CSM_RH Paper 32

Residual-Weighted EMBF Equivalence, Abel Endogeneity Return, and the PACPSA Difference-Stability Bottleneck

Project: CSM_RH
Paper: 32
Version: v0.1
Date: 2026-09-07
Parent state: CSM_RH v1.22 / Paper 31
Campaign: 31 — ENDOGENOUS_PARITY_BILINEAR_REAUDIT
Status: residual parity-bilinear re-audit / coupled-sieve stability localization; not a proof or disproof of RH

0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

RH_PROVED = FALSE
RH_DISPROVED = FALSE
GLOBAL_RH_CERTIFICATE = FALSE
CSM_RH_ROOT_STATUS = OPEN

Paper 31 identified the Friedlander–Iwaniec parity-breaking bilinear axiom with the EMBF geometry obtained from the PESC positivity lifts.

Campaign 31 reaudits that branch using later CSM_RH results:

Lambda-sharp has subpolynomial centered primitive;
the prime residual has strong almost-all local uniformity;
PESC remains fixed-exponent equivalent after sieve subtraction.

The campaign obtains one positive bridge and three closure results.

EMBF is fixed-exponent equivalent, throughout the first-strip range, to the same Möbius bilinear form weighted by the residual primitive
$F=\sum(\Lambda-\Lambda^\sharp).$
Abel summation does not externalize the endogenous weight for free: differentiating $F$ returns block sums of the prime residual $f=\Lambda-\Lambda^\sharp$ .
Current almost-all short-interval theorems do not control the multiplicative grids sampled by EMBF.
Removing the outer absolute value destroys the dual sign-uniformity built into the Friedlander–Iwaniec parity axiom.

The remaining parity-side mechanism is therefore PACPSA difference stability.

No live GLM-5.3-Flash run is claimed.

1. Canonical EMBF

Let

B_\vartheta(x)=\vartheta(x)-x.

For a dyadic $n$ -range

L<n\le2L

and parameter $C$ , define

\gamma(n,C) = \sum_{\substack{d\mid n\\d\le C}}\mu(d).

Paper 13 defines the endogenous Möbius bilinear fidelity seminorm

\boxed{ \mathfrak B_{N,L,C}[B_\vartheta] = \sum_m \left| \sum_{\substack{ L<n\le2L\\ mn<2N }} \gamma(n,C)\mu(mn)w_N(mn) B_\vartheta(mn-1) \right|. }

The endpoint multiplicity satisfies

0\le w_N(k)\le N.

2. Residual primitive

Use the modern sieve approximant

\Lambda^\sharp(n) = \frac{P(R)}{\varphi(P(R))} 1_{\gcd(n,P(R))=1},

with

R = \exp \left( (\log N)^{1/10} \right).

Define

f(n) = \Lambda(n)-\Lambda^\sharp(n),

and

\boxed{ F(x) = \sum_{n\le x}f(n). }

Define also

B^\sharp(x) = \sum_{n\le x} [ \Lambda^\sharp(n)-1 ].

Paper 28 gives

\boxed{ B^\sharp(x) = N^{o(1)} }

uniformly for $x\le2N$ .

Since

F(x)+B^\sharp(x) = \psi(x)-x,

while

B_\vartheta(x) = \vartheta(x)-x,

we have

\boxed{ B_\vartheta(x) = F(x) + B^\sharp(x) - [ \psi(x)-\vartheta(x) ]. }

3. Prime-power correction

The prime-power difference satisfies

\psi(x)-\vartheta(x) = \sum_{\substack{p^j\le x\\j\ge2}} \log p = x^{1/2+o(1)}.

Therefore:

Theorem 3.1 — Residual Primitive Approximation

Uniformly for $x\le2N$ ,

\boxed{ B_\vartheta(x) = F(x) + O \left( N^{1/2+o(1)} \right). }

The $N^{1/2}$ term is the prime-power barrier already present in Paper 10.

4. Pair-count lemma

The EMBF range contains only $O(N)$ pairs.

