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CSM_RH Paper 03 — Principal Zeta-Packet Bottleneck, Positive-Gate No-Bypass, and Campaign 02 Closure

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CSM_RH Paper 03

Principal Zeta-Packet Bottleneck, Positive-Gate No-Bypass, and Campaign 02 Closure

Project: CSM_RH
Paper: 03
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v0.3 / Paper 02
Status: structural reduction / gate-strength theorem; not a proof or disproof of RH
中文標題： CSM_RH 論文 03：主 Zeta 封包瓶頸、正閘門不可繞道與 Campaign 02 閉合
作者： Neo.K
機構： EveMissLab／一言諾科技有限公司
Language: English

0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

Canonical root state:

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

The purpose of this paper is narrower:

  1. isolate the q=1q=1 principal zeta packet inside the positive EMAE gate;
  2. prove an exact equivalence between its fixed-power bound and a fixed zeta zero strip;
  3. show that conductor weights, character-family averaging, zero-density estimates, and Deuring–Heilbronn mechanisms cannot by themselves bypass this q=1q=1 necessary condition;
  4. close CSM_RH Campaign 02 as a reduction campaign rather than a proof campaign;
  5. define the next gate-minimality search.

No live GLM-5.3-Flash run is claimed.

1. Starting state

Paper 02 proved that the smooth q=1q=1 zeta-zero packet has exact exponential type.

Let

Θζ=sup⁡ρℜρ\Theta_\zeta = \sup_{\rho} \Re\rho

over nontrivial zeta zeros, and let

Δζ=Θζ−12.\Delta_\zeta = \Theta_\zeta-\frac12.

By functional-equation symmetry,

Θζ=12+Δζ.\Theta_\zeta = \frac12+\Delta_\zeta.

Paper 02 established, for the Hilbert packet F(T)F(T),

inf⁡{a:∥F(T)∥=O(eaT)}=Θζ.\inf \left\{ a: \|F(T)\| = O(e^{aT}) \right\} = \Theta_\zeta.

For the associated positive q=1q=1 pole-zero energy,

Eζ(eT;U)=eT∥F(T)∥2,\mathcal E_\zeta(e^T;U) = e^T \|F(T)\|^2,

the exact type is

τζ=1+2Θζ=2+2Δζ.\boxed{ \tau_\zeta = 1+2\Theta_\zeta = 2+2\Delta_\zeta. }

This paper studies what that exact identity does to the full arithmetic EMAE program.

2. The q=1q=1 packet is an unavoidable positive summand

The v3.18 character major-arc energy is

Z2(x;Q,U)=∑q≤Qq squarefree∣μ(q)∣2ϕ(q)3∑χ mod q∣τ(χ)∣2∫∣ϵ∣≤U/x∣Wx,ϵ(1)∣2∣Zχ(x,ϵ)∣2dϵ.\mathfrak Z_2(x;Q,U) = \sum_{\substack{ q\le Q\\ q\ {\rm squarefree} }} \frac{ |\mu(q)|^2 }{ \phi(q)^3 } \sum_{\chi\bmod q} |\tau(\chi)|^2 \int_{|\epsilon|\le U/x} |W_{x,\epsilon}(1)|^2 |Z_\chi(x,\epsilon)|^2 d\epsilon.

At

q=1,q=1,

all arithmetic weights are equal to one.

The unique character is the trivial character, and the corresponding primitive LL -function is the Riemann zeta function.

Therefore the q=1q=1 term is exactly

Eζ(x;U)=∫∣ϵ∣≤U/x∣Wx,ϵ(1)∣2∣Zζ(x,ϵ)∣2dϵ.\boxed{ \mathcal E_\zeta(x;U) = \int_{|\epsilon|\le U/x} |W_{x,\epsilon}(1)|^2 |Z_\zeta(x,\epsilon)|^2 d\epsilon. }

Since every term in Z2\mathfrak Z_2 is nonnegative,

Z2(x;Q,U)≥Eζ(x;U).\boxed{ \mathfrak Z_2(x;Q,U) \ge \mathcal E_\zeta(x;U). }

This inequality is independent of every nonprincipal character estimate.

3. Principal Zeta-Packet Fixed Power

For fixed

0≤η≤1,0\le\eta\le1,

define:

ZPPF (η)(\eta)

For every

ε>0,\varepsilon>0, Eζ(x;U)≪ε,w,Ux3−η+ε.\boxed{ \mathcal E_\zeta(x;U) \ll_{\varepsilon,w,U} x^{3-\eta+\varepsilon}. }

Equivalently,

Eζ(x;U)≪x3−η+o(1).\mathcal E_\zeta(x;U) \ll x^{3-\eta+o(1)}.

