# Second SWC attempt: certify progress, then seek an infinite argument

6 September 2026. **RH remains unresolved.** This continues the [first report](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/REPORT.md) in response to the request to prove the missing step.

The [next attempt](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/LARGE_J.md) extends the finite contraction range through j = 10 and audits a proposed Möbius block update. The eventual contraction remains unproved.

This continuation proves three kinds of intermediate result: exact lower bounds on finite-basis optima, the proposed contraction estimate at seven finite transitions, and an unconditional obstruction to geometric-only witness support. It also proves a sufficient SWC condition based on accumulated mass of certified improvements, and rules out a natural finite dilation recursion. The condition needed at infinitely many scales has not been established.

No novelty priority or proof-assistant formalization is claimed for these results.

## 1. The previous computations were not bounds on the optima

Keep

\[
 g_k(x)=\{1/(kx)\}-k^{-1}\{1/x\},\quad
 d_N^2=\min_{c_2,\ldots,c_N}\left\|1-\sum_{k=2}^N c_k g_k\right\|_2^2,
 \quad e_j=d_{2^j}^2.
\]

The minimum exists because this is distance to a finite-dimensional subspace. The first report certified errors of particular coefficient vectors. An upper bound for one vector's error is an upper bound on d_N², but a lower bound for that same vector's error is **not** a lower bound on d_N². Inferring contraction of the optima from two such vector enclosures would be invalid. The first report did not make that inference; the distinction becomes essential when trying to prove its proposed estimate.

Here is an exact dual certificate that supplies the missing lower bound.

Let I_m = (1/(m+1),1/m), of length w_m = 1/[m(m+1)]. Choose finitely many integers α₁,…,α_M and define h(x) = α_m/w_m on I_m, and zero elsewhere. If

\[
 \sum_{m=1}^M\alpha_m(m\bmod k)=0
 \quad\text{for every }2\le k\le N, \tag{17}
\]

then h is orthogonal to every g_k in the basis. Hence every candidate F satisfies

\[
 \langle 1-F,h\rangle=\sum_m\alpha_m,\qquad
 \|h\|_2^2=\sum_m m(m+1)\alpha_m^2.
\]

Cauchy–Schwarz proves

\[
 \boxed{d_N^2\ge
 L(\alpha):=\frac{(\sum_m\alpha_m)^2}
 {\sum_m m(m+1)\alpha_m^2}.} \tag{18}
\]

This proves a statement about **all** coefficient vectors in the declared finite basis. There is no infinite-tail approximation in this lower bound: the test function h is exactly zero outside its stated support.

### Producing exact orthogonality without trusting a floating-point inverse

Let A_k = Σ_(m≥k) α_m and S = Σ_m mα_m. Summing the floor function by levels shows that (17) is equivalent to

\[
 S=k\sum_{r\ge1} A_{rk}\quad(2\le k\le N). \tag{19}
\]

Choose S divisible by lcm(2,…,N), and choose the integers A_(N+1),…,A_M freely. Descending from k = N to 2, set

\[
 A_k=S/k-\sum_{r\ge2}A_{rk}.
\]

Then set A₁ = S−Σ_(k=2)^M A_k, A_(M+1) = 0, and α_m = A_m−A_(m+1). This construction enforces both S = Σ_m mα_m and all constraints (19), exactly.

The producer uses an approximate least-squares residual only to choose the free tail entries. The separate checker reconstructs cumulative sums from the submitted integer α vector and checks (19). Its acceptance uses neither the producer's inverse matrix nor a floating-point tolerance.

## 2. The proposed contraction is now proved at seven transitions

The dual certificates use M = 8192 cells. Combining their lower bounds L_N with the first report's upper bounds U_(2N) gives

\[
 d_{2N}^2\le U_{2N}\le\frac{U_{2N}}{L_N}\,d_N^2.
\]

For N = 2^j define the certified number

\[
 \kappa_j=(j+1)\left(1-\frac{U_{2N}}{L_N}\right).
\]

All seven saved certificates satisfy κ_j > 1/2. Consequently the exact checker proves

\[
 \boxed{e_{j+1}\le\left(1-\frac{1}{2(j+1)}\right)e_j
 \quad\text{for }j=2,3,4,5,6,7,8.} \tag{20}
\]

This is a finite theorem about the true optimal errors, stronger than reporting decreasing errors for selected approximants. Decimal bounds in the table are rounded conservatively; the checker uses the exact rational values.

