# Transferring the clipped neural outputs into the original RH basis

Private continuation of ZERO_MINIMUM_ATTEMPT.md. This attempt constructs and verifies finite transfers. It does not prove transfer with error tending to zero, membership of the required clipped family in the closed span, or RH.

## 1. Actual finite transfers

For h_k(m)=1 when k does not divide m and zero otherwise, I fitted finite original-basis combinations F=sum_{i=2}^N c_i g_i. Numerical least squares on 32,768 arithmetic cells proposed the coefficients. They were rationalized as c_i=i b_i/2^40, then checked using integer arithmetic on one million cells and a proved bound on the entire remaining tail.

The following are conservative upper bounds for **squared transfer error ||h_k-F||_2^2**:

| Largest basis index N | Target h_3 | Target h_8 | Target h_32 |
|---|---:|---:|---:|
| 16 | 0.021410234 | 0.021298023 | 0.020095238 |
| 64 | 0.014343561 | 0.013678673 | 0.012768943 |
| 256 | 0.010901797 | 0.010365750 | 0.009098175 |

These are genuine allowed finite combinations approximating the clipped functions. They are not errors for the original constant target and do not improve the previously saved bound for that target. The displayed decrease supplies finite examples, not a theorem about arbitrarily large N or k.

The same checker revalidates existing finite-support dual annihilators and pairs them with each new target. This gives lower bounds for the best possible error at the specified N, separately from lower bounds for the error of the chosen candidate. All exact values, candidate parameters, and dependency fingerprints are retained in clipped-transfer-candidates.json and clipped-transfer-results.json.

## 2. Exact certificate for the changed target

Write u_i(m)=(m mod i)/i and F=sum c_i u_i. Let mu_1 be the existing exact periodic mean of (1-F)^2. The mean for the clipped target is

    mu_k = mu_1 - 1/k + sum_i b_i(i-gcd(i,k))/(D k),

where c_i=i b_i/D. Indeed h_k=1-I_k, with I_k=1_(k divides m), so

    (h_k-F)^2 = (1-F)^2 - I_k + 2 I_k F.

Over a full common period, E[I_k]=1/k. Conditional on I_k=1, the residue m mod i is uniform over the multiples of gcd(i,k), giving E[I_k u_i]=(i-gcd(i,k))/(2ik). This proves the mean identity.

Let K_1 be the previous partial-sum discrepancy bound for (1-F)^2. A valid bound for centered partial sums of the clipped residual square is

    K_k = K_1 + k
          + (2/D) sum_i |b_i|(lcm(i,k)-i)(i-1).

To verify this, expand h_k^2-2h_k F+F^2. The h_k term has period k and absolute bound one. Each mixed product h_k u_i has period lcm(i,k) and absolute bound (i-1)/i. The quadratic F^2 terms use the unchanged earlier bound. A centered periodic term has partial sums bounded by its period times its range. Summation by parts then gives the infinite-tail enclosure

    |sum_{m>M} (h_k(m)-F(m))^2/[m(m+1)] - mu_k/(M+1)|
        <= K_k/[(M+1)(M+2)].

The exact head is evaluated using the earlier integer residual sieve, subtracting D at every multiple of k. Each term is rounded outward to the common grid 2^-96. A separate pointwise amplitude bound is also available; the checker uses the smaller upper tail bound.

For a finite-support integer dual vector alpha annihilating every original g_i through N, Cauchy-Schwarz gives

    inf_F ||h_k-F||_2^2 >=
       (sum_{m: k does not divide m} alpha_m)^2 /
       sum_m m(m+1) alpha_m^2.

Orthogonality is rechecked exactly. No floating-point estimate is used as a proof premise in these bounds.

## 3. An exact head-matching construction was also tested

For target h_k and N>=2, define a_i=Lambda_i(h_k) for 2<=i<N, using the earlier Möbius coefficient-extraction functional. Set

    c_i=a_i for i<N,
    c_N=N(1-sum_{i=2}^{N-1} a_i/i).

The first-cell slope is one, and the coefficient identities give F_m=h_k(m) for every 1<=m<N. Thus this sequence converges pointwise to h_k as N grows. Coefficients and the matching identities were checked rationally.

