# Attempt to extend the contraction to all sufficiently large j

6 September 2026. **The eventual contraction and RH remain unproved.** The finite certified range has been extended from j = 2,…,8 to **j = 2,…,10**, reaching the transition from basis cutoff 1024 to 2048. A proposed Möbius-only update was also refuted at one scale by exact bounds. This continues the [previous attempt](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/CONTINUATION.md), focusing specifically on

\[
 e_{j+1}\le\left(1-\frac{1}{2(j+1)}\right)e_j,
 \qquad e_j=d_{2^j}^2. \tag{29}
\]

Here d_N² is the minimum squared error when approximating 1 in L²(0,1) using every g_k(x) = {1/(kx)}−k^−1{1/x}, for 2 ≤ k ≤ N. A power-of-two cutoff includes all intervening integer basis indices.

The work below identifies the exact additional estimate needed for a block argument. It also explains why a leading asymptotic of order 1/log N would not, on its own, establish the particular adjacent-scale inequality (29).

The subsequent [recursive-induction investigation](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/RECURSIVE_INDUCTION.md) tests an infinite operator loop and the induction premise needed to force its limiting error to zero.

## 1. Exact block improvement formula

Let P_N be orthogonal projection onto H_N = span(g₂,…,g_N), and let r_N = 1−P_N1. Form the Gram matrix of the old and new basis vectors:

\[
 \begin{pmatrix}G&C\\C^\top&H\end{pmatrix},
 \qquad
 u=(\langle1,g_k\rangle)_{2\le k\le N},
 \quad v=(\langle1,g_k\rangle)_{N<k\le2N}.
\]

These Gram matrices are positive definite. One direct independence proof uses arithmetic cells: if Σc_kg_k vanishes, its value on cell I₁ implies t := Σc_k/k = 0. Its difference between cells I_(m−1) and I_m is t−Σ_(k|m,k≥2)c_k. Induction on m = 2,…,N then gives every c_k = 0.

Define

\[
 S=H-C^\top G^{-1}C,
 \qquad z=v-C^\top G^{-1}u. \tag{30}
\]

Thus S is the Gram matrix of the new directions h_k = (I−P_N)g_k, and z_k = ⟨r_N,h_k⟩. Completing the square in the old coefficients and then the new coefficients gives

\[
 d_N^2=1-u^\top G^{-1}u,\qquad
 \boxed{d_N^2-d_{2N}^2=z^\top S^{-1}z.} \tag{31}
\]

Consequently (29) is exactly the statement

\[
 \boxed{z^\top S^{-1}z\ge
 \frac{1-u^\top G^{-1}u}{2(1+\log_2N)}}
 \quad\text{for sufficiently large dyadic }N. \tag{32}
\]

Positive definiteness of S proves that the gain is nonnegative. It does not supply the quantitative lower bound in (32). Cauchy–Schwarz supplies an upper bound on correlations, whereas (32) needs sufficiently large correlation with the specific residual r_N.

The numerical probes made this distinction visible: replacing the inverse quadratic form by the bound ||z||²/λ_max(S) lost most of the gain. In probes through N = 512 this elementary spectral bound gave normalized constants below 0.05, substantially below the required 0.5. These are floating-point diagnostics, not certified bounds or an impossibility theorem for better spectral arguments.

## 2. A concrete Möbius block candidate

A candidate with the same arithmetic definition at every scale is

\[
 \Phi_N=-\sum_{N<k\le2N}\mu(k)g_k,\qquad
 h_N=(I-P_N)\Phi_N.
\]

If h_N ≠ 0, the best approximation in H_N + span(Φ_N) improves the squared error by exactly

\[
 \frac{|\langle r_N,\Phi_N\rangle|^2}{\|h_N\|_2^2}. \tag{33}
\]

This follows by minimizing ||r_N−t h_N||² over t. It is a specific candidate for the full gain (31). A sufficient large-N estimate for this candidate would be

\[
 |\langle r_N,\Phi_N\rangle|^2
 \ge \frac{\|r_N\|_2^2\|h_N\|_2^2}
 {2(1+\log_2N)}. \tag{34}
\]

The following normalized constants were measured in a finite-head numerical model with cutoff 65,536. They guided the subsequent exact checks; they are not values of certified infinite integrals.

| N | Möbius-only block constant | Unrestricted optimal block constant |
|---:|---:|---:|
| 32 | 0.9636 | 1.1450 |
| 64 | 0.9511 | 1.0600 |
| 128 | 1.0114 | 1.1864 |
| 256 | 0.6803 | 0.9384 |
| 512 | 0.9484 | 1.1928 |
| 1024 | 0.4101 | 0.8085 |

The last row is why the initial pattern could not be promoted to a theorem. I also tested adding directions with coefficients μ(k/d) when d divides k, for d = 1,…,2,4,8. At N = 1024 the corresponding numerical constants were approximately 0.4214, 0.4457 and 0.4841. This finite experiment does not rule out an eventual arithmetic construction, but it does require more than an extrapolation of the first successful blocks.

