CSM_RH Paper 06

Critical Selberg Contraction, Cumulative-Gap Classification, and Multiplicative Gram Neutrality

Project: CSM_RH
Paper: 06
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v0.6 / Paper 05
Campaign: 05 — ZERO_FREQUENCY_MULTIPLICATIVE_CONTRACTION
Status: contraction-classification / multiplicative-decomposition audit; not a proof or disproof of RH

0. Trust boundary

This paper does not prove or disprove the Riemann Hypothesis.

Canonical state:

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

This paper does not prove a fixed-power CSSA estimate.

Its purpose is to determine exactly what kind of contraction a successful multiplicative argument must produce.

The main conclusions are:

FIXED POINTWISE GAP
  sufficient but not necessary

LINEAR CUMULATIVE CONTRACTION MASS
  exact scale-level criterion for a power-law contraction mechanism

SELBERG BASIC SYMMETRY
  critical at first absolute-value closure

SELBERG SIGNED IMPROVEMENT
  nonlinear amplitude contraction with vanishing relative gap

EXACT VAUGHAN / HEATH-BROWN STYLE DECOMPOSITION
  contraction-neutral until a new Gram/correlation estimate is proved

NEW FRONTIER
  MULTIPLICATIVE_GRAM_GAP_CERTIFICATE

No theorem below claims that elementary or multiplicative methods can never prove a fixed zero strip.

The result is a typed audit of what the decomposition itself does and does not provide.

1. Canonical normalized frontier

The centered signed shift aggregate is

\mathcal A_N = \sum_{h=1}^{2N-2} \mathcal R_N(h).

Define

\boxed{ X(N) = \frac{ |\mathcal A_N| }{ N^3 }. }

The fixed-power target is:

\boxed{ X(N) \ll N^{-\kappa+o(1)} }

for some fixed

\kappa>0.

Paper 05 supplied one sufficient mechanism:

X(N) \le \lambda X(\theta N) + O(N^{-\kappa_0}),

with fixed

0<\lambda<1, \qquad 0<\theta<1.

The present paper replaces this by a more general cumulative criterion.

2. Geometric scale normalization

Fix

q>1.

Let

N_k = N_0q^k,

and write

Y_k = X(N_k).

Suppose an arithmetic argument yields

\boxed{ Y_k \le \lambda_kY_{k-1} + C N_k^{-\kappa_0}, }

where

0<\lambda_k\le1

and

\kappa_0>0.

Define the log-contraction increment

g_k = -\log\lambda_k \ge0.

For

0\le m<k,

define cumulative contraction mass

\boxed{ G_{m,k} = \sum_{r=m+1}^{k}g_r. }

3. Exact iteration formula

Repeated substitution gives:

Lemma 3.1

\boxed{ Y_k \le e^{-G_{0,k}}Y_0 + C \sum_{m=1}^{k} N_m^{-\kappa_0} e^{-G_{m,k}}. }

Proof

Iterating once gives

Y_k \le \lambda_k\lambda_{k-1}Y_{k-2} + C\lambda_kN_{k-1}^{-\kappa_0} + CN_k^{-\kappa_0}.

Continuing to scale $N_0$ gives the stated product formula.

Because

\prod_{r=m+1}^{k}\lambda_r = \exp \left( - G_{m,k} \right),

the result follows. $\square$

The recurrence is therefore controlled by cumulative log-contraction, not by one individual scale.

4. Linear cumulative-gap theorem

Theorem 4.1 — Linear cumulative contraction gives a fixed power

Assume there exist constants

\delta>0

and

C_0\ge0

such that for every

0\le m<k,

\boxed{ G_{m,k} \ge \delta \log \left( \frac{N_k}{N_m} \right) - C_0. }

Then

\boxed{ Y_k \ll N_k^{-\min(\delta,\kappa_0)+o(1)}. }

Equivalently,

\boxed{ X(N) \ll N^{-\kappa+o(1)} }

along the geometric scale sequence, with

\kappa = \min(\delta,\kappa_0)>0.

Proof

The hypothesis gives

e^{-G_{m,k}} \ll \left( \frac{N_m}{N_k} \right)^\delta.

Insert this into Lemma 3.1:

Y_k \ll N_k^{-\delta} + N_k^{-\delta} \sum_{m=1}^{k} N_m^{\delta-\kappa_0}.

