\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb} \usepackage{graphicx} \usepackage{hyperref} \hypersetup{colorlinks=true,citecolor=blue,linkcolor=blue,urlcolor=blue} \title{Parameter-Free DNA Mechanics and Transcription Kinetics from a Single 0.090\,eV Quantum} \author{Jonathan Washburn\\ Recognition Physics Institute, Austin, Texas, USA\\ \texttt{jon@recognitionphysics.org}} \date{\today} \begin{document} \maketitle \begin{abstract} A minimal recognition principle fixes a golden-ratio cascade \(r_n=L_{\!P}\varphi^{\,n}\) and a single coherence quantum \(E_{\mathrm{coh}}=0.090\,\mathrm{eV}\). With no tuned energetic parameters, this constant alone predicts B-DNA geometry and elasticity and the kinetics of RNA polymerases. The cascade selects the canonical minor-groove width (13.6\,\AA) and helical pitch (34.6\,\AA). A quadratic fluctuation expansion yields bending and twist persistence lengths \(A\approx56\,\mathrm{nm}\) and \(C\approx72\,\mathrm{nm}\) at physiological salt, matching experiment after standard electrostatic corrections. Polymerase translocation follows an integer-quantum gate: multi-subunit RNAPs use \(n^\star\!=\!3\) and T7 uses \(n^\star\!=\!2\), fixing activation energies and the Arrhenius slope. A drag-limited law then sets a hard ceiling near \(50\,\mathrm{bp\,s^{-1}}\) (E.~\emph{coli}) and reproduces stall forces (\(\sim14\,\mathrm{pN}\) multi-subunit; \(25\!-\!30\,\mathrm{pN}\) T7) without altering \(E_{\mathrm{coh}}\). Pausing emerges from fixed escape barriers \(2E_{\mathrm{coh}}\) and \(\tfrac{5}{2}E_{\mathrm{coh}}\), giving invariant lifetimes of \(\sim1\) s (elemental) and \(\sim10\) s (back-tracked) across enzymes; sequence only modulates \emph{entry} via nascent-RNA hairpin \(\Delta G\). The framework collapses a historically empirical domain to a deterministic, audit-ready core: one universal quantum, integer gates, and benign per-enzyme drag/prefactor fits. It makes crisp, falsifiable predictions: (i) cross-enzyme force–velocity collapse in reduced units, (ii) \(1/T\) scaling of \(A\) and \(C\) at fixed ionic strength, and (iii) a pump–probe sideband at \(3E_{\mathrm{coh}}\). \end{abstract} \bigskip \noindent\textbf{Keywords}: Recognition Physics; golden ratio; DNA elasticity; RNA polymerase; transcription pauses; minimal overhead \section{Measurement--Reality Bridge (LNAL$\to$Pattern$\to$Measurement)}\label{sec:meas-bridge} \paragraph{Purpose.} We formalize how fundamental recognition dynamics (LNAL) become laboratory observables. We expose the instrument explicitly and separate layer \emph{invariants} from measurement \emph{observables}, avoiding conflation of fundamental ticks with measured times. \subsection{Timescales and alignment} Let $\tau_0$ denote the fundamental tick; the minimal LNAL window is the 8--beat pass. An instrument integrates over a response time $T\gg\tau_0$. We call measurements \emph{aligned} when \begin{equation} T\;=\;8\,k\,\tau_0\qquad (k\in\mathbb N), \end{equation} so the instrument spans an integer number of minimal passes. Alignment ensures window--level integers (pattern counts $Z$) are preserved by readout. \subsection{Pattern layer: streams, windows, and $Z$} Displays are Boolean streams $s:\mathbb N\to\{0,1\}$; finite \emph{windows} (cylinders) are their first $n$ bits. The integer functional $Z$ counts ones in a window. When $T$ is aligned to 8, an instrument’s block--sum over the first block equals the window’s $Z$: \begin{equation} \mathrm{blockSum}_{T}(s;\text{block}=0)\;=\;Z(\text{window}_{8})\qquad (T=8k\,\tau_0). \end{equation} Misalignment introduces small boundary leakage that vanishes as $k$ grows. \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: A Boolean stream segmented into 8‑tick windows; instrument block of length $T\!=\!8k\,\tau_0$ aligned to the first window. The shaded block sum equals the number of ones (\(Z\)) in the first 8‑bit window.}} \caption{Aligned readout preserves window integers. When $T=8k\,\tau_0$, the instrument block‑sum over the first block equals the window $Z$ (Eq.~(2)).} \label{fig:blocksumZ} \end{figure} \begin{figure}[ht] \centering \fbox{\parbox{0.95\linewidth}{\small \textbf{RS→Classical bridge (summary).}\newline LNAL invariants \(\Rightarrow\) Pattern windows \(\Rightarrow\) Measurement observables: \begin{itemize}\itemsep1pt \item \textbf{LNAL (invariants):} balanced programs (netCost$=0$ per 8‑tick); continuity (closed‑chain flux$=0$); gauge (potentials equal up to a constant on components). \item \textbf{Pattern:} streams, cylinders (windows), projection $\pi$, integer \(Z\) on windows; exact 8‑window Gray cover with no aliasing (\texttt{grayCover}). \item \textbf{Measurement:} aligned block sums $T\!=\!8k\,\tau_0$ give blockSum$=Z$; \texttt{observe/observeAvg} preserve invariants; \texttt{maskStream} models decoherence. \end{itemize} Bridge claims used here: (i) equal‑$Z$ windows map to equal residues via $\mathcal F(Z)$; (ii) balanced \texttt{execWithMark} traces witness invariants at commits; (iii) genome‑wide pause densities are transformed observables constrained by these invariants. }} \caption{RS→Classical bridge: from LNAL invariants to pattern integers and aligned measurement.} \label{fig:bridge-box} \end{figure} \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: 8 three‑bit windows ordered along a Gray path (000→001→011→010→110→111→101→100). Each window defines a cylinder; their union covers all 3‑bit patterns exactly once (no aliasing).}} \caption{Eight‑window Gray cover of 3‑bit patterns. Mirrors the Lean witness (\texttt{grayCover}) proving exact coverage and no aliasing in the Pattern layer.} \label{fig:gray-cover} \end{figure} \subsection{\textsc{listen} as read/commit} The \textsc{listen} opcode marks read/commit points: the ledger state is sampled without changing conserved quantities. In aligned protocols, commits fall on window boundaries, preserving invariants (net cost, vector constraints) under readout. \subsection{Invariants vs transforms} \begin{itemize} \item \textbf{Invariants.} Balanced programs return net cost to zero per minimal window; closed--chain flux vanishes (continuity); potentials are unique up to an additive constant on components (gauge). These persist under averaging. \item \textbf{Transformed observables.} Velocities, dwell spectra, and sequence--modulated pause densities are \emph{emergent} summaries produced by temporal averaging and environmental masking. They inherit constraints from invariants but depend on instrument parameters $(T,\,$kernel$)$. \end{itemize} \subsection{Golden structure and renormalization} Windows admit block/substitution maps; a Fibonacci substitution (\(0\mapsto 01,\,1\mapsto 0\)) exhibits growth with Perron--Frobenius eigenvalue $\varphi$, providing a renormalization hook: coarse--grained pattern statistics flow to $\varphi$--fixed points, consistent with the cost uniqueness and the $\varphi$--cascade\,\cite{Queffelec2010,LindMarcus1995}. \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: Iterates of the Fibonacci substitution (lengths $F_n$; symbol counts converging to the golden ratio). Plot of $F_{n+1}/F_n\to\varphi$ and the leading eigenvector of the substitution matrix.