The acronym is:

ZPPF
PRINCIPAL ZETA-PACKET FIXED POWER

4. Exact ZPPF / zero-strip equivalence

Theorem 4.1 — Principal Zeta-Packet Equivalence

For fixed

0≤η≤1,0\le\eta\le1,

the following are equivalent:

ZPPF⁡(η)\boxed{ \operatorname{ZPPF}(\eta) }

and

Θζ≤1−η2.\boxed{ \Theta_\zeta \le 1-\frac{\eta}{2}. }

Equivalently,

Δζ≤12−η2.\boxed{ \Delta_\zeta \le \frac12-\frac{\eta}{2}. }

Proof: ZPPF implies the strip

ZPPF (η)(\eta) gives

τζ≤3−η.\tau_\zeta \le 3-\eta.

Paper 02 gives

τζ=1+2Θζ.\tau_\zeta = 1+2\Theta_\zeta.

Therefore

1+2Θζ≤3−η,1+2\Theta_\zeta \le 3-\eta,

so

Θζ≤1−η2.\Theta_\zeta \le 1-\frac{\eta}{2}.

Proof: the strip implies ZPPF

Paper 02 also gives the direct upper bound

∥F(T)∥≤eΘζT∑ρ∥vρ∥.\|F(T)\| \le e^{\Theta_\zeta T} \sum_\rho \|v_\rho\|.

The coefficient sum is finite.

Hence

Eζ(eT;U)≪e(1+2Θζ)T.\mathcal E_\zeta(e^T;U) \ll e^{(1+2\Theta_\zeta)T}.

If

Θζ≤1−η2,\Theta_\zeta \le 1-\frac{\eta}{2},

then

1+2Θζ≤3−η.1+2\Theta_\zeta \le 3-\eta.

Therefore

Eζ(x;U)≪x3−η.\mathcal E_\zeta(x;U) \ll x^{3-\eta}.

Thus ZPPF (η)(\eta) holds. □\square

5. RH endpoint

Set

η=1.\eta=1.

Theorem 4.1 gives

ZPPF⁡(1)⟺Θζ≤12.\operatorname{ZPPF}(1) \Longleftrightarrow \Theta_\zeta \le \frac12.

Functional-equation symmetry gives

Θζ≥12.\Theta_\zeta \ge \frac12.

Therefore

ZPPF⁡(1)⟺RH.\boxed{ \operatorname{ZPPF}(1) \Longleftrightarrow RH. }

The q=1q=1 positive packet is therefore an exact RH diagnostic at the endpoint.

6. EMAE necessity theorem

Recall

EMAE⁡(η):Z2(x;Q,U)+Z4(x;Q,U)≪x3−η+o(1).\operatorname{EMAE}(\eta): \qquad \mathfrak Z_2(x;Q,U) + \mathfrak Z_4(x;Q,U) \ll x^{3-\eta+o(1)}.

Because

Z4≥0\mathfrak Z_4\ge0

and

Z2≥Eζ,\mathfrak Z_2 \ge \mathcal E_\zeta,

we obtain:

Theorem 6.1 — EMAE necessarily contains a zeta strip

For every fixed

η>0,\eta>0, EMAE⁡(η)⟹ZPPF⁡(η)⟹Θζ≤1−η2.\boxed{ \operatorname{EMAE}(\eta) \Longrightarrow \operatorname{ZPPF}(\eta) \Longrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. }

Thus a proof of positive EMAE at any fixed power already proves a new fixed zero strip for zeta before any nonprincipal character issue is considered.

This is a necessary-condition theorem.

It is stronger than the earlier single-zero heuristic because the packet interference question was closed in Paper 02.

7. Positive-Gate No-Bypass theorem

The previous theorem extends beyond the exact EMAE definition.

Let

G(x)≥0\mathcal G(x)\ge0

be any proposed positive sufficient gate.

Assume that for some nonnegative weight c(x)c(x),

G(x)≥c(x)Eζ(x;U).\mathcal G(x) \ge c(x) \mathcal E_\zeta(x;U).

Theorem 7.1 — Subpolynomial q=1q=1 retention

Suppose

c(x)=x−o(1).c(x) = x^{-o(1)}.