Since $N_m$ is geometric:

if $\delta<\kappa_0$ , the sum is bounded;
if $\delta>\kappa_0$ , the last scale dominates and gives $N_k^{\delta-\kappa_0}$ ;
if $\delta=\kappa_0$ , the sum contributes only a factor $k=O(\log N_k)$ .

Thus

Y_k \ll N_k^{-\min(\delta,\kappa_0)} \log N_k,

with the logarithm required only at the endpoint.

This is

N_k^{-\min(\delta,\kappa_0)+o(1)}.

$\square$

5. Converse audit: sublinear cumulative mass cannot force a fixed power by recurrence alone

The previous condition is also the correct recurrence-level obstruction.

Theorem 5.1 — Homogeneous countermodel

Suppose only the recurrence class is specified and

G_{0,k} = o \left( \log N_k \right).

Then that recurrence class alone cannot force

Y_k = O(N_k^{-\kappa})

for any fixed

\kappa>0.

Proof

Set the additive error equal to zero and take equality:

Y_k = \lambda_kY_{k-1}.

Then

Y_k = Y_0e^{-G_{0,k}}.

G_{0,k} = o(\log N_k),

then

Y_k = N_k^{-o(1)}.

For every fixed

\kappa>0,

this is asymptotically larger than a generic

N_k^{-\kappa}

power.

Therefore no fixed power follows from the recurrence shape alone. $\square$

This is not a counterexample to the arithmetic truth of CSSA.

It is a countermodel to an insufficient contraction certificate.

6. Canonical success criterion

The correct Campaign 05 criterion is therefore not merely

\lambda_k<1.

It is:

\boxed{ G_{0,k} \asymp \log N_k }

at least from below, with the stronger tail-uniform version from Theorem 4.1 when additive errors are present.

In words:

a fixed power requires contraction mass linear in logarithmic scale depth.

A fixed

\lambda<1

is one way to obtain this.

It is not the only way.

7. Model vanishing-gap classification

Let

t_k = \log N_k = t_0+kL,

where

L=\log q.

Consider

\lambda_k = 1- \frac{c}{t_k^a},

for sufficiently large $k$ , with

c>0, \qquad a\ge0.

For

a>0,

-\log\lambda_k = \frac{c}{t_k^a} + O \left( t_k^{-2a} \right).

Summation gives the following classification.

Theorem 7.1 — Contraction spectrum

Case A: $a=0$

The gap is fixed:

\lambda_k=1-c.

Then

G_{0,k} = \frac{ -\log(1-c) }{ L } t_k + O(1).

Hence a fixed power follows.

Case B: $0<a<1$

G_{0,k} = \frac{ c }{ L(1-a) } t_k^{1-a} + o \left( t_k^{1-a} \right).

Thus the homogeneous decay is

\boxed{ Y_k = \exp \left[ - \frac{ c+o(1) }{ L(1-a) } (\log N_k)^{1-a} \right]. }

This is subpower:

Y_k = N_k^{-o(1)}.

Case C: $a=1$

G_{0,k} = \frac cL \log t_k + O(1),

\boxed{ Y_k \asymp (\log N_k)^{-c/L} }

at homogeneous scale.

Case D: $a>1$

The series

\sum_k t_k^{-a}

converges.

Therefore

G_{0,k}

remains bounded.

The recurrence does not force $Y_k$ to tend to zero.

8. Why subexponential-in-log error is still subpower

For

0<a<1,

the decay

\exp \left[ -c (\log N)^{1-a} \right]

may be much stronger than any fixed power of

\log N.

But for every fixed

\kappa>0,

\exp \left[ -c (\log N)^{1-a} \right] \gg N^{-\kappa}

for sufficiently large $N$ .

Thus:

stretched exponential in log N
  !=
fixed power of N

This distinction is exactly the distinction relevant to the CSSA frontier.

9. Selberg symmetry as a critical contraction calibrator

Write

R(x) = \Psi(x)-x.

A standard Selberg symmetry formula has the form

\boxed{ R(x) + \sum_{n\le x} R(x/n) \frac{ \Lambda(n) }{ \log x } = O \left( \frac{x}{\log x} \right). }

Suppose one has the inductive bound

|R(t)| \le \beta t + O(1)

for smaller arguments.

Applying the triangle inequality to the first symmetry relation gives

|R(x)| \le \beta x + O \left( \frac{x}{\log x} \right).