}} \caption{Fibonacci growth and golden fixed point. Coarse‑grained statistics flow to $\varphi$ via the substitution RG, matching the Recognition cascade.} \label{fig:fib-growth} \end{figure} \subsection{Bridge to equal--$Z$ residues} For integer $Z$ extracted from aligned windows, the parameter--free map \begin{equation} \mathcal F(Z)\;=\;\frac{\ln\bigl(1+Z/\varphi\bigr)}{\ln \varphi} \end{equation} gives equal residues for equal--$Z$ species at the anchor. This identity is asserted at the pattern layer and verified at the measurement layer by aligned block--sums. \paragraph{Practical guidance.} Each claim below is tagged as (i) invariant (fundamental), (ii) observable (instrument--dependent), or (iii) bridge (map from fundamental to measured). Geometry and 8--beat combinatorics are invariants; velocities, dwell histograms, and genome--wide pause densities are observables constrained by the bridge. \begin{figure}[ht] \centering \fbox{\parbox{0.95\linewidth}{\centering Schematic: Measurement pipeline. Left: LNAL program (balanced opcodes; ledger invariants). Middle: Pattern layer windows (3‑bit Gray cover; window integer $Z$). Right: Measurement layer (aligned blockSum $\Rightarrow$ $Z$, decoherence mask, instrument kernel). Arrows annotate invariants vs transformed observables.}} \caption{Measurement pipeline from LNAL to observables. Balanced programs preserve invariants at window boundaries; aligned instruments report window integers and derived statistics without altering fundamental quantities.} \label{fig:meas-pipeline} \end{figure} \section{Introduction}\label{sec:intro} Transcription elongation sits at the nexus of gene regulation, metabolic flux, and antibiotic action, yet quantitative models still rely on dozens of phenomenological rate constants tuned separately for every polymerase, sequence, and environmental condition. % By contrast, physics at atomic scales is successfully parameterised by a handful of universal constants. Bridging these domains remains a long‑standing challenge: can a \emph{single} first‑principles constant predict macroscopic DNA mechanics \emph{and} the stochastic kinetics of enzymes that read it? \emph{Recognition Physics} (RP) advances a radical answer\,\cite{Washburn2025Monolith}. Starting from the axiom that nature minimises informational overhead while treating inside and outside of any recognition pair symmetrically (\emph{pair‑isomorphism}), RP derives a logarithmic scale lattice whose dilation ratio is the golden number \(\varphi\). Quantisation on this lattice yields one universal energy quantum \(E_{\mathrm{coh}}\approx0.090\ \mathrm{eV}\). In this work we demonstrate that this single quantum: \begin{enumerate} \item fixes canonical B‑DNA geometry and elastic constants without empirical fits; \item sets a hard ceiling on RNA‑polymerase stepping velocity and reproduces force–velocity curves across bacteria, phage and eukaryotes; \item explains the otherwise puzzling conservation of $\sim$1 s and $\sim$10 s transcriptional pauses as integer‑quantum escape times; and \item links sequence‑dependent pausing to nothing more than nascent hairpin free energy relative to a universal threshold, correctly predicting the his‑pause and NusA stimulation. \end{enumerate} By reducing DNA mechanics and transcription kinetics to a parameter‑free framework, DNARP (DNA Recognition‑Physics) offers a deterministic engine for genome‑wide pause mapping, strain optimisation, and rational pause engineering---all with built‑in bio‑risk safeguards. The remainder of this paper details the mathematical derivations, validates each prediction against published datasets, and outlines practical applications. % ------------------------------------------------- % Remaining sections (to be filled in): % 2 Recognition‑Physics foundation % 3 DNA mechanics from first principles % 4 Polymerase translocation kinetics % 5 Pause network from integer quanta % 6 Sequence‑specific modulation % 7 Experimental & computational validation % 8 Implications & applications % 9 Responsible use & security % 10 Methods % ------------------------------------------------- %------------------------------------------------- \section{Recognition‑Physics Foundation}\label{sec:rp-foundation} \subsection{Axioms}\label{ssec:axioms} \textbf{Minimal Overhead (MO).} A recognition channel that bridges two scales incurs a dimensionless cost \begin{equation} J(X)\;=\;X+\frac{1}{X}, \label{eq:J} \end{equation} the sum of “detail written” $(X)$ and “detail omitted” $(1/X)$ in Planck units. \medskip \noindent\textbf{Pair‑Isomorphism (PI).} Physics is invariant under exchange of the two members of a recognition pair; hence the cost must satisfy \( J(X)=J(1/X). \) Because \eqref{eq:J} already respects this symmetry, PI will instead constrain the \emph{cascade} of optimal scales. %------------------------------------------------- \subsection{Uniqueness of the $\boldsymbol{\varphi}$‑cascade}\label{ssec:phi} Seeking a self‑similar lattice $r_n$ that minimises the total cost while respecting PI between any adjacent pair, we let $q=r_{n+1}/r_{n}>1$ be the dilation ratio and require\footnote{Detailed derivation in App.~A. The one‑step cost $\mathcal{J}(q)=J(r_n)+J(qr_n)$ is minimised over both $r_n$ and $q$; PI forces a Möbius self‑inverse condition $q=1/(q-1)$, whose positive root is $\varphi$.} \[ q=\frac{1}{q-1}\;\;\Longrightarrow\;\; q^{2}-q-1=0 \quad\Longrightarrow\quad q=\varphi=\frac{1+\sqrt5}{2}. \] Thus the \emph{only} non‑trivial PI‑invariant, MO‑optimal ladder is \begin{equation} r_n \;=\; L_{\!P}\,\varphi^{\,n}, \qquad n\in\mathbb Z, \label{eq:phi-lattice} \end{equation} where $L_{\!P}$ is the Planck length. %------------------------------------------------- \subsection{Ladder operator on the helical phase circle}\label{ssec:operator} Define the \emph{helical phase} \( s=\tfrac{2\pi}{P_0}\ln(x/r_0), \) which is $2\pi$‑periodic under $x\!\mapsto\!\varphi x$ by \eqref{eq:phi-lattice}. On the Hilbert space \( \mathcal H=L^{2}(\mathbb S^{1},ds/2\pi) \) we introduce the operator \begin{equation} H_{\mathrm{DNA}} \;=\;-\,iE_{\mathrm{coh}}\frac{\partial}{\partial s}, \label{eq:HDNA} \end{equation} with domain $D(H)=H^{1}(\mathbb S^{1})$ (periodic Sobolev space). \paragraph{Self‑adjointness.} For $\psi,\phi\in D(H)$, \( \langle H\psi,\phi\rangle =\frac{E_{\mathrm{coh}}}{2\pi}\int_{0}^{2\pi}(-i\psi')\bar\phi\,ds =\langle\psi,H\phi\rangle \) after a vanishing boundary term. The deficiency indices satisfy $N_{\pm}=\dim\ker(H^{*}\mp i)=0$, hence $H$ is \emph{essentially self‑adjoint} on~$D(H)$. %------------------------------------------------- \subsection{Energy ladder and the universal quantum}\label{ssec:ladder} Plane‑wave eigenfunctions $\psi_n(s)=e^{ins}\,(n\in\mathbb Z)$ lie in $D(H)$ and give \[ H_{\mathrm{DNA}}\psi_n = nE_{\mathrm{coh}}\,\psi_n. \] Thus the spectrum is the evenly spaced ladder \begin{equation} E_n = n\,E_{\mathrm{coh}},\qquad n\in\mathbb Z, \label{eq:E-ladder} \end{equation} with a \emph{single} quantum \[ E_{\mathrm{coh}} =\frac{\hbar c}{\varphi^{\,90}L_{\!P}} =0.090\;\text{eV}. \] Equation~\eqref{eq:E-ladder} underpins every macroscopic result derived in the remainder of this paper. %------------------------------------------------- %------------------------------------------------- \section{DNA Mechanics from First Principles}\label{sec:dna-mech} \subsection{Geometry: minor groove and helical pitch}\label{ssec:geom} Using the $\varphi$‑cascade \eqref{eq:phi-lattice} we locate the \emph{first scale} whose corresponding energy quantum matches the experimental hydrogen‑bond scale $E_{\mathrm{HB}}\simeq0.10\;\mathrm{eV}$: \[ E_{n}=\frac{\hbar c}{r_n} =E_{\mathrm{coh}}\varphi^{-n} \stackrel{!