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\hypersetup{pdfauthor={Angel E Navidad},pdftitle={On the Origins of Belize},pdfsubject={},pdfkeywords={}} % pdf metadata
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\begin{document}
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\title{On the Origins of Belize}
\author{A Navidad}
\address{Harvard College, Cambridge, MA, US}
\curraddr{Benque Viejo, Cayo, Belize}
\email{navidad@college.harvard.edu}
\date{12 Feb 2025}
\thanks{} % incl non-grant support per ams
\begin{abstract}
\end{abstract}
\keywords{}
\maketitle
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% s intro
\section{Introduction}
\label{s:intro}
	The origins of Belize have been an open problem in historical literature since at least the 18th century, and remain so today, with over two dozen theories proposed to date, and little consensus to boot.\footnote{Belize is the only pirate settlement in the Caribbean, and only logwood settlement, to have survived past the 19th century. As such, it is the only country in the Commonwealth Caribbean (or Commonwealth?) for which this problem is open.}
	
	This paper seeks to add fuel to the fire by xxx
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% s claim
\section{Claim}
\label{s:claim}
	We present this paper's main claim here, and map out the support it will be given later on. Let \ment{question*} be `What are Belize's origins?' And let \ment{stories*} be answers to question*.\footnote{Existing or possible, subject to some suitability criteria, eg are historical, are not flimsy, so on.} Then,
	%
	% main claim
	\begin{mainclm}
	\label{clm:maindecency}
	\ref{eqn:namedmap} and \ref{eqn:namedlog} are a decent way of building a story*.\footnote{Qualified, eg up to similarity. Tools used jointly only, plus historical data.}
	\end{mainclm}
	%
	\begin{equation}
	\tag{Map}
	\label{eqn:namedmap}
		\begin{tikzcd}[arrows=dash]
		\nt &\bc \rar \ar[rr,bend left] &\mt \rar &\id &\et
		\end{tikzcd}
	\end{equation}
	%
	For vertices as certain places in the Bay, and edges as a certain relation between them. The Bay is mapped from northwest to southeast, beginning in northern Yucatan (\(\nt\)), flowing down to northern Belize (\(\bc\)), then the mouth of the Bay (\(\mt\)), next the Bay Islands (\(\id\)), and finally eastern Honduras (\(\et\)). We say a place \(\ab\) is connected to place \(\cd\) (\(\ab-\cd\)) if \(\ab\) is \ment{close} to \(\cd\), and not otherwise.\footnote{Ignoring trivial cases eg where \(\ab=\cd\).} Basically, spaces are close if they are geographically close.
	%
	\begin{equation}
	\tag{Log}
	\label{eqn:namedlog}
		\begin{tikzcd}[sep=huge]
		t_0 \rar["\pre" description] &t_c \rar["\cpd" description] &t_d \rar[bend left,"\pst_0" description] \rar[bend right,"\pst_1" description] &t_1
		\end{tikzcd}
	\end{equation}
	%
	For vertices as certain points in time, and arcs as certain series of events from one point to the other. A story* is logged from start to finish, beginning at the earliest point (\(t_0\)), then a certain \ment{coupling} point (\(t_c\)), next an ensuing \ment{de}coupling point (\(t_d\)), and ending at the latest point in the story* (\(t_1\)). These four points split a story* into three parts, with a coupl\ment{ed} period (\(\cpd\)) in the middle, and a preceding \ment{pre}coupled period (\(\pre\)). The succeeding \ment{post}coupled period (\(\pst\)) is plotted as two series of events (\(\pst_0\) and \(\pst_1\)) rather than one, to account for two possibly distinct sequences of events. Basically, the coupling refers to the coincidence of piracy and logging in the Bay.
	
	%
	% paper flow
	We bolster Main Claim~\ref{clm:maindecency} by spelling out what makes for a decent way of building a story*, and then demonstrating that \ref{eqn:namedmap} and \ref{eqn:namedlog} are decent. So we first gloss \ment{decency} here.
	%
	\begin{gloss}
	\label{gloss:decency}
		A way of building a story* is decent if the story* so built does \emph{not} answer any relevant \ment{thick} question and \emph{does} at least one relevant \ment{thin} one, where relevancy is to the question*, and thick questions are interpretation-heavy and thin ones less so.\footnote{The latter possibly in such a way as to only partially help answer at least one thick question. Might want further features too, eg is new, offers granular stories*, so on.}
	\end{gloss}
	%
	The general idea being that a maximally useful way of building a story* ought to be open to iterpretation.
	%
	\begin{minrclm}
	\label{clm:minorquestions}
	Relevant thick questions are only `When was Belize settled?' and `When was Belize founded?' Relevant thin ones are at least `When was Belize used for piracy?' and `When was Belize used for logging?'\footnote{Again for relevance to the question*. Up to similarity. Taken as granted, though support for Main Claim~\ref{clm:maindecency} hinges on the first part here.}
	\end{minrclm}
	%
	In Section~\ref{s:support}, we first build \ref{eqn:namedmap} and \ref{eqn:namedlog}, then show they jointly answer neither thick question, and lastly show they do answer at least one thin one.
%
%
%
% s support
\section{Support}
\label{s:support}
	%
	% ss:build
	\subsection{Build}
	\label{ss:build}
		We build \ref{eqn:namedmap} and \ref{eqn:namedlog} from the ground up here.
		
		We start by noting that almost all existing stories* begin with piracy and end with logging, and noting that there is almost no consensus on the path charted from one to the other, nor on the time it took to traverse said path. Nonetheless, we now have thematic start and end points, call them \ment{first-theme} (piracy) and \ment{last-theme} (logging).
		
