METHODOLOGY_2PAGE.tex

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\begin{center}
{\Large\bfseries Formation Engine --- Methodology Summary}\\[3pt]
{\small Version 0.1 \quad|\quad 13 Sections \quad|\quad 35 Validation Tests}
\end{center}
\vspace{-0.5em}\hrule\vspace{0.5em}

\section{Problem \& Scope (\S1)}

The Formation Engine is a classical atomistic simulation instrument
that generates molecular and crystalline structures from elemental
identity and thermodynamic boundary conditions.  It targets the
intermediate regime (100--10,000 atoms) where quantum methods are too
expensive and continuum models assume structure rather than
generating it.

\textbf{Design axioms:}
explicit units everywhere ({\AA}, fs, kcal/mol, amu);
no hidden normalization;
no silent model switching;
deterministic core with stochastic exploration controlled by seed.

\textbf{Sole data source:} the periodic table (Z=1--102).
LJ parameters from the Universal Force Field (UFF).
No molecular databases.

\textbf{Domain:} atoms with $Z \ge 6$, temperatures $T > 50$\,K,
system sizes $N = 2$--$10{,}000$.
Excludes excited states, quantum tunnelling, and reaction kinetics.

\section{State Ontology (\S0, \S2)}

The canonical state is:
\[
\mathcal{S} = (N,\,\mathbf{M},\,\boldsymbol{\tau},\,\mathbf{X},\,
\mathbf{V},\,\mathbf{Q},\,\mathbf{B},\,\mathbf{F},\,\mathcal{E},\,
\mathcal{L},\,\mathcal{B})
\]
partitioned into \textbf{Identity} ($N, \mathbf{M}, \boldsymbol{\tau}$
--- immutable), \textbf{Phase} ($\mathbf{X}, \mathbf{V}, \mathbf{Q},
\mathbf{B}$ --- evolve), and \textbf{Scratch} ($\mathbf{F},
\mathcal{E}, \mathcal{L}, \mathcal{B}$ --- recomputed).

Per-particle identity vector (\S0):
$\mathbf{I}_i = (Z_i, A_i, Q_i, \Sigma_i, \Lambda_i, \Theta_i)$.

Energy ledger: $\mathcal{E} = (U_\text{bond}, U_\text{angle},
U_\text{torsion}, U_\text{vdW}, U_\text{Coul}, U_\text{ext})$.

File hierarchy: \texttt{.xyz} (geometry) $\to$ \texttt{.xyza}
(trajectory) $\to$ \texttt{.xyzc} (checkpoint with velocities +
thermodynamic state + SHA-256 provenance hash).

\section{Interaction Model (\S3)}

\textbf{Nonbonded:} Lennard-Jones 12-6
$U_\text{LJ} = 4\varepsilon[(\sigma/r)^{12} - (\sigma/r)^6]$
with Lorentz-Berthelot combining rules and quintic switching
$r_\text{on}{=}9$, $r_\text{cut}{=}10$\,{\AA}.
Coulomb $U_\text{C} = k_e q_i q_j / r$ with
$k_e = 332.0636$\,kcal$\cdot${\AA}/(mol$\cdot$e$^2$).

\textbf{Bonded:} Harmonic bonds $k_b(r{-}r_0)^2$, harmonic angles
$k_\theta(\theta{-}\theta_0)^2$, periodic torsions
$V_n[1{+}\cos(n\phi{-}\gamma)]$, improper torsions.
Parameters from UFF.

All forces computed as $\mathbf{F}_i = -\nabla_{\mathbf{x}_i} U$
through a pure-function interface: same state always yields same
forces.

\section{Thermodynamics \& Units (\S4)}

Unit system: {\AA}, fs, kcal/mol, amu, K, $e$.

\begin{tabular}{@{}ll@{}}
$k_B$ & $= 0.001987204$ kcal/(mol$\cdot$K)\\
$C_\text{KE}$ & $= 2390.057$ (amu$\cdot${\AA}$^2$/fs$^2 \to$ kcal/mol)\\
$k_e$ & $= 332.0636$ kcal$\cdot${\AA}/(mol$\cdot$e$^2$)
\end{tabular}

Temperature from equipartition:
$T = 2K / (N_\text{df}\, k_B)$, \;
$N_\text{df} = 3N - 3$.

