# Foundation contract — Fixed Grid String Lab 0.2.0

This contract governs the interpretation of `FOUNDATION_PROTOCOL_0.2.json` and
the accompanying Theory of Reality revision. It establishes requirements for
claims; it does not certify every historical proof, implementation or dataset.
New interpretations must preserve the distinctions below.

## 1. Five objects that must remain separate

| Object | Required content | What success establishes |
|---|---|---|
| Mathematical representation | Typed sets, observables, projections, measures and stated assumptions | Conditional relationships between defined objects |
| Fixed physical model | A bounded domain, physical states, units, dynamics, interactions and parameters | A candidate description with consequences that can be tested |
| Measurement likelihood | Preparation, intervention, detector response, accessible history, noise and nuisance parameters | Probabilities for recorded observations under the candidate |
| Implementation | Algorithms, finite precision, boundaries, numerical tolerances and reproducible inputs | Agreement of executed calculations with specified mathematics within checked limits |
| Evidence | Provenance, uncertainties, calibration/validation separation and declared decisions | Support or rejection within the tested domain and comparison class |

Mathematical validity (**M**), computation (**C**), measurement (**E**), physical
hypothesis (**H**) and theological interpretation (**T**) retain their separate
status. A result may contain several types; each component must be identified.
Neither a test count nor an independent implementation is independent physical
replication. Imported measured inputs are not predictions of those inputs.

The seven descriptive roles organize records; they are not seven axioms of
nature or seven spatial axes. Prefix coherence and terminal-diagram completeness
hold for any suitable finite tower. H7 requires an independently specified
physical domain and faithful representation; finite success cannot establish
universal terminal completeness. Record size, field order and checksums supply
no physical dimensional evidence.

## 2. Dynamics and observational sufficiency

The deterministic descent theorem remains conditional: an autonomous observed
map exists exactly when projected successors are constant on each observation
fiber. Finite partition refinement computes sufficient distinctions for the
specified transition family. It does not recover nature's complete transition
table. The fixed-width hidden-register bound uses the **maximum actual number**
of future-equivalence classes in any original fiber, not an arbitrary upper
bound substituted for that maximum.

A weighted one-step conditional need not define autonomous stochastic dynamics.
Let \(X_t\in\{0,1,2,3\}\), \(X_{t+1}=X_t+1\pmod4\), with uniformly distributed
initial phase, and let \(Y(0)=Y(1)=0\), \(Y(2)=Y(3)=1\). This stationary process
has visible cycle \(0,0,1,1\). Its one-step kernel is exactly

\[
P(Y_{t+1}=b\mid Y_t=a)=\tfrac12\qquad(a,b\in\{0,1\}).
\]

Nevertheless,

\[
P(Y_{t+1}=1\mid Y_{t-1}=0,Y_t=0)=1,
\qquad
P(Y_{t+1}=1\mid Y_{t-1}=1,Y_t=0)=0.
\]

Both conditioning events have probability \(1/4\). Iterating the one-step kernel
therefore loses predictive history. A stochastic closure claim must supply
conditions ensuring the required conditional independence, or retain the
relevant memory. Stationary weights alone do not suffice.

Ensemble experiments compare distributions, not equalities between arbitrary
individual states. A protocol must declare the joint accessible observation
record, histories, preparation labels and interventions, plus tolerances,
measurement uncertainty and statistical decision rule. Equal separate
marginals do not guarantee equal joint distributions. Different future
distributions under those controls establish insufficiency relative to that
record; they do not identify one missing mechanism.

## 3. Computational spacing and the frozen benchmark

Computational \(\Delta x\) and \(\Delta t\) specify representation or evaluation
resolution. Physical spacings would be parameters of a fixed physical law with
identifiable consequences. Decreasing numerical error does not establish a
physical lattice, and a passing tiny tick is not evidence of quantized time.

The frozen benchmark is the radial Coulomb S-wave finite-difference Hamiltonian
named `radial-coulomb-dirichlet-v1`. It imports nonrelativistic Schrödinger
dynamics, a fixed point proton, Coulomb potential, complex amplitudes and Born
probabilities. Its units, grid, boundaries, reference spectrum, comparisons and
tolerances are fixed in the protocol. It fits no parameters and consumes no
measured validation data.

This benchmark is not derived from five-dimensional strings. Its radial mesh
is not a translationally invariant three-dimensional spatial lattice. Energy
gap ratios are numerical comparisons; predicting observed spectral lines also
requires the measurement interaction and relevant selection rules. Finite-box,
spatial discretization and numerical errors must be distinguished from physical
corrections.

For the same finite Hamiltonian \(H\),
\(U=\exp(-i\tau H)\) satisfies \(U^m=\exp(-im\tau H)\) exactly in mathematics.
Fixed ticks therefore sample an existing continuous-time evolution of that
matrix. This is not exact continuum-space hydrogen dynamics and does not
identify a fundamental time tick. Floating-point agreement checks the
implementation of this identity within tolerance.

## 4. Physical promotion requires a separate gate

A residual is a discrepancy relative to a model and observation procedure.
An unrestricted term capable of fitting every discrepancy supplies no causal
identification. It must never be promoted to a force, hidden dimension, God or
other mechanism without independent restrictions and discriminating evidence.

Before a physical promotion test, freeze all of the following:

1. One model identity, physical domain, equations, units and allowed parameters,
   with a matched conventional model and relevant competing explanations.
2. An explicit measurement map or likelihood, including detector response,
   preparation uncertainty, nuisance correlations and accessible history.
3. A **numerical contrast** between candidate and conventional predictions for
   a named measured observable, with uncertainty and sufficient resolving power.
   Exact observational equivalence blocks discrimination through that observer.
4. A calibration split and genuinely unused measured validation; no fitting or
   model repair after opening validation. Declare no-fitting cases explicitly.
5. A precision budget including numerical errors and relevant standard physical
   corrections. Spectroscopy requires the appropriate reduced-mass, relativistic,
   radiative and nuclear contributions before attributing residuals to new physics.
6. A scoring and rejection rule, thresholds, search multiplicity and stopping
   policy. Preserve failed outcomes; revisions require a new protocol and fresh
   validation. Specify an independent replication criterion for positive claims.

Statuses progress only when their own evidence warrants it: defined model,
verified calculation, identifiable test, measured comparison, replicated scoped
support. There is no automatic promotion between stages and no final stage
that deductively proves unrestricted ontology. This release remains a frozen
mathematical/numerical benchmark; its physical promotion gate is blocked.

## 5. Retained corrections

- Complete deterministic closure for one fixed transition family is idempotent;
  repeated closure cannot produce seven strict complete refinements.
- Integer-tick capture does not establish continuous non-crossing. Bond creation
  and topology change remain distinct claims.
- Angular impulse is \(\Delta\mathbf L=\mathbf r\times\mathbf J\), with no extra
  time factor; restricted contact helpers do not establish general impact laws.
- Finite prefixes do not identify unrestricted generators. The retained \(2K\)
  continuation guarantee requires exact consecutive samples in the declared
  field and a noiseless homogeneous constant-coefficient recurrence of order
  at most \(K\), valid from the initial state.
- Appendix AR's finite-precision tick non-identification and Appendix BO's
  two-oscillator projection equivalence remain in force. Neither discretization
  nor hidden-state language independently derives Born probabilities.
- No result here establishes seven physical dimensions, universal spacetime
  increments, a Millennium solution, \(P=NP\), or a theological identification.