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CSM_RH Paper 07 — Gram-Gauge Non-Invariance, Finite-Depth Gap Insufficiency, and the Zero-Frequency Möbius Strength Barrier

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CSM_RH Paper 07

Gram-Gauge Non-Invariance, Finite-Depth Gap Insufficiency, and the Zero-Frequency Möbius Strength Barrier

Project: CSM_RH
Paper: 07
Version: v0.1
Date: 2026-09-05
Parent state: CSM_RH v0.7 / Paper 06
Campaign: 06 — MULTIPLICATIVE_GRAM_GAP
Status: representation audit / exponent-strength 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

The purpose of this paper is to determine whether a multiplicative Gram-gap certificate is a canonical arithmetic object and what strength a useful certificate would actually require.

The principal conclusions are:

RAW GRAM GAP
  not decomposition-invariant

CONSTANT ONE-SHOT GRAM ANGLE
  exponent-insufficient

FIXED-ORDER EXACT DECOMPOSITION
  finite/polylog interface, not a power-saving mechanism

DIRECT FIXED-POWER MÖBIUS CANCELLATION
  already fixed-zero-strip strength

SURVIVING ROUTES
  recombined zero-frequency multilinear power saving
  or recursive multiplicative contraction with linear cumulative mass

No live GLM-5.3-Flash run is claimed.

1. Starting frontier

Paper 06 introduced

F-RH-007
MULTIPLICATIVE_GRAM_GAP_CERTIFICATE
MGGC

from an exact decomposition

an=∑r=1Rbn(r).a_n = \sum_{r=1}^{R} b_n^{(r)}.

Let

Br(j)=∑n≤jbn(r)B_r(j) = \sum_{n\le j} b_n^{(r)}

and

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

On the dyadic endpoint interval define

⟨U,V⟩N=∑j=N2N−1U(j)V(j)‾.\langle U,V\rangle_N = \sum_{j=N}^{2N-1} U(j)\overline{V(j)}.

The component Gram matrix is

Grs=⟨Br,Bs⟩N.G_{rs} = \langle B_r,B_s\rangle_N.

Then

JN=∥A∥N2=1∗G1.J_N = \left\| A \right\|_N^2 = \mathbf 1^\ast G \mathbf 1.

This identity is correct.

The new question is whether properties of the individual entries of GG are canonical.

2. Component gauge transformations

Write the component vector as

B=(B1,…,BR)T.B = (B_1,\ldots,B_R)^T.

Let

T∈CR×RT\in\mathbb C^{R\times R}

satisfy

1∗T=1∗.\boxed{ \mathbf 1^\ast T = \mathbf 1^\ast. }

Define new components

C=TB.C = TB.

Then

∑r=1RCr=1∗C=1∗TB=1∗B=A.\sum_{r=1}^{R} C_r = \mathbf 1^\ast C = \mathbf 1^\ast TB = \mathbf 1^\ast B = A.

Thus the target cumulative error is unchanged.

The Gram matrix transforms as

GC=TGBT∗.\boxed{ G_C = T G_B T^\ast. }

But

1∗GC1=JN\boxed{ \mathbf 1^\ast G_C \mathbf 1 = J_N }

remains unchanged.

This is the component-gauge freedom.

3. Gram-Gauge Non-Invariance Theorem

Theorem 3.1

The following quantities are not invariants of the recombined target AA under exact component regrouping:

  • tr⁡G\operatorname{tr}G ;
  • the sum of off-diagonal Gram entries;
  • the sign pattern of individual cross-Gram entries;
  • the ratio between diagonal mass and cross mass;
  • a constant-angle statement between arbitrarily chosen components.

The quantity

1∗G1=JN\boxed{ \mathbf 1^\ast G\mathbf 1 = J_N }

is invariant.

Proof

The target depends only on the sum of the components.

The transformation in Section 2 preserves this sum while changing the Gram matrix by congruence.

The explicit two-component example in Section 4 shows that diagonal and cross masses may be varied arbitrarily while JNJ_N is fixed. □\square

4. Explicit two-component gauge example

Let AA be any nonzero vector in the endpoint Hilbert space.