Indeed,

\begin{aligned} \#\{ (m,n): L<n\le2L,\, mn<2N \} &\le \sum_{L<n\le2L} \frac{2N}{n} \\ &= O(N). \end{aligned}

Also

|\gamma(n,C)| \le \tau(n) = N^{o(1)}

uniformly, with no restriction that $C$ be fixed.

5. Residual-weight EMBF equivalence

Define

\boxed{ \mathfrak B_{N,L,C}[F] = \sum_m \left| \sum_{\substack{ L<n\le2L\\ mn<2N }} \gamma(n,C)\mu(mn)w_N(mn) F(mn-1) \right|. }

The seminorm is Lipschitz in its weight.

Using Theorem 3.1, Section 4, and $w_N\le N$ :

Theorem 5.1 — Residual-Weighted EMBF Equivalence

\boxed{ \left| \mathfrak B_{N,L,C}[B_\vartheta] - \mathfrak B_{N,L,C}[F] \right| \ll N^{5/2+o(1)}. }

Therefore, for every fixed

0<\kappa<\frac12,

a target of size

N^{3-\kappa+o(1)}

is unchanged by the replacement

B_\vartheta \rightsquigarrow F.

Create:

B-RH-007
SIEVE_RESIDUAL_EMBF_EQUIVALENCE
status:
  CERTIFIED

This joins the parity branch to the residual branch of Papers 28–30.

6. What the replacement means

After Theorem 5.1, the parity-side critical object is

\boxed{ \sum_m \left| \sum_n \gamma(n,C)\mu(mn)w_N(mn) F(mn-1) \right|. }

Thus the modern sieve model removes the local small-prime structure from the endogenous weight.

The remaining weight is exactly the primitive of the true prime residual

f=\Lambda-\Lambda^\sharp.

The PESC-hard low-frequency drift survives the model subtraction.

7. Attempted Möbius externalization

For fixed $m$ , write the inner residual sum as

\boxed{ S_m = \sum_{a<n\le b} \alpha_m(n) F(mn-1), }

where

\alpha_m(n) = \gamma(n,C)\mu(mn)w_N(mn)

and the interval is the relevant truncation of $(L,2L]$ .

Define the partial sum

\boxed{ A_m(t) = \sum_{a<n\le t}\alpha_m(n). }

Discrete Abel summation gives

\boxed{ S_m = A_m(b)F(mb-1) - \sum_{t=a}^{b-1} A_m(t) [ F(m(t+1)-1)-F(mt-1) ] }

up to the harmless lower-endpoint convention.

8. Differentiating the endogenous weight returns the prime residual

Because $F$ is the primitive of $f$ ,

\boxed{ F(m(t+1)-1)-F(mt-1) = \sum_{mt\le k<m(t+1)} f(k). }

Thus any attempt to exploit cancellation in the Möbius coefficient partial sums $A_m(t)$ by Abel summation creates short block sums of

\boxed{ f=\Lambda-\Lambda^\sharp. }

The endogenous weight cannot be treated as an arbitrary smooth external coefficient.

Create:

O-RH-069
ABEL_EXTERNALIZATION_RETURNS_PRIME_RESIDUAL
status:
  CERTIFIED

This is the exact endogenous-fidelity mechanism behind the earlier abstract warning.

9. Modern Möbius short-interval input

Current higher-uniformity theory gives, for

H\ge X^{1/3+\varepsilon},

Möbius discorrelation against bounded-complexity nilsequences of size

\boxed{ H\log^{-A}X }

for all but an exceptional set of starting points of measure

\boxed{ O_A \left( X\log^{-A}X \right). }

The same theorem gives the corresponding logarithmic residual estimate for

\Lambda-\Lambda^\sharp.

These are powerful local statements.

They are not automatically compatible with the multiplicative sampling geometry in Section 8.

10. Multiplicative-grid exceptional-set blindness

Consider the balanced EMBF geometry

L=N^{1/2+o(1)}, \qquad m=N^{1/2+o(1)}.