		Now we note that, while existing stories* are agreed on first-theme, there is yet no agreement on when nor where to begin, with values for the former ranging over three centuries, and for the latter spanning two landmasses.\footnote{Sixteenth to eighteenth cneturies, and Central America including Yucatan, plus the Greater Antilles.} But despite not having them, call them \ment{first-time} (when story* begins) and \ment{first-place} (where story* begins), and similarly for \ment{last-time} and \ment{last-place}.\footnote{These spatial and temporal points are for stories*, \emph{not} for the start/end of our themes ie phenomena ie piracy and logging.}
		%
		% sss:map
		\subsubsection{Map}
		\label{sss:map}
			Now, to get first-places from existing stories*, we either stop and consider each proferred first-place individually, or else sort them into classes and consider each class instead. We do the latter here.
		
			First, group all first-places located within the Bay, and group all others.\footnote{Discard null and `Bay' first-places.} We now have the Bay Islands and Belize on the one hand, and Campeachy, Catoche, Jamaica, Mosquito Shore, Providence and Tortuga on the other.\footnote{Possibly ignoring minor/unreviewed first-places. For Belize, mostly north-central. For Campeachy, including Terminos and Tris. For Catoche, including Cozumel and Ascension, possibly. For Tortuga, including north-west Hispaniola. For Mosquito Shore, assuming/mostly south-central. Providence is \emph{Old} Providence.}
		
			Before proceeding, we note that these first-places look, of course, like a list of pirate haunts in the greater vicinity of Belize, just as we would expect from first-theme. To have a comprehensive \ref{eqn:namedmap}, then, we would want a \emph{complete} list of pirate haunts in said vicinity.
		
			Now, before checking that we have a complete list of pirate haunts, we here note that nautical entry to the Bay is either through the north or the east, such that for first-places outside the Bay, regardless of their geographic location, the relevant agents would all enter the Bay through one of two points.\footnote{Counting north-east entrance via Caymans as east.} In which case, we might map the north (\(\nt\)) and east (\(\et\)) entrances, and take these as proxies for \emph{all} pirate haunts outside the Bay. So, \(\nt\) and \(\et\) give us a complete list of pirate haunts for the second group.
		
			We have no such shortcut for the first group of first-places (pirate haunts in the Bay) though, so this check is done by hand. This done, we find we might want to add one first-place to this group, namely, the mouth of the Bay (\(\mt\)), giving us three first-places (with the Bay Islands (\(\id\)) and Belize (\(\bc\))). We assume our hand check was thorough, and so conclude that \(\mt\), \(\id\), and \(\bc\) give us a complete list of pirate haunts for the first group.\footnote{This step to be supported later on ie that hand check was thorough, and that \(\mt\) was a pirate haunt (at least up to criteria used to say \(\bc\) was one.}
		
			So, we now have a list of five first-places which name all pirate haunts in the vicinity of Belize. As \ref{eqn:namedmap} charts only these, it too is complete in this respect.
		
			Lastly, as we would like to chart our first-places in some relevant manner, and not just list them, we look for relevant ways these pirate haunts were related. This is done by hand. This done, the first and most obvious relation is distance, of course. But in addition to this, we have cost, ie the difficulty of getting from one first-place to the other. And we find no further salient relations.\footnote{Ignoring non-salient/missed relations, but assume no others, or assume these are subsumed by cost. May want to exclude from or keep constant in cost the nautical knowledge/practice/expereience weight. Thoroughness of hand check supported later.}
			
			As cost subsumes distance, we consider only cost. As those were the only salient relations among our first-places, cost gives us \emph{all} the ways our first-places are related.\footnote{All ways we care about, assuming hand check was thorough.} So say some first-place is \ment{close} to another if it's \emph{not} costly to get from the one to the other and vice versa.\footnote{Giving us undirected closeness from directed cost, for simplicity, but could keep directed. By \ment{not costly} we mean for some cutoff of cost which we deem sufficiently cheap.} Then, we have that \(\bc\), \(\mt\), and \(\id\) are close to each other, while \(\nt\) and \(\et\) are close to none.\footnote{Ignoring self-closeness.}
			
			So, we now have one relation which subsumes all salient or relevant ways in which our first-places are related. As \ref{eqn:namedmap} charts only this relation, it too is complete in this respect.
			
			This fully gives us \ref{eqn:namedmap}, so our work here is done.
		%
		% sss:log
		\subsubsection{Log}
		\label{sss:log}
			We start by noting that in existing stories* we have very many first-times and last-times (when stories* start and end). We further have two relevant phenomena, ie our first-theme and last-theme ie piracy and logging. Call their temporal start and end \ment{first-piracy}, \ment{last-piracy}, \ment{first-logging}, and \ment{last-logging}.
			
			Now, we note that none of these first-times pre-dates first-piracy, and none of these last-times post-dates last-piracy. So, first-theme is present at every point in time in existing stories*. We further note that \emph{only} some of these first-times pre-date first-logging, and none of these last-times post-dates last-logging. So last-theme is either present at every point, or else present only in later points in time in existing stories*.
			
			So, of the temporal points of first-theme and last-theme, \emph{none} show up in \emph{all} existing stories*.
			
			We salvage this sad state by looking at the stories* which \emph{do} include first-logging. In almost all of them, this point is not given prominence but rather minimised in favour of a second point, that when the incidence of last-theme reached some significant threshhold, call it \ment{first-significant-logging}.\footnote{There is a further \ment{first-significant-piracy} in some existing stories*, but not needed here.}
			
			If we now look for first-significant-logging in existing stories*, we do find it in all of them.
			
			Instead of plotting first-significant-logging, however, we take stock of \emph{why} first-logging is only in some stories*, and why first-significant-logging is favoured. The immediate answer is that we can't see the incidence of last-theme, of course.\footnote{No records on logging.} But additionally, we further presume that it was \emph{in}significant for some time after first-logging, and that it first reached significance only after first-significant-logging.\footnote{Significance regarding frequency of logging, or amount logged, or number of loggers, and so on.}
			
			So, all existing stories* presume that there was some non-null time between first-logging and first-significant-logging. Again, instead of plotting, we take stock of why this is.
			