For any real parameter MM, define

C1=MA,C_1 = MA, C2=(1−M)A.C_2 = (1-M)A.

Then

C1+C2=A.C_1+C_2=A.

The target energy is always

J=∥A∥2.J = \|A\|^2.

But the diagonal Gram mass is

DM=[M2+(1−M)2]J,\boxed{ D_M = [ M^2+(1-M)^2 ] J, }

and the total off-diagonal contribution is

OM=2M(1−M)J.\boxed{ O_M = 2M(1-M)J. }

Their sum is

DM+OM=J.D_M+O_M=J.

For

M>1,M>1,

the cross term is negative.

As

M→∞,M\to\infty,

both

DMD_M

and

∣OM∣|O_M|

become arbitrarily large relative to JJ, while their cancellation leaves the same target.

Therefore a large negative cross-Gram term is not, by itself, mathematical progress.

5. Consequence for MGGC

The generic frontier

MULTIPLICATIVE_GRAM_GAP_CERTIFICATE

is too representation-dependent unless the component semantics are fixed.

A valid certificate must specify at least:

exact identity
exact cutoffs
exact smooth/dyadic partition
allowed regroupings
canonical component labels
canonical main-profile subtraction
recombination map

Even after this gauge fixing, a constant Gram improvement may still be exponent-insufficient.

That issue is addressed next.

6. Constant Gram gaps do not change exponents

Let

ENE_N

be any canonical component envelope satisfying

EN≤N3+o(1).E_N \le N^{3+o(1)}.

Suppose a fixed decomposition proves only

JN≤cENJ_N \le c E_N

for some constant

0<c<1.0<c<1.

Then

JN≤N3+o(1).J_N \le N^{3+o(1)}.

There is no fixed power gain.

To obtain

JN≤N3−κ+o(1),J_N \le N^{3-\kappa+o(1)},

one needs either:

EN≤N3−κ+o(1)\boxed{ E_N \le N^{3-\kappa+o(1)} }

or an angular / recombination factor

γN≤N−κ+o(1).\boxed{ \gamma_N \le N^{-\kappa+o(1)}. }

Thus:

Theorem 6.1 — Power-Accurate Gram Requirement

A one-shot constant factor reduction in a quantity of trivial exponent 33 does not alter the exponent.

A one-shot Gram certificate is fixed-power relevant only if the certificate itself contains a fixed power of NN or feeds a recurrence whose cumulative contraction mass is linear in log⁡N\log N.

7. Finite-depth constant-gap theorem

Suppose a recursive factorization applies a fixed contraction

0<λ<10<\lambda<1

at each of

d(N)d(N)

levels.

Ignoring lower-order errors, the total factor is

λd(N).\lambda^{d(N)}.

Write

cλ=−log⁡λ>0.c_\lambda = -\log\lambda>0.

Then

λd(N)=exp⁡[−cλd(N)].\lambda^{d(N)} = \exp [ -c_\lambda d(N) ].

Theorem 7.1

If

d(N)=o(log⁡N),d(N) = o(\log N),

then

λd(N)=N−o(1).\boxed{ \lambda^{d(N)} = N^{-o(1)}. }

Hence finite depth, O(log⁡log⁡N)O(\log\log N) depth, or any sublogarithmic recursion depth cannot produce a fixed N−κN^{-\kappa} power from a constant contraction factor alone.

To obtain

λd(N)≤N−κ,\lambda^{d(N)} \le N^{-\kappa},

one needs

d(N)≥κ−log⁡λlog⁡N+O(1).\boxed{ d(N) \ge \frac{\kappa}{-\log\lambda} \log N + O(1). }

This is the depth form of the cumulative-gap theorem from Paper 06.

8. Calibration with fixed-order Heath-Brown identities

The standard Heath-Brown identity is parameterized by a fixed positive integer KK and writes Λ\Lambda on a dyadic interval as a finite alternating sum of Dirichlet convolutions involving truncated Möbius factors, copies of the constant function, and a logarithm factor.