The residual blocks produced by Section 8 have:

length:
  m ~ N^{1/2}

starting points:
  mt

number of sampled starts for fixed m:
  about L ~ N^{1/2}.

But the current almost-all theorem permits an exceptional set of size

N\log^{-A}N.

For every fixed $A$ ,

\boxed{ N\log^{-A}N \gg N^{1/2}. }

Therefore the theorem permits, without contradiction, every point in one fixed multiplicative grid

\{ mt: L<t\le2L \}

to be exceptional.

Hence:

Theorem 10.1 — Exceptional-Set Grid Blindness

The current almost-all short-interval theorem does not, by cardinality alone, provide any nontrivial uniform control on a fixed balanced EMBF multiplicative grid.

Create:

O-RH-070
ALMOST_ALL_EXCEPTIONAL_SET_MULTIPLICATIVE_GRID_BLINDNESS
status:
  CERTIFIED AS TRANSFERENCE OBSTRUCTION

This does not say the residual is actually large on such a grid.

It says the current theorem does not rule it out.

11. Why arbitrary choice of the log exponent does not fix the grid issue

The short-interval theorem permits every fixed $A>0$ .

But the implied constants depend on $A$ .

One may not take

A=A(N)\to\infty

without a theorem uniform in that parameter.

For each fixed $A$ ,

\frac{ N\log^{-A}N }{ N^{1/2} } = N^{1/2}\log^{-A}N \to\infty.

Thus arbitrary fixed log-power precision remains insufficient for deterministic control of the multiplicative grid.

12. Outer absolute value as dual sign uniformity

For each outer variable $m$ , write the EMBF inner sum as

I_m.

Then the exact duality identity is

Theorem 12.1 — Outer-Absolute Duality

For complex $I_m$ ,

\boxed{ \sum_m|I_m| = \sup_{\substack{|\varepsilon_m|\le1}} \left| \sum_m \varepsilon_m I_m \right|. }

The supremum is attained by choosing $\varepsilon_m$ to align the phases of the nonzero $I_m$ .

Therefore the outer absolute value is equivalent to uniformity against an arbitrary outer sign/phase sequence.

Create:

B-RH-008
EMBF_OUTER_ABSOLUTE_DUAL_UNIFORMITY
status:
  CERTIFIED

13. Why simply removing the outer absolute is not a certified shortcut

Removing the outer absolute controls only the single phase choice

\varepsilon_m\equiv1.

The Friedlander–Iwaniec axiom requires the entire dual family.

A toy example already shows the gap:

I_m=(-1)^m.

Then the unsigned aggregate is $O(1)$ while

\sum_m|I_m|

is the full number of outer variables.

Thus cancellation across $m$ can hide large parity-sensitive inner sums.

A signed aggregate may still be useful if a new direct PESC bridge is proved.

No such bridge is obtained in Campaign 31.

14. PACPSA difference-stability problem

The two PESC positivity lifts are

a_n^\pm = w_N(n) [ M_N\pm B_\vartheta(n-1) ].

The Friedlander–Iwaniec theorem is formulated for real nonnegative sequences.

Its prime-detection asymptotic has an error proportional to the total positive mass of the input sequence.

Applying such a theorem separately to $a^+$ and $a^-$ gives no certified cancellation between the two theorem errors.

Even if the two lifts share a huge common background

M_Nw_N(n),

separate estimates only give

|\mathrm{Err}(a^+)-\mathrm{Err}(a^-)| \le |\mathrm{Err}(a^+)| + |\mathrm{Err}(a^-)|.

The common positive mass is therefore paid twice unless a new coupled stability theorem is proved.

15. Size of the common positivity background

The endpoint weights satisfy

\sum_{n<2N}w_N(n) \asymp N^2.

Positivity requires

M_N \ge \max_{j<2N}|B_\vartheta(j)|.

Current unconditional PNT bounds make $M_N$ smaller than $N$ by a subpower factor, but not by a fixed power.