			Here, we find that all such stories* presume that first-theme \emph{lead to} last-theme in a \emph{specific} way, namely, by pirates' logging a little bit from first-logging, then more and more, until logging a lot from first-significant-logging.\footnote{For some significance threshhold.} Which is to say, all existing stories* presume the increasing \emph{co}incidence of our first-theme and last-theme.
			
			This, finally, would make for a useful organising principle for \ref{eqn:namedlog}, due not just to existing consensus, but additionally to the greater insight we have into the incidence of first-theme.
			
			Now, we have \emph{strong} significant positive coincidence between first-theme and last-theme starting from first-significant-logging. But even \emph{weak} significant positive coincidence might be useful when using first-theme incidence as proxy for last-theme incidence.\footnote{For some value of storng or weak.}
			
			So, call the point in time when first-theme and last-theme first positively, significantly coincided \(t_c\), and add to \ref{eqn:namedlog}.
			
			Naturally, in the period from start-logging to \(t_c\), first-theme and last-theme did not significantly coincide at all.\footnote{No examples of significant negative coincidence in stories*.} For stories* with first-times at or after first-logging, this covers the entire period. For stories* with first-times pre-dating first-logging, there would still be no significant coincidence at all, given no incidence of last-theme. So call this period of insignifcant coincidence \(\pre\), and call first-times \(t_0\), and add both to \ref{eqn:namedlog}.
			
			Now, in the period after \(t_c\), first-theme and last-theme did significantly coincide at all points, but in all stories* this is only so up to a point.\footnote{Due to various reasons including fall of last-theme incidence below significance, fall of first-theme incidence below significance, so on.} This is, of course, what we would expect, given that last-theme outlived first-theme by a lot.
			
			So, at some point after \(t_c\), first-theme and last-theme first \emph{stopped} positively or significantly coinciding. Call it \(t_d\) and add to \ref{eqn:namedlog}. Furthermore, call the period between \(t_c\) and this point \(\cpd\), and likewise add to \ref{eqn:namedlog}.\footnote{Only one possible path between these points, ie significant coincidence all the time.}
			
			After \(t_d\), all stories* agree that the incidence of first-theme fell below some significance threshhold, while only some say as much of last-theme. Nonetheless, the first consensus is all we need to see all possible paths to all last-times.
			
			So, from \(t_d\) to last-time, the incidence of last-theme either was significant at all points, or it was not, while the coincidence of first-theme and last-theme was either insignificant at all points, or it was not.
			%
			% table
			\begin{table}
			\caption{Paths from \(t_d\) to last-time.}
			\label{tab:pathsforlog}
			\begin{tabular}{cccc}
			first-theme &last-theme &coinc &coinc direcn\\
			insig	&sig	&insig	&irlvt\\
			insig	&sig	&sig		&+ or --\\
			insig	&insig	&insig	&irlvt\\
			insig	&insig	&sig		&+ or --
			\end{tabular}
			\end{table}
			%
			
			The paths possible from \(t_d\) to last-time are listed in Table~\ref{tab:pathsforlog}. We note that coincidence is insignificant at \emph{all} points in time in the first and third, while logging is significant at \emph{all} points in first and second.
			
			For simplicity, we merge the first and third paths and call them \(\pst_0\), and merge the second and fourth call them \(\pst_1\), and call last-times \(t_1\), and add all to Table~\ref{tab:pathsforlog}. 
			
			This fully gives us \ref{eqn:namedlog}, so our work here is done.
	%
	% ss:decency
	\subsection{Decency}
	\label{ss:decency}
		We show \ref{eqn:namedmap} and \ref{eqn:namedlog} as built are decent here. We first note that \ref{eqn:namedlog} requires at least all data on the incidence of piracy in the Bay, so say we have that. We must then (somehow) identify \(t_c\) and \(t_d\) in these data, and get a picture direct picture of piracy, and proxy picture of logging, in the Bay from \(t_0\) to \(t_1\).\footnote{\(t_c\) and \(t_d\) are the only defined temporal points in \ref{eqn:namedlog}.} This obviously falls way short of answering either thick question, and yet directly answers the first listed thin one, while at least partially answering the second thin question.\footnote{Assuming good data for the thin ones.}
		