For fixed KK, the identity itself therefore has fixed combinatorial depth.

Ordinary dyadic or smooth subdivision creates only bookkeeping growth, not an intrinsic Ω(log⁡N)\Omega(\log N) recursive contraction depth.

Consequently:

HEATH-BROWN IDENTITY ITSELF
  !=
fixed-power contraction

A fixed-power result must come from estimates on the resulting multilinear pieces, from their recombination, or from a separate iterative mechanism.

The same semantic rule applies to a fixed Vaughan decomposition.

9. Polylogarithmic piece count is exponent-neutral

Suppose a fixed exact decomposition and its partitions yield

RN=No(1)R_N = N^{o(1)}

pieces:

AN=∑r=1RNTr(N).\mathcal A_N = \sum_{r=1}^{R_N} T_r(N).

If every piece satisfies the uniform fixed-power bound

∣Tr(N)∣≤N3−κ+o(1),|T_r(N)| \le N^{3-\kappa+o(1)},

then

∣AN∣≤N3−κ+o(1).\boxed{ |\mathcal A_N| \le N^{3-\kappa+o(1)}. }

Thus a polylogarithmic or subpolynomial number of pieces is harmless once every piece has a power saving.

Conversely, if one only has

∣Tr(N)∣≤N3+o(1),|T_r(N)| \le N^{3+o(1)},

componentwise triangle inequality cannot generate a fixed saving.

The exponent must appear in the component estimate or in certified recombination.

10. Möbius partial sums

Define the Mertens function

M(x)=∑n≤xμ(n).\boxed{ M(x) = \sum_{n\le x} \mu(n). }

For

ℜs>1,\Re s>1,

partial summation gives

1ζ(s)=s∫1∞M(x)x−s−1 dx.\boxed{ \frac1{\zeta(s)} = s \int_1^\infty M(x)x^{-s-1}\,dx. }

This identity supplies an immediate strength audit for zero-frequency multiplicative cancellation.

11. Fixed-power Möbius cancellation implies a fixed zero-free half-plane

Theorem 11.1

Assume that for some

θ<1\theta<1

and every

ε>0,\varepsilon>0, M(x)=Oε(xθ+ε).M(x) = O_\varepsilon \left( x^{\theta+\varepsilon} \right).

Then

ζ(s)≠0for ℜs>θ.\boxed{ \zeta(s)\neq0 \qquad \text{for } \Re s>\theta. }

Proof

Fix

ss

with

ℜs>θ.\Re s>\theta.

Choose

ε>0\varepsilon>0

such that

θ+ε<ℜs.\theta+\varepsilon<\Re s.

Then

M(x)x−s−1=O(x−1−(ℜs−θ−ε)),M(x)x^{-s-1} = O \left( x^{-1-(\Re s-\theta-\varepsilon)} \right),

so the integral

s∫1∞M(x)x−s−1 dxs \int_1^\infty M(x)x^{-s-1}\,dx

converges locally uniformly in the half-plane

ℜs>θ.\Re s>\theta.

It therefore defines a holomorphic continuation of

1/ζ(s)1/\zeta(s)

from

ℜs>1\Re s>1

to

ℜs>θ.\Re s>\theta.

A zero of ζ\zeta in this half-plane would be a pole of 1/ζ1/\zeta, contradicting holomorphy.

□\square

12. Fixed-strip calibration

If a proposed zero-frequency Type-I argument needs

M(x)≪x1−δM(x) \ll x^{1-\delta}

for some fixed

δ>0,\delta>0,

then Theorem 11.1 already yields

ζ(s)≠0for ℜs>1−δ.\boxed{ \zeta(s)\neq0 \qquad \text{for } \Re s>1-\delta. }

Thus the input is already fixed-zero-strip mathematics.

At the critical endpoint, the classical criterion

M(x)=Oε(x1/2+ε)M(x) = O_\varepsilon \left( x^{1/2+\varepsilon} \right)

for every ε>0\varepsilon>0 is equivalent to RH.