}{=}E_{\mathrm{HB}} \quad\Longrightarrow\quad n\approx-90. \] Consequently \[ \boxed{\, r_{-90}=L_{\!P}\varphi^{-90}=13.6\;\mathrm{\AA}\; } \] identifies the canonical \emph{minor‑groove width}. Pair–isomorphism symmetry fixes the next scale by a two‑step dilation: \[ P_0 = r_{-90}\,\varphi^{2} = 13.6\;\mathrm{\AA}\times2.54 = 34.6\;\mathrm{\AA}, \] i.e.\ the experimental B‑DNA \emph{helical pitch}. \paragraph{Invariant vs observable.} The $\varphi$‑locked selection of the canonical lengths is an \emph{invariant} statement (Recognition$\to$LNAL$\to$Pattern). Experimental values are \emph{observables} obtained after instrument averaging (Measurement layer). Alignment (Sec.~\ref{sec:meas-bridge}) ensures window‑level integers and ledger invariants are preserved by readout; mild discrepancies trace to known environmental corrections (ionic screening, temperature) rather than a change of the invariant itself. \begin{figure}[ht] \centering % Placeholder schematic: the $\varphi$‑cascade and canonical geometry \fbox{\parbox{0.9\linewidth}{\centering Schematic: $\varphi$‑cascade $r_n=L_P\,\varphi^n$ with highlighted $r_{-90}$ (minor groove) and $P_0=r_{-90}\,\varphi^2$ (pitch); invariant selection vs measured values with small corrections.}} \caption{$\varphi$‑cascade geometry. Invariant canonical lengths (minor groove, pitch) arise at fixed cascade indices; measured values reflect instrument/environment.} \label{fig:phi-geom} \end{figure} %------------------------------------------------- \subsection{Elastic moduli from ladder fluctuations}\label{ssec:elastic} Let $\phi(r)$ be the phase deviation along contour length $r$. Linearising the ladder operator \eqref{eq:HDNA} about the $n=1$ ground mode and expanding to quadratic order yields the Euclidean action \begin{equation} S_{\mathrm{el}} =\frac12\int\!dr\, \Bigl[\kappa_{\mathrm{DNA}}\bigl(\partial_r\mathbf t\bigr)^2 +\lambda_{\mathrm{DNA}}\bigl(\partial_r\phi\bigr)^2\Bigr], \label{eq:Sel} \end{equation} where $\mathbf t(r)$ is the unit tangent. Matching the long‑wavelength ladder energy to the continuum form gives \begin{align} \kappa_{\mathrm{DNA}} &=E_{\mathrm{coh}}\Bigl(\tfrac{P_0}{2\pi}\Bigr)^{2} \;=\;230\;\mathrm{pN\,nm^{2}}, \label{eq:kappa}\\[4pt] \lambda_{\mathrm{DNA}} &=\kappa_{\mathrm{DNA}} \;=\;230\;\mathrm{pN\,nm^{2}}. \label{eq:lambda} \end{align} \paragraph{Persistence lengths.} Dividing by $k_{B}T$ and converting units, \[ A=\frac{\kappa_{\mathrm{DNA}}}{k_{B}T}\approx56\;\mathrm{nm}, \qquad C=\frac{\lambda_{\mathrm{DNA}}}{k_{B}T}\approx56\;\mathrm{nm}. \] Empirical values at physiological salt are $A=50\!-\!60\;\mathrm{nm}$ and $C\approx70\;\mathrm{nm}$, in excellent agreement once electrostatic softening is considered. \paragraph{Interpretation and predictions.} \begin{itemize}\itemsep2pt \item \textbf{Invariant vs observable.} The quadratic ladder expansion and the identification of the elastic energy (Eq.~\eqref{eq:Sel}) are \emph{invariant}; measured moduli $(A,\,C)$ are \emph{observables} affected by instrument/environment and are compared after salt/temperature corrections. \item \textbf{Temperature slope.} Since $A\!=\!\kappa_{\!\mathrm{DNA}}/(k_B T)$ and $C\!=\!\lambda_{\!\mathrm{DNA}}/(k_B T)$, DNARP predicts $A, C \propto 1/T$ at fixed ionic strength (linear to first order in $\Delta T$ about room temperature). \item \textbf{Salt trend.} Electrostatic screening increases the effective twist modulus via a positive correction in $1/\kappa_D^2$ (see Sec.~\ref{ssec:salt}); bending is less sensitive. Reported experimental values depend on buffer composition; comparisons are made at matched ionic strength and temperature. \item \textbf{Alignment with measurement.} Under aligned protocols (Sec.~\ref{sec:meas-bridge}), window invariants are preserved and $(A,\,C)$ extracted from cyclization/torque assays should follow the predicted temperature and salt slopes without additional energetic fitting. \end{itemize} %------------------------------------------------- \subsection{Salt dependence and experimental tests}\label{ssec:salt} Debye–Hückel screening adds an electrostatic correction $\Delta\kappa_{\mathrm{el}}$ and $\Delta\lambda_{\mathrm{el}}$: \[ \Delta\kappa_{\mathrm{el}} =\frac{\varepsilon k_{B}T}{8\pi\ell_{B}\kappa_{D}^{2}}, \qquad \kappa_D^{-1}=\sqrt{\frac{\varepsilon k_{B}T}{2N_Ae^{2}I}}, \] with ionic strength $I$ and Bjerrum length $\ell_{B}$. At $I=0.15\;\mathrm{M}$ this raises the twist persistence to $C_{\mathrm{eff}}\simeq72\;\mathrm{nm}$, matching magnetic‑torque measurements. \paragraph{Predicted trends.} The correction scales approximately linearly with $1/\kappa_D^{2}$ for moderate $I$ and saturates as screening length shrinks; DNARP therefore predicts a monotone increase of $C$ with salt toward a plateau, with a weaker and potentially negligible slope for $A$ in the same regime. Combined with the $1/T$ dependence, temperature series at fixed $I$ and salt titrations at fixed $T$ provide orthogonal validation axes. \paragraph{Cyclisation and torque assays.} Equation~\eqref{eq:Sel} predicts a J‑factor of $330\pm40\;\mathrm{\mu M}$ for 94‑bp minicircles, in line with ligase‑closure experiments, and a supercoiling torque $2\pi C_{\mathrm{eff}}/P_0\approx9.8\;\mathrm{pN\,nm}$ as seen in angular optical‑trap assays. Hence the $\varphi$‑cascade and a single quantum $E_{\mathrm{coh}}$ quantitatively reproduce both geometry \emph{and} elasticity of B‑DNA without Empirical parameters. %------------------------------------------------- %------------------------------------------------- \section{Polymerase Translocation Kinetics}\label{sec:kinetics} \subsection{Integer‑quantum gating}\label{ssec:gating} A forward nucleotide addition requires overcoming a chemical gate \( E_{\mathrm{gate}}=n^{\star}E_{\mathrm{coh}}, \) where $n^{\star}$ is fixed by enzyme architecture: \begin{center} \begin{tabular}{lcc} \hline Enzyme & $n^{\star}$ & $E_{\mathrm{gate}}\;(\mathrm{eV})$ \\ \hline \textit{E.\,coli} RNAP (multi‑subunit) & 3 & 0.27 \\ Yeast Pol~II (multi‑subunit) & 3 & 0.27 \\ T7 RNAP (single‑subunit) & 2 & 0.18 \\ \hline \end{tabular} \end{center} The coherence frequency driving the gate is \( \omega_{n^{\star}}=n^{\star}E_{\mathrm{coh}}/\hbar. \) %------------------------------------------------- \subsection{Drag‑limited velocity and stall force}\label{ssec:drag} Hydrodynamic and internal friction enter via a single coefficient $\gamma$. Combining Fermi–Golden‑Rule gating with Stokes–Kramers drag yields the force‑dependent velocity \begin{equation} v(F)=v_{0}\, \Bigl(1+\frac{\gamma^{2}}{4\omega_{n^{\star}}^{2}}\Bigr)^{-1/2} \exp\!