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TAV_RGC_World_DiscoverPhysics_Submission/N4/rgc_paper.tex

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+\documentclass[11pt,a4paper]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb}
+\usepackage{booktabs}
+\usepackage{longtable}
+\usepackage{listings}
+\usepackage{xcolor}
+\usepackage[hidelinks]{hyperref}
+\usepackage{caption}
+
+\lstset{
+  basicstyle=\ttfamily\small,
+  frame=single,
+  framesep=4pt,
+  columns=fullflexible,
+  breaklines=true,
+  keywordstyle=\color{black}\bfseries,
+}
+
+\title{\bf Basin Concentration in the Fixed-Point Logistic Map:\\
+A 30-Day, 49-Core FPGA Dataset and a Bit-Exact Reference Model}
+
+\author{Ernest C. Gatlin III\\
+\small Independent Researcher, Tuscaloosa, Alabama\\
+\small \texttt{claygatlin@gmail.com}}
+
+\date{August 2026}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+We report a 30-day continuous hardware capture of 49 parallel fixed-point logistic
+maps implemented on a Xilinx Artix-7 FPGA, spanning the period-doubling route to chaos at
+$r_k = 3.5 + k/96$, $k = 0,\dots,48$. The run produced $1.81\times10^{7}$
+telemetry frames with no frame loss. We construct a software reference model of
+the 32-bit Q1.31 arithmetic and verify that it reproduces the measured cycle
+length of all 49 cores exactly, establishing the capture as a bit-exact,
+regenerable dataset whose entire $1.28$\,GB content is specified by 396 bytes of
+parameters. Using the verified model we then run an ensemble of $4.9\times10^{4}$
+random seeds and characterize the cycle-length distribution. We find that the
+quantized map is \emph{not} statistically a random map on its state space: mean
+cycle lengths are $0.240$ times the random-map prediction
+($95\%$ CI $[0.188, 0.296]$, core-level bootstrap), and the Rayleigh law implied
+by random-map theory is rejected. The cause is basin concentration --- a median of
+4 attractors per parameter value, with the dominant basin capturing a median
+$91.8\%$ of sampled initial conditions, against $\sim\!11$ expected cycles with
+dispersed basins under the random-map model. We additionally document a
+representability failure at $r = 4$, where the ROM constant saturates and
+produces an anomalous 272-iteration cycle that escapes the absorbing fixed point
+of the exact map. All data, the reference model, and the analysis tools are
+public.
+\end{abstract}
+
+\section{Introduction}
+
+Chaotic maps realized in finite-precision arithmetic do not remain chaotic. Every
+orbit is eventually periodic, and the resulting cycles are typically far shorter
+than the state space would suggest. This phenomenon --- \emph{dynamical
+degradation of digital chaos} --- has been studied since Grebogi, Ott and
+Yorke~\cite{grebogi1988} established that average period length scales with
+computational precision and the correlation dimension of the attractor. It is of
+practical consequence in chaos-based cryptography, where short cycles compromise
+the randomness a cipher depends on~\cite{li2005,persohn2012}.
+
+The standard analytical baseline is Harris's random-map model: if the quantized
+map is treated as a uniformly random function on $N$ states, the cycle length
+$L$ reached from a random start satisfies
+\begin{equation}
+\Pr(L > \lambda\sqrt{N}) = \exp(-\lambda^2/2),
+\qquad
+\mathbb{E}[L] = \sqrt{\pi N/8},
+\label{eq:randommap}
+\end{equation}
+and the expected number of distinct cycles is $\tfrac{1}{2}\ln N$.
+
+Most published measurements are simulation studies at a single control parameter,
+reported as summary statistics. This work contributes three things that are less
+common. First, a long-duration hardware capture with the complete raw telemetry
+released publicly. Second, a reference model verified bit-exactly against that
+hardware on all 49 cores, which makes the dataset independently regenerable.
+Third, a seed ensemble large enough to replace single-orbit measurements with
+distributions, from which we find that the random-map baseline fails --- and
+identify basin concentration as the reason.
+
+We make no claim of novelty for the phenomenon or for the verification method;
+both are mature. The contribution is measurement.
+
+\section{Apparatus and arithmetic}
+
+\subsection{Platform}
+
+The array is implemented on a Digilent Arty A7-100T development board
+(Xilinx Artix-7 \texttt{XC7A100T-1CSG324C}), clocked at $100$\,MHz, with
+telemetry delivered over the on-board USB--UART bridge at $230{,}400$\,baud.