Thus the obvious contraction coefficient is exactly critical:

\boxed{ \beta \mapsto \beta. }

No strict improvement occurs.

This is a classical feature of the elementary PNT proof: the first weighted identity removes almost enough mass, but not enough to close by naïve induction.

10. Selberg signed improvement is nonlinear amplitude contraction

The second signed formula in the Selberg argument contains two sums with opposite signs.

A standard proof sketch obtains an improvement of the form

\boxed{ \beta_{\rm new} \le \beta - c_0\beta^2 }

for an absolute

c_0>0

within the iteration regime.

One explicit exposition uses

c_0=0.007.

The relative contraction factor is

\boxed{ \frac{ \beta_{\rm new} }{ \beta } \le 1-c_0\beta. }

\beta\to0,

this factor tends to one.

Therefore the signed Selberg mechanism is a genuine contraction, but its relative gap vanishes with the current error amplitude.

11. Amplitude iteration law

Assume

0<c_0\beta_m<1

and

\beta_{m+1} \le \beta_m-c_0\beta_m^2.

Then

\beta_{m+1} \le \beta_m ( 1-c_0\beta_m ).

Hence

\frac1{\beta_{m+1}} \ge \frac1{\beta_m} + c_0,

up to a harmless stronger denominator correction.

Therefore:

Proposition 11.1

\boxed{ \beta_m = O \left( \frac1m \right). }

This is enough to drive the normalized PNT error amplitude to zero after indefinitely many valid iterations.

It is not a fixed multiplicative contraction.

Even under the idealized assumption that one such iteration were available per fixed logarithmic scale step, so that

m\asymp\log x,

the resulting direct amplitude profile would only be of logarithmic type.

The actual Selberg proof does not supply such a uniform fixed-scale iteration schedule; this observation is only a strength calibration.

12. Historical elementary error terms and the contraction spectrum

Classical refinements of the Selberg mechanism go beyond the first amplitude iteration.

Historically:

Bombieri and Wirsing obtained $$ \psi(x)
x
O_A \left( \frac{x}{\log^A x} \right) $$ for arbitrary fixed $A>0$ by generalized elementary methods;
later elementary work obtained stretched-exponential-in-log errors, with a record of the shape $$ \psi(x)
x
O \left( x \exp [ -c(\log x)^{1/6} ] \right) $$ in the historical literature.

These are much stronger than the original qualitative PNT.

But they remain

x^{1-o(1)}

rather than

x^{1-\delta}

for fixed

\delta>0.

This is consistent with the contraction-spectrum distinction:

sublinear cumulative contraction mass
  -> subpower error

linear cumulative contraction mass
  -> fixed power

No impossibility claim for future elementary methods is made.

13. Multiplicative decompositions are exact rewrites before estimation

Now consider any exact finite decomposition on the required range:

\boxed{ a_n = \sum_{r=1}^{R} b_n^{(r)}. }

This abstractly includes the role played by truncated Vaughan-, Heath-Brown-, Selberg-, or other convolution decompositions after all coefficients and cutoffs have been fixed.

Define cumulative component errors

B_r(j) = \sum_{n\le j} b_n^{(r)}.

Then

A(j) = \sum_{r=1}^{R} B_r(j).

14. Multiplicative Gram identity

Define the Hilbert inner product on the dyadic endpoint interval:

\langle U,V\rangle_N = \sum_{j=N}^{2N-1} U(j)\overline{V(j)}.

Let

G_{rs}(N) = \langle B_r, B_s \rangle_N.

Then:

Theorem 14.1 — Exact Gram reconstruction

\boxed{ J_N = \sum_{r,s=1}^{R} G_{rs}(N). }

Equivalently, if

\mathbf 1 = (1,\ldots,1)^T,

then

\boxed{ J_N = \mathbf 1^\ast G_N \mathbf 1. }

Because $G_N$ is a Gram matrix,

G_N\succeq0.

This is exact.

15. Triangle / Cauchy closes the decomposition without contraction

By the Hilbert triangle inequality,

\|A\|_N = \left\| \sum_r B_r \right\|_N \le \sum_r \|B_r\|_N.

Therefore

\boxed{ J_N \le \left( \sum_r \sqrt{ G_{rr}(N) } \right)^2. }

This inequality contains no strict contraction coefficient.