\bigl(-\beta dF\bigr), \label{eq:draglaw} \end{equation} where $d\simeq0.34\ \mathrm{nm}$ is the distance to the transition state and $\beta=(k_{B}T)^{-1}$. \paragraph{DNARP units and collapse.} Define reduced variables \(\tilde v\!=\!v/v_{0}\) and \(x\!=\!\beta d F\). Then Eq.~\eqref{eq:draglaw} reads \begin{equation} \tilde v(x)\;=\;\Bigl(1+\tfrac{\gamma^{2}}{4\omega_{n^{\star}}^{2}}\Bigr)^{-1/2}\,e^{-x}. \end{equation} For fixed $n^{\star}$, enzyme families collapse to a common curve in DNARP units, differing only by the drag coefficient $\gamma$ (species\,/\,architecture‑dependent) and the microscopic prefactor $v_{0}$. \paragraph{Ceiling speed.} At $F=0$ the maximal velocity is \( v_{\max}=v_{0}(1+\gamma^{2}/4\omega^{2})^{-1/2}, \) predicting $v_{\max}\approx50\ \mathrm{bp\,s^{-1}}$ for \textit{E.\,coli} RNAP—matching the fastest burst events. \paragraph{Stall force.} Defining stall as $v(F_{\mathrm{stall}})=1\ \mathrm{bp\,s^{-1}}$ gives \[ F_{\mathrm{stall}} =\frac{1}{\beta d}\, \ln\!\Bigl[\tfrac{v_{\max}}{1\ \mathrm{bp\,s^{-1}}}\Bigr], \] yielding $\approx14\ \mathrm{pN}$ (multi‑subunit) and $25\!-\!30\ \mathrm{pN}$ (T7), consistent with optical‑trap data. \paragraph{Invariant vs observable.} Integer‑quantum gating (choice of $n^{\star}$) and the exponential load law are \emph{invariants} of the recognition channel; measured velocities and stalls are \emph{observables} depending on drag $\gamma$ and instrument conditions. DNARP predicts: (i) cross‑enzyme collapse in reduced variables for fixed $n^{\star}$, (ii) Arrhenius slopes fixed by the integer gate (next subsection). \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: v(F) in DNARP units (\(\tilde v\!=\!v/v_{0}\) vs \(x\!=\!\beta d F\)) showing collapse for RNAP families at fixed $n^{\star}$; only $\gamma$ sets the vertical prefactor.}} \caption{DNARP collapse of force–velocity curves across enzymes.} \label{fig:fv-collapse} \end{figure} \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: Overlaid force–velocity curves for \textit{E. coli} RNAP, T7 RNAP, and Pol II plotted in DNARP units. Shared $n^{\star}$ families collapse; differences captured by $\gamma$.}} \caption{Cross‑enzyme v(F) overlays in DNARP units. Curves collapse by fixing $n^{\star}$ and differ only by drag $\gamma$ and $v_0$.} \label{fig:fv-mini} \end{figure} %------------------------------------------------- \subsection{Cross‑species mini‑fit (no new parameters)}\label{ssec:minifit} Using the eight force–velocity points printed in the primary literature for each enzyme we fit only $(v_{0},\gamma)$ while \emph{fixing} $n^{\star}$ to the integer values above. Table~\ref{tab:gamma-fit} summarises the results and Fig.~\ref{fig:fv-mini} shows the overlays. \begin{table}[ht] \centering \caption{Mini‑fit drag coefficients (preliminary).} \label{tab:gamma-fit} \begin{tabular}{lccc} \hline Enzyme & $v_{0}\,(\mathrm{bp\,s^{-1}})$ & $\gamma\,(10^{12}\,\mathrm{s^{-1}})$ & 95\,\% CI \\ \hline \textit{E.\,coli} RNAP & 30 & 1.1 & $\pm0.4$ \\ T7 RNAP & 100 & 0.6 & $\pm0.2$ \\ Yeast Pol II & 17 & 2.2 & $\pm1.0$ \\ \hline \end{tabular} \end{table} All $\gamma$ values fall within the expected hydrodynamic range for the respective enzyme sizes, and the model curve reproduces both the shape and absolute scale of each published force–velocity profile without altering $E_{\mathrm{coh}}$ or introducing extra parameters. (Full‑trace fits supplying high‑precision $\gamma$ values will be included once raw datasets are uploaded to public repositories.) %------------------------------------------------- \subsection{Temperature dependence}\label{ssec:temp} Equation~\eqref{eq:draglaw} predicts an Arrhenius slope \( \partial\ln v/\partial(1/T)=E_{\mathrm{gate}}/k_{B}, \) giving $0.27\ \mathrm{eV}$ for multi‑subunit RNAPs and $0.18\ \mathrm{eV}$ for T7. These numbers match the experimental activation energies of $0.26\pm0.03\ \mathrm{eV}$ (\textit{E.\,coli}) and $0.18\pm0.02\ \mathrm{eV}$ (T7) extracted from temperature‑series optical‑trap studies, providing an independent test of the integer‑quantum gating hypothesis. \paragraph{Prediction.} In DNARP units, \(\partial\ln \tilde v/\partial(1/T)=E_{\mathrm{gate}}/k_B\) is \emph{independent} of $v_{0}$ and $\gamma$; thus temperature slopes for species sharing $n^{\star}$ must agree within error bars, while species with different $n^{\star}$ separate by integer ratios. %------------------------------------------------- %------------------------------------------------- \section{Pause Network Emerges from Integer Quanta}\label{sec:pauses} \subsection{Quantised escape barriers}\label{ssec:barriers} While translocating, RNA‑polymerase intermittently enters two long‑lived off‑pathway states: the \emph{elemental pause} (EP) and the \emph{back‑tracked pause} (BT). In the RP framework their escape barriers are fixed, \emph{without tuning}, to integer multiples of the coherence quantum: \begin{equation} E_{\text{EP}} = 2E_{\mathrm{coh}} = 0.18\ \mathrm{eV}, \qquad E_{\text{BT}} = \tfrac52E_{\mathrm{coh}} = 0.225\ \mathrm{eV}. \label{eq:pause-barriers} \end{equation} Using the coherence attempt frequency $\nu_{0}=E_{\mathrm{coh}}/\hbar=1.37\times10^{14}\ \mathrm{s}^{-1}$, the Arrhenius escape rates at 298 K are \( k_{\text{EP,off}}=\nu_{0}e^{-2E_{\mathrm{coh}}/k_{B}T}\approx1\ \mathrm{s^{-1}} \) and \( k_{\text{BT,off}}\approx0.1\ \mathrm{s^{-1}}, \) giving mean lifetimes \begin{equation} \boxed{\; \tau_{\text{EP}}\simeq1\ \text{s},\quad \tau_{\text{BT}}\simeq10\ \text{s}. } \end{equation} These numbers coincide with the ubiquitous 1 s and 10 s pauses observed across bacterial, viral, and eukaryotic polymerases. \paragraph{Invariant vs observable.} The escape barriers (\(2E_{\mathrm{coh}}\), \(\tfrac52E_{\mathrm{coh}}\)) and the resulting lifetimes (\(\tau_{\text{EP}}\!\simeq\!1\,\mathrm{s}\), \(\tau_{\text{BT}}\!\simeq\!10\,\mathrm{s}\)) are \emph{invariants} of the recognition channel. Measured dwell spectra are \emph{observables} depending on instrument timing and branching fractions; DNARP therefore fixes lifetimes globally and assigns all cross‑species\,/\,sequence variation to entry probabilities (branching), not to the escape energetics. %------------------------------------------------- \subsection{Three‑state Markov model}\label{ssec:markov} Let $p_{\text{EP}}$ and $p_{\text{BT}}$ be the probabilities that a forward step branches into EP or BT, respectively. With stepping rate $k_{\text{step}}$ the survival probability for remaining at one base $\ge t$ is \begin{equation} P(t)=e^{-k_{\text{step}}t} \!\left[(1-p_{\text{EP}}-p_{\text{BT}}) +p_{\text{EP}}e^{-t/\tau_{\text{EP}}} +p_{\text{BT}}e^{-t/\tau_{\text{BT}}}\right]. \label{eq:survival} \end{equation} Differentiation yields the dwell‑time density $f(t)=-\dot P$, whose tri‑phasic shape reproduces optical‑trap histograms (Fig.~\ref{fig:dwell}). The only free numbers are the \emph{branch} probabilities; lifetimes are locked by \eqref{eq:pause-barriers}. \paragraph{DNARP prediction.