+
+\subsection{Core}
+
+Each core iterates the logistic map in 32-bit fixed point. The state $x$ is
+Q1.31; the control parameter $r$ is Q2.30. With $\mathrm{ONE} = 2^{31}$ and
+$M = 2^{32}-1$, one iteration is
+\begin{equation}
+t = \frac{x \cdot \big((\mathrm{ONE} - x)\ \&\ M\big)}{2^{31}},
+\qquad
+x' = \frac{r \cdot t}{2^{30}},
+\label{eq:step}
+\end{equation}
+with both results masked to 32 bits and both divisions realized as arithmetic
+right shifts with truncation (not rounding). The 49 cores are instantiated in
+identical logic differing only in the ROM constant
+\begin{equation}
+R_k = \min\!\left(\mathrm{round}\!\left[(3.5 + k/96)\cdot 2^{30}\right],\ 2^{32}-1\right),
+\qquad k = 0,\dots,48,
+\label{eq:rom}
+\end{equation}
+and in the seed $x_0 = 2^{30} + k$.
+
+The saturation in Eq.~\eqref{eq:rom} is not decorative. At $k=48$ the intended
+value $r = 4.0$ requires $4.0 \times 2^{30} = 2^{32}$, one bit wider than the
+Q2.30 word. The hardware saturates to \texttt{0xFFFFFFFF}, so core~48 in fact runs
+$r = 4 - 2^{-30} = 3.999999999069$. Section~\ref{sec:core48} shows this is
+directly observable in the data.
+
+\subsection{Telemetry}
+
+Cores are sampled round-robin. Each frame carries a 24-bit projection of the
+state, $x_{24} = x \gg 8$, together with a global frame counter. Consecutive
+samples of the same core are separated by $s = 49$ map iterations; this stride
+was recovered post hoc by matching telemetry against known periodic orbits and
+confirmed against the frame-counter field. A sampled period $p$ therefore implies
+a true iteration period $L$ satisfying $L/\gcd(L,s) = p$, which is not in general
+a bijection --- an ambiguity the reference model resolves.
+
+The export discards the low 8 bits of every frame ($1.45\times10^{8}$ bits over
+the full run). Because the model is bit-exact, this information is recoverable by
+re-running the map and is not lost in any operational sense.
+
+\subsection{Run N4}
+
+The capture ran from 2026-06-06 14:59:47 to 2026-07-06 14:59:48 (exactly
+$2.592\times10^{6}$\,s), producing $18{,}143{,}974$ frames, approximately
+$3.7\times10^{5}$ per core. A watchdog terminated the acquisition process at the
+30-day mark and recorded the shutdown. Across the sampled portion of the capture,
+the inter-frame counter difference equalled the expected stride in every row; no
+frames were dropped. A $10$-byte frame occupies $0.434$\,ms of line time at $230{,}400$\,baud
+against a median inter-frame interval of $163$\,ms, so the link operated at
+$0.27\%$ utilization; the host reader was deliberately paced to eliminate
+overrun risk and to reserve inbound bandwidth, split reads at the
+driver level were reassembled by the acquisition software without loss. Three
+earlier runs (N1--N3) under identical parameters produced bit-identical
+per-core statistics, as expected of a deterministic synchronous pipeline with
+fixed seeds.
+
+\section{Bit-exact reference model}
+
+We implemented Eq.~\eqref{eq:step} in arbitrary-precision integer arithmetic and
+located each core's cycle using Brent's algorithm, obtaining the exact iteration
+period $L$ and transient length $\mu$ without reference to the telemetry. The
+predicted sampled period $L/\gcd(L,49)$ was then compared against the period
+measured in the hardware stream.
+
+\begin{quote}
+\textbf{All 49 cores agree exactly. There are no mismatches.}
+\end{quote}
+
+The agreement is non-trivial in two respects. Five cores have $\gcd(L,49) > 1$,
+so their telemetry period is a proper fraction of the true cycle: core~22 has
+$L = 16{,}709 = 49 \times 341$ and reports a sampled period of 341, concealing a
+factor of 49. The model recovers these correctly. Transient lengths, which the
+telemetry cannot measure at all, range from $\mu = 1$ (eighteen cores land
+immediately on their cycle) to $\mu = 30{,}225$.
+
+The practical consequence is that the run's irreducible information content is
+49 ROM constants, 49 seeds and a frame count --- 396 bytes. The published
+$1.28$\,GB is a deterministic unfolding of these, and can be regenerated
+bit-for-bit.