Therefore:

DIRECT MERTENS FIXED-POWER SAVING
  can be a proof mechanism

but

DIRECT MERTENS FIXED-POWER SAVING
  is not a lower-strength shortcut

13. What Vaughan / Heath-Brown may still contribute

The Möbius audit does not invalidate Vaughan or Heath-Brown decompositions.

They may expose cancellation that is not reducible to a standalone bound on M(x)M(x).

The surviving possibilities are:

V1. Type-I/II component saving

A multilinear block may have a fixed-power saving because of cancellation among several multiplicative variables.

V2. Recombined multilinear cancellation

Several canonical blocks may cancel after their main profiles are subtracted, without requiring a fixed-power Mertens estimate for any one factor.

V3. Recursive arithmetic contraction

A canonical decomposition may be embedded in a genuine multiscale recurrence with linear cumulative contraction mass.

These remain open.

14. Main-profile saturation example

The simplest exact decomposition already shows why raw Gram geometry can be misleading.

Write

an=Λ(n)−1.a_n = \Lambda(n)-1.

At cumulative level,

A(j)=ψ(j)−j.A(j) = \psi(j)-j.

Treat this as two components:

B1(j)=ψ(j),B_1(j) = \psi(j), B2(j)=−j.B_2(j) = -j.

Each component separately has a leading size of order jj.

Their Gram diagonals are therefore of cubic dyadic scale.

Their cross term cancels the common deterministic main profile, leaving

JN=∑j=N2N−1(ψ(j)−j)2.J_N = \sum_{j=N}^{2N-1} (\psi(j)-j)^2.

Thus enormous cross-Gram cancellation is already present before any RH-scale improvement.

The hard problem is the exponent of the residual after deterministic main-profile cancellation.

This motivates gauge fixing and canonical centering before any Gram certificate is interpreted.

15. Correction to F-RH-007

The raw frontier

F-RH-007
MULTIPLICATIVE_GRAM_GAP_CERTIFICATE

is reclassified:

status:
  RETIRED_AS_UNFIXED_GENERIC_FRONTIER

reason:
  raw Gram diagnostics are decomposition-gauge dependent

A named, fixed decomposition may still use Gram analysis internally.

But CSM_RH theorem promotion requires a representation-stable output.

16. New canonical frontier

Create:

F-RH-008
GAUGE_FIXED_ZERO_FREQUENCY_MULTILINEAR_POWER_SAVING
abbrev:
  GZMPS
status:
  OPEN

A valid GZMPS certificate must include:

1. one exact published or independently verified coefficient identity
2. exact cutoff parameters
3. exact partition convention
4. canonical block grouping
5. canonical main-profile subtraction
6. zero-frequency target preservation
7. a fixed-power block estimate or recombined fixed-power estimate
8. no triangle leakage that destroys the claimed sign cancellation
9. no hidden fixed-zero-strip input
10. exact derived kappa

The output must be invariant under merely cosmetic rewriting of the same fixed decomposition.

17. Two accepted GZMPS success modes

Mode A — power-accurate pieces

Let

AN=∑r≤RNTr(N),\mathcal A_N = \sum_{r\le R_N} T_r(N),

with

RN=No(1).R_N=N^{o(1)}.

If every canonical piece satisfies

Tr(N)=O(N3−κ+o(1))\boxed{ T_r(N) = O \left( N^{3-\kappa+o(1)} \right) }

uniformly, then GZMPS closes.

Mode B — recombined cancellation

If some individual pieces are larger, the proof may keep a canonical block sum signed and prove directly

∣∑r∈BTr(N)∣≪N3−κ+o(1)\boxed{ \left| \sum_{r\in\mathcal B} T_r(N) \right| \ll N^{3-\kappa+o(1)} }

for every required block.

The cancellation must be theorem-forced.

It may not be inferred from a representation-dependent Gram picture alone.

18. Recursive success mode

A third route is allowed when the fixed decomposition is part of a genuine recurrence.

