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\fancyhead[L]{\small\textit{Computation of Jacobsthal's Function on Primorials}}
\fancyhead[R]{\small\textit{A.~Weeks}}
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\begin{center}
 {\LARGE\bfseries Computing Jacobsthal's Function on Primorials:\\[4pt] Exact Values and Constructive Bounds}\\[12pt]
 {\large Alben Weeks}\\[4pt]
 \href{mailto:weeks.alben@gmail.com}{\texttt{weeks.alben@gmail.com}}\\[6pt]
 October 25, 2026
\end{center}


\begin{abstract}
Let $h(x)=j(P_x)$, where $j$ is Jacobsthal's function and $P_x$ is the product of the first $x$ primes. We describe a parallel residue-covering search. Exhaustive exclusion runs, together with independently checked covering witnesses, establish $h(65)=1134$, $h(66)=1160$, $h(67)=1192$, $h(68)=1216$, $h(69)=1248$, $h(70)=1274$, and $h(71)=1298$. Method 5B gives explicit certificates for $h(100)\ge2054$, $h(150)\ge3558$, and $h(200)\ge5172$, using partial coverings with five reserve primes. Exhaustive searches give $h(100)\le2798$, $h(150)\le6898$, and $h(200)\le11998$. The search combines parity compression, descending endpoint restrictions, canonical deduplication, a sliding-window bootstrap, and dynamic prime selection related to Ziller and Morack's Greedy Permutation Algorithm. We state the conditions needed for capacity bounds and conditional propagation, and organize extensions of complete or partial coverings in a five-tier framework. A catalog of 62 coverings at index 65 supports a verified multi-certificate envelope. The solver source, run evidence, certificates, and arithmetic verifiers are publicly available in the accompanying \href{https://github.com/albenweeks/JacobsthalFunction/releases/tag/v1.0-paper}{GitHub release \texttt{v1.0-paper}}. Upper-bound conclusions depend on exhaustive execution of the search; the record audits do not independently repeat those searches or supply standalone unsatisfiability certificates.
\end{abstract}

\noindent\textbf{2020 Mathematics Subject Classification:} Primary 11Y16; Secondary 11N05, 05B40, 68W05.\\
\textbf{Keywords:} Jacobsthal function, primorial, residue covering, branch-and-bound, parallel computation.

\section{Introduction}
For a positive integer $n$, Jacobsthal's function $j(n)$ is the least positive integer $m$ such that every interval of $m$ consecutive integers contains an integer coprime to $n$ \cite{Jacobsthal1960}. With $p_i$ denoting the $i$th prime, define
\begin{equation}
 P_x=\prod_{i=1}^{x}p_i,\qquad h(x)=j(P_x).
\end{equation}
Thus $h(x)$ is the largest distance between consecutive integers coprime to $P_x$, and $h(x)-1$ is the longest interval covered by one residue class modulo each of the first $x$ primes. These sequences are OEIS A048670 and A058989, respectively \cite{Haugland1999}. Our upper bounds concern these particular primorials, not all integers with $x$ distinct prime factors. Primorials need not maximize Jacobsthal's function among such integers: Hajdu and Saradha disproved that conjecture at $x=24$ \cite{HajduSaradha2012}.

\subsection{Relation to prime gaps and earlier computations}
If residue classes $a_i\pmod{p_i}$ cover $[1,L]$, the Chinese remainder theorem (CRT) gives a starting point $n_0$ satisfying $n_0\equiv-a_i\pmod{p_i}$. Taking a sufficiently large representative makes all $n_0+1,\ldots,n_0+L$ composite. Such a block lies between consecutive primes whose difference is at least $L+1$. In particular, an extremal covering with $L=h(x)-1$ produces a prime gap of at least $h(x)$.

Write $\log_r$ for the $r$-fold iterated natural logarithm. The covering constructions underlying large-prime-gap results give asymptotic lower bounds for $h$. In prime-index notation, Pintz's bound and the later result of Ford, Green, Konyagin, Maynard, and Tao imply, respectively,
\begin{align}
 h(x)&\ge (2e^\gamma+o(1))\,
 x(\log x)^2\frac{\log_3 x}{(\log_2 x)^2},\label{eq:pintz}\\
 h(x)&\gg x(\log x)^2\frac{\log_3 x}{\log_2 x}.
 \label{eq:fgkmt}
\end{align}
Here the conversion from the prime cutoff $p_x$ uses $p_x\sim x\log x$ \cite{Pintz1997,FGKMT2018}. Iwaniec's sieve bound gives $h(x)\ll x^2(\log x)^2$ \cite{Iwaniec1978}. These are asymptotic statements; their displayed leading terms do not by themselves give numerical lower bounds at a specified finite index. We make no finite crossover comparison with them. A useful finite benchmark is $h(x)\ge2p_{x-1}$ for $x\ge2$ \cite[Proposition~1.1]{Hagedorn2009}; we compare fixed-base extensions with this bound in Section~\ref{sec:h70_extensions}.