If

G(x)≪x3−η+o(1)\mathcal G(x) \ll x^{3-\eta+o(1)}

for a fixed

η>0,\eta>0,

then

Θζ≤1−η2.\boxed{ \Theta_\zeta \le 1-\frac{\eta}{2}. }

Proof

Since

c(x)−1=xo(1),c(x)^{-1} = x^{o(1)},

we have

Eζ(x;U)≤c(x)−1G(x)≪x3−η+o(1).\mathcal E_\zeta(x;U) \le c(x)^{-1}\mathcal G(x) \ll x^{3-\eta+o(1)}.

Apply Theorem 4.1. □\square

8. Fixed-power weight-loss law

More generally, suppose

c(x)≥x−λ+o(1)c(x) \ge x^{-\lambda+o(1)}

for fixed

λ≥0.\lambda\ge0.

Then

G(x)≪x3−η+o(1)\mathcal G(x) \ll x^{3-\eta+o(1)}

implies

Eζ(x;U)≪x3−(η−λ)+o(1).\mathcal E_\zeta(x;U) \ll x^{3-(\eta-\lambda)+o(1)}.

Hence, if

η>λ,\eta>\lambda,

then

Θζ≤1−η−λ2.\boxed{ \Theta_\zeta \le 1-\frac{\eta-\lambda}{2}. }

Consequently, a positive gate can avoid any fixed-strip consequence only by losing at least the full fixed power in its q=1q=1 retention.

But such a loss creates a separate target-fidelity question:

does the weakened gate still control the actual q=1q=1 prime-pair mismatch?

That question cannot be assumed.

9. Why conductor weighting cannot solve the full EMAE gate

For

q>1,q>1,

v3.18 identified useful weights such as

q−2+o(1)q^{-2+o(1)}

in the pole-zero channel.

These weights can genuinely suppress high-conductor characters.

However, at

q=1,q=1,

the conductor weight is exactly one.

Therefore no conductor-weight argument can reduce the zeta exponent

1+2Θζ.1+2\Theta_\zeta.

Conductor weighting remains useful for the nonprincipal remainder after the q=1q=1 problem has been separately handled.

It is not a bypass of the principal bottleneck.

10. Why height weights cannot change fixed exponential type

For each fixed zeta zero

ρ=β+iγ,\rho=\beta+i\gamma,

the smooth Mellin coefficient

vρv_\rho

may be very small for large

∣γ∣.|\gamma|.

But Paper 02 proved

vρ≠0v_\rho\neq0

for every zero and

τF=Θζ.\tau_F=\Theta_\zeta.

Therefore any xx -independent nonzero coefficient suppression in zero height changes constants but not the exponential type.

In particular, rapid vertical decay is sufficient for convergence, but it does not turn

xβx^\beta

into

xβ−δx^{\beta-\delta}

for a fixed zero.

This closes the following false inference:

very small high-zero coefficient
->
fixed-power suppression of every near-extremal zero

The implication is invalid at the asymptotic exponent level.

11. Current prime exponential-sum input at the q=1q=1 core

Maynard–Pandey–Radziwiłł (2026) prove that if

α=aq+ϵ\alpha=\frac aq+\epsilon

and

B=max⁡(q,qN∣ϵ∣),B=\max(q,qN|\epsilon|),

then

∣∑n<NΛ(n)e(nα)∣≤No(1)(NB1/2+N19/24).\left| \sum_{n<N} \Lambda(n)e(n\alpha) \right| \le N^{o(1)} \left( \frac{N}{B^{1/2}} + N^{19/24} \right).

On the principal core arc,

q=1,ϵ=uN,∣u∣≤U,q=1, \qquad \epsilon=\frac{u}{N}, \qquad |u|\le U,

so

B=max⁡(1,∣u∣)=OU(1).B = \max(1,|u|) = O_U(1).

Hence the theorem gives only

∣SN(u/N)∣≤N1+o(1)\boxed{ |S_N(u/N)| \le N^{1+o(1)} }

at this core scale.

The new

N19/24N^{19/24}

generic term is dominated by the unsaved

N/B1/2N/B^{1/2}

term.

Thus the strongest current generic exponential-sum advance is fully consistent with the CSM_RH principal-bottleneck classification.

12. Current zero-density progress does not remove the principal bottleneck

Guth–Maynard (Annals of Mathematics, 2026) prove

N(σ,T)≤T30(1−σ)/13+o(1).N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)}.