| N → 2N | Lower bound for d_N² | Upper bound for d_(2N)² | Certified κ_j at least |
|---|---:|---:|---:|
| 4 → 8 | 0.065923047990 | 0.024244538801 | 1.896689 |
| 8 → 16 | 0.024225041717 | 0.017936393392 | 1.038371 |
| 16 → 32 | 0.017915442004 | 0.014078439170 | 1.070864 |
| 32 → 64 | 0.014050196872 | 0.011408758282 | 1.128000 |
| 64 → 128 | 0.011355611076 | 0.009703283589 | 1.018553 |
| 128 → 256 | 0.009614837599 | 0.008274944905 | 1.114854 |
| 256 → 512 | 0.008136924940 | 0.007525030344 | 0.676797 |

The finite list does not imply (20) for j ≥ 9. Neither an induction transformer nor a bound uniform over those later scales has been proved.

There is also an asymptotic constraint on the proposed rate. The established lower bound of [Burnol](https://arxiv.org/abs/math/0103058) implies d_N² ≥ c/log N for some c > 0 and all sufficiently large N; restricting to balanced approximants cannot decrease this lower bound. Thus e_j ≥ c′/j eventually. An eventual contraction with a constant κ > 1 would instead give e_j = O(j^−κ), a contradiction. The large finite κ values in the table therefore cannot be extrapolated as an eventual rate above 1. The proposed κ = 1/2 is not excluded by this argument.

## 3. A weaker infinite target expressed directly in SWC mass

Requiring a fixed contraction at every later scale is stronger than necessary. A rigorous accumulation rule permits gaps.

For every finite N, let L = lcm(2,…,N). On I_L every basis function is zero, so every approximant has squared error at least 1/[L(L+1)]. Therefore e_j > 0. Monotonicity of the approximation spaces also gives e_(j+1) ≤ e_j. Define

\[
 q_j=1-\frac{e_{j+1}}{e_j}\in[0,1).
\]

**Accumulated-gain criterion.**

\[
 \boxed{\mathrm{RH}\quad\Longleftrightarrow\quad
 \sum_{j=2}^{\infty}q_j=\infty.} \tag{21}
\]

**Proof.** The approximation criterion makes RH equivalent to e_j → 0, since the dyadic sequence is cofinal. Also e_J/e₂ = Π_(j=2)^(J−1)(1−q_j). If Σq_j diverges, log(1−q_j) ≤ −q_j forces this product to zero. If Σq_j converges, then eventually q_j ≤ 1/2 and −log(1−q_j) ≤ 2q_j, so the product has a positive limit. ∎

This elementary infinite-product equivalence exposes exactly what finite progress lacks. The q_j are unknown true gains; (21) is not an evaluation of their sum.

**SWC block-mass sufficient condition.** Fix κ > 0. For sufficiently large integers r, let

\[
 B_r=\{2^r,\ldots,2^{r+1}-1\}.
\]

Suppose an SWC generator first chooses j uniformly from B_r, then proposes a certificate whose soundness implies q_j ≥ κ/(j+1). Suppose its **proved**, not estimated, accepted probability is at least δ_r and

\[
 \sum_r\delta_r=\infty. \tag{22}
\]

Then RH holds.

**Proof.** An accepted pair can occur only at a good index j. Since the conditional probability of acceptance at any j is at most 1, there must be at least δ_r|B_r| good indices. Each has 1/(j+1) ≥ 2^−(r+1), hence

\[
 \sum_{j\in B_r}q_j\ge
 \frac{\kappa\delta_r|B_r|}{2^{r+1}}
 =\frac{\kappa\delta_r}{2}.
\]

Summing disjoint blocks and applying (21) proves the claim. ∎

This gives SWC a precise alternative to proving every individual transition: prove enough total mass of certified improvements. Pointwise positivity δ_r > 0 alone is insufficient; for example δ_r = 2^−r has a finite sum. The analytic accumulation proof is what permits a density or mass statement here to imply the universal conclusion. There is no analogous implication from a density-one count of critical-line zeros alone.

Our seven finite transitions certify all indices in B₁ and B₂ and one index in B₃. This supplies block acceptance masses 1, 1 and 1/8 for a generator that returns the stored certificate when available. It supplies no divergent series in (22). No positive lower mass bound at all later blocks has been established.

## 4. A proved obstruction to geometric-only support

Consider a shortcut that permits basis indices 2,4,8,… only. This must not be confused with the experiments above: a cutoff N = 2^j there includes **every** integer k from 2 through N.