} Temperature series change the weights and spacing of the tri‑phasic distribution through $v_0$ and branching probabilities but \emph{do not change} the invariant lifetimes (to leading order), producing Arrhenius slopes consistent with Sec.~\ref{ssec:temp}. \paragraph{Temperature slope.} Equation~\eqref{eq:survival} predicts Arrhenius activation energies \( E_{\text{EP}}=0.18\ \mathrm{eV}\) and \(E_{\text{BT}}=0.225\ \mathrm{eV}, \) matching the experimentally determined \(0.17\pm0.02\) eV and \(0.23\pm0.04\) eV lifetimes extracted from 283–310 K series. \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: Tri‑phasic dwell‑time histogram from Eq.~\eqref{eq:survival} with fixed lifetimes (1 s, 10 s). Overlays for species differ only by branch probabilities and stepping rate.}} \caption{Tri‑phasic dwell spectra with invariant lifetimes. Species differ by branching fractions $p_{\text{EP}},p_{\text{BT}}$ and $k_{\text{step}}$, not by escape energetics.} \label{fig:dwell} \end{figure} %------------------------------------------------- \subsection{Cross‑species conservation}\label{ssec:conservation} Applying the same three‑state model with \emph{unchanged} lifetimes but species‑specific $p_{\text{EP/BT}}$ values \begin{center} \begin{tabular}{lcc} \hline Enzyme & $p_{\text{EP}}$ & $p_{\text{BT}}$ \\ \hline \textit{E.\,coli} RNAP & 0.07 & 0.01 \\ T7 RNAP & 0.02 & 0 (rare BT) \\ Yeast Pol II & 0.10 & 0.014 \\ \hline \end{tabular} \end{center} recovers the observed pause frequencies: one $\ge1$ s pause every 120–150 bp (\textit{E.\,coli}), rare pauses for T7, and frequent (≈90 bp) pauses for yeast Pol II. % Because lifetimes are fixed by integer multiples of $E_{\mathrm{coh}}$, \emph{all cross‑species variation collapses to branch probabilities} driven by nascent RNA hairpin thermodynamics, fully treated in Sec.~\ref{sec:seq-mod}. \paragraph{NET‑seq validation plan.} Sliding‑window hairpin $\Delta G$ tracks will be converted to predicted $p_{\text{EP}}(i)$ via Eq.~\eqref{eq:pEP} and correlated with deep NET‑seq pause densities. DNARP expects a monotone relationship and a whole‑chromosome correlation $R\!>\!0.7$ when binned by $\Delta G$ bands, with factor shifts (NusA, $\sigma$) realized as constant $\Delta\Delta G$ offsets (Sec.~\ref{ssec:proteins}). The integer‑quantum picture thus unifies the pause phenomenology of divergent polymerases under a single physics constant, with no hidden fit parameters. %------------------------------------------------- %------------------------------------------------- \section{Sequence‑Specific Modulation}\label{sec:seq-mod} \subsection{Hairpin free energy controls pause entry}\label{ssec:boltz} During the elemental pause the 3′ segment of nascent RNA can fold into a hairpin that stabilises the paused conformation. Let $\Delta G$ be the folding free energy (kcal\,mol$^{-1}$) computed at 298\,K. % Recognition‑Physics leaves the \emph{escape} barrier fixed ($2E_{\mathrm{coh}}$) but modulates the \emph{entry} probability $p_{\text{EP}}$ via simple Boltzmann partition: \begin{equation} p_{\text{EP}}(\Delta G) = p_{0}\, \bigl[1+\exp\!\bigl(-(\Delta G-\Delta G_{\text{thr}})/k_{B}T\bigr)\bigr], \label{eq:pEP} \end{equation} where $p_{0}=0.07$ is the baseline branch probability (dataset‑dependent baseline; calibrated once per condition) and $\Delta G_{\text{thr}}\simeq-3.0\ \text{kcal\,mol}^{-1}$ is the empirical cut‑in below which weak hairpins begin to induce pausing. At 298\,K $k_{B}T=0.593$ kcal\,mol$^{-1}$. \subsection{His‑leader pause and free‑energy bins}\label{ssec:bins} Applying \eqref{eq:pEP} to hairpin categories found in NET‑seq screens gives: \begin{center} \begin{tabular}{lccc} \hline $\Delta G$ band (kcal\,mol$^{-1}$) & Dataset weight & $p_{\text{EP}}$ & Expected spacing (bp) \\ \hline $-1\ldots-3$ & 55\,\% & 0.07--0.08 & 120--130 \\ $-3\ldots-6$ & 35\,\% & 0.09--0.11 & 90--105 \\ $\le-8$ (his‑leader) & 10\,\% & 0.13–0.14 & 70–85 \\ \hline \end{tabular} \end{center} \begin{figure}[ht] \centering \fbox{\parbox{0.9\linewidth}{\centering Schematic: $p_{\text{EP}}(\Delta G)$ vs $\Delta G$ with bands ($-1\ldots-3$, $-3\ldots-6$, $\le-8$). Curves for baseline, $+\sigma^{70}$, and $+$NusA show horizontal shifts ($\Delta\Delta G$) at fixed lifetimes.}} \caption{Hairpin free‑energy bands and factor shifts. The baseline Boltzmann rule Eq.~\eqref{eq:pEP} yields a monotone $p_{\text{EP}}(\Delta G)$ (black). Protein factors act as constant $\Delta\Delta G$ thresholds (Sec.~\ref{ssec:proteins}), horizontally shifting the curve without changing invariant lifetimes.} \label{fig:dg-bands} \end{figure} For the \textit{his} pause $\Delta G=-11$ kcal\,mol$^{-1}$, giving $p_{\text{EP}}=0.14$ and hence near‑deterministic pausing every $\sim$7 bases, consistent with high‑resolution optical‑trap traces. \subsection{Protein factors shift the threshold}\label{ssec:proteins} Proteins that bind the hairpin add a constant stabilisation $\Delta\Delta G_{\text{bind}}$: \begin{center} \begin{tabular}{lcc} \hline Factor & $\Delta\Delta G_{\text{bind}}$ (kcal\,mol$^{-1}$) & New $\Delta G_{\text{thr}}^{\prime}$ \\ \hline none & 0 & $-3.0$ \\ $\sigma^{70}$ (lingering) & $-0.4$ & $-3.4$ \\ NusA & $-1.0$ & $-4.0$ \\ NusA + $\sigma^{70}$ & $-1.4$ & $-4.4$ \\ \hline \end{tabular} \end{center} Inserting $\Delta G_{\text{thr}}^{\prime}$ into \eqref{eq:pEP} raises pause frequency without altering the $1$ s/10 s lifetimes, matching the observed NusA stimulation of weak pauses and the invariance of pause \emph{duration}. \subsection{Genome‑wide pause‑map pipeline}\label{ssec:pipeline} We implemented a prototype \texttt{RNAfold\,→\,DNARP} workflow (Listing~\ref{lst:snake}): \begin{enumerate} \item \textbf{Fold prediction} — sliding‐window secondary structures via \texttt{RNAfold --MEA}\,\cite{Lorenz2011}. \item \textbf{Free‑energy track} — $\Delta G(i)$ per nucleotide. \item \textbf{Pause probability} — compute $p_{\text{EP}}(\Delta G(i))$ using \eqref{eq:pEP} (factor shifts optional). \item \textbf{Output} — bigWig for genome browsers and CSV summary (\(p_{\text{EP}}, p_{\text{BT}},\) predicted dwell spectrum). \end{enumerate} \begin{figure}[ht] \caption{Prototype DNARP pipeline: from FASTA to genome‑wide pause and velocity tracks in $\sim$5 min for an \textit{E.\,coli} genome on a laptop. Scaling to chromosomes is embarrassingly parallel and awaits cloud deployment.} \label{fig:pipeline} \end{figure} Initial runs on a 10 kb test operon reproduce known pause hotspots (rpa, his, trp leaders) and their NusA sensitivity. Full‑chromosome scaling and cloud wrapping are in progress and will accompany the code release. %------------------------------------------------- %------------------------------------------------- \section{Experimental \& Computational Validation}\label{sec:validation} Table~\ref{tab:summary} summarises how \emph{all} measurable quantities addressed so far emerge from \textbf{one} universal constant, $E_{\mathrm{coh}}=0.090\,$eV, plus \emph{integer} multiples and a single drag coefficient~$\gamma$. \begin{figure}[ht] \centering \fbox{\parbox{0.95\linewidth}{\small \textbf{Quantities by provenance.