+
+\section{The \texorpdfstring{$r=4$}{r=4} representability failure}
+\label{sec:core48}
+
+Core~48 is the only core whose behaviour cannot be predicted from the nominal
+parameter. With the saturated constant \texttt{0xFFFFFFFF}, the model gives
+$L = 272$, $\mu = 1$, and exactly 270 distinct 24-bit export codes. All three
+match the hardware, as does a fourth signature: core~48 is the only core in the
+array whose telemetry ever attains the boundary values $0$ and $2^{23}-1$. This
+is dynamically correct, since $r = 4$ is the unique parameter for which the map's
+image is the full interval; for $r < 4$ the attractor is strictly interior.
+
+The 270-versus-272 discrepancy is the export truncation: two pairs of distinct
+32-bit states on the cycle share a 24-bit prefix. This also explains an apparent
+paradox in the raw telemetry, where the exported value $0$ is followed by three
+different successors --- impossible for a deterministic function of the exported
+state, and direct evidence that the projection is lossy.
+
+We emphasize that the exact map at $r = 4$ has \emph{no} attracting periodic
+orbit; in double precision the orbit falls into the absorbing fixed point at
+$0$. The 272-cycle is created by quantization, not preserved by it. It is also
+extremely fragile: perturbing the ROM constant by one LSB
+($\Delta r = 2^{-30}$) leaves it intact, but two LSBs destroy it
+($L = 13{,}051$). The literature customarily writes $r \in (3.5699, 4]$ as though
+the endpoint were attainable; in Q2.30 it is not.
+
+\section{Seed ensemble}
+
+A single hardware seed per core yields one draw per parameter value, which cannot
+test a distributional claim. We therefore used the verified model to sample
+$10^{3}$ uniformly random 32-bit seeds per core ($4.9\times10^{4}$ total), burning
+in $2\times10^{5}$ iterations to guarantee arrival on a cycle before measuring
+the exact period. No seed failed to resolve.
+
+Table~\ref{tab:main} summarizes the result. We restrict quantitative comparison
+to the 38 cores whose orbits are aperiodic in double precision; the remaining 11
+lie in genuine periodic windows of the exact map and are reproduced correctly by
+the hardware (periods 3, 4, 5, 6, 8, 10) as a fidelity check.
+
+\subsection{Cycle lengths}
+
+Pooled over the 38 chaotic-band cores, the mean cycle length is $9{,}870$ against
+the random-map prediction $\sqrt{\pi \cdot 2^{32}/8} = 41{,}069$, a ratio of
+$0.240$.
+
+Because seeds within a core are strongly dependent --- $10^{3}$ seeds typically
+resolve to only $\sim\!4$ distinct cycle lengths --- a seed-level confidence
+interval is badly anticonservative. Bootstrapping over cores instead gives
+\begin{equation}
+\frac{\mathbb{E}[L]_{\text{measured}}}{\mathbb{E}[L]_{\text{random-map}}}
+= 0.240 \quad (95\%\ \mathrm{CI}\ [0.188,\, 0.296]).
+\end{equation}
+The interval excludes unity by a wide margin. A maximum-likelihood Rayleigh fit
+to the pooled sample gives an effective state space of $2^{26.21}$ against a
+hardware word of $2^{32}$: the map explores roughly $1.5\%$ of its available
+states.
+
+\subsection{Basin concentration}
+
+The deviation is not merely one of scale. Under the random-map model a state
+space of $2^{32}$ should carry $\tfrac{1}{2}\ln(2^{32}) \approx 11.1$ distinct
+cycles with basins distributed according to a Poisson--Dirichlet law, so a sample
+of $10^{3}$ starts would recover many distinct lengths. We observe instead:
+
+\begin{center}
+\begin{tabular}{lcc}
+\toprule
+Statistic & Measured & Random-map \\
+\midrule
+Distinct cycles per core (median) & 4 & 11.1 \\
+Distinct cycles per core (range)  & 1--10 & --- \\
+Dominant basin share (median)     & $91.8\%$ & dispersed \\
+Dominant basin share (mean)       & $81.7\%$ & dispersed \\
+Cores with a basin $> 90\%$       & 21 / 38 & $\approx 0$ \\
+Cores with a basin $> 99\%$       & 5 / 38  & $\approx 0$ \\
+\bottomrule
+\end{tabular}
+\end{center}
+
+The quantized logistic map is therefore better described as a map with a small
+number of attractors, one of which typically captures over $90\%$ of the state
+space, than as a random function.
+
+\subsection{Two distinct effects, separated by rounding mode}
+\label{sec:rounding}
+
+A natural conjecture is that both the short cycles and the concentrated basins
+follow from the truncating right shifts in Eq.~\eqref{eq:step}, which are not
+measure-preserving: each iteration discards low-order bits, so the image of the
+map is strictly smaller than its domain. We tested this by replacing truncation
+with round-to-nearest (adding a half-LSB before each shift) and repeating the
+ensemble on a matched subsample of 14 chaotic-band cores with identical seeds.