		So, \ref{eqn:namedmap} and \ref{eqn:namedlog} are decent.
	%
	% ss:ends
	\subsection{Ends}
	\label{ss:ends}
		We finish work left for later in Section~\ref{ss:build} here.
		%
		% sss:onmt
		\subsubsection{On \(\mt\)}
		\label{sss:onmt}
		For thoroughness of first hand check in Section~\ref{sss:map}, we note the literature and sources reviewed, and those missed.\footnote{List here.} For the inclusion of \(\mt\) as a pirate haunt, we note only the greater incidence of piracy near it than near \(\bc\). As the latter is deemed a haunt, so ought the former.
		%
		% sss:onmt
		\subsubsection{On closeness}
		\label{sss:oncloseness}
		For the second hand check in Section~\ref{sss:map}, we again note literature reviewed and missed.\footnote{List here.}
		%
		% sss:onstories
		\subsubsection{On stories*}
		\label{sss:onstories}
		For Section~\ref{sss:log}, read \ment{existing} stories* as those reviewed in literature noted here.\footnote{List here.}
%
%
%
%
% s use
\section{Use}
\label{s:use}
	We use \ref{eqn:namedmap} and \ref{eqn:namedlog} here, first with help from only particular existing stories*, then in a generalised manner. For the particular stories*, we use those with the earliest and latest answers to question*, ie Campbell and Restall, plus a few in between, ie the Fénix column and Read's letter and Bulmer-Thomases paper. \footnote{\citelist{\cite{cam11} \cite{cam03} \cite{res19} \cite{rds32} \cite{fen49} \cite{bul16}}. A non-exhaustive list of stories* to consider.}
	%
	% ss:particular
	\subsection{Particular}
	\label{ss:particular}
		Qualify all claims regarding stories* here as being \emph{for} a given reading. We contend only that the given readings are \emph{possible}.
		%
		% sss:bulmers
		\subsubsection{Bulmer-Thomases}
		\label{sss:bulmers}
		The Bulmer-Thomases offer a haven-camp story* set spatially in \(\bc\), \(\id\), and \(\et\) only.\footnote{Buccaneers possibly use as haven in 1642--1670 and turn to logging in 1670s \cite{bul16}*{pp 137--138, 145, 151--156}.} They explicitly set \(t_0\) in 1642 and \(t_d\) in 1670s and \(\pst\) via first path in \(\pst_0\). They do not explicitly set \(t_1\), but we read \emph{prior} to 1705 for it, and further set \(t_c\) as mid 17th century from explicit values.\footnote{Implicit in \cite{bul16}*{p 149}.} This completes \ref{eqn:namedlog}.
		%
		% sss:campbell
		\subsubsection{Campbell}
		\label{sss:campbell}
		Campbell offers a haven-camp-settlement story* set spatially in \(\bc\) only.\footnote{Loose pirate communities from probably mid 16th cent \cite{cam11}*{pp 95--96, 100 \forn{pass}}, pirate haven on St George's and surrounding cayes from probably mid 1550s \cite{cam03}*{pp 176, 179}, buccaneer haven from 17th cent \cite{cam03}*{p 174}, buccaneer haven definitely from 1670s \citelist{\cite{cam11}*{pp 82--83} \cite{cam03}*{p 176, 178}}, partial Providencer presence from 1641 \cite{cam11}*{p 129}, first important settlement on St George's definitely from 1765 \citelist{\cite{cam11}*{pp 121--122, 129} \cite{cam03}*{pp 175, 179--180}}, and non-buccaneering ie sedentary ie logging settlement from mid 17th cent \cite{cam03}*{p 171}. The Providencer answer possibly a misreading.} They explicitly set \(t_0\) in mid 1550s, \(t_d\) in mid 17th century, and \(\pst\) via first path in \(\pst_0\) ie first in Table~\ref{tab:pathsforlog}.\footnote{Explicit in \cite{cam03}*{p 171}. Also implicit in \cite{cam11}*{pp 91--92, 104, 108}. But Campbell does not mean \emph{only} \forn{H campech} by \ment{logwood} \cite{cam11}*{pp 104--105}, whereas we do, so a bit tricky to read.} % p91-92 = by 1708, p104 = by 1680s, p108 = by early 1670s
		They do \emph{not} quite so explicitly set \(t_c\) nor \(t_1\), but we read early 17th and early 18th centuries for them.\footnote{That is, \(t_c\) implicitly set to Elizabethan era in \cite{cam11}*{pp 106--107}, and to prior to 1695 and 1697 and 1655 in \cite{cam11}*{pp 84, 91, 106}, respectively for latter. And \(t_1\) implicitly to 1708 in \cite{cam11}*{pp vii, 93}, and possibly more. Again, for \ment{logwood} meaning \ment{dyewood}.} This completes \ref{eqn:namedlog}.
		% For Campbell, if our reading is right, then we get coincident piracy and logging from early to mid 17th century ie during \(\cpd\), with significant logging starting at some point therein.\footnote{And attendant significant piracy, given coincidence. Read \emph{significant} \emph{positive} coincidence for coincidence \forn{simpliciter} hereon.} And as after \(t_d\) we take the first \(\pst_0\), we further get significant logging at all points to early 18th century.\footnote{Modulo interruptions, which Campbell tolerates.}
		%
		% sss:fenix
		\subsubsection{Fénix}
		\label{sss:fenix} Sierra O'Reilly offers a haven-settlement story* set spatially in \(\bc\) only, and temporally in mid 17th century for \(t_1\) only.\footnote{Pirates settle on Old from mid 17th cent \cite{fen49}*{p 3}. This is a Wallace story*. \citelist{\cite{cam09}*{pp 72--110} \cite{bul16}*{pp 137--151} \cite{res19}*{pp 19--24}} have thoroughly shown that \ment{Wallace} in such stories* is most likely apocryphal. This, however, is also a Meridian story*, that is, one from Merida built on or after the 18th cent, when that city was heavily focussed on the Bay. (\citelist{\cite{car71}*{pp 55, 210} \cite{car78}*{pp 260--261} \cite{anc78}*{pp 370--376} \cite{pen69}*{pp 217--219}} are also Meridian stories*.) By the former, these stories* are at least partly apocryphal. By the latter, these stories* are at least possibly better-sourced than some others, given the non-zero chance of unique primary sources in Merida in and after the 18th cent. So, we ignore most Wallace details here, and read \ment{one of the first Baymen} for \ment{Wallace}. Note this means we disregard the move by \cite{bul16}*{pp 137--138, 145, 151--156} from \ment{Wallace}'s being apocryphal to temporal restrictions on acceptable answers to question*, and further disregard charges of dishonesty in Wallace stories* in \citelist{\cite{bul16}*{pp 138, 140--141} \cite{cam09}*{pp 87--88, 105--106} \cite{res19}*{pp 23--24}}, as the former follows and the latter regards only \ment{Wallace} properly (not symbolically) read.}
		We find nothing further, so do not complete \ref{eqn:namedlog}.
		%
		% sss:reads
		\subsubsection{Read's}
		\label{sss:reads} This Bayman offers a camp-settlement story* set spatially in \(\bc\) only.\footnote{Buccaneer-loggers camp-settle on Old then camp-settle up to Hondo in mid 17th cent \cite{rds32}*{p 3}. This is one of the earliest stories* available in print (at least), and one of the only to ascribe to later Bacalar strikes a reason over piracy (possibly). The unsigned letter, dated Campeachy, 24 Nov 1731, reads in part:\begin{quote}As to the State of the Bay of Honduras, I shall give it you as briefly as possible. The ancient City of Bacalar, situate in that Part of the Province of Yucatan, which lies on the Bay of Honduras, was twice sack'd, and at last totally ruined by the English many Years ago; on which the Logwood-Cutters of that Nation, who had settled on the River of Valis, possessed themselves of the New River and that of the Hondo; which last is distant from the Ruins of Bacalar about five Leagus. Here they built a great many Houses and Hutts, and employ'd Multitudes of Negroes in cutting Logwood, which was transported to Jamaica and Europe by Numbers of Vessels trading from thence to the Bay.\end{quote} \cite{res19}*{pp 13, 16} deems a similar contemporaneous story* `a politically motivated rhetorical exaggeration' (ie the 1734 Pitt story* that logging camps on Old `had been possessed by the English for more than a hundred years,' in AGI Mexico 3099 ff 5--10). We partially disagree on this.} % agi not digitised http://pares.mcu.es/ParesBusquedas20/catalogo/description/374685
		They explicitly set \(\pst\) as the first path in \(\pst_0\). They do not explicitly set \(t_1\), but we read \emph{prior} to mid 17th century for it. Nothing further found, so no complete \ref{eqn:namedlog}.
		%
		% sss:restall
		\subsubsection{Restall}
		\label{sss:restall}
		Restall offers a camp-settlement story* set spatially in \(\bc\) and \(\nt\) only, and temporally in 1662 for \(t_0\) and 1717 for \(t_1\) only.\footnote{Haphazard haven-camp possibly from 1662, and settled logging camp definitely from 1716 \cite{res19}*{pp 2--3, 12--13, 17, 20--21, 30}.} Nothing further found, so no complete \ref{eqn:namedlog}.
	%
	% ss:noteson
	\subsection{Notes}
	\label{ss:noteson}
		We make a number of observations here regarding the preceding stories*.
		%
		% sss:onnolog
		\subsubsection{On \ref{eqn:namedlog} completion}
		\label{sss:onnolog}
		The first thing we note from reviewed stories* above is our having three \ref{eqn:namedlog} not complete.
		