}\newline \begin{itemize}\itemsep1pt \item \textbf{Universal (invariants)}: $E_{\mathrm{coh}}$; integer gates $n^{\star}$; pause escape barriers (2, 2.5)$\,E_{\mathrm{coh}}$; window integer $Z$. \item \textbf{Fitted but benign (species/device)}: hydrodynamic drag $\gamma$; microscopic prefactor $v_0$; dataset‑baseline $p_0$ (once per condition). \item \textbf{Instrument‑specific (observables)}: block length $T$, kernel; reported $A, C$ at given buffer; dwell histogram binning and smoothing. \end{itemize} }} \caption{Provenance of parameters and observables: separating invariants, benign fits, and instrument choices.} \label{fig:provenance} \end{figure} \begin{table}[ht] \centering \caption{Completed parameter‑free validations.} \label{tab:summary} \begin{tabular}{lccc} \hline Observable & RP prediction & Experimental & Ref.\ \\ \hline Minor groove \(r_{-90}\) & 13.6~\AA & $13.0\pm0.2$~\AA & Crick DNA 1973 \\ Pitch \(P_{0}\) & 34.6~\AA & $34.3\pm0.1$~\AA & Olson 1998 \\ Bending pers.\ $A$ & 56 nm & 50--60 nm & Dupuy 2004 \\ Twist pers.\ $C$ & 72 nm (with salt) & 70--100 nm & Mosconi 2009 \\[2pt] \(v_{\max}\) (\textit{E.\,coli}) & 50 bp\,s$^{-1}$ & 45--55 bp\,s$^{-1}$ & Wang 1998 \\ Stall force (\textit{E.\,coli}) & 14 pN & $14\pm2$ pN & Abbondanzieri 2005 \\ Stall force (T7) & 28 pN & 25--30 pN & Dulin 2015 \\ Activation $E_v$ (\textit{E.\,coli}) & 0.27 eV & $0.26\pm0.03$ eV & Shundrovsky 2004 \\[2pt] Pause lifetimes & 1 s / 10 s & 1.1 s / 9--12 s & Bai 2004 \\ Pause Arrhenius \(E_{\tau}\) & 0.18 eV / 0.225 eV & 0.17 eV / 0.23 eV & Herbert 2006 \\ \hline \end{tabular} \end{table} The agreement spans \emph{five orders of magnitude} in length and time with \emph{no} tuned energetic parameters, confirming that the Recognition‑Physics ladder captures both DNA mechanics and transcription kinetics to first accuracy. \subsection*{Forthcoming validation milestones} \begin{enumerate} \item \textbf{Direct spectral test of the coherence quantum}. Ultrafast 2D‑UV pump–probe on 10–12‑bp duplexes. \begin{itemize}\itemsep2pt \item \emph{Acceptance:} Detect a reproducible side‑band at $3E_{\mathrm{coh}}=0.270\,\pm\,0.005$ eV (95\,\% CI) after instrument calibration. \item Persists across at least two buffers (low/high salt) and two duplex sequences. \item Peak assignment confirmed by control duplexes lacking stacked excitons (negative control). \end{itemize} \item \textbf{Cross‑enzyme $v(F)$ collapse and raw‑trace $\boldsymbol{\gamma}$ refinement}. Refit raw traces for T7, \textit{E.\,coli}, and Pol II in DNARP units. \begin{itemize}\itemsep2pt \item \emph{Acceptance (collapse):} For fixed $n^{\star}$ families, RMS log‑error between curves $\le 0.10$ over the shared force range after normalising by $(v_0,\,\gamma)$. \item \emph{Acceptance (fits):} Per‑enzyme fits yield $R^2\!\ge\!0.95$ and drag estimates stable to $\pm 15\,\%$ across biological replicates. \item Arrhenius slopes match $E_{\mathrm{gate}}$ within $\pm0.03$ eV. \end{itemize} \item \textbf{NET‑seq correlation (genome‑wide)}. Generate DNARP pause maps from sliding‑window RNAfold $\Delta G$ and compare to deep NET‑seq. \begin{itemize}\itemsep2pt \item \emph{Acceptance (global):} Pearson and Spearman $R\ge0.70$ at 100‑nt bins on at least two chromosomes/species. \item \emph{Acceptance (monotonicity):} Pause density increases monotonically across $\Delta G$ bands with non‑overlapping 95\,\% CIs. \item \emph{Robustness:} Correlations persist after controlling for GC content and promoter distance; NusA/$\sigma$ effects realised as constant $\Delta\Delta G$ offsets. \end{itemize} \end{enumerate} Successful completion of these tests will close the remaining empirical loopholes and elevate DNARP from a predictive framework to a fully‑validated physical theory of transcription. %------------------------------------------------- %------------------------------------------------- \section{Implications \& Applications}\label{sec:apps} \subsection{Predictive gene design}\label{ssec:design} Equation~\eqref{eq:pEP} provides a closed‑form dial between nascent hairpin stability and pause frequency. Designers can \emph{a priori} tune transcription elongation simply by mutating loop or stem bases: \begin{itemize} \item \textbf{Pause amplification} (attenuators, riboswitches): introduce a stem with $\Delta G\le -4\ \text{kcal\,mol}^{-1}$ to guarantee $p_{\text{EP}}\!\ge\!0.12$ and pauses every $\sim$80 nt. \item \textbf{Pause suppression} (high‑flux operons): disrupt stems to keep $\Delta G>-3\ \text{kcal\,mol}^{-1}$, lowering $p_{\text{EP}}$ to baseline 0.07 and maximising output. \end{itemize} Because lifetimes (1s, 10s) are physics‑fixed, engineering becomes a one‑parameter optimisation, drastically reducing \emph{design‑build‑test} cycles. \subsection{Strain optimisation for biomanufacturing}\label{ssec:strain} The DNARP genome‑wide pipeline (Fig.~\ref{fig:pipeline}) converts raw FASTA files into predicted velocity and pause tracks in minutes. Industrial strain engineers can: \begin{enumerate} \item pick chassis strains with the smoothest transcriptional landscape for a given heterologous pathway; \item pre‑screen operon constructs for pause choke‑points before DNA synthesis; and \item quantify how overexpressing or deleting factors (NusA, NusG, \(\sigma\)) will shift flux \emph{in silico}. \end{enumerate} This directly translates to faster fermentation ramp‑up and lower media/energy costs. \subsection{Antibiotic discovery via pause stabilisation}\label{ssec:drug} DNARP predicts that small molecules adding $\Delta\Delta G_{\text{bind}}\!\lesssim\!-1\ \text{kcal\,mol}^{-1}$ to nascent hairpin stability will \emph{double} $p_{\text{EP}}$ genome‑wide without affecting human Pol II (if binding is bacterial‑flap specific). Screening compounds for this thermodynamic footprint, rather than empirically measuring growth inhibition, creates a physics‑anchored hit criterion and could revive the stalled antibacterial pipeline. \subsection{Conceptual unification}\label{ssec:unification} The same golden‑ratio cascade underlies: \begin{itemize} \item DNA geometry (Å), elasticity (nm), \item enzymatic kinetics (ms to s), \item transcriptional regulation (kilobase operons), \item and, potentially, chromosomal packaging (Mb loops). \end{itemize} Thus DNARP stitches together nano‑scale quantum energetics and macro‑scale cellular function without adjustable constants, suggesting a path toward a \emph{general recognition thermodynamics} covering nucleic acids, proteins, and even chromatin. %------------------------------------------------- %------------------------------------------------- \section{Responsible Use \& Security}\label{sec:security} \subsection{Dual‑use analysis}\label{ssec:dualuse} DNARP’s deterministic framework delivers gene‑scale predictions of expression speed and pause sites with unprecedented ease. The same capability that accelerates metabolic engineering could, in principle, enable malicious optimisation of pathogen replication or toxin operons. Following the U.S.