It is sharp when the component vectors are positively collinear.

Therefore:

Theorem 15.1 — Exact-decomposition neutrality

An exact multiplicative decomposition, followed only by componentwise triangle inequality or Cauchy-Schwarz, does not itself provide a fixed contraction gap for $J_N$ or CSSA.

Any fixed-power gain must enter through additional arithmetic information about:

the diagonal component norms;
the off-diagonal Gram terms;
a two-scale transfer law;
or another noncircular structural estimate.

The decomposition is an interface.

It is not the saving.

16. Where a real multiplicative gain may occur

A successful Vaughan / Heath-Brown / Selberg-style worker result must therefore prove at least one genuinely new statement of the following types.

Type M1 — component power saving

For a decomposition piece,

\|B_r\|_N^2 \ll N^{3-\eta}

with enough uniformity to sum all pieces without losing the gain.

Type M2 — Gram cross-cancellation

Prove a structurally forced estimate such as

2 \sum_{r<s} \Re G_{rs}(N) \le -\delta \sum_r G_{rr}(N) + \mathcal E_N

with

\delta>0

and a lower-order error.

Type M3 — two-scale contraction

Prove

X(N_k) \le \lambda_k X(N_{k-1}) + O(N_k^{-\kappa_0})

with contraction mass satisfying Theorem 4.1.

Type M4 — direct recombined cancellation

Control

\mathbf 1^\ast G_N\mathbf 1

directly without taking absolute values on the individual Gram entries.

All four are genuine arithmetic estimates.

None is supplied by the convolution identity alone.

17. New obstruction: exact decomposition neutrality

Create:

O-RH-010
EXACT_MULTIPLICATIVE_DECOMPOSITION_NEUTRALITY
status:
  CERTIFIED

Statement:

Replacing $\Lambda$ by an exact Vaughan-, Heath-Brown-, Selberg-, or other finite convolution decomposition does not create contraction by itself. After cumulative summation the PNT mean square is exactly the all-ones quadratic form of the component Gram matrix. A saving requires new norm, cross-Gram, or scale-transfer information.

This does not reject those methods.

It identifies their real proof obligation.

18. New obstruction: sublinear contraction mass

Create:

O-RH-011
SUBLINEAR_CONTRACTION_MASS
status:
  CERTIFIED_AS_RECURRENCE_OBSTRUCTION

Statement:

A scale recurrence whose cumulative log-contraction is only $o(\log N)$ cannot, by recurrence structure alone, force a fixed $N^{-\kappa}$ saving.

This is a statement about the recurrence certificate, not about the arithmetic function outside that certificate.

19. New canonical frontier

Create:

F-RH-007
MULTIPLICATIVE_GRAM_GAP_CERTIFICATE
abbrev:
  MGGC
status:
  OPEN

A valid MGGC may take any of the forms M1–M4 from Section 16, provided it yields either:

X(N) \ll N^{-\kappa}

directly, or a recurrence whose cumulative contraction mass is linear in

\log N.

20. New survivors

Create:

S-RH-012
LINEAR_CUMULATIVE_CONTRACTION_MASS
status:
  OPEN / SURVIVOR

and:

S-RH-013
SIGNED_MULTIPLICATIVE_GRAM_CANCELLATION
status:
  OPEN / SURVIVOR

These replace the overly narrow requirement that every scale individually have one fixed contraction ratio.

21. Campaign 05 candidate audit

C05-A — Basic Selberg symmetry

status:
  CRITICAL / NO FIXED GAP

certificate:
  naive beta -> beta

It is sufficient as a starting point for PNT.

It is not a fixed-power contraction.

C05-B — Signed Selberg second relation

status:
  GENUINE AMPLITUDE CONTRACTION
  VANISHING RELATIVE GAP

profile:
  beta -> beta - c beta^2

It proves qualitative decay.

It does not provide a fixed relative contraction as $\beta\to0$ .

C05-C — Higher-weight Selberg / Bombieri / Wirsing style recursion

status:
  HISTORICALLY STRONG SUBPOWER MECHANISM
  NO FIXED POWER CURRENTLY CERTIFIED

Known elementary remainder improvements are consistent with sublinear cumulative contraction mass.

This is not a no-go theorem against future higher-weight identities.

