMIC_BROWNIAN_TIME_CHANGE_WITH_CONSTANT_DISTORTION
CERTIFIED

Create:

B-RH-039
PESC_CENTERED_MELLIN_ROOT_ENERGY_WITH_CONSTANT_DISTORTION
CERTIFIED

Create:

B-RH-040
PESC_FIXED_POWER_EQUIVALENT_TO_CENTERED_MELLIN_MIDDLE_BAND
CERTIFIED

Create:

B-RH-041
PESC_CENTERED_GAUSSIAN_TRANSLATED_WINDOW_KERNEL_WITH_ENDPOINT_CORRECTION
CERTIFIED

Create:

O-RH-128
PESC_SUBMACROSCOPIC_POSITIVE_SYNTHESIS_PAYS_POWER_RECONSTRUCTION_TAX
CERTIFIED AS PK3 CONDITIONING BARRIER

No new RH certificate is created.

18. External calibration

The mathematical identities used for the deterministic synthesis are self-contained in this paper. The following literature is calibration rather than a substituted proof.

  1. The Brownian covariance kernel min⁡(s,t)\min(s,t) and its basis expansions are classical. The Levy--Ciesielski construction represents Brownian motion in the integrated Haar/Schauder basis; see T. Kleyntssens and S. Nicolay, From the Brownian motion to a multifractal process using the Levy--Ciesielski construction, Statistics & Probability Letters 186 (2022), 109450.

  2. The connection between frequency mean squares of exponential/Dirichlet sums and short-sum or Selberg-type energies is classical Gallagher territory. For a modern weighted formulation, see G. Coppola and M. Laporta, A generalization of Gallagher's lemma for exponential sums, arXiv:1411.1739.

  3. Paper 42 already certified the Gaussian translated ratio kernel internally. Theorem 12.1 does not reuse that result as a root theorem; it derives the centered root window independently and then identifies the Paper 42 ratio geometry as its first term.

19. State transition

Candidate project state:

CSM_RH v1.40

Campaign 44:

ACTIVE

Track states:

PK1 CLOSED_AS_EXACT_LAG_POSITION_NORMALIZATION_WITH_PREFIX_GRAM_COMPLETION
PK2 CLOSED_AS_EXACT_BROWNIAN_SPECTRAL_REPRESENTATION_WITH_LOW_MODE_PRESERVED
PK3 CLOSED_AS_POWER_SAFE_LOG_BROWNIAN_TO_CENTERED_MELLIN_ROOT_SYNTHESIS
PK4 OPEN_CENTERED_LAMBDA_PRIME_ONLY_BRIDGE
PK5 OPEN_FIXED_POWER_PESC_ADMISSION

Root frontiers remain

F-RH-010 PESC  OPEN
F-RH-016 MLEPG OPEN

and

RH_PROVED = FALSE
RH_DISPROVED = FALSE
GLOBAL_RH_CERTIFICATE = FALSE

20. Final status

Paper 49 resolves the deterministic synthesis ambiguity left by Paper 48.

The exact Brownian root kernel has a genuine N2N^2 spectral spread, and positive submacroscopic synthesis cannot make that stiffness disappear. Nevertheless an exact local/coarse construction exists, and a separate logarithmic time change provides a power-safe route into Mellin frequency.

The central identity is the constant-distortion root bridge

JNϑ≍N3∫R∣MN(t)∣2t2 dt,\boxed{ J_N^\vartheta \asymp N^3 \int_{\mathbb R} \frac{|\mathfrak M_N(t)|^2}{t^2}\,dt, }

with

MN(t)=B(N)N(2it−1)+∑N<n<2Ncnn[(2Nn)it−1].\boxed{ \mathfrak M_N(t) = \frac{B(N)}{N}(2^{it}-1) + \sum_{N<n<2N} \frac{c_n}{n} \left[ \left(\frac{2N}{n}\right)^{it}-1 \right]. }

The centering is essential. It regularizes t=0t=0, preserves the endpoint aggregate, and produces the exact correction terms missing from a ratio-only translated kernel.

The very low and very high Mellin frequencies are deterministically admissible at the fixed-power scale, leaving the polynomial middle band as the next root arithmetic battlefield.

The resulting continuation is therefore no longer

find a frame for PESC

but

prove a fixed-power arithmetic contraction for the centered prime-error Mellin middle band

inside Campaign 44 / PK4.