Chen–Gupta–Li, in the 2026 revision of their character large-values paper, prove

∑χ mod qN(σ,T,χ)≪ε(qT)7(1−σ)/3+ε.\sum_{\chi\bmod q} N(\sigma,T,\chi) \ll_\varepsilon (qT)^{7(1-\sigma)/3+\varepsilon}.

These are major zero-population estimates.

But neither statement logically excludes one zeta zero with

ℜρ>1−η2\Re\rho>1-\frac{\eta}{2}

for a prescribed fixed

η>0.\eta>0.

By Theorem 4.1, one such zero is enough to rule out ZPPF (η)(\eta).

Therefore:

ZERO POPULATION CONTROL
does not imply
ZPPF FIXED POWER.

This is now a direct consequence of the exact packet-type theorem.

13. Current zero-free regions are not fixed strips

Bellotti–Trudgian–Yang (2026) prove an explicit zeta zero-free region of the form

ζ(σ+it)≠0\zeta(\sigma+it)\neq0

when

t≥3t\ge3

and

σ≥1−14.896log⁡t.\sigma \ge 1-\frac{1}{4.896\log t}.

This is valuable explicit progress.

However the width

14.896log⁡t\frac{1}{4.896\log t}

tends to zero with height.

It does not imply a fixed

δ>0\delta>0

such that

Θζ≤1−δ.\Theta_\zeta \le1-\delta.

Therefore it does not close ZPPF (η)(\eta) for any fixed

η>0.\eta>0.

14. Deuring–Heilbronn does not bypass q=1q=1

Modern explicit Deuring–Heilbronn results give zero repulsion for Dirichlet LL -functions under the existence of a Landau–Siegel zero.

This mechanism is relevant to exceptional real characters.

But the necessary principal subgate in Theorem 6.1 is the Riemann zeta packet at

q=1.q=1.

Therefore Deuring–Heilbronn may improve the nonprincipal character remainder but cannot, by itself, establish the q=1q=1 ZPPF gate.

This is a scope obstruction, not a criticism of the theorem.

15. Correction to the v3.18 weighted-extreme-zero optimism

v3.18 correctly observed that the full character major-arc energy contains:

  • conductor weights;
  • arc widths;
  • height localization;
  • Mellin decay;
  • Gauss-sum weights.

It therefore suggested that a weighted theorem might permit some zeros closer to one without requiring a uniform Dirichlet fixed strip.

Paper 03 refines that statement.

For nonprincipal characters, this opening remains valid.

For the complete positive EMAE gate, however, the q=1q=1 zeta packet has:

conductor weight = 1
character averaging = none
Gauss-sum discount = none
fixed nonzero Mellin coefficient per zero

Therefore:

weighted character flexibility≠principal zeta flexibility.\boxed{ \text{weighted character flexibility} \neq \text{principal zeta flexibility}. }

Any full positive EMAE theorem still contains a fixed zeta zero-strip theorem as a necessary subresult.

16. Campaign 02 candidate audit

Candidate C02-A — Density + conductor + height boxes

status:
  REJECTED_AS_FULL_EMAE_BYPASS

reason:
  useful for q>1 boxes;
  cannot discharge q=1 ZPPF

Candidate C02-B — Nonexceptional-modulus prime exponential-sum improvement

status:
  REJECTED_AS_PRINCIPAL_CORE_BYPASS

reason:
  q=1 core has B=O(1);
  N/B^(1/2) term retains N-scale

Candidate C02-C — Deuring–Heilbronn exceptional-zero repulsion

status:
  PARTIAL / NONPRINCIPAL ONLY

reason:
  does not solve q=1 zeta packet

Candidate C02-D — Positive conductor / height reweighting

status:
  REJECTED_IF_q1_WEIGHT_IS_x^(-o(1))

certificate:
  Theorem 7.1

Candidate C02-E — Remove / subtract the principal zeta packet

status:
  REJECTED_WITHOUT_TARGET_FIDELITY_BRIDGE

reason:
  deleting the blocker from the sufficient gate is not closure
  unless the resulting gate still controls the original prime-pair target

Candidate C02-F — Direct zeta fixed-strip arithmetic theorem

status:
  VALID TARGET
strength:
  BREAKTHROUGH_LEVEL

comment:
  not a simplification;
  exactly the required new mathematics

Candidate C02-G — Signed principal mismatch / non-positive gate

status:
  SURVIVOR

reason:
  Theorem 7.1 blocks positive gates retaining q=1,
  but does not rule out a genuinely signed target-faithful cancellation mechanism.

proof status:
  OPEN

17. Campaign 02 closure verdict

Campaign 02 asked:

find the weakest genuinely arithmetic theorem that yields EMAE (η)(\eta) for some fixed η>0\eta>0.