		Now, we read all those stories* as presuming piracy lead to logging.\footnote{And as stories* not missing parts, which might not be the case for Read's in Section~\ref{ss:particular}, but we ignore that here.} This still seems correct on second thought. But per Section~\ref{sss:log}, we expect complete \ref{eqn:namedlog} from such stories*. So we probably erred somewhere in Section~\ref{sss:log}. We now review the steops in that section to find the error(s).
		%
		\begin{steps}[In Section~\ref{sss:log}]
		We built \ref{eqn:namedlog} (up to \(t_c\)) like so:
		\begin{enumerate}
		\item First-logging is \emph{not} in all stories*.\footnote{Or is minimised.}
		\item First-significant-logging \emph{is} in all stories*.\footnote{Or is not similarly minimised.}
		\item So, all stories* presume insignificant logging to first-significant-logging.
		\item So, all stories* presume piracy lead to logging in a specific way.\footnote{By pirates logging a little bit from first-logging, then more and more, until logging a lot from first-significant-logging.}
		\item So, all stories* presume increasing coincidence of piracy and logging.
		\item So, for \(t_c\) the point when piracy and logging first positively significantly coincided, \(t_c\) is in all stories.
		\qedhere % apparantly needed when ending in list
		\end{enumerate}
		\end{steps}
		%
		We now find (1) still holds for all stories* in Section~\ref{ss:particular}, and find (2) holds for all but Fénix. Now, this is a temporally compressed land-and-settle story*, while the rest are drawn out squat-squat-settle ones.\footnote{Distinction first from \cite{cam11}*{pp 95--96}. The associated distinction (intentional vs accidental) is not imported here.} We ignore stories of the former ilk hereon.
		
		So both (1) and (2) hold for all our stories*. We further find (3) holds too, but the move to (4) is tricky, so we now spell it out more fully.
		
		First, let \ment{piracy-win*} be \ment{successful piracy}, and \ment{piracy-fail*} be \ment{failed piracy}.\footnote{For simple or minimal \ment{success} eg where pirates surprised a ship or sacked a port, even if they got no/nil booty. And similar \ment{failure} eg where they found no ships nor came to port.} And let \ment{piracy*} be either.
		
		That done, we now review step (4). We find that swapping in \ment{piracy-win*} for \ment{piracy} does not work, whereas \ment{piracy-fail*} \emph{does}. That is, for the particular way of \ment{leading to} spelt out in (4), we want to say that all stories* presume piracy-fail* lead to logging, \emph{not} that piracy-win* did so.
		
		Now let (4a*) be `All stories* presume piracy-fail* lead to logging in \emph{some} way.' Then we find (4a*) holds for all our stories*, independently of (3), and ignore stories* where it doesn't hereon.\footnote{Eg where successful pirates discovered logging then afterwards intentionally/en-masse set out only to log ie sudden conversion. This might be the case eg in stories* where social/legal pressure forced successful pirates en masse to logging, but maybe only if pressures suddenly/rapidly not gradually materialise. So, we ignore stories* where pirates suddenly/en masse dropped piracy for logging.}
		