Thus the common positivity mass remains

\boxed{ M_NN^2 = N^{3-o(1)} }

at current unconditional strength.

A relative asymptotic-sieve error of logarithmic size therefore remains

\boxed{ N^{3-o(1)}. }

It does not yield PESC $(\kappa)$ .

16. Why standard asymptotic-sieve output does not give difference stability

The classical Friedlander–Iwaniec prime asymptotic has relative error

O \left( \frac{\log\delta(x)}{\log\Delta(x)} \right).

For the practical parameter class this is

O \left( \frac{\log\log x}{\log x} \right).

The theorem supplies an estimate for each nonnegative input sequence.

It does not state a Lipschitz bound for the difference of theorem errors in terms of the signed perturbation

a^+-a^- = 2w_NB_\vartheta.

Such a theorem would be exactly the missing coupled difference-stability input.

Create:

O-RH-071
SEPARATE_POSITIVE_LIFT_SIEVE_ERROR_FLOOR
status:
  CERTIFIED AS CURRENT-THEOREM AUDIT

17. PACPSA restated after Campaign 31

Paper 15 introduced

F-RH-014
PACPSA
POWER_ACCURATE_COUPLED_PARITY_SIEVE_ASSEMBLY

Campaign 31 sharpens its meaning:

PACPSA is not merely "apply an asymptotic sieve to two lifts." It requires a theorem whose error functional is stable at the difference level, so that the common $N^{3-o(1)}$ positivity background cancels before the error estimate is paid.

This is now the only parity-side mechanism not already closed by the re-audit.

18. Campaign 31 track audit

V1 — Lambda-sharp subtraction inside EMBF

status:
  SUCCESS

result:
  B-RH-007

first-strip replacement error:
  N^(5/2+o(1))

V2 — modern Möbius / Liouville short-interval uniformity

status:
  DOES NOT TRANSFER DIRECTLY

reasons:
  endogenous Abel differentiation;
  multiplicative-grid exceptional-set blindness;
  log/subpower precision

V3 — low-frequency projection of the endogenous weight

status:
  NO CANONICAL FIXED-POWER PROJECTION FOUND

smooth external projection:
  Möbius control remains log/subpower

V4 — remove the outer absolute value

status:
  OUTER ABSOLUTE = DUAL SIGN UNIFORMITY

removal:
  strictly weaker

direct PESC bridge:
  not found

V5 — PACPSA difference stability

status:
  ONLY SURVIVING PARITY-SIDE MECHANISM

current theorem:
  no coupled difference-error estimate

19. Campaign 31 verdict

The parity branch has now been reaudited using all later residual and short-interval information.

The updated closure diagram is:

PESC positivity lifts
    |
    v
FI parity bilinear
    |
    v
EMBF
    |
Lambda-sharp subtraction
    |
    v
residual-weight EMBF
    |
    +--> Abel externalization -> residual f-blocks
    |                         -> multiplicative-grid debt
    |
    +--> remove outer abs -> loses dual sign uniformity
    |
    v
PACPSA difference stability

No lower-strength fixed-power theorem is obtained.

20. New certified package

Create:

B-RH-007
SIEVE_RESIDUAL_EMBF_EQUIVALENCE
CERTIFIED

B-RH-008
EMBF_OUTER_ABSOLUTE_DUAL_UNIFORMITY
CERTIFIED

O-RH-069
ABEL_EXTERNALIZATION_RETURNS_PRIME_RESIDUAL
CERTIFIED

O-RH-070
ALMOST_ALL_EXCEPTIONAL_SET_MULTIPLICATIVE_GRID_BLINDNESS
CERTIFIED

O-RH-071
SEPARATE_POSITIVE_LIFT_SIEVE_ERROR_FLOOR
CERTIFIED AS CURRENT-THEOREM AUDIT

No new frontier is created.