**Geometric-support obstruction.** For any integer b ≥ 2 and any finite coefficients,

\[
 \boxed{\left\|1-\sum_{j=1}^Jc_jg_{b^j}\right\|_2^2
 \ge\frac{1}{2((b+1)^2+1)}.} \tag{23}
\]

In particular, a power-of-two-only sampler has zero accepted mass at every strict error target ε ≤ 1/20, regardless of how its coefficients are trained, weighted or conditioned.

**Proof.** Choose dual cell masses +1,+1,−1 at m = 1,b,b+1, respectively, and zero elsewhere. For k = b, their residues are 1,0,1, which cancel. For k = b^j with j ≥ 2, all three cell indices are less than k, and the cancellation is 1+b−(b+1) = 0. Thus the corresponding dual function annihilates every allowed basis function, including all later powers. Its integral is 1 and its squared norm is

\[
 2+b(b+1)+(b+1)(b+2)=2((b+1)^2+1).
\]

Cauchy–Schwarz gives (23). ∎

For b = 2 the dual function has the values 2,6,−12 on I₁,I₂,I₃. This is an exact finite certificate for an obstruction to an entire infinite support family. It does not obstruct SWC with the full integer basis: g₃ already fails the annihilation relation, and the unrestricted first-report witnesses have errors below 1/20.

## 5. Attempted recursive proof: the finite dilation identity cannot exist

SWC's reduction and transport rules suggest looking for a contraction that transforms an existing approximation into a better one. Extend the g_k and χ = 1_(0,1) by zero on x > 1. Define D_k f(x) = f(kx). Then

\[
 \|D_k f\|_2=k^{-1/2}\|f\|_2,\qquad
 D_k g_n=g_{kn}-\frac1n g_k. \tag{24}
\]

The second identity follows immediately by expanding the fractional parts. It shows that integer dilations preserve the closure of the integer basis.

A tempting proof would find a finite basis combination F and finitely many numbers a_k, k ≥ 2, such that

\[
 \chi=F+T\chi,\qquad T=\sum_{k=2}^K a_kD_k,
 \qquad\sum_{k=2}^K\frac{|a_k|}{\sqrt k}<1. \tag{25}
\]

If this held, the convergent Neumann expansion χ = Σ_(r≥0) T^rF would put χ in the required closed span and prove RH. This would be a concrete recursive witness transport.

**Obstruction.** There is no identity χ = F + Σ_(k=2)^K a_kD_kχ with F any finite integer-basis combination, even without the last inequality in (25).

**Proof.** On cell I_m, F is a periodic function of the integer m, with period dividing the least common multiple of its basis indices. For all m ≥ K, every D_kχ equals 1. The asserted identity therefore makes F(m) equal to the constant 1−Σa_k on all sufficiently large m. At large multiples of the common period all basis functions g_n vanish, so this constant is zero. Periodicity now forces F(m) = 0 on every cell. On I₁, however, each D_kχ with k ≥ 2 is zero, so the asserted identity becomes 1 = 0. ∎

Allowing an infinite dilation expansion avoids the finite-identity contradiction, but does not automatically give a norm contraction. For example, 1−2g₂ has alternating cell values 0,1,0,1,…, giving the pointwise expansion

\[
 \chi-2g_2=\sum_{k=2}^{\infty}(-1)^kD_k\chi.
\]

At each x > 0 only finitely many terms on the right are nonzero. Nevertheless Σ_(k≥2) k^−1/2 diverges, so the absolute operator-norm bound required for the proposed contraction is unavailable. Pointwise finiteness does not establish convergence in operator norm. More generally, a nonconstant periodic residual produces nonzero periodic differences as dilation coefficients, again defeating this absolute bound.

This rules out this specific recursive proof architecture. It does not rule out approximate recursions with a controlled remainder, scale-dependent operators or other SWC transports. Those would require additional estimates.

## 6. Attempted analytic bridge: the reciprocal-zeta terms remain uncontrolled

The error can also be written on the critical line. Set

\[
 P_c(s)=\sum_{k=2}^N c_k(k^{-1}-k^{-s}).
\]

The Mellin identity and Plancherel give

\[
 E(c)=\frac1{2\pi}\int_{-\infty}^{\infty}
 \left|1-\zeta(1/2+it)P_c(1/2+it)\right|^2
 \frac{dt}{1/4+t^2}. \tag{26}
\]

Thus a possible analytic route is a controlled approximation to 1/ζ by Dirichlet polynomials in this weighted norm. The uniform control is essential; simply writing a formal reciprocal is not enough.

