# 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: ```text 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: ```text 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 ```text F-RH-007 MULTIPLICATIVE_GRAM_GAP_CERTIFICATE MGGC ``` from an exact decomposition $$ a_n = \sum_{r=1}^{R} b_n^{(r)}. $$ Let $$ B_r(j) = \sum_{n\le j} b_n^{(r)} $$ and $$ A(j) = \sum_{r=1}^{R} B_r(j). $$ On the dyadic endpoint interval define $$ \langle U,V\rangle_N = \sum_{j=N}^{2N-1} U(j)\overline{V(j)}. $$ The component Gram matrix is $$ G_{rs} = \langle B_r,B_s\rangle_N. $$ Then $$ J_N = \left\| A \right\|_N^2 = \mathbf 1^\ast G \mathbf 1. $$ This identity is correct. The new question is whether properties of the individual entries of $G$ are canonical. --- # 2. Component gauge transformations Write the component vector as $$ B = (B_1,\ldots,B_R)^T. $$ Let $$ T\in\mathbb C^{R\times R} $$ satisfy $$ \boxed{ \mathbf 1^\ast T = \mathbf 1^\ast. } $$ Define new components $$ C = TB. $$ Then $$ \sum_{r=1}^{R} C_r = \mathbf 1^\ast C = \mathbf 1^\ast TB = \mathbf 1^\ast B = A. $$ Thus the target cumulative error is unchanged. The Gram matrix transforms as $$ \boxed{ G_C = T G_B T^\ast. } $$ But $$ \boxed{ \mathbf 1^\ast G_C \mathbf 1 = J_N } $$ 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 $A$ under exact component regrouping: - $\operatorname{tr}G$ ; - 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 $$ \boxed{ \mathbf 1^\ast G\mathbf 1 = J_N } $$ 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 $J_N$ is fixed. $\square$ --- # 4. Explicit two-component gauge example Let $A$ be any nonzero vector in the endpoint Hilbert space. For any real parameter $M$, define $$ C_1 = MA, $$ $$ C_2 = (1-M)A. $$ Then $$ C_1+C_2=A. $$ The target energy is always $$ J = \|A\|^2. $$ But the diagonal Gram mass is $$ \boxed{ D_M = [ M^2+(1-M)^2 ] J, } $$ and the total off-diagonal contribution is $$ \boxed{ O_M = 2M(1-M)J. } $$ Their sum is $$ D_M+O_M=J. $$ For $$ M>1, $$ the cross term is negative. As $$ M\to\infty, $$ both $$ D_M $$ and $$ |O_M| $$ become arbitrarily large relative to $J$, 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 ```text MULTIPLICATIVE_GRAM_GAP_CERTIFICATE ``` is too representation-dependent unless the component semantics are fixed. A valid certificate must specify at least: ```text 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 $$ E_N $$ be any canonical component envelope satisfying $$ E_N \le N^{3+o(1)}. $$ Suppose a fixed decomposition proves only $$ J_N \le c E_N $$ for some constant $$ 00. $$ Then $$ \lambda^{d(N)} = \exp [ -c_\lambda d(N) ]. $$ ## Theorem 7.1 If $$ d(N) = o(\log N), $$ then $$ \boxed{ \lambda^{d(N)} = N^{-o(1)}. } $$ Hence finite depth, $O(\log\log N)$ depth, or any sublogarithmic recursion depth cannot produce a fixed $N^{-\kappa}$ power from a constant contraction factor alone. To obtain $$ \lambda^{d(N)} \le N^{-\kappa}, $$ one needs $$ \boxed{ d(N) \ge \frac{\kappa}{-\log\lambda} \log N + O(1). } $$ 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 $K$ and writes $\Lambda$ 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 $K$, the identity itself therefore has fixed combinatorial depth. Ordinary dyadic or smooth subdivision creates only bookkeeping growth, not an intrinsic $\Omega(\log N)$ recursive contraction depth. Consequently: ```text 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 $$ R_N = N^{o(1)} $$ pieces: $$ \mathcal A_N = \sum_{r=1}^{R_N} T_r(N). $$ If every piece satisfies the uniform fixed-power bound $$ |T_r(N)| \le N^{3-\kappa+o(1)}, $$ then $$ \boxed{ |\mathcal A_N| \le N^{3-\kappa+o(1)}. } $$ Thus a polylogarithmic or subpolynomial number of pieces is harmless once every piece has a power saving. Conversely, if one only has $$ |T_r(N)| \le N^{3+o(1)}, $$ 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 $$ \boxed{ M(x) = \sum_{n\le x} \mu(n). } $$ For $$ \Re s>1, $$ partial summation gives $$ \boxed{ \frac1{\zeta(s)} = s \int_1^\infty M(x)x^{-s-1}\,dx. } $$ 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 $$ \theta<1 $$ and every $$ \varepsilon>0, $$ $$ M(x) = O_\varepsilon \left( x^{\theta+\varepsilon} \right). $$ Then $$ \boxed{ \zeta(s)\neq0 \qquad \text{for } \Re s>\theta. } $$ ### Proof Fix $$ s $$ with $$ \Re s>\theta. $$ Choose $$ \varepsilon>0 $$ such that $$ \theta+\varepsilon<\Re s. $$ Then $$ M(x)x^{-s-1} = O \left( x^{-1-(\Re s-\theta-\varepsilon)} \right), $$ so the integral $$ s \int_1^\infty M(x)x^{-s-1}\,dx $$ converges locally uniformly in the half-plane $$ \Re s>\theta. $$ It therefore defines a holomorphic continuation of $$ 1/\zeta(s) $$ from $$ \Re s>1 $$ to $$ \Re s>\theta. $$ A zero of $\zeta$ in this half-plane would be a pole of $1/\zeta$, contradicting holomorphy. $\square$ --- # 12. Fixed-strip calibration If a proposed zero-frequency Type-I argument needs $$ M(x) \ll x^{1-\delta} $$ for some fixed $$ \delta>0, $$ then Theorem 11.1 already yields $$ \boxed{ \zeta(s)\neq0 \qquad \text{for } \Re s>1-\delta. } $$ Thus the input is already fixed-zero-strip mathematics. At the critical endpoint, the classical criterion $$ M(x) = O_\varepsilon \left( x^{1/2+\varepsilon} \right) $$ for every $\varepsilon>0$ is equivalent to RH. Therefore: ```text 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 $M(x)$. 