# Fixed Grid String Lab 0.2.0 — bounded results

The release supplies a frozen radial Coulomb benchmark and corrects two
foundational interpretations. Its results are conditional mathematics and
finite computation. There are **zero fitted parameters and no measured
validation data**. The physical promotion gate remains blocked.

Protocol SHA-256:

    af00364dd565503bb6c3fb256eb7c36e16e00f7bb04aac77a5b624bf6eda619a

## Corrected memory and stochastic claims

For a finite deterministic system, let \(s_y\) be the actual number of
future-equivalence classes inside observation fiber \(F_y\). With the present
observation retained, the minimum fixed-width hidden register is

\[
b_{\min}=\left\lceil\log_2\max_y s_y\right\rceil.
\]

A loose upper bound gives a sufficient capacity, not necessarily the minimum.
The checked four-state identity map with constant observation has one future
class and needs zero hidden bits, although four is an upper bound. A second
example has fiber class counts \(1,2\): examining only the first fiber misses
the required global maximum and the one-bit requirement.

For a uniformly initialized four-cycle observed as \(0,0,1,1\), both current
observations give \(P(Y_{t+1}=1\mid Y_t)=1/2\). However,
\(P(Y_{t+1}=1\mid Y_{t-1}Y_t=00)=1\), whereas the value for history \(10\)
is zero. The actual probability of observing one two steps after zero is one;
iterating the one-step kernel gives \(1/2\). This exact counterexample clarifies
that one-step conditionals alone do not establish autonomous Markov closure.

## Frozen spatial result

In atomic units the finite matrix has diagonal \(1/a^2-1/(ja)\),
off-diagonal \(-1/(2a^2)\), and Dirichlet boundaries at zero and \(R\).
The exact negative spectrum of the corresponding infinite half-line operator is

\[
E_n(a)=-\frac{1}{n^2(1+\sqrt{1+(a/n)^2})}.
\]

The derivation and prior-work attribution are in
[FOUNDATION_DERIVATION.md](FOUNDATION_DERIVATION.md). These exact half-line
energies are analytical references, not exact finite-box eigenvalues or
measured hydrogen energies. The leading continuum error is \(a^2/(8n^4)\).

| Spacing \(a/a_0\) | Half-line \(E_1\), Hartree | Half-line gap-ratio shift from \(32/27\), ppm |
|---:|---:|---:|
| 0.40 | -0.481456008918130 | 5251.576580 |
| 0.20 | -0.495097567963924 | 1368.828470 |
| 0.10 | -0.498756211208903 | 345.950407 |
| 0.05 | -0.499687890015715 | 86.725779 |

The ratio is \((E_3-E_1)/(E_2-E_1)\). At the primary spacing and \(R=160a_0\),
the computed ratio is 1.186807500408563. Across the four spacings and first
four levels at that radius, the largest production discrepancy from the
half-line reference is \(4.7463\times10^{-14}\) Hartree, below the frozen
\(10^{-8}\) threshold. The largest eigenvector residual across all eight
grids is \(1.3491\times10^{-13}\), below \(10^{-9}\).

The fourth-level change between \(R=80a_0\) and \(160a_0\) is approximately
\(1.22\)–\(1.29\times10^{-9}\) Hartree. Two box sizes alone are not a proof of an
infinite-domain error bound. The half-line formula permits a direct model
comparison; decreasing spatial error is not evidence for physical discreteness.

## Time equivalence and alias control

For this finite Hamiltonian, \(U=\exp(-i\tau H)\) gives
\(U^m=\exp(-im\tau H)\). These are exact samples of continuous-time evolution
of the **same finite matrix**, not exact continuum-space hydrogen dynamics.
At \(\tau=0.1\), over 129 ticks, production norm drift is
\(8.993\times10^{-15}\), and the largest amplitude difference between the two
evaluations is \(1.296\times10^{-15}\).

Adding \(2\pi/\tau\) to one eigenvalue leaves all integer-tick evolution
unchanged, while changing intermediate-time predictions. For the equal
four-level packet, the half-tick fidelity is 0.25 and radial total variation
is approximately 0.1537764. This alias need not retain the Coulomb stencil:
it demonstrates what unrestricted tick observations cannot identify.

## Independent replay and claim ceiling

The saved independent audit reports **102 checks passed, zero failed**. It
imports no production functions and uses scalar extended-precision Sturm-count
bisection, exact finite counterexamples, and a separate small dense evolution.
Its largest finite-spectrum disagreement with production is
\(4.5853\times10^{-14}\) Hartree. Floating-point brackets are not formally
certified interval arithmetic. Shared equations and a shared runtime remain
common dependencies; this is implementation independence, not physical replication.

The complete launcher passed all **23 unit tests** and the **102 independent
audit checks**. The executed log is `outputs/foundation_release_run.txt`;
`outputs/foundation_tests.txt` records the separate unit-suite run.

The radial S-wave, Coulomb and Born assumptions are imported quantum mechanics.
No five-dimensional string interaction, complete gauge dynamics, optical
measurement likelihood or precision-QED spectroscopy validation is supplied.
The new spatial/time parameters remain benchmark settings. Existing tick,
projection and historical corrections remain intact under
[FOUNDATION_CONTRACT.md](FOUNDATION_CONTRACT.md).

The immediate next bounded test is to add the \(l=1\) radial sector with its
declared centrifugal term and a dipole operator. Before running, freeze the
1S–2P transition-strength target, selection-rule checks, spacings, boundaries,
analytic nonrelativistic references and tolerances. This develops a concrete
measurement interaction within imported quantum mechanics; it does not derive
that interaction from higher-dimensional strings.

The later physical gate requires a fixed distinct model, conventional precision
corrections, an explicit observation likelihood, a numerical contrast and a
frozen rejection rule on unused measured validation. Passing the next
implementation benchmark cannot replace that gate.