# SWC energy landscape: admissibility, descent, and the remaining limit

Polished formulation, 7 September 2026. **The original zero minimum and RH remain unproved.** This version incorporates the verified N=4096 run and a new attempt to justify the residual correction using an admissibility penalty.

## 1. Fix the problem before optimizing

Let S_N=span{g_2,...,g_N}, let S be its increasing closed-span limit, and let the target be chi=1. The original energy and limiting defect are

    E_N(F)=||chi-F||_2^2,  F in S_N,
    d_N^2=min_{F in S_N} E_N(F),
    D=lim_N d_N^2=dist(chi,S)^2.

The [Báez-Duarte criterion](https://arxiv.org/abs/math/0202141), in the normalization used throughout this investigation, makes D=0 equivalent to RH. The zero of E_N has a fixed meaning: exact agreement with the target. Subtracting a fitted minimum or changing the allowed class does not preserve that meaning.

In coefficients, E_N(c)=1-2v^T c+c^T Gc, with G positive definite. Completing the square gives

    E_N(c)=d_N^2+||G^(1/2)(c-c*)||^2,   c*=G^-1 v.

Whitening rounds the quadratic geometry and can improve conditioning. It preserves the floor d_N^2. A better optimizer reduces excess energy above this floor; it does not independently establish D=0.

## 2. A valid replacement for the literal reset

For an existing allowed F in S_N and a larger M>=N, put r=chi-F. The ideal allowed correction is

    Delta=P_(S_M) r,
    F_new=F+Delta=P_(S_M) chi.

Orthogonal projection gives

    ||chi-F_new||^2=||r||^2-||P_(S_M)r||^2.

This is a valid residual update entirely within the allowed enlarged basis. In computation, we approximate it with rational coefficients and certify the resulting full error. By contrast, Delta=r is the unrestricted reset and requires the unproved membership r in S.

The recent gradient run supplies an actual finite, admissible replacement: subtracting the old N=2048 coefficients from the new N=4096 coefficients gives Delta in S_4096. The previously checked error bounds prove improvement. It is not the exact infinite-space projection and does not have zero residual.

## 3. New attempt: penalize leaving the space

Allow a general cell function H, but charge for its distance from the original finite space:

    J_lambda(H)=||chi-H||^2+lambda dist(H,S_N)^2,  lambda>0.

Write chi=p+r, where p=P_(S_N)chi and r is orthogonal to S_N, and H=s+z with s in S_N and z perpendicular to S_N. Then

    J_lambda(H)=||p-s||^2+||r-z||^2+lambda||z||^2
       =||p-s||^2+(1+lambda)||z-r/(1+lambda)||^2
          +[lambda/(1+lambda)]d_N^2.

Therefore the minimizer and minimum are exactly

    H*=p+r/(1+lambda),
    min J_lambda=[lambda/(1+lambda)]d_N^2.

At this relaxed minimizer,

    fidelity error ||chi-H*||^2=[lambda/(1+lambda)]^2 d_N^2,
    inadmissibility dist(H*,S_N)^2=[1/(1+lambda)]^2 d_N^2.

Weakening the penalty to zero makes the fidelity error vanish while the inadmissibility approaches d_N^2. Enforcing the constraint by sending lambda to infinity makes inadmissibility vanish while fidelity error returns to d_N^2. For positive lambda, a zero minimum of the penalized problem would already require d_N^2=0. The same decomposition holds with the infinite closed span S and D in place of S_N and d_N^2.

**Outcome of this attempt:** softening admissibility does not prove that the original minimum is zero. It allows the residual correction back into the model. Hardening admissibility restores the unresolved original defect.

## 4. The two quantities a certificate must control

For every H in the ambient Hilbert space, the triangle inequality gives

    dist(chi,S_N) <= ||chi-H|| + dist(H,S_N).

A small raw loss is therefore insufficient when the output has a nonzero component outside the allowed space.

A valid SWC transfer certificate can instead provide an explicit pair (H,F), with F in the allowed finite span, and verified bounds

    ||chi-H||^2 <= a,   ||H-F||^2 <= b.

It then certifies ||chi-F||^2 <= (sqrt(a)+sqrt(b))^2. The accepted mass must apply to the joint verified pair, or be derived from sound conditional-mass bounds. Positivity for two unrelated events does not establish positivity for their intersection. A family with both a and b tending to zero would settle the original approximation problem; no such all-accuracy family has been established here.

## 5. Present numerical status

For the original target at N=4096, independent exact upper and lower certificates give

    0.0045410457 < d_4096^2 < 0.0054968440.

The chosen candidate's error is more tightly enclosed between approximately 0.005439829386 and 0.005496843987. These latter decimals are display approximations; the machine-readable rational endpoints are authoritative. Thus the exact finite optimum is not known, but it is strictly positive and above 1/256. Neither extra gradient iterations nor coordinate preconditioning can violate the lower bound at this same basis size.

The recent modeled loss was approximately 0.005375663273. It is a numerical head-plus-average-tail objective, not the verified full error. The displayed geometry must not treat that number as a certified global minimum.

## 6. Exact geometry used in the interactive explanation

The interactive penalty plot uses the analytically known N=2 case, not an invented landscape for N=4096. Here p=2g_2, r=1-2g_2, and ||r||^2=1-log 2. Along H_t=p+t r,

    fidelity=(1-t)^2(1-log 2),
    inadmissibility=t^2(1-log 2),
    penalized energy=fidelity+lambda*inadmissibility.

Only t=0 is in the original one-function span. At t=1 fidelity is zero, but the entire missing residual has been added. The relaxed minimizer is t=1/(1+lambda). The separate N=4096 panel displays the actual saved certificate intervals and is not a landscape reconstruction.

## Reproduction and scope

`energy_landscape_v2.py` verifies the new scalar identities, checks the saved certificate fingerprints, and records conservative penalty bounds in `energy-landscape-v2-results.json`. It reuses the earlier large computations; it does not rerun 192 million cells or claim a new optimized candidate. `GRADIENT_EXTENSION.md` gives the full error and dual-check procedures.

The earlier energy note is retained as derivation history. The PDF and ZIP remain older snapshots. This formulation is a clearer optimization and verification framework, with an explicit unsuccessful penalty attempt, not an RH proof or a probability assigned to RH.