### An exact obstruction to the Möbius-only update at N = 1024

Let d_(N,μ)² be the minimum error over H_N + span(Φ_N), allowing every real coefficient in H_N and every real scalar multiplying Φ_N. An exact finite-support dual vector proves

\[
 d_{1024,\mu}^2>0.0062508258660.
\]

A separately certified full old-basis approximant gives

\[
 d_{1024}^2<0.0065450610590.
\]

The exact rational gap is positive:

\[
 L_{1024,\mu}-\frac{21}{22}U_{1024}
 >0.0000032675825.
\]

Therefore

\[
 \boxed{d_{1024,\mu}^2>\frac{21}{22}d_{1024}^2.} \tag{36}
\]

This is a counterexample to using the single Möbius block to achieve (29) at j = 10. It is not a counterexample to (29) with the unrestricted next space H₂₀₄₈, and it does not exclude a Möbius rule that might hold only beyond some larger starting index.

The dual vector is supported on the first 65,536 arithmetic cells. As in the previous report, its integer cell masses α_m annihilate every old basis function. They additionally satisfy

\[
 \sum_{k=N+1}^{2N}-\mu(k)
 \left(\frac{S}{k}-\sum_{r\ge1}A_{rk}\right)=0,
 \quad A_k=\sum_{m\ge k}\alpha_m,\quad S=\sum_m m\alpha_m.
\]

This is exactly orthogonality to Φ_N. Both families of equalities are checked using rational arithmetic, after which Cauchy–Schwarz gives the lower bound on the entire restricted subspace. To construct the vector, I choose S divisible by lcm(2,…,2N) and correct one free tail entry A_k with N < k ≤ 2N and μ(k) = ±1 before enforcing the old constraints. The extra relation is preserved because it only involves free entries with indices greater than N and the fixed S.

The upper certificate includes the infinite tail and a head cutoff of 128,000,000 cells. The large cutoff is used solely to resolve the small strict gap in (36). It adds no assumption about larger basis indices or larger j.

## 3. Why a fitted or leading asymptotic does not settle this inequality

Even a nonincreasing sequence with e_j ~ C/j need not obey (29) eventually. An explicit example, for j ≥ 2, is

\[
 a_j=\frac1{2\lfloor j/2\rfloor}.
\]

It is positive, nonincreasing, tends to zero, and satisfies a_j = 1/j + O(j^−2). Nevertheless a_(2m+1) = a_(2m), so its adjacent gain is zero at infinitely many indices. The proposed contraction fails there.

Such behavior is possible for abstract projection errors as well. In an orthonormal basis, choose a target with squared coordinates a_j−a_(j+1); its tail norms are exactly a_j. Thus nested projection geometry, monotonicity and the leading 1/j asymptotic do not alone prevent these plateaus. This example is not a counterexample involving the zeta function.

A useful **sufficient regularity condition** can be proved exactly. If a constant C > 0 and an index J satisfy

\[
 \frac Cj-\frac{C}{4j^2}\le e_j\le
 \frac Cj+\frac{C}{4j^2}\qquad(j\ge J), \tag{35}
\]

then (29) holds for every j ≥ max(J,1). More generally, a second-order expansion

\[
 e_j=C/j+B/j^2+o(j^{-2})
\]

implies (j+1)(1−e_(j+1)/e_j) → 1 and hence the desired inequality eventually.

For (35), substitute the lower envelope at j and the upper envelope at j+1. For any η ≤ 1/4 the exact difference, divided by C, is

\[
 \left(1-\frac1{2(j+1)}\right)
 \left(\frac1j-\frac\eta{j^2}\right)
 -\left(\frac1{j+1}+\frac\eta{(j+1)^2}\right)
 =\frac{(1-4\eta)j^2+(1-3\eta)j-\eta}
 {2j^2(j+1)^2}.
\]

At η = 1/4 this is (j−1)/[8j²(j+1)²], which is nonnegative for j ≥ 1. For the second-order expansion, subtract consecutive terms and divide by e_j to obtain the stated limit.

**Neither (35) nor the stated second-order expansion has been proved for our errors.** Their formulation exposes a sufficient control on the remainder, rather than supplying it. They already force e_j → 0, so assuming them would assume a conclusion strong enough for RH.

## 4. What the literature does and does not supply here

The [Bettin–Conrey–Farmer theorem](https://arxiv.org/abs/1211.5191) gives an optimal asymptotic for particular smoothed Möbius approximants under RH and a condition on reciprocal zeta derivatives. It supplies neither an unconditional proof of (32) nor the remainder control in (35) for these optimal balanced errors. The [Ehm Gram-matrix analysis](https://arxiv.org/abs/2405.06349v2) gives relevant decompositions and explicitly leaves major asymptotic terms to estimate.