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 $$ a_n = \Lambda(n)-1. $$ At cumulative level, $$ A(j) = \psi(j)-j. $$ Treat this as two components: $$ B_1(j) = \psi(j), $$ $$ B_2(j) = -j. $$ Each component separately has a leading size of order $j$. Their Gram diagonals are therefore of cubic dyadic scale. Their cross term cancels the common deterministic main profile, leaving $$ J_N = \sum_{j=N}^{2N-1} (\psi(j)-j)^2. $$ 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 ```text F-RH-007 MULTIPLICATIVE_GRAM_GAP_CERTIFICATE ``` is reclassified: ```text 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: ```text F-RH-008 GAUGE_FIXED_ZERO_FREQUENCY_MULTILINEAR_POWER_SAVING abbrev: GZMPS status: OPEN ``` A valid GZMPS certificate must include: ```text 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 $$ \mathcal A_N = \sum_{r\le R_N} T_r(N), $$ with $$ R_N=N^{o(1)}. $$ If every canonical piece satisfies $$ \boxed{ T_r(N) = O \left( N^{3-\kappa+o(1)} \right) } $$ 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 $$ \boxed{ \left| \sum_{r\in\mathcal B} T_r(N) \right| \ll N^{3-\kappa+o(1)} } $$ 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 $$ X(N_k) \le \lambda_k X(N_{k-1}) + O(N_k^{-\kappa_0}). $$ Then Paper 06 applies. The worker must prove $$ \boxed{ \sum_{r=m+1}^{k} -\log\lambda_r \ge \delta \log \left( \frac{N_k}{N_m} \right) - O(1) } $$ for some fixed $$ \delta>0. $$ 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 ```text status: REJECTED AS NONCANONICAL reason: Gram sign and magnitude depend on component gauge ``` ## C06-B — fixed constant Gram angle ```text 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 ```text 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 ```text 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 ```text status: SURVIVOR ``` ## C06-F — recursive multiplicative contraction ```text status: SURVIVOR gate: linear cumulative log-contraction mass ``` --- # 20. New obstruction: Gram gauge non-invariance Create: ```text 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: ```text O-RH-013 FINITE_DEPTH_CONSTANT_GAP_INSUFFICIENCY status: CERTIFIED ``` Statement: > A constant contraction repeated only $o(\log N)$ times produces at most $N^{-o(1)}$ decay. A fixed-order decomposition therefore cannot obtain a fixed power from constant-factor improvement alone. --- # 22. New obstruction: Möbius strength conservation Create: ```text O-RH-014 MOBIUS_FIXED_STRIP_STRENGTH status: CERTIFIED ``` Statement: > Any zero-frequency proof input of the form $M(x)=O(x^{\theta+\varepsilon})$ for every $\varepsilon>0$ already excludes zeta zeros in $\Re s>\theta$. Such an input may prove the desired result, but it is not a theorem-strength bypass. --- # 23. New survivors Create: ```text S-RH-014 POWER_ACCURATE_TYPE_I_II_COMPONENT_ESTIMATE status: OPEN ``` Create: ```text S-RH-015 RECOMBINED_ZERO_FREQUENCY_MULTILINEAR_CANCELLATION status: OPEN ``` Create: ```text S-RH-016 RECURSIVE_MULTIPLICATIVE_LOG_DEPTH_CONTRACTION status: OPEN ``` --- # 24. Campaign 07 The next campaign is: ```text 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: ```text 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 $$ \mathcal A_N = \sum_{n<2N} w_N(n) a_nA(n-1) - \mathcal M_N. $$ 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: ```text 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 $0.9$ 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 $N^{-\kappa}$. --- # 27. External calibration The fixed-order Heath-Brown identity is a standard finite convolution identity for $\Lambda$ involving truncated Möbius functions and a fixed integer parameter $K$. 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: ```text identity != estimate ``` The Mertens criterion provides the corresponding strength calibration for direct Möbius cancellation. --- # 28. State transition The canonical transition is: ```text CSM_RH v0.7 -> CSM_RH v0.8 ``` with: ```text 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 ```text 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: $$ \boxed{ \text{decomposition geometry is not theorem authority}. } $$ The next proof attempt must pin one exact decomposition and demonstrate where an actual fixed power enters the recombined zero-frequency arithmetic estimate.