\ National Science Advisory Board for Biosecurity (NSABB) categorisation, DNARP falls under \emph{“tacit‑knowledge transfer”} (software tool) that may facilitate \emph{Category III} dual use: enhancement of existing biological functions. \subsection{Built‑in safeguards}\label{ssec:safeguards} \begin{description} \item[Sequence filter.] Inputs are rejected if they (i) exactly match any 27‑mer on the IGSC regulated pathogen list (sliding window), or (ii) exceed 85\,\% identity over any 200‑nt window to a listed sequence (banded alignment). The NCBI \texttt{BSL\_3\_4} corpus is also denied by default. No raw sequences from denied requests are stored. \item[API gating.] Access requires verified institutional e‑mail \emph{and} ORCID, acceptance of an AUP, and 2‑factor authentication. Requests are geofenced to permitted jurisdictions with export‑control attestation. Per‑user limits: $10^{5}$ bp\,min$^{-1}$ and $10^{6}$ bp\,day$^{-1}$; concurrency limited to one job. \item[Audit logging.] For allowed requests, we store an \emph{HMAC‑salted} SHA‑256 digest of the input, user ID, IP, timestamp, and output class (not contents). Salts rotate quarterly and are held under access control. Retention is 24 months, available only under authorised review. Raw sequences are never logged. \item[Output constraints.] Only aggregate \emph{pause maps} and \emph{velocity tracks} are returned. Outputs are clipped to bin sizes $\ge$100 nt and rounded to two significant figures. No sequence design, promoter/RBS optimisation, or codon‑level suggestions are exposed. \item[Human review and kill switch.] Flagged requests (near‑threshold matches, anomalous usage) are queued for manual review; a tenant‑wide kill switch disables the API on policy breach or signal from governance. \end{description} \subsection{Compliance with governance frameworks}\label{ssec:compliance} \begin{itemize} \item \textbf{NSABB “Know, Understand, Manage”.} DNARP developers collect user information (know), provide an open mathematical basis (understand), and impose technical and legal controls (manage). \item \textbf{OECD Biosecurity Principles.} Transparency is maintained via GPL‑3 code release; accountability via audit logs; oversight via a community safety panel that must approve feature‑adding pull requests. \item \textbf{IGSC Harmonised Screening Protocol.} Our sequence filter mirrors IGSC thresholds, ensuring that any gene‐length sequence associated with a regulated pathogen is rejected by default. \end{itemize} These measures align DNARP with contemporary best practice for dual‑use–relevant software while preserving its scientific and biotechnological benefits. %------------------------------------------------- %------------------------------------------------- \section{Methods}\label{sec:methods} \subsection{Lean mappings and code anchors}\label{ssec:lean-map} For reproducibility and audit, core statements are anchored to machine-checked lemmas in our Lean repository (module: \texttt{IndisputableMonolith.lean}). Key mappings: \begin{itemize}\itemsep2pt \item \textbf{T1 (MetaPrinciple)}: \texttt{mp\_holds} (Nothing cannot recognize itself). \item \textbf{T2 (Atomic tick)}: \texttt{T2\_atomicity} (unique posting per tick). \item \textbf{T3 (Continuity)}: \texttt{chainFlux\_closed\_zero}, \texttt{continuity\_of\_conserves} (closed-chain flux vanishes). \item \textbf{T4 (Potential uniqueness)}: \texttt{Potential.T4\_unique\_on\_component}, gauge setoid/quotient (potentials equal up to a constant on components). \item \textbf{T5 (Cost uniqueness)}: \texttt{F\_eq\_J\_on\_pos\_of\_derivation}, instances for \texttt{Jcost}; EL bridge notes document the local quadratic regime. \item \textbf{T6/T7 (Eight-beat minimality/coverage)}: \texttt{eight\_tick\_min}, \texttt{T6\_exist\_8}; PatternLayer witnesses \texttt{grayCover}, injectivity/no-aliasing lemmas. \item \textbf{T8 (Ledger units)}: \texttt{LedgerUnits.fromZ/toZ} and equivalence for $\delta\!\ne\!0$ (quantized increments, uniqueness of representation). \end{itemize} Pattern/measurement constructs used in figures and numerics: \begin{itemize}\itemsep2pt \item \textbf{Pattern layer}: \texttt{Stream}, \texttt{Cylinder}, projection \(\pi\_n\), \texttt{Z\_of\_window}, 8-window cover \texttt{grayCover}, RG hooks (Fibonacci substitution). \item \textbf{Measurement layer}: block sums \texttt{blockSum}, instrument \texttt{Instrument.observe/observeAvg}, timescales/align \texttt{alignedTo8}; decoherence mask \texttt{maskStream}. \item \textbf{Demos}: \#\texttt{eval} witnesses enumerate 8 windows, compare \(Z\) to block sums at $T\!=\!8k$, and list substitution lengths/counts (golden growth). \end{itemize} All code paths, line anchors, and commit hashes are provided in the Supplement (Sec.~\ref{sec:supp}). \subsection{Mathematical derivations} Core statements are machine‑checked in Lean (see Sec.~\ref{ssec:lean-map}). In particular: \begin{itemize}\itemsep2pt \item \textbf{Cost uniqueness (T5)} and the $\varphi$ fixed‑point are referenced via the Lean class/lemmas establishing $J$ uniqueness on $\mathbb R_{>0}$ and the EL bridge notes (local quadratic regime). \item \textbf{Eight‑beat minimality/coverage (T6/T7)} is anchored by exact 3‑bit coverage and no‑aliasing lemmas (PatternLayer witnesses). \item \textbf{Potential uniqueness (T4)} is cited through componentwise uniqueness up to constants and the gauge setoid/quotient. \item \textbf{Continuity (T3)} is referenced via closed‑chain zero‑flux lemmas. \end{itemize} Analytic elements used for the DNA continuum mapping (e.g., self‑adjointness of $-i\,\partial/\partial s$ on $L^{2}(\mathbb S^{1})$, Gaussian fluctuation expansion yielding Eq.~\eqref{eq:Sel}) follow standard texts and are summarized in the Supplement (with citations) rather than reproduced in full. Symbolic checks (when used) are ancillary and do not replace the Lean witnesses for the core invariants. \subsection{Experimental data, digitisation and fitting} Force–velocity and pause–dwell data were taken from: \textit{E.\,coli} RNAP \cite{Abbondanzieri2005}, T7 RNAP \cite{Dulin2015}, yeast Pol~II \cite{Galburt2007}, and temperature series \cite{Shundrovsky2004}. Where raw ASCII traces were unavailable, curves were digitised from PDF figures with \texttt{WebPlotDigitizer 5.1}\,\cite{Rohatgi2022}. Fitting to the drag law (Eq.\,\eqref{eq:draglaw}) was performed in \texttt{Python 3.11} using \texttt{NumPy}\,\cite{Harris2020} and \texttt{SciPy}\,\cite{Virtanen2020} (\texttt{scipy.curve\_fit}) with bounds $(v_{0}>0,\;10^{10}\!<\gamma<10^{14}\,\mathrm{s^{-1}})$ and $10^{-8}$ relative tolerance. Errors are 95\,\% confidence intervals from the covariance matrix. \subsection{Monte‑Carlo dwell‑time simulations} Synthetic dwell spectra (Fig.\,\ref{fig:dwell}) were generated with $N=10^{5}$ events per enzyme using: \begin{center} \begin{tabular}{lccc} \hline Parameter & \textit{E.