+
+\begin{center}
+\begin{tabular}{lccc}
+\toprule
+Statistic & Truncation & Round-to-nearest & Random-map \\
+\midrule
+Mean cycle length            & $16{,}652$ & $7{,}431$ & $41{,}069$ \\
+Ratio to $\mathbb{E}[L]$     & $0.405$ & $0.181$ & $1.000$ \\
+Distinct cycles per core     & $3.4$ & $5.1$ & $11.1$ \\
+Dominant basin share         & $0.91$ & $0.65$ & dispersed \\
+\bottomrule
+\end{tabular}
+\end{center}
+
+The conjecture is confirmed for basin structure and refuted for cycle length.
+Rounding disperses the basins substantially and raises the cycle count, moving
+both statistics toward the random-map baseline: basin concentration is therefore
+a truncation artifact. Mean cycle length, however, moves \emph{further} from the
+random-map prediction under rounding, so cycle shortening is a separate effect
+that truncation does not explain.
+
+A third measurement isolates the two further. Exhaustive computation of the
+reachable set $A = \bigcap_{n\ge 0} f^{n}(\{0,\dots,2^{W}-1\})$ at reduced word
+widths gives $|A| \sim 2^{0.45W}$, and this contraction is essentially unaffected
+by rounding mode ($|A| = 61, 134, 282, 507$ under truncation at
+$W = 16, 18, 20, 22$ against $102, 59, 452, 607$ under rounding, with no
+systematic difference). The collapse of the reachable set to $O(\sqrt{N})$ states
+is thus intrinsic to quantizing a non-injective map, not a consequence of the
+rounding rule.
+
+Three effects are therefore distinguishable: reachable-set contraction
+(rounding-independent), basin concentration (a truncation artifact), and cycle
+shortening (worsened by rounding, and unexplained here). Since $|A|$ is unchanged
+between modes while the cycle count rises, rounding must partition the same
+reachable set into more and smaller cycles; why it does so we do not know.
+
+The subsample used for this comparison was selected toward long-cycle cores, so
+its absolute truncation ratio ($0.405$) is higher than the full 38-core value
+($0.240$) reported above. The comparison between modes is nonetheless matched:
+same cores, same seeds, rounding rule the only difference.
+
+\subsection{Validity of the single-seed hardware run}
+
+The ensemble also permits a retrospective check on N4. For 36 of the 38
+chaotic-band cores, the hardware seed landed on the largest basin, and no
+hardware cycle was absent from the $10^{3}$-seed sample. The two exceptions are
+core~16 ($L = 1478$, fourth-largest basin, $10.5\%$ of seeds) and core~45
+($L = 13{,}643$, second-largest, $47.8\%$). Given the basin concentration
+documented above, this outcome was near-certain rather than fortunate --- but it
+could not have been established from the hardware run alone.
+
+\section{Discussion}
+
+Three limitations bound these results. The measurement is of one arithmetic
+realization: truncating shifts, Q1.31 state, Q2.30 parameter. Section~\ref{sec:rounding} varies the rounding rule and finds the
+basin structure to be truncation-specific while the cycle shortening is not; word
+width and multiplier structure remain unvaried. The
+parameter lattice, $r_k = 3.5 + k/96$, is coarse relative to the fine structure
+of the bifurcation diagram, and steps over narrow periodic windows entirely.
+Finally, the comparison baseline is the random-map model, which is a heuristic
+rather than a theorem for structured maps.
+
+The dataset's principal utility is as a benchmark. Because the reference model
+is verified and the parameters are 396 bytes, any claim made here can be checked
+independently and cheaply, and the hardware capture provides a ground truth
+against which alternative period-search methods can be validated at scale.
+The functional-graph and state-mapping-network
+approaches~\cite{li2019,fpsa2019} in particular compute transient lengths, fixed
+points and limit-cycle distributions directly, and would resolve at lower cost
+several quantities we obtained here by exhaustive search.
+
+\section{Data and code availability}
+
+The complete N4 capture ($1.28$\,GB telemetry, $837$\,MB acquisition log), the
+earlier runs, the reference model, and the analysis tools
+(\texttt{rgc\_cycle\_probe2.py}, \texttt{rgc\_seed\_ensemble.py}) are available at
+\url{https://huggingface.co/parti-wave/TAV_RGC_World_DiscoverPhysics}.