		At this point, it might be helpful to see how (the incidence of) piracy-win* and piracy-fail* might look like plotted together. We first imagine a bell curve for piracy*.\footnote{Note, we presume both piracy-win* and piracy-fail* incidence are multimodal distributions with various peaks and valleys, and so too piracy*. So the following is just a slice of these curves.} Then, under the piracy* curve, piracy-win* and piracy-fail* look like either 1 or 2 in Figure~\ref{fig:yaynayinterference}.\footnote{Figure~\ref{fig:yaynayinterference} presumes pirates were more successful than not (more piracy-win* than piracy-fail* incidents), but even if otherwise, the following still works. Additionally, cases where piracy-fail* peak precedes piracy-win* peak are not shown, but their curves are just 1 and 2 in Figure~\ref{fig:yaynayinterference} flipped, so following still works.} In the first case, we say they \emph{closely} tracked or \emph{positively} coincided, and say they loosely tracked or negatively coincided in the second case.
		% small fig 
		\input{ppfiginter} % \label{fig:yaynayinterference}
		
		Note, in Section~\ref{sss:log} we said we had a clearer picture of piracy than logging. While this still holds, we now say we have a clearer picture of piracy-win* than logging, and further a clearer picture of piracy-win* than of piracy-fail*.\footnote{Or possibly no picture of piracy-fail* at all, unless by proxy from piracy-win*. We further note our relative lack of non-Spanish records for piracy* in the Bay, and further, their tendency to label any foreigner \ment{pirate} (inflating piracy-win* incidence), and the (we think likely) chance of their being so overwhelmed or desentised over time to piracy* so as to stop recording/reporting individual piracy-win* incidents (especially minor ones, thereby deflating incidence).}
		
		Moving on though, let (4b*) be `All stories* presume piracy-fail* lead to logging in a \emph{particular} way, namely, via increasingly-significant (positive) coincidence.'\footnote{Logging a little bit from first-logging, then more and more, until logging a lot from first-significant-logging, with all of this occurring almost only during piracy-fail* incidents. That is, logging and piracy-fail* coincide only a few times from first-logging, then more and more, until coinciding very many/almost at all times from first-significant-logging.} Then, the move from (4a*) to (4b*) is natural or likely, and so made.\footnote{Though (4b*) does not follow from (4a*), but cannot see how else (4a*) might obtain. May explore later.}
		
		This, of course, immediately gets us to (5*) and (6*) (steps (5) and (6) with \ment{piracy-fail*} swapped in for \ment{piracy}). But we now have a not-so-useful \ref{eqn:namedlog}.
		
		At this point, it might be useful to get a clearer view of 2 in Figure~\ref{fig:yaynayinterference}. The possible orders of peaks are sketched in Figure~\ref{fig:yaynayorder}.\footnote{Also possible for 1 in Figure~\ref{fig:yaynayinterference}, with closer peaks, but irrelevant here.} That is, if piracy-win* and piracy-fail* only loosely track each other, rather than closely, then periods of heightened piracy-fail* incidence might precede, or succeed, or both precede and succeed periods of heightened piracy-win* incidence.\footnote{Probably 2 in Figure~\ref{fig:yaynayorder} is likelier than the others eg pirates being successful lead to more pirates trying (in a delayed manner).}
		% small fig 
		\input{ppfigorder} % \label{fig:yaynayorder}
		
		So, if/when we have 1 in Figure~\ref{fig:yaynayinterference}, piracy-win* data from historical records might serve as a proxy picture for piracy-fail*, regardless of the order of their peaks. However, if/when we have 2 in Figure~\ref{fig:yaynayinterference}, then unless we know which of 1--3 in Figure~\ref{fig:yaynayorder} we have, piracy-win* data would \emph{not} make for good proxy for piracy-fail*.\footnote{With options 1 and 2 in Figure~\ref{fig:yaynayinterference} not exclusive, of course.}
		
		As that's the case, then \ref{eqn:namedlog} might be jointly decent with \ref{eqn:namedmap}, but it's not of much use as regards getting a picture of logging incidence.
		
		But as regards our stories* in Section~\ref{ss:particular}, if the above is correct, then we ought to have gotten at least \(t_c\) (and therefore \(t_c^*\)) in Read's and Restall.\footnote{For \(t_c^*\) as the temporal point when piracy-fail* and logging first positively significantly coincided. Note Fénix was discarded above.} However, we get said point in neither. For Read's, we say the story* is abridged, and so supply a \(t_c^*\) ourselves. For Restall, the story* is \emph{not} abridged, so something else it at play.
		
		We consider three possibilities for Restall. 
		
		First, say some phenomenon \ment{leads} to another \ment{coarsely} if they (as a \emph{whole}) are appropriately related or \ment{tied}, and say it leads to another \ment{finely} if enough of their (\emph{individual}) incidents are appropriately related ie tied. Then one might presume that piracy coarsely lead to logging sans presuming one finely lead to the other, eg via an en masse/swift pirates-to-loggers conversion. For (4a*) we previously discarded all the swift conversion stories*. We now revise that, and say we discard all stories* that do \emph{not} presume piracy \emph{finely} lead to logging. We read Restall as presuming fine leading to, and so keep.
		
		Next, say piracy finely lead to logging, but with most of the tied incidents \emph{outside} the Bay. Now, this might follow simply from higher piracy incidence outside the Bay, or a stronger pirac-logging tie outside the Bay (or both).
		
		The first case (of higher piracy outside the Bay), all else equal, is granted, and its implications disregarded as causing us no trouble.\footnote{In this case, we would have lower logging incidence in Bay than outside, resulting possibly in insignificant logging only, and so in a delayed first-significant-logging. But the move to insignificant logging imports interpretation (how high one's standards for significant logging are, or use of relative vs absolute standards), and so not relevant here.}
		
		The second case (of a stronger tie outside the Bay), all else equal, is \emph{not} granted.\footnote{Though this is a live possibility in either direction, and in fact, we probably have either a location- and time-sensitive tie or coincidence between piracy and logging, eg stronger tie in Bay when pirates had less luck there, or stronger tie in Campeachy after Jamaica invaded, and so on.} Depending on standards used, this might naturally lead to a too-weak-to-count tie in the Bay, and so to only-insignificant piracy-logging coincidence in Bay. For now, all such stories* are discarded, giving us location insensitive tie and coincidence.\footnote{Possibly to review later.}
		
		The third case (both of the above) is not granted, as the second case is not.
		