C05-D — Vaughan decomposition

status:
  INTERFACE SURVIVES
  DECOMPOSITION-ONLY GAP = NONE

Required new input:

Type I / II cumulative norm saving
or
cross-Gram cancellation
or
two-scale contraction

C05-E — Heath-Brown decomposition

status:
  INTERFACE SURVIVES
  DECOMPOSITION-ONLY GAP = NONE

The additional multilinear resolution may expose more cancellation opportunities.

But the exact identity alone remains MGGC-neutral.

C05-F — Fixed-gap recurrence

status:
  VALID SUFFICIENT SPECIAL CASE

It is now subsumed by the more general linear cumulative-gap theorem.

22. Campaign 05 verdict

Campaign 05 asked whether a standard multiplicative decomposition could itself provide the missing fixed-gap contraction.

The answer is:

BASIC SELBERG
  critical

SIGNED SELBERG
  contracting but asymptotically critical

KNOWN HIGHER-WEIGHT ELEMENTARY METHODS
  strong subpower, not fixed power

VAUGHAN / HEATH-BROWN EXACT DECOMPOSITION
  contraction-neutral until new arithmetic estimates are supplied

FIXED POINTWISE GAP
  sufficient but too restrictive as a search specification

LINEAR CUMULATIVE GAP
  correct generalized success criterion

MULTIPLICATIVE GRAM GAP
  next irreducible frontier

No fixed-power estimate has been proved.

But the next proof obligation is now materially smaller and more falsifiable.

23. Campaign 06

The next campaign is:

CSM_RH Campaign 06
MULTIPLICATIVE_GRAM_GAP

The worker must choose one exact decomposition and output an explicit Gram-gap certificate.

The campaign must not accept:

"Vaughan identity gives cancellation"

or:

"Type II sums should be square-root size"

without an explicit theorem and exponent ledger.

24. Campaign 06 required output

For every candidate decomposition, the worker must provide:

1. exact coefficient identity
2. cutoff parameters
3. cumulative component functions B_r(j)
4. Gram matrix entries G_rs(N)
5. which entries carry the saving
6. proof of the saving
7. recombination without triangle leakage
8. scale-transfer law if used
9. cumulative contraction mass
10. resulting kappa
11. hidden-zero-strip audit
12. failure mode

If a proposed recurrence has coefficients $\lambda_k$ , the worker must compute:

\boxed{ G_{m,k} = \sum_{r=m+1}^{k} -\log\lambda_r }

and explicitly test whether it is linear in

\log(N_k/N_m).

25. Hard rejection filters for Campaign 06

Reject a candidate if:

R1. Decomposition-only rhetoric

An exact identity is presented as if it were already an estimate.

R2. Triangle leakage

Cross terms are individually absolutized before the proposed cancellation.

R3. Critical contraction hidden as strict contraction

A coefficient

\lambda_k\to1

is called a fixed gap without cumulative-mass analysis.

R4. Subpower advertised as fixed power

A bound such as

\exp[-c(\log N)^\alpha], \qquad 0<\alpha<1,

is not

N^{-\kappa}.

R5. Hidden zero-strip input

The new estimate assumes a fixed zero-free half-plane.

R6. Finite Gram numerics promoted globally

Numerics may reject a conjectured sign law.

They cannot certify asymptotic MGGC.

26. External calibration

The standard Selberg symmetry formula and its elementary PNT use are documented in modern analytic-number-theory lecture notes.

One standard proof sketch explicitly observes that the first absolute-value induction only reproduces the same constant $\beta$ , and then uses a second signed identity to improve

\beta

\beta-c\beta^2.

Historical surveys of elementary PNT error terms record progressively stronger estimates:

x\log^{-A}x

for arbitrary fixed $A$ , followed by stretched-exponential-in-log errors of the form

x\exp[-c(\log x)^{1/6}].

These provide calibration for the distinction between:

qualitative / subpower self-improvement
and
fixed-power contraction.

They are not used to prove any impossibility theorem.