For h_3 the whole-domain squared errors satisfy the following conservative intervals:

| N | Exact cells matched | Squared error interval |
|---|---:|---:|
| 16 | 15 | (0.05473425, 0.05473439) |
| 64 | 63 | (0.05551771, 0.05555646) |
| 256 | 255 | (0.05932572, 0.06529911) |

The exact lower bound at each later listed size exceeds the exact upper bound at the preceding size. Therefore this particular sequence does not have monotonically improving norm error. This finite comparison does not prove eventual nonconvergence, but it invalidates an induction claiming that matching more cells automatically reduces the total error.

Pointwise convergence alone does not justify the required L2 limit. The general danger is already documented for related Beurling approximation sequences in [Báez-Duarte's arithmetical analysis](https://arxiv.org/abs/math/0011254).

## 4. A one-cell shift exposes the missing closure assumption

Consider the bounded backward shift R on cell values:

    (RF)_1=0,    (RF)_m=F_(m-1) for m>=2.

It is contractive since

    ||RF||_2^2 = sum_{n>=1} |F_n|^2/[(n+1)(n+2)] <= ||F||_2^2.

For the allowed function g_2, an exact identity is

    Rg_2 = (1/2)1 - g_2.

If S is the original closed span, it follows immediately that

    dist(Rg_2,S) = (1/2)dist(1,S).

Thus the seemingly mild assertion Rg_2 belongs to S is already equivalent to RH in the [established approximation formulation](https://arxiv.org/abs/math/0202141). Boundedness or contraction of an AI operation on the ambient space does not establish that it preserves the required subspace. Assuming unrestricted shift closure would insert the unresolved conclusion into the argument.

## 5. An analytic condition for transferring h_3

There is a more arithmetic description of the obstacle. For complex s with Re(s)>0, use the Mellin transform M f(s)=integral_0^1 f(x)x^(s-1) dx. A cellwise summation, first absolutely justified for Re(s)>1 and then continued analytically, gives

    M g_i(s) = [zeta(s)/s](1/i-i^-s).

Let chi_3(n) be zero on multiples of 3, one on n=1 mod 3, and minus one on n=2 mod 3; write L_3(s)=sum chi_3(n)n^-s. Bounded partial sums make this analytic for Re(s)>0. The jumps of h_3 are +1 at n=1 mod 3 and -1 at n=0 mod 3. Consequently

    M h_3(s) = [(1-3^(1-s)) zeta(s)+L_3(s)]/(2s).

The apparent pole at s=1 is removable. Now suppose rho=beta+i gamma is a hypothetical zero of zeta with beta>1/2. Mellin evaluation at rho is a bounded functional on L2(0,1), of norm 1/sqrt(2 beta-1). It vanishes on every allowed combination, hence on S. Therefore

    dist(h_3,S)^2 >= (2 beta-1)|L_3(rho)|^2/(4|rho|^2).

In particular, transferring h_3 into S would require L_3(rho)=0 at every such zeta zero (and further conditions would be needed for a converse). No off-line zeta zero is asserted to exist here, and no assertion is made that common zeros of these two functions are impossible. This is a conditional necessary condition, not a proof or disproof of RH.

## 6. What remains

The distance to a closed set is 1-Lipschitz. Since ||1-h_k||_2<sqrt(2)/k,

    |dist(h_k,S)-dist(1,S)| < sqrt(2)/k.

Thus the limiting transfer distance is exactly the original unknown approximation distance. To finish this route, one must obtain original-basis approximants F_k with ||h_k-F_k||_2 tending to zero along an unbounded sequence of k. The finite fits, exact head interpolation, and contractive-shift argument did not establish that statement.

The concrete outcome is twelve verified finite candidates, a checker for the clipped-target infinite tail, and explicit limitations of two proposed transfer shortcuts. The zero minimum and RH remain unresolved.

## Reproduction

Run `python3 research/swc-rh/clipped_transfer.py --verify` to check the saved rational candidates and the old dual inputs. This uses the standard library. Use `--produce` to regenerate numerical proposals with NumPy; `--produce --verify` does both. Without either flag, the script only performs the small exact algebra checks.

The original PDF and supplement ZIP remain earlier snapshots. No result in this note assigns a statistical probability to RH or claims a new RH breakthrough.