Consider the logarithmically smoothed Möbius polynomial

\[
 V_N(s)=\sum_{n\le N}\mu(n)
 \left(1-\frac{\log n}{\log N}\right)n^{-s}.
\]

For N > 1, Re(s) = 1/2, and c > 1/2, absolute convergence on the initial contour gives

\[
 V_N(s)=\frac1{2\pi i\log N}
 \int_{c-i\infty}^{c+i\infty}
 \frac{N^w}{\zeta(s+w)w^2}\,dw. \tag{27}
\]

This follows by substituting the absolutely convergent Dirichlet series for 1/ζ(s+w) and applying the elementary Perron integral with kernel w^−2. To fit (26), one may balance this polynomial by replacing it with V_N(s)−V_N(1).

Now attempt to shift the contour in (27). At w = 0, when ζ(s) ≠ 0, the residue of the normalized integrand is

\[
 \frac1{\zeta(s)}-
 \frac{\zeta'(s)}{\zeta(s)^2\log N}.
\]

At a simple zero ρ ≠ s crossed by the contour, the residue is

\[
 \frac{N^{\rho-s}}
 {\zeta'(\rho)(\rho-s)^2\log N}. \tag{28}
\]

These are local residue calculations. They do not assert convergence of an unrestricted sum over zeros; a finite rectangular contour still has its boundary integrals, and all limiting operations require estimates. Multiple zeros introduce higher-order poles.

The obstruction is explicit: a possible zero with Re(ρ) > 1/2 contributes the factor N^(Re(ρ)−1/2) in (28). Such terms cannot be dropped as decaying errors. Even with every zero on the line, controlling the aggregate terms requires information involving reciprocal derivatives of ζ. An estimate for the frequency of large values of ζ or Dirichlet polynomials does not by itself bound these reciprocal quantities near zeros.

The pertinent theorem of [Bettin, Conrey and Farmer](https://arxiv.org/abs/1211.5191) proves the optimal asymptotic for V_N under **both RH and** a growth condition on Σ|ζ′(ρ)|^−2. Its hypotheses therefore cannot be imported to prove RH here. The related [Ehm Gram-matrix analysis](https://arxiv.org/abs/2405.06349v2) gives useful exact decompositions, while explicitly leaving difficult Möbius-weighted inversion and asymptotic terms to estimate. This continuation does not treat those terms as zero or assume their required bounds.

The 2026 literature checked in the first report remains relevant context, but none of the inspected statements provides the missing estimates in (22), (25) or this contour argument without an additional assumption. No such estimate was derived in this continuation.

## What is proved, and what is still missing

| Statement | Status |
|---|---|
| Exact finite upper error and SWC mass certificates from the first report | Checked; preserved |
| Dual lower bound (18), including its arithmetic verification rule | Proved above; implemented and checked |
| Contraction with κ = 1/2 at j = 2,…,8 | Seven exact certificates verified |
| Accumulated-gain equivalence (21) | Proved above from the established approximation criterion |
| Block-mass condition (22) implies RH | Conditional theorem proved above |
| Geometric-support obstruction (23) | Unconditional theorem proved above |
| Impossibility of the finite dilation identity in (25) | Unconditional theorem proved above |
| Divergent accepted-mass budget, or contraction at all later scales | **Not proved** |
| Riemann hypothesis | **Not proved** |

The first report's infinite-tail and mass proofs were rechecked algebraically during this continuation; no mathematical error was found in those derivations. The stronger distinction between selected errors and optimal errors is now handled by the dual certificates. The combined suite has 20 passing tests. Independent direct residue sums check a generated dual witness, and altered coefficients, inflated lower bounds and enlarged scopes are rejected. These implementation checks do not constitute external review or a proof-assistant formalization of the analytic theorems.

The unresolved work is now concentrated on a precise alternative: establish a divergent budget of **certified** normalized improvements, or supply another analytic mechanism forcing the optimal error to zero. The stored finite certificates do neither on their own.

## Reproduce this continuation

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

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

Both verification commands use only the standard library. The optional producer requires NumPy:

```bash
python3 research/swc-rh/dual_search.py
```

- [Dual verifier and finite contraction checker](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/dual_verify.py)
- [Dual certificate producer](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/dual_search.py)
- [Exact finite contraction results](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/contraction-results.json)
- [Additional dual and support tests](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/test_dual.py)
- [Largest dual certificate](/Users/kt/Documents/ChatGPT/mathstuff/research/swc-rh/dual-certificates/dual-N0256.json.gz)