27. State transition

The canonical transition is:

CSM_RH v0.6
  ->
CSM_RH v0.7

with:

Campaign 05
  CLOSED_AS_CONTRACTION_CLASSIFICATION_AUDIT

F-RH-007
  MULTIPLICATIVE_GRAM_GAP_CERTIFICATE
  CREATED / OPEN

O-RH-010
  EXACT_MULTIPLICATIVE_DECOMPOSITION_NEUTRALITY
  CREATED / CERTIFIED

O-RH-011
  SUBLINEAR_CONTRACTION_MASS
  CREATED / CERTIFIED_AS_RECURRENCE_OBSTRUCTION

S-RH-012
  LINEAR_CUMULATIVE_CONTRACTION_MASS
  CREATED / OPEN

S-RH-013
  SIGNED_MULTIPLICATIVE_GRAM_CANCELLATION
  CREATED / OPEN

Campaign 06
  MULTIPLICATIVE_GRAM_GAP
  READY

28. Final status

RH = OPEN

CSSA FIXED POWER = OPEN

FIXED-GAP SEARCH
= GENERALIZED TO CUMULATIVE-GAP SEARCH

SELBERG BASIC CONTRACTION
= CRITICAL

SELBERG SIGNED CONTRACTION
= REAL BUT VANISHING-GAP

EXACT MULTIPLICATIVE DECOMPOSITION
= NEUTRAL UNTIL ESTIMATED

MULTIPLICATIVE GRAM GAP
= OPEN

NEXT CAMPAIGN
= 06

The principal criterion is now:

\boxed{ G_{m,k} \gtrsim \delta \log \left( \frac{N_k}{N_m} \right) }

for some fixed

\delta>0.

The missing theorem is no longer "find a clever identity."

It is:

prove enough arithmetic cancellation inside an exact multiplicative Gram expansion that the cumulative contraction mass is linear in logarithmic scale depth, or obtain the fixed CSSA power directly.

CSM_RH Paper 06 — Critical Selberg Contraction, Cumulative-Gap Classification, and Multiplicative Gram Neutrality

CSM_RH Paper 06

Critical Selberg Contraction, Cumulative-Gap Classification, and Multiplicative Gram Neutrality

0. Trust boundary

1. Canonical normalized frontier

2. Geometric scale normalization

3. Exact iteration formula

Lemma 3.1

Proof

4. Linear cumulative-gap theorem

Theorem 4.1 — Linear cumulative contraction gives a fixed power

Proof

5. Converse audit: sublinear cumulative mass cannot force a fixed power by recurrence alone

Theorem 5.1 — Homogeneous countermodel

Proof

6. Canonical success criterion

7. Model vanishing-gap classification

Theorem 7.1 — Contraction spectrum

Case A: a=0a=0a=0

Case B: 0<a<10<a<10<a<1

Case C: a=1a=1a=1

Case D: a>1a>1a>1

8. Why subexponential-in-log error is still subpower

9. Selberg symmetry as a critical contraction calibrator

10. Selberg signed improvement is nonlinear amplitude contraction

11. Amplitude iteration law

Proposition 11.1

12. Historical elementary error terms and the contraction spectrum

Bombieri and Wirsing obtained $$ \psi(x)

later elementary work obtained stretched-exponential-in-log errors, with a record of the shape $$ \psi(x)

13. Multiplicative decompositions are exact rewrites before estimation

14. Multiplicative Gram identity

Theorem 14.1 — Exact Gram reconstruction

15. Triangle / Cauchy closes the decomposition without contraction

Theorem 15.1 — Exact-decomposition neutrality

16. Where a real multiplicative gain may occur

Type M1 — component power saving

Type M2 — Gram cross-cancellation

Type M3 — two-scale contraction

Type M4 — direct recombined cancellation

17. New obstruction: exact decomposition neutrality

18. New obstruction: sublinear contraction mass

19. New canonical frontier

20. New survivors

21. Campaign 05 candidate audit

C05-A — Basic Selberg symmetry

C05-B — Signed Selberg second relation

C05-C — Higher-weight Selberg / Bombieri / Wirsing style recursion

C05-D — Vaughan decomposition

C05-E — Heath-Brown decomposition

C05-F — Fixed-gap recurrence

22. Campaign 05 verdict

23. Campaign 06

24. Campaign 06 required output

25. Hard rejection filters for Campaign 06

R1. Decomposition-only rhetoric

R2. Triangle leakage

R3. Critical contraction hidden as strict contraction

R4. Subpower advertised as fixed power

R5. Hidden zero-strip input

R6. Finite Gram numerics promoted globally

26. External calibration

27. State transition

28. Final status

Case A: $a=0$

Case B: $0<a<1$

Case C: $a=1$

Case D: $a>1$