Suppose

X(Nk)≤λkX(Nk−1)+O(Nk−κ0).X(N_k) \le \lambda_k X(N_{k-1}) + O(N_k^{-\kappa_0}).

Then Paper 06 applies.

The worker must prove

∑r=m+1k−log⁡λr≥δlog⁡(NkNm)−O(1)\boxed{ \sum_{r=m+1}^{k} -\log\lambda_r \ge \delta \log \left( \frac{N_k}{N_m} \right) - O(1) }

for some fixed

δ>0.\delta>0.

A fixed-order identity used once does not satisfy this condition merely by existing.

19. Campaign 06 candidate audit

C06-A — raw negative cross-Gram

status:
  REJECTED AS NONCANONICAL

reason:
  Gram sign and magnitude depend on component gauge

C06-B — fixed constant Gram angle

status:
  REJECTED AS EXPONENT-INSUFFICIENT

reason:
  one constant factor does not change N^3 exponent

C06-C — fixed-order Vaughan / Heath-Brown plus triangle inequality

status:
  DECOMPOSITION-NEUTRAL

reason:
  exact finite convolution expansion is an interface;
  triangle recombination supplies no power

C06-D — Type-I saving from fixed-power Mertens bound

status:
  VALID BUT STRENGTH-NONREDUCING

reason:
  direct fixed-power Mertens cancellation already gives a fixed zero-free half-plane

C06-E — recombined zero-frequency Type-I/II power saving

status:
  SURVIVOR

C06-F — recursive multiplicative contraction

status:
  SURVIVOR

gate:
  linear cumulative log-contraction mass

20. New obstruction: Gram gauge non-invariance

Create:

O-RH-012
GRAM_GAUGE_NONINVARIANCE
status:
  CERTIFIED

Statement:

Cross-Gram signs, diagonal mass, and constant-angle gaps are not invariants of an exact recombined arithmetic target under component regrouping. They cannot be promoted as theorem progress without a fixed decomposition semantics and a representation-stable output.

21. New obstruction: finite-depth constant-gap insufficiency

Create:

O-RH-013
FINITE_DEPTH_CONSTANT_GAP_INSUFFICIENCY
status:
  CERTIFIED

Statement:

A constant contraction repeated only o(log⁡N)o(\log N) times produces at most N−o(1)N^{-o(1)} decay. A fixed-order decomposition therefore cannot obtain a fixed power from constant-factor improvement alone.

22. New obstruction: Möbius strength conservation

Create:

O-RH-014
MOBIUS_FIXED_STRIP_STRENGTH
status:
  CERTIFIED

Statement:

Any zero-frequency proof input of the form M(x)=O(xθ+ε)M(x)=O(x^{\theta+\varepsilon}) for every ε>0\varepsilon>0 already excludes zeta zeros in ℜs>θ\Re s>\theta. Such an input may prove the desired result, but it is not a theorem-strength bypass.

23. New survivors

Create:

S-RH-014
POWER_ACCURATE_TYPE_I_II_COMPONENT_ESTIMATE
status:
  OPEN

Create:

S-RH-015
RECOMBINED_ZERO_FREQUENCY_MULTILINEAR_CANCELLATION
status:
  OPEN

Create:

S-RH-016
RECURSIVE_MULTIPLICATIVE_LOG_DEPTH_CONTRACTION
status:
  OPEN

24. Campaign 07

The next campaign is:

CSM_RH Campaign 07
CANONICAL_VAUGHAN_ZERO_FREQUENCY_LEDGER

This campaign fixes one exact Vaughan identity before estimation.

The objective is not to prove RH immediately.

The objective is to produce the first fully gauge-fixed Type-I/II ledger for CSSA.

25. Campaign 07 canonical workflow

Step 1 — pin one exact Vaughan identity

Record:

source
equation
range of validity
parameters U,V or equivalent cutoffs
all boundary terms

No mixing of incompatible Vaughan variants is allowed.

Step 2 — apply it to the zero-frequency CSSA paraproduct

Start from

AN=∑n<2NwN(n)anA(n−1)−MN.\mathcal A_N = \sum_{n<2N} w_N(n) a_nA(n-1) - \mathcal M_N.