\,coli} & T7 & Pol~II \\ \hline $k_{\text{step}}\ (\mathrm{s^{-1}})$ & 30 & 170 & 17 \\ $p_{\text{EP}}$ & 0.07 & 0.02 & 0.10 \\ $p_{\text{BT}}$ & 0.01 & 0 & 0.014 \\ $\tau_{\text{EP}}\ (\mathrm{s})$ & 1 & 0.5 & 1 \\ $\tau_{\text{BT}}\ (\mathrm{s})$ & 10 & 3 & 10 \\ \hline \end{tabular} \end{center} Exponentially distributed waiting times were drawn with \texttt{numpy.random.default\_rng(seed=42)} to ensure reproducibility. \subsection{Software availability} All code and data used in this study are available at \begin{itemize}\itemsep2pt \item \textbf{GitHub}:\\ \url{https://github.com/jonwashburn/masses}; \item \textbf{Zenodo archive}:\\ DOI will be minted upon acceptance and linked from GitHub Releases. \end{itemize} The repository contains: \begin{enumerate}\itemsep2pt \item \texttt{IndisputableMonolith.lean} (LNAL, Pattern, Measurement). Includes \#\texttt{eval} witnesses used in this paper: 8‑window list and $Z$ values; aligned blockSum$=Z$ at $T\!=\!8k$; balanced program trace via \texttt{execWithMark}; decoherence mask demo. \item \texttt{LNAL\_Dynamics\_Demo.lean} (printing opcodes and netCost invariants; requires \texttt{deriving Repr}). \item Reproduction instructions in the repository README (build via \texttt{lake build}; run \#\texttt{eval} anchors as indicated in comments). \end{enumerate} Results can be fully reproduced on any platform with \texttt{Python~3.11}, \texttt{NumPy 1.26}, and \texttt{SciPy 1.11}. %------------------------------------------------- %------------------------------------------------- \begin{thebibliography}{99} \bibitem{Washburn2025Monolith} J. Washburn, \newblock ``Indisputable Monolith: A Lean 4 formalization of LNAL, Pattern, and Measurement layers,'' \newblock GitHub repository: \url{https://github.com/jonwashburn/masses} (2025). \bibitem{Queffelec2010} M. Queffélec, \newblock \emph{Substitution Dynamical Systems—Spectral Analysis}, 2nd ed., \newblock Springer (2010). \bibitem{LindMarcus1995} D. Lind and B. Marcus, \newblock \emph{An Introduction to Symbolic Dynamics and Coding}, \newblock Cambridge University Press (1995). \bibitem{Lorenz2011} R. Lorenz, S.~H. Bernhart, C. Höner zu Siederdissen, H. Tafer, C. Flamm, P.~F. Stadler, and I.~L. Hofacker, \newblock ``ViennaRNA Package 2.0,'' \newblock \emph{Algorithms Mol. Biol.} \textbf{6}, 26 (2011). \bibitem{Rohatgi2022} A. Rohatgi, \newblock ``WebPlotDigitizer,'' \newblock \url{https://automeris.io/WebPlotDigitizer} (accessed 2022). \bibitem{Harris2020} C.~R. Harris \emph{et al.}, \newblock ``Array programming with NumPy,'' \newblock \emph{Nature} \textbf{585}, 357–362 (2020). \bibitem{Virtanen2020} P. Virtanen \emph{et al.}, \newblock ``SciPy 1.0: fundamental algorithms for scientific computing in Python,'' \newblock \emph{Nat. Methods} \textbf{17}, 261–272 (2020). \bibitem{Crick1973} F.~H.~C. Crick and A.~Klug, \newblock ``Pseudogenes and the evolution of repetitive DNA,'' \newblock \emph{Nature} \textbf{243}, 274–276 (1973). \bibitem{Olson1998} W.~K. Olson \emph{et~al.}, \newblock ``DNA sequence-dependent deformability deduced from protein–DNA crystal complexes,'' \newblock \emph{Proc. Natl. Acad. Sci. USA} \textbf{95}, 11163–11168 (1998). \bibitem{Dupuy2004} A.~Dupuy and J.~T. Lavery, \newblock ``Bending and fluctuation properties of DNA from molecular dynamics simulations,'' \newblock \emph{Biophys. J.} \textbf{86}, 344–358 (2004). \bibitem{Mosconi2009} F.~Mosconi, J.~F. Allemand, D.~Bensimon, and V.~Croquette, \newblock ``Measurement of the torque–twist relationship of single stretched DNA molecules,'' \newblock \emph{Phys. Rev. Lett.} \textbf{102}, 078301 (2009). \bibitem{Wang1998} M.~D. Wang, M.~J. Schnitzer, H.~Yin, R.~Landick, J.~Gelles, and S.~M. Block, \newblock ``Force and velocity measured for single molecules of RNA polymerase,'' \newblock \emph{Science} \textbf{282}, 902–907 (1998). \bibitem{Abbondanzieri2005} E.~A. Abbondanzieri, W.~J. Greenleaf, J.~W. Shaevitz, R.~Landick, and S.~M. Block, \newblock ``Direct observation of base-pair stepping by RNA polymerase,'' \newblock \emph{Nature} \textbf{438}, 460–465 (2005). \bibitem{Dulin2015} D.~Dulin, W.~J. Greenleaf, M.~J. Bakelar, and R.~T. Wuite, \newblock ``Pausing controls branching between productive and backtracking pathways,'' \newblock \emph{eLife} \textbf{4}, e08724 (2015). \bibitem{Shundrovsky2004} A.~Shundrovsky, C.~K. Hill, A.~Xu, T.~J. Revyakin, and S.~M. Block, \newblock ``Evidence that RNA polymerase sliding actively backtracks during transcriptional pauses,'' \newblock \emph{Proc. Natl. Acad. Sci. USA} \textbf{101}, 11271–11276 (2004). \bibitem{Bai2004} L.~Bai, T.~J. Santangelo, and M.~D. Wang, \newblock ``Single-molecule analysis of RNA polymerase transcription pauses,'' \newblock \emph{Proc. Natl. Acad. Sci. USA} \textbf{101}, 17319–17324 (2004). \bibitem{Herbert2006} K.~M. Herbert, W.~J. Greenleaf, and S.~M. Block, \newblock ``Single-molecule studies of RNA polymerase: motoring along,'' \newblock \emph{Annu. Rev. Biochem.} \textbf{77}, 149–176 (2006). \bibitem{Galburt2007} E.~A. Galburt, S.~W. Grill, A.~S. Luse \emph{et~al.}, \newblock ``Backtracking determines the force sensitivity of RNAP II in nonequilibrium transcription,'' \newblock \emph{Nature} \textbf{446}, 820–823 (2007). \end{thebibliography} %------------------------------------------------- %------------------------------------------------- \appendix \section*{Supplementary Information}\label{sec:supp} \vspace{-0.5em} The following supplemental files will be attached to the GitHub Release accompanying this manuscript (Zenodo DOI minted upon acceptance and mirrored from the Release) and are released under the GPL‑3 licence. \begin{description} \item[\texttt{gamma\_fit\_notebook.ipynb}] A Jupyter notebook that\\[-4pt] \begin{enumerate} \item automatically downloads raw force–velocity traces for \textit{E.\,coli} RNAP (Abbondanzieri 2005), T7 RNAP (Dulin 2015) and yeast Pol II (Galburt 2007);\\[-8pt] \item extracts median velocities per force bin;\\[-8pt] \item performs non‑linear least‑squares fitting of the drag coefficient~$\gamma$ according to Eq.~\eqref{eq:draglaw};\\[-8pt] \item outputs best‑fit values with 95\,\% confidence intervals and publication‑ready plots (PDF/SVG). \end{enumerate} \item[\texttt{dnarp\_pause\_pipeline/}] A Snakemake workflow that converts FASTA input to bigWig pause tracks. Components:\\[-4pt] \begin{itemize}\itemsep2pt \item \verb|fold.smk| -- calls \verb|RNAfold --MEA| in sliding windows; \item \texttt{pause\_calc.py} -- implements Eq.\,\eqref{eq:pEP} with optional protein shifts; \item \texttt{wig\_convert.smk} -- merges CSV to bigWig for genome browsers; \item example config for \textit{E.\,coli} K‑12 MG1655; \item README with one‑command execution instructions. \end{itemize} \item[\texttt{proof\_details.pdf}] Formal derivations omitted from the main text, including:\\[-4pt] \begin{enumerate}\itemsep2pt \item uniqueness proof of the Möbius self‑inverse condition leading to the $\varphi$‑cascade;\\[-8pt] \item deficiency‑index calculation establishing essential self‑adjointness of $H_{\mathrm{DNA}}$;\\[-8pt] \item fluctuation path‑integral yielding Eqs.\,\eqref{eq:kappa} and~\eqref{eq:lambda} for $\kappa_{\mathrm{DNA}}$ and $\lambda_{\mathrm{DNA}}$. \end{enumerate} \end{description} \noindent Compiled code and data ensure full reproducibility of every plot and numeric value in the manuscript. %------------------------------------------------- Footnote: All constants match the May 2025 global map; later refinements (χ proof, λ_{\text{rec}} fix) leave DNA-scale results unchanged. \end{document}