\section{Integration (\S5)}

\textbf{Velocity Verlet (NVE):} Symplectic, 2nd-order,
$\Delta t = 1$\,fs default.  Energy drift
$< 3 \times 10^{-5}$\,kcal/mol/atom over $10^4$ steps.

\textbf{Langevin (NVT):} Euler--Maruyama discretisation with friction
$\gamma$ and random force
$\sqrt{2\gamma m_i k_B T}\;\xi_i$.
Temperature error $< 0.6\%$.

\textbf{FIRE minimisation:} Damped MD with adaptive $\Delta t$.
Convergence criterion $F_\text{max} < 0.01$\,kcal/(mol$\cdot${\AA}).

\section{Formation Physics (\S6)}

A \textbf{formation} is a local PES minimum found via:
(1) VSEPR geometry prediction,
(2) MD exploration at elevated $T$,
(3) periodic FIRE quenching,
(4) scoring and classification.

Bond inference uses covalent radii with tolerance factor $f=1.2$.
Bonds are mutable caches, never serialised as truth.

\section{Statistics \& Scoring (\S7)}

Welford's algorithm for numerically stable online mean/variance.
Stationarity gate: 10 consecutive passes of coefficient-of-variation
$< \epsilon$ triggers convergence.
Kabsch alignment (SVD) for RMSD comparison.
Multiplicative 6-factor scoring function for ranking formations.

\section{Reaction \& Electronic (\S8--9)}

QEq charge equilibration: minimise
$E = \sum_i(\chi_i q_i + \eta_i q_i^2) + \sum_{i<j} k_e q_i q_j / r_{ij}$
subject to $\sum q_i = Q_\text{total}$.
Fukui functions, HSAB matching, Bell--Evans--Polanyi barrier
estimates.  Interface defined; full MD integration pending.

\section{Multiscale (\S10)}

Four regimes: molecular (2--50), cluster (50--500), bulk
(500--10k, PBC + supercell), mesoscale (10k+, CG placeholder).
Supercell: $\mathbf{r}_{i,pqr} = \mathbf{r}_i + p\mathbf{a} +
q\mathbf{b} + r\mathbf{c}$; bonds re-inferred post-replication.

\section{Self-Audit (\S11)}

\textbf{Determinism contract:} same (formula, seed, params) $\Rightarrow$
bit-identical output.  Enforced via seeded RNG, index-ordered force
evaluation, IEEE~754 compliance, no fast-math.

Three Python tools:
\textbf{Failure Classifier} (6 categories: NUMERICAL, PHYSICS, OOD,
CONVERGENCE, TIMEOUT, UNKNOWN);
\textbf{Gap Targeter} ($6{\times}10{\times}10$ grid, 95\% coverage
target);
\textbf{Regression Detector} (4 invariants, binary pass/fail).

\section{Validation (\S12)}

35 hierarchical tests across 5 levels:

\begin{center}\small
\begin{tabular}{@{}lcc@{}}
\toprule
Level & Tests & Status \\
\midrule
0 --- Unit system & 12 & 100\% \\
1 --- Force evaluation & 8 & 100\% \\
2 --- Integration & 5 & 100\% \\
3 --- Thermodynamics & 8 & 87\% \\
4 --- Reproducibility & 3 & 100\% \\
\bottomrule
\end{tabular}
\end{center}

Production certified for LJ-dominated systems (noble gases,
hydrocarbons, small organics).
Known limitation: Coulomb--integrator coupling instability for
ionic MD.

\section{Future Work (\S13)}

Priority: neighbour lists ($50\times$ speedup) $\to$ NPT barostat
$\to$ reactive bond orders $\to$ quantum corrections (delta-learning)
$\to$ ML potentials (SchNet/MACE).

\vspace{0.5em}\hrule\vspace{0.3em}
{\footnotesize Full methodology: 11 LaTeX source files in
\texttt{docs/}.  Compile with \texttt{pdflatex}.  MIT License.}

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