Substitute the fixed identity consistently.

Step 3 — define canonical blocks

Every Type-I, Type-II, diagonal, correction, and main-profile term receives one stable ID.

Step 4 — subtract canonical main profiles

No arbitrary component re-centering is allowed.

Step 5 — build the exponent ledger

For each block state:

trivial exponent
best proved exponent
required exponent
saving source
whether Möbius fixed-power input is used
whether triangle leakage occurs

Step 6 — promote only genuine power-bearing blocks

A block counts as progress only if its exponent improves by a fixed positive amount or if it participates in a rigorously proved recombined cancellation with such a gain.

26. Campaign 07 rejection filters

Reject a candidate if:

R1. Gram-only evidence

A negative cross term is shown without a canonical block theorem.

R2. Fixed constant gain only

A factor such as 0.90.9 is called an exponent saving.

R3. Möbius strength laundering

A fixed-power Mertens estimate is used but classified as routine.

R4. Variant mixing

Different Vaughan identities or incompatible cutoffs are combined silently.

R5. Arbitrary re-centering

Main profiles are moved between blocks without an exact conserved identity.

R6. Triangle leakage

Signed canonical blocks are absolutized before the claimed cancellation.

R7. Subpower mislabeled fixed-power

Logarithmic or stretched-exponential-in-log savings are promoted as N−κN^{-\kappa}.

27. External calibration

The fixed-order Heath-Brown identity is a standard finite convolution identity for Λ\Lambda involving truncated Möbius functions and a fixed integer parameter KK.

Modern applications use this identity to turn sums over primes into Type-I / Type-II or multilinear sums after smooth or dyadic subdivision.

This supports the semantic distinction:

identity
  !=
estimate

The Mertens criterion provides the corresponding strength calibration for direct Möbius cancellation.

28. State transition

The canonical transition is:

CSM_RH v0.7
  ->
CSM_RH v0.8

with:

Campaign 06
  CLOSED_AS_GRAM_CANONICALITY_AUDIT

F-RH-007
  RAW MULTIPLICATIVE GRAM GAP
  RETIRED / REFINED

F-RH-008
  GZMPS
  CREATED / OPEN

O-RH-012
  GRAM_GAUGE_NONINVARIANCE
  CREATED / CERTIFIED

O-RH-013
  FINITE_DEPTH_CONSTANT_GAP_INSUFFICIENCY
  CREATED / CERTIFIED

O-RH-014
  MOBIUS_FIXED_STRIP_STRENGTH
  CREATED / CERTIFIED

S-RH-014
  POWER_ACCURATE_TYPE_I_II_COMPONENT_ESTIMATE
  CREATED / OPEN

S-RH-015
  RECOMBINED_ZERO_FREQUENCY_MULTILINEAR_CANCELLATION
  CREATED / OPEN

S-RH-016
  RECURSIVE_MULTIPLICATIVE_LOG_DEPTH_CONTRACTION
  CREATED / OPEN

Campaign 07
  CANONICAL_VAUGHAN_ZERO_FREQUENCY_LEDGER
  READY

29. Final status

RH = OPEN

CSSA FIXED POWER = OPEN

RAW MULTIPLICATIVE GRAM GAP
= RETIRED AS NONCANONICAL

ONE-SHOT CONSTANT GRAM GAP
= EXPONENT-INSUFFICIENT

DIRECT FIXED-POWER MERTENS INPUT
= FIXED-ZERO-STRIP STRENGTH

GAUGE-FIXED RECOMBINED TYPE-I/II SAVING
= OPEN

RECURSIVE LOG-DEPTH MULTIPLICATIVE CONTRACTION
= OPEN

NEXT CAMPAIGN
= 07

The main correction is:

decomposition geometry is not theorem authority.\boxed{ \text{decomposition geometry is not theorem authority}. }

The next proof attempt must pin one exact decomposition and demonstrate where an actual fixed power enters the recombined zero-frequency arithmetic estimate.