I checked these statements against their hypotheses rather than inserting their conclusions into the proposed RH proof. The recent literature reviewed in the earlier reports did not provide the missing lower correlation bound or a suitable unconditional adjacent-scale remainder estimate. No such bound was derived in this attempt.

For SWC, a valid large-scale proof could certify positive mass of new directions that meet (32), with the correct scope or with the divergent improvement budget from the previous report. The elementary mass calculation would then be sound. The unresolved part is the analytic lower bound on those successful directions.

## 5. Two additional finite contractions are certified

After the Möbius-only update failed at N = 1024, I used the unrestricted new-block optimizer to propose a candidate for H₂₀₄₈. Every proposal was then rationalized and passed through the finite-head and infinite-tail arithmetic bounds. Rounding means the N = 1024 candidate generated from the successful N = 512 Möbius update has approximately, rather than exactly, that coefficient shape. Its claimed error and resulting full-space contraction remain checked exactly.

| Transition | Old minimum lower bound L_N | New candidate upper bound U_(2N) | Certified (j+1)(1−U_(2N)/L_N) at least |
|---|---:|---:|---:|
| 512 → 1024, j = 9 | 0.0073399441230 | 0.0067607171897 | 0.7891435 |
| 1024 → 2048, j = 10 | 0.0064928671105 | 0.0061879318472 | 0.5166112 |

The table rounds lower bounds downward and upper bounds upward. The contraction constants use the full stored rational bounds, not these rounded decimals. Together with the earlier seven transitions this proves

\[
 \boxed{e_{j+1}\le\left(1-\frac1{2(j+1)}\right)e_j
 \quad\text{for every }j\in\{2,3,\ldots,10\}.} \tag{37}
\]

For the last upper certificate, the head cutoff is 128,000,000 cells. The 64,000,000-cell certificate attempt had an upper error bound about 0.0062268093 and only certified a normalized constant about 0.45075. The tighter tail bound lowers the final upper error to about 0.00618793185, which is sufficient. The failed 64-million-cell attempt remains recorded as unresolved; it was a limitation of that error enclosure, not a counterexample to (29).

The verifier now processes the divisor sieve in fixed-size segments, so its memory use does not grow with the head cutoff. Segment boundaries were checked against direct residue enumeration, including block sizes smaller than the basis indices. The unchanged exact formulas still use arbitrary-precision integers and rational comparisons.

The meaning of j in (37) is the logarithm of the **basis cutoff**. It is not the number of verified zeta zeros or a list of new accuracy targets. These calculations do not establish the universal quantifier in (29), and the bound 0.00618793185 remains larger than the next dyadic error target 1/256.

## 6. What remains to prove

The extension to sufficiently large j would require a uniform argument for (32), a suitable family of directions replacing the failed simple update, or another theorem giving the required local control of e_j. Neither positivity of the Gram matrices, a fit to the finite data, nor the leading 1/j asymptotic supplies that argument.

This attempt establishes two additional finite cases and an exact obstruction to one proposed transfer rule. It also proves conditional regularity lemmas explaining a possible route from a sufficiently sharp asymptotic to the desired inequality. It does **not** establish the asymptotic or the eventual inequality itself.

The standalone verifiers completed all nine finite contraction checks and the restricted-subspace obstruction check, including recomputation of the two 128-million-cell bounds. The implementation suite has 27 passing tests. The exact finite certificate computations and the analytic arguments are not a proof-assistant formalization or external peer review.

## Reproduction and artifacts

From `/Users/kt/Documents/ChatGPT/mathstuff`:

```bash
python3 research/swc-rh/dual_verify.py
python3 research/swc-rh/mobius_dual_verify.py
python3 -m unittest discover -s research/swc-rh -p 'test_*.py' -v
```

The two verification commands use only the Python standard library. The largest head computations can take several minutes. To regenerate the proposals, with NumPy installed, run these in order:

```bash
python3 research/swc-rh/block_search.py
python3 research/swc-rh/mobius_dual_search.py
```

- [Full finite contraction results](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/contraction-results.json)
- [Uncertified numerical block probes, clearly labeled](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/block-probes.json)
- [Largest accepted upper certificate](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/certificates/nb-N2048.json)
- [Largest unrestricted dual certificate](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/dual-certificates/dual-N1024.json.gz)
- [Restricted-subspace dual certificate](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/block-audit/mobius-dual-N1024.json.gz)
- [Restricted-subspace verifier](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/mobius_dual_verify.py)
- [Exact restricted-rule obstruction result](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/mobius-obstruction.json)
- [Recorded unresolved 64-million-cell enclosure](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/attempts/block-N1024-M64000000.json)