+The per-seed ensemble used in Section~5 is included as \texttt{ensemble.csv}
+($4.9\times10^{4}$ rows).
+
+\section*{Acknowledgements}
+
+Analysis assistance and independent verification of the reference model were
+performed with Claude (Anthropic).
+
+\begin{thebibliography}{9}
+\bibitem{grebogi1988}
+C.~Grebogi, E.~Ott and J.~A.~Yorke,
+\emph{Roundoff-induced periodicity and the correlation dimension of chaotic
+attractors}, Phys. Rev. A \textbf{38}, 3688--3692 (1988).
+\href{https://doi.org/10.1103/PhysRevA.38.3688}{doi:10.1103/PhysRevA.38.3688}
+
+\bibitem{li2005}
+S.~Li, G.~Chen and X.~Mou,
+\emph{On the dynamical degradation of digital piecewise linear chaotic maps},
+Int. J. Bifurcat. Chaos \textbf{15}(10), 3119--3151 (2005).
+\href{https://doi.org/10.1142/S0218127405014052}{doi:10.1142/S0218127405014052}
+
+\bibitem{persohn2012}
+K.~J.~Persohn and R.~J.~Povinelli,
+\emph{Analyzing logistic map pseudorandom number generators for periodicity
+induced by finite precision floating-point representation},
+Chaos Solitons Fractals \textbf{45}(3), 238--245 (2012).
+\href{https://doi.org/10.1016/j.chaos.2011.12.006}{doi:10.1016/j.chaos.2011.12.006}
+
+\bibitem{nagaraj2008}
+N.~Nagaraj, M.~C.~Shastry and P.~G.~Vaidya,
+\emph{Increasing average period lengths by switching of robust chaos maps in
+finite precision}, Eur. Phys. J. Spec. Top. \textbf{165}, 73--83 (2008).
+\href{https://doi.org/10.1140/epjst/e2008-00850-4}{doi:10.1140/epjst/e2008-00850-4}
+
+\bibitem{li2019}
+C.~Li, B.~Feng, S.~Li, J.~Kurths and G.~Chen,
+\emph{Dynamic analysis of digital chaotic maps via state-mapping networks},
+IEEE Trans. Circuits Syst. I \textbf{66}(6), 2322--2335 (2019).
+\href{https://doi.org/10.1109/TCSI.2018.2888688}{doi:10.1109/TCSI.2018.2888688}
+
+\bibitem{fpsa2019}
+C.~Fan and Q.~Ding,
+\emph{Analysing the dynamics of digital chaotic maps via a new period search
+algorithm}, Nonlinear Dyn. \textbf{97}(1), 831--841 (2019).
+\href{https://doi.org/10.1007/s11071-019-05015-4}{doi:10.1007/s11071-019-05015-4}
+\end{thebibliography}
+
+\clearpage
+\appendix
+\section{Per-core results}
+
+\begin{longtable}{rrrrrrr}
+\caption{Per-core results. $L_{\mathrm{hw}}$ is the true iteration period of the
+N4 hardware seed, recovered by the reference model and consistent with the
+measured telemetry period in every case. $L_{\max}$ is the cycle with the largest
+basin in the $10^{3}$-seed ensemble; $n_{\mathrm{cyc}}$ the number of distinct
+cycles found; $\tilde L$ the ensemble median; $\beta$ the dominant basin share.}
+\label{tab:main}\\
+\toprule
+$k$ & $r_k$ & $L_{\mathrm{hw}}$ & $L_{\max}$ & $n_{\mathrm{cyc}}$ & $\tilde L$ & $\beta$ \\
+\midrule
+\endfirsthead
+\toprule
+$k$ & $r_k$ & $L_{\mathrm{hw}}$ & $L_{\max}$ & $n_{\mathrm{cyc}}$ & $\tilde L$ & $\beta$ \\
+\midrule
+\endhead
+\bottomrule
+\endfoot
+\input{table_rows.tex}
+\end{longtable}
+
+\clearpage
+\section{Reference model}
+
+\begin{lstlisting}[language=Python]
+ONE, M32 = 1 << 31, (1 << 32) - 1
+
+def r_rom(k):
+    "Hardware ROM constant; saturates at k=48."
+    return min(round((3.5 + k/96) * (1 << 30)), M32)
+
+def step(x, r):
+    t = ((x * ((ONE - x) & M32)) >> 31) & M32
+    return ((r * t) >> 30) & M32
+
+# core k: seed x0 = (1 << 30) + k, telemetry stride 49, export x >> 8
+\end{lstlisting}
+
+\end{document}