		Of the three cases above, we read Restall as of the first variety. That is, we say they have high standards for logging significance, and interpret \(t_c^*=t_1\).
		
		In sum, \ref{eqn:namedlog} isn't of much use, and there were a number of errors in building it which ought to be corrected.\footnote{We further note coincidence defined \emph{only} spatiotemporally, vs coincidence as importing more than proximity of incidents, or else not including spatiotemporal proximity criteria at all, eg requiring additionally/only a tie/appropriate relation between them. It seems we might only need the minimal one in this paper. For minimal coincidence, we further distinguish undirected boolean (values \code{true} or \code{false} for `for any piracy incident, there is at least one logging incident nearby; and for any logging incident, there is at least one piracy incident nearby') from directed categorical (values of \(-1\) or \(0\) or \(1\) with \(1\) for boolean coincidence and \(-1\) for `for any piracy incident, there is no logging incident nearby; and for any logging incident, there is no piracy incident nearby' and \(0\) for `for any piracy incident, there may/may not be at least one logging incident nearby; or for any logging incident, there may/may not be at least one piracy incident nearby'). We can even further make the directed categorical into a graded/quantitative measure with vlaues in interval \([-1,1]\) by mapping coincident \ment{ratio} (eg `for any piracy incident in some given year, there are on average \(n\) logging incidents nearby'). So, coincidence gets very messy, and we might just discard and instead count logging as a strict subset of piracy-fail* incidents. So we might look at only eg pure-piracy-fail*, piracy-fail-et-log*, and pure-log incidence curves.} %\input{ppfigcoinc} % \label{fig:naylogcoincidence} - shows only perfect coinc rn - shows only minimal/ie only spatiotemporal coincidence rn
		%
		% sss:onmapclusters
		\subsubsection{On \ref{eqn:namedmap} clustering}
		\label{sss:onmapclusters}
		We note the popularity of \(\bc\) as a setting in stories* in Section~\ref{ss:particular}, compared to the other places of the inner Bay, namely \(\id\) and \(\mt\). The former is used (minimally) only by Campbell, while the latter is used by none.
		%
		% sss:ontrendsuse
		\subsubsection{On trends use}
		\label{sss:ontrendsuse}
		We further note the underuse of patterns in building reviewed stories*, with only Restall offering a story* so built (from cartographic patterns, in their case) rather than one from discrete datapoints.\footnote{Missing patterns, in addition to piracy incidence, include dwindling Spanish commerce in the Bay (and its effect on piracy), a growing resident pirate population near Bay, growing social/legal pressures against piracy (though this one \emph{is} generally noted), and growing textile industry (and thus logwood demand) in England (and Netherlands?).}
		%
		% sss:onmissinginterpret
		\subsubsection{On missing interpretations}
		\label{sss:onmissinginterpret}
		We lastly note possible interpretations missing from reviewed stories*.\footnote{Not a thorough list, and in no particular order.} 
		
		Firstly, \ment{Bay}'s referring to \emph{all} of the inner Bay seems unpopular, despite this possibly or likely being the case for early resident pirates.\footnote{\ment{All} meaning \(\id\), \(\mt\), \(\bc\). \ment{Bay}'s meaining \(\id\) and \(\bc\) is also possible. We imagine the inner Bay may have functioned as an early \ment{Bay Triangle}, eg if piracy and logging were coupled before the invasion of Jamaica. (The \ment{Bay Triangle} of Bay, Jamaica, and Mosquito Shore gotten from \citelist{\cite{cam11}*{p 130} \cite{cam03}*{p 186}}, but first proposed by \cite{nay89}*{pp 46--53}, it seems.) We further think this might be important not only to understanding early and later Baymen's sense of place, but to interpreting their answers to question* (\forn{mut mut}). For instance, say they derived their sense of ownership from use. And say they first used \(\id\) as a haven at \(t_i\), and only later used \(\mt\) and \(\bc\) as havens or camps at \(t_k\). Then if they already felt like they owned \(\id\) at some intermediate \(t_j\), later use of \(\mt\) and \(\bc\) might count to them as \emph{expansion} of their original territory, such that they might claim first ownership of \emph{all} the inner Bay from \(t_i\) or \(t_j\), despite only gaining two-thirds of it from \(t_k\). (This seems analogous to saying `the US was founded in 1776,' despite a vast majority of the country not existing at the time, and despite the War not being won and the Union not being constituted but for several years later.) Further, say any of these places fell into disuse from \(t_m\) for some extended time. If the Baymen came back at some later \(t_n\) to find no one using said place, then they might consider this simply a \emph{continuation} of their original use and ownership from \(t_i\) or \(t_j\), rather than a \emph{renewal} of use or ownership. (This seems analogous to answering question* with any pre-17xxx date, as Baymen's presence here has \emph{only} been uninterrupted since this date.) Finally, say it was the \emph{original} place (\(\id\) here) which fell into disuse from \(t_m\), this time \emph{permanently}. Then the Baymen might nonetheless claim ownership from \(t_i\) or \(t_j\), most especially if they viewed themselves as constituting a polity. (Analogously, if the Thirteen Colonies' successor states seceded from the US tomorrow, it would \emph{still} be the case that the US was founded in 1776.) We lastly note that the later political expediency of recognising or rehashing such claims (or not) does \emph{not} seem relevant to their truth or reasonableness.}
		