The result is:

DIRECT FULL EMAE BY CURRENT CHARACTER-FAMILY TECHNOLOGY
= NOT CLOSED

PRINCIPAL NECESSARY SUBGATE
= EXACTLY IDENTIFIED

q=1 POSITIVE GATE STRENGTH
= EXACTLY FIXED ZERO STRIP

CURRENT DENSITY / CONDUCTOR / HEIGHT / REPULSION INPUTS
= INSUFFICIENT TO BYPASS q=1

SIGNED TARGET-FAITHFUL ALTERNATIVE
= SURVIVOR / OPEN

Therefore Campaign 02 is closed as a reduction and no-go audit.

It is not closed as a proof of EMAE.

18. New canonical frontier object

Create:

F-RH-005
PRINCIPAL_ZETA_PACKET_FIXED_POWER
abbrev:
  ZPPF
status:
  OPEN
type:
  EXACT_EQUIVALENCE_FRONTIER

For fixed η\eta:

F-RH-005(η)⟺Θζ≤1−η2.\boxed{ F\text{-}RH\text{-}005(\eta) \Longleftrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. }

This frontier is intentionally marked as an exact-equivalence frontier.

It is not advertised as an easier reformulation.

19. New obstruction

Create:

O-RH-006
POSITIVE_GATE_PRINCIPAL_NO_BYPASS
status:
  CERTIFIED

Statement:

Any positive fixed-power sufficient gate which retains the q=1q=1 principal zeta packet with only subpolynomial loss already contains a fixed zeta zero-strip theorem.

This obstruction applies to:

positive character-family energies
positive conductor reweightings
positive height reweightings
positive large-sieve envelopes
positive zero-density envelopes

whenever they retain the principal packet at x−o(1)x^{-o(1)} strength.

20. New survivor

Create:

S-RH-008
SIGNED_TARGET_FAITHFUL_PRINCIPAL_GATE
status:
  OPEN / SURVIVOR

The survivor asks whether the original prime-pair target admits a sufficient gate that:

  1. preserves the actual signed structure;
  2. does not dominate Eζ\mathcal E_\zeta as a positive summand;
  3. still controls the target pair-spectrum deviation;
  4. has a weaker fixed-power strength profile than ZPPF;
  5. does not subtract zeta zeros by assumption;
  6. does not insert RH into the model.

This is not known to exist.

It is the only structural bypass left by the present positive-gate audit.

21. Campaign 03 — Principal Gate Minimality Audit

The next worker campaign is:

CSM_RH Campaign 03
PRINCIPAL_GATE_MINIMALITY

Primary question:

Is the current positive q=1q=1 energy gate stronger than necessary for the actual prime-pair target, or is every target-faithful fixed-power principal gate still forced to have zero-strip strength?

This is a gate-design question before it is a new estimate question.

22. Campaign 03 candidate families

Workers may generate candidates from:

A
SIGNED PRINCIPAL MISMATCH

B
LINEARIZED POLE-RESIDUAL OBSERVABLE

C
POLARIZED CROSS-CORRELATION

D
SCALE-FREQUENCY SIGNED TRANSFORM

E
DIRECT PAIR-SPECTRUM DIFFERENCE WITHOUT POSITIVE DOMINATION

F
CERTIFIED CANCELLATION BETWEEN PRINCIPAL SUBTERMS

These are search classes, not theorem claims.

23. Campaign 03 rejection tests

Every candidate must answer:

T1. Target fidelity

Does the proposed gate still imply the original prime-pair / pair-spectrum target?

T2. Principal retention

Has the zeta packet merely been deleted, renamed, or assumed small?

T3. Hidden positivity

Does the proof eventually dominate a positive object containing

x−o(1)Eζ?x^{-o(1)} \mathcal E_\zeta?

If yes, O-RH-006 applies.

T4. Hidden fixed strip

Does any intermediate estimate already imply

Θζ≤1−δ\Theta_\zeta \le1-\delta

for fixed δ>0\delta>0?

If yes, classify it honestly as breakthrough-strength.

T5. Signed cancellation certificate

If cancellation is used, is it structurally forced or only heuristic?