21. Canonical status

F-RH-010
PESC
OPEN / ROOT TARGET

F-RH-016
MLEPG
OPEN

F-RH-012
EMBF
AUXILIARY / RESIDUAL-EQUIVALENT IN FIRST-STRIP RANGE

F-RH-014
PACPSA
OPEN / ONLY SURVIVING PARITY-ASSEMBLY MECHANISM

22. Campaign 32

The next campaign is:

CSM_RH Campaign 32
PACPSA_DIFFERENCE_STABILITY_ATTACK

The target is not to improve the ordinary asymptotic sieve.

It is to determine whether the proof can be linearized around the common positive background of the two PESC lifts.

23. Campaign 32 tracks

D1 — common-background linearization

Write

a^\pm=a_0\pm h,

with

a_0=M_Nw_N, \qquad h=w_NB_\vartheta.

Track every asymptotic-sieve error term under the perturbation parameter $t$ in

a_t=a_0+th.

Seek a derivative bound depending on $h$ , not on $a_0$ .

D2 — signed remainder axioms

Linearize the divisor-distribution remainders

r_d(a_t)

and test whether the derivative sequence satisfies a signed analogue of the sieve remainder axioms at a power-accurate norm.

D3 — bilinear derivative stability

Differentiate the Friedlander–Iwaniec bilinear form before taking the outer absolute value.

Determine whether phase alignment can be controlled uniformly in $t$ .

D4 — Jordan / polarization formulation

Seek a theorem for signed perturbations by polarization of the nonnegative asymptotic sieve.

Reject the route if total variation of the perturbation remains $N^{3-o(1)}$ .

D5 — difference-level prime-detection functional

Work directly with

\mathfrak E(a^+)-\mathfrak E(a^-)

and derive a sieve identity in which the common positivity background cancels algebraically before all error estimates.

This is the preferred route.

24. Campaign 32 rejection filters

Reject a candidate if:

R1. It applies the ordinary asymptotic sieve separately to $a^+$ and $a^-$ .

R2. Its signed norm is total variation of $a^+-a^-$ at $N^{3-o(1)}$ scale.

R3. It obtains only logarithmic relative error.

R4. It assumes EMBF/PESC fixed power as an input.

R5. It drops the outer absolute without replacing the lost dual sign uniformity.

R6. It claims cancellation between two big- $O$ errors without a joint theorem.

25. External calibration

Current relevant theorems:

The 2026 higher-uniformity theorem gives both $\mu$ and $\Lambda-\Lambda^\sharp$ logarithmic discorrelation on almost all intervals of length at least $X^{1/3+\varepsilon}$ , with logarithmically small exceptional sets.
The Friedlander–Iwaniec asymptotic sieve is explicitly formulated for nonnegative sequences, and its prime-detection output has relative error $\log\delta/\log\Delta$ , which is $\log\log x/\log x$ in the practical parameter regime.

These facts explain why modern local uniformity and classical parity breaking still do not supply a power-accurate coupled difference theorem.

26. State transition

CSM_RH v1.22
  ->
CSM_RH v1.23

with:

Campaign 31
  CLOSED_AS_ENDOGENOUS_PARITY_BILINEAR_REAUDIT

B-RH-007
  SIEVE_RESIDUAL_EMBF_EQUIVALENCE
  CREATED / CERTIFIED

B-RH-008
  EMBF_OUTER_ABSOLUTE_DUAL_UNIFORMITY
  CREATED / CERTIFIED

O-RH-069
  ABEL_EXTERNALIZATION_RETURNS_PRIME_RESIDUAL
  CREATED / CERTIFIED

O-RH-070
  ALMOST_ALL_EXCEPTIONAL_SET_MULTIPLICATIVE_GRID_BLINDNESS
  CREATED / CERTIFIED

O-RH-071
  SEPARATE_POSITIVE_LIFT_SIEVE_ERROR_FLOOR
  CREATED / CERTIFIED AS CURRENT-THEOREM AUDIT

F-RH-014
  PACPSA
  REMAINS OPEN / ONLY SURVIVING PARITY-ASSEMBLY MECHANISM

Campaign 32
  PACPSA_DIFFERENCE_STABILITY_ATTACK
  READY

27. Final status

RH = OPEN

PESC = OPEN

MLEPG = OPEN

EMBF = FIRST-STRIP EQUIVALENT AFTER RESIDUAL WEIGHT SUBSTITUTION

MOBIUS SHORT-INTERVAL UNIFORMITY = STRONG BUT NOT ENDOGENOUS-GRID STABLE

OUTER ABSOLUTE = ESSENTIAL DUAL SIGN UNIFORMITY IN FI AXIOM

SEPARATE POSITIVE-LIFT ASYMPTOTIC SIEVE = N^(3-o(1)) ERROR FLOOR

PACPSA DIFFERENCE STABILITY = ONLY SURVIVING PARITY MECHANISM

NEXT CAMPAIGN = 32

The principal new bridge is

\boxed{ \mathfrak B_{N,L,C}[B_\vartheta] = \mathfrak B_{N,L,C}[F] + O \left( N^{5/2+o(1)} \right) }

at seminorm-distance level.

The principal remaining question is no longer whether parity can be broken.

It is whether the asymptotic-sieve error can be made stable under the signed endogenous perturbation before the common positive background is charged.

CSM_RH Paper 32 — Residual-Weighted EMBF Equivalence, Abel Endogeneity Return, and the PACPSA Difference-Stability Bottleneck

CSM_RH Paper 32

Residual-Weighted EMBF Equivalence, Abel Endogeneity Return, and the PACPSA Difference-Stability Bottleneck

0. Trust boundary

1. Canonical EMBF

2. Residual primitive

3. Prime-power correction

Theorem 3.1 — Residual Primitive Approximation

4. Pair-count lemma

5. Residual-weight EMBF equivalence

Theorem 5.1 — Residual-Weighted EMBF Equivalence

6. What the replacement means

7. Attempted Möbius externalization

8. Differentiating the endogenous weight returns the prime residual

9. Modern Möbius short-interval input

10. Multiplicative-grid exceptional-set blindness

Theorem 10.1 — Exceptional-Set Grid Blindness

11. Why arbitrary choice of the log exponent does not fix the grid issue

12. Outer absolute value as dual sign uniformity

Theorem 12.1 — Outer-Absolute Duality

13. Why simply removing the outer absolute is not a certified shortcut

14. PACPSA difference-stability problem

15. Size of the common positivity background

16. Why standard asymptotic-sieve output does not give difference stability

17. PACPSA restated after Campaign 31

18. Campaign 31 track audit

V1 — Lambda-sharp subtraction inside EMBF

V2 — modern Möbius / Liouville short-interval uniformity

V3 — low-frequency projection of the endogenous weight

V4 — remove the outer absolute value

V5 — PACPSA difference stability

19. Campaign 31 verdict

20. New certified package

21. Canonical status

22. Campaign 32

23. Campaign 32 tracks

D1 — common-background linearization

D2 — signed remainder axioms

D3 — bilinear derivative stability

D4 — Jordan / polarization formulation

D5 — difference-level prime-detection functional

24. Campaign 32 rejection filters

R1. It applies the ordinary asymptotic sieve separately to a+a^+a+ and a−a^-a−.

R2. Its signed norm is total variation of a+−a−a^+-a^-a+−a− at N3−o(1)N^{3-o(1)}N3−o(1) scale.

R3. It obtains only logarithmic relative error.

R4. It assumes EMBF/PESC fixed power as an input.

R5. It drops the outer absolute without replacing the lost dual sign uniformity.

R6. It claims cancellation between two big- OOO errors without a joint theorem.

25. External calibration

26. State transition

27. Final status

R1. It applies the ordinary asymptotic sieve separately to $a^+$ and $a^-$ .

R2. Its signed norm is total variation of $a^+-a^-$ at $N^{3-o(1)}$ scale.

R6. It claims cancellation between two big- $O$ errors without a joint theorem.