		Additionally, a couple of discrete events are not prominently featured, despite their possible or likely importance, including the Tipu rebellion and Santo Tomas discovery.\footnote{The first for shifting the Maya population away from coastal \(\bc\), and the latter for shifting Spanish shipping even closer to \(\mt\) from Puerto Caballos. On the first, we note that some literature implies that Maya flight took a few \emph{years}, despite its being a live likelihood that it took only a few \emph{months}, at most, from \forn{k'atun 1 ahaw} (6 Jun 1638).}
		
		Finally, only early Spanish martitime control has been emphasised, despite the inverse judgment being equally possible or even likelier.
	%
	% ss:general
	\subsection{General}
	\label{ss:general}
	We use \ref{eqn:namedmap} and \ref{eqn:namedlog} to build a minimal story* here, with help from Section~\ref{ss:particular} stories*. Then we offer possible interpretations to answer question*.
		%
		% sss:storybuild
		\subsubsection{Build}
		\label{sss:storybuild}
		We first set places in \ref{eqn:namedmap}. For entrance, we set \(\et\), given this was the historical entrance in the 16th century, and given the eastern location of resident pirate population in the 17th. Now, of non-entrance places in \ref{eqn:namedmap}, \(\id\) is closest to \(\et\), and is coincidentally is the most prominent haven in the Bay in literature, and in incidence data, so we set \(\id\). Of course, \(\bc\) is close to \(\id\), and the most popular place in reviewed stories*, so we further set \(\bc\). But now we have chosen places in the Bay which are both close to \(\id\) and either good logging or haven locations. If this is our criteria, then we can't really leave out \(\mt\), which is equally as close to \(\id\) as \(\bc\), and would make for good logging, and is the second most prominent haven in the Bay after \(\id\) in incidence data. So we lastly set \(\mt\), which means we have set \emph{all} of the inner Bay in \ref{eqn:namedmap}.
		
		We now set points in \ref{eqn:namedlog}, starting with \(t_c\).\footnote{This part of our story* poorly built, Section~\ref{sss:onnolog} having shown we have more choices to make, and thus likely need more data, than \ref{eqn:namedlog} strictly requires. Regarding the latter, we'd probably need at least piracy incidence data for Campeachy/Yucatan. Relevant commercial and demographic trends, and piracy in the Greater Antilles, might also prove useful.} For this, we might first find incidence peaks and work out \(t_c\) from there.\footnote{We see three major peaks in our data, ie in the 1600s, 1630s to early 1650s, and late 1670s to 1680s.} Or we might follow reviewed stories* to set \(t_c\). For novelty, we do the former here, and focus on the second incidence peak in data.\footnote{There seem to be more reasons to choose the second peak over the first or third, but the latter are still live options. Reasons likely or possibly include its size and scope, plus its matching a boom in resident pirate population and logwood demand, and matching a bust in Spanish commerce, defence, available ready-for-market logwood, and coastal native population.} Then piracy-fail* either closely or loosely tracked this piracy-win* peak, per Section~\ref{sss:onnolog}. For simplicity, we say closely, and set \(t_c\) during this peak's early years, ie early 1630s.\footnote{Again, there might be more reasons to choose these rather than other years within the peak, but all are still possible. Here, we imagine \emph{no} loose tracking before or after this peak, despite this still being possible on top of our presumed close tracking.} Now, we know logging decoupled from piracy by at least 1662 in Campeachy, so say it similarly decoupled in the Bay, and set \(t_d\) in early 1660s.\footnote{Again, this and following choices very likely not this straightforward. Note, some literature gives 1662 as \emph{the} start date of logging in Campeachy. Strictly, however, logging there started at least \emph{by} said year.} We lastly follow reviewed stories* to set \(t_0\) in early 1590s, and \(t_1\) in early 1700s, and \(\pst\) as the first path in \(\pst_0\), thereby completing \ref{eqn:namedlog}.
		%
		% sss:storyinterpret
		\subsubsection{Interpretations}
		\label{sss:storyinterpret}
		There are a number of ways to interpret this story*, but we offer only a few here. First, say we adopt resident pirates' possible sense of founding and settlement. Then even a seasonal pirate haven or camp might count as polities, making them the earliest predecessor states available, and giving us an answer to question* circa \(t_c\). Or say we first scrutinise their initial senses of founding and settlement. Then it is \emph{still} possible to find them reasonable, again, giving a question* answer in circa \(t_c\). Of course, we might just as possibly find them wanting, and adopt stricter senses, giving us a question* answer in circa \(t_1\).\footnote{The point here being not that any of these is the uniquely suitable interpretation, but rather that \emph{none} of these is \emph{un}suitable, modulo the story's* being well built (which it is not, in our case) and interpretation fleshed out.}
%
%
%
% s concl
\section{Conclusion}
\label{s:concl}
	xxx 
%
%
%
% s data
\section{Data}
\label{s:data}
	We present the data this paper uses.\footnote{Deposited at \url{https://github.com/aenavidad/pp-0}.}
	%
	% ss:datapiracy
	\subsection{Piracy}
	\label{ss:datapiracy}
	Incidence data for the Bay in the 17th century shown in Figure~\ref{fig:piracyinbay}.\footnote{From literature (all years) and AGI Guatemala and Mexico series (to 1669 inclusive). Note the review process very likely missed some literature and records. Further, incidents do not seem consistently carved out, and further include some defence incidents, and ones of dubious piracy. The AGI review includes a tally (and partial calendar) of defence related records.}
	%
	% ss:dataliterature
	\subsection{Literature}
	\label{ss:dataliterature}
	Answers to question* in literature since 18th century shown in Table~\ref{tab:literature}.\footnote{As interpreted, of course.}
%
% large fig and tab
\input{ppfigpir} % \label{fig:piracyinbay}
\input{pptablit} % \label{tab:literature}
%
%
%
% s references
\begin{bibdiv}
\label{s:references}
	\begin{biblist}
	\bibselect{pprefs}
	\end{biblist}
\end{bibdiv}
%
%
%
\end{document}