T6. Quantifier audit

Does the argument work uniformly as

x→∞x\to\infty

for all fixed zero witnesses?

T7. Supremum audit

Does it avoid assuming that Θζ\Theta_\zeta is attained?

24. Worker architecture

CSM_RH remains the protocol.

GLM-5.3-Flash may be used as a current high-volume worker provider.

Canonical roles:

Designer
Builder
Verifier

The provider remains replaceable.

Worker agreement is not theorem authority.

Campaign artifacts must be promoted only after:

strength audit
target-fidelity audit
counterexample attempt
quantifier audit
independent frontier review

25. External evidence snapshot

As of 2026-09-05:

Maynard–Pandey–Radziwiłł

Exponential sums over primes, arXiv:2608.14777, submitted 2026-08-14.

Main current pointwise estimate:

∣SN(α)∣≤No(1)(NB1/2+N19/24).|S_N(\alpha)| \le N^{o(1)} \left( \frac{N}{B^{1/2}} + N^{19/24} \right).

The authors explicitly interpret the first term morally as reflecting possible exceptional zeros close to one.

Guth–Maynard

New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 (2026), 623–675.

Zero-density consequence:

N(σ,T)≤T30(1−σ)/13+o(1).N(\sigma,T) \le T^{30(1-\sigma)/13+o(1)}.

Chen–Gupta–Li

Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet L-Functions, arXiv:2507.08296v2, revised 2026-07-27.

Character-family zero-density estimate:

∑χ mod qN(σ,T,χ)≪ε(qT)7(1−σ)/3+ε.\sum_{\chi\bmod q} N(\sigma,T,\chi) \ll_\varepsilon (qT)^{7(1-\sigma)/3+\varepsilon}.

Bellotti–Trudgian–Yang

Zero-free regions inspired by work of Heath-Brown, arXiv:2603.21490.

Explicit region:

σ≥1−14.896log⁡t,t≥3.\sigma \ge 1-\frac{1}{4.896\log t}, \qquad t\ge3.

Benli–Goel–Twiss–Zaman

Explicit Deuring-Heilbronn phenomenon for Dirichlet L-functions, arXiv:2410.06082v3, revised 2026-01-08.

Scope:

Landau-Siegel-zero conditional repulsion
for Dirichlet L-functions modulo q

These results are external inputs or consistency checks only.

None is promoted here into a fixed zeta zero strip.

26. State transition

The canonical transition is:

CSM_RH v0.3
  ->
CSM_RH v0.4

with:

Campaign 02:
  CLOSED_AS_REDUCTION_AUDIT

F-RH-003:
  EMAE
  OPEN
  but now has certified necessary subgate F-RH-005

F-RH-005:
  PRINCIPAL_ZETA_PACKET_FIXED_POWER
  CREATED
  OPEN / EXACT_EQUIVALENCE_FRONTIER

O-RH-006:
  POSITIVE_GATE_PRINCIPAL_NO_BYPASS
  CREATED / CERTIFIED

S-RH-008:
  SIGNED_TARGET_FAITHFUL_PRINCIPAL_GATE
  CREATED / OPEN

next:
  Campaign 03
  PRINCIPAL_GATE_MINIMALITY

27. Final status

RH = OPEN

MAJOR-ARC PACKET GLOBALIZATION
= CLOSED_AT_EXPONENTIAL_TYPE

FULL POSITIVE EMAE
= OPEN

q=1 PRINCIPAL POSITIVE FIXED POWER
= EXACTLY EQUIVALENT TO A FIXED ZETA ZERO STRIP

CURRENT CHARACTER-FAMILY ADVANCES
= DO NOT BYPASS q=1

POSITIVE-GATE BYPASS
= CLOSED / NO

SIGNED TARGET-FAITHFUL BYPASS
= OPEN / SURVIVOR

NEXT CAMPAIGN
= PRINCIPAL_GATE_MINIMALITY

The central result is:

ZPPF⁡(η)⟺Θζ≤1−η2.\boxed{ \operatorname{ZPPF}(\eta) \Longleftrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. }

and therefore:

EMAE⁡(η)⟹Θζ≤1−η2.\boxed{ \operatorname{EMAE}(\eta) \Longrightarrow \Theta_\zeta \le 1-\frac{\eta}{2}. }

The remaining design choice is now explicit:

either prove genuinely new fixed-strip mathematics, or abandon the current positive-dominating principal gate and prove that a weaker signed target-faithful gate is sufficient.

That is the canonical output of Campaign 02.