Hagedorn computed the values through $x=49$ \cite{Hagedorn2009}, and Ziller and Morack extended the range through $x=54$ using residue-covering algorithms \cite{ZillerMorack2016}. OEIS A048670 attributes the next extensions to Gerbicz ($55\le x\le57$, 2017) and Bozek ($58\le x\le64$, 2021); the entry records Google Cloud support specifically for $x=62,63,64$ \cite{Haugland1999}. Costello and Watts developed a different computational approach to upper bounds, covering indices through $10{,}000$ \cite{CostelloWatts2012}. Ziller's restricted coverings provide the endpoint condition used below \cite{Ziller2020}.

\subsection{Contributions and organization}
The computational contributions are two-gap verifications at all indices $65$--$70$, a descending computation establishing $h(71)=1298$, the Method~5B lower bounds $h(100)\ge2054$, $h(150)\ge3558$, and $h(200)\ge5172$, the upper bounds $h(100)\le2798$, $h(150)\le6898$, and $h(200)\le11998$, and further constructive lower-bound certificates. Section~\ref{sec:new_results} distinguishes independently checked witnesses from exhaustive-search evidence. The algorithm has four components:
\begin{enumerate}
 \item Descending endpoint restrictions, applied selectively to preserve efficient handling of large primes (Section~\ref{sec:down_endpoint}).
 \item Small-prime anchor decomposition, canonical survivor-board deduplication, and a sliding-window bootstrap for initial upper-bound searches (Section~\ref{sec:canonical}).
 \item Bit-parallel capacity bounds, conflict deductions, and branching heuristics (Sections~\ref{sec:watched}--\ref{sec:mrv}).
 \item Constructive extension of complete or partial coverings, including envelopes over several certificates (Section~\ref{subsec:four_tier_lower_bounds}).
\end{enumerate}
Method~5B adapts the classical partial-sieving and completion strategy to a finite search with a prescribed reserve-prime budget; its relation to earlier singleton and pair completion constructions is detailed in Section~\ref{sec:tier5}. Section~\ref{sec:experiments} gives the numerical results, explicit witnesses, and reported timings. The endpoint and sliding arguments are mathematical reductions. Exactness additionally depends on complete execution of the finite searches; a covering witness alone proves only a lower bound.

\section{Parity Compression and Descending Endpoint Pruning}
\label{sec:down_endpoint}

\subsection{Parity Compression to the Odd Interior Board}

Let $x \ge 2$ and let $P_x = 2 \cdot 3 \cdot 5 \cdots p_x$. Since $p_1 = 2$ is always the first prime, every even integer is divisible by $2$. For $x \ge 2$, $h(x)$ is always an even integer $G \ge 4$: an extremal interval of $G - 1$ consecutive integers covered by $\{p_1, \dots, p_x\}$ is bounded by two consecutive coprimes to $P_x$ at positions $n_0$ and $n_0 + G$, which must both be odd. Consequently, $n_0 + 1, n_0 + 3, \dots, n_0 + G - 1$ are even integers automatically covered by $p_1 = 2$, and the covering problem reduces exclusively to the $T$ \emph{odd interior positions}:
\begin{equation}
 n_0 + 2,\; n_0 + 4,\; \dots,\; n_0 + G - 2, \qquad \text{where } T = \frac{G - 2}{2}.
\end{equation}
Indexing these $T$ odd interior spots by $s \in \{0, 1, \dots, T - 1\}$, where spot $s$ corresponds to the integer $n_0 + 2s + 2$, the left and right odd coprime endpoints $n_0$ and $n_0 + G$ correspond exactly to the boundary indices
\begin{equation}
 s_{\text{left}} = -1 \qquad \text{and} \qquad s_{\text{right}} = T.
\end{equation}
For each odd prime $p_i \in \{p_2, \dots, p_x\}$, since $\gcd(2, p_i) = 1$, choosing a residue class $n_0 \pmod{p_i}$ is bijectively equivalent to choosing the unique residue class $r_i \in \{0, 1, \dots, p_i - 1\}$ of the odd spot index $s \equiv r_i \pmod{p_i}$ that $p_i$ covers. On the $T$-bit board $U_0 = \{0, 1, \dots, T - 1\}$, choosing residue $r_i \pmod{p_i}$ covers the bitmask
\begin{equation}
 M(p_i, r_i) = \left\{ s \in \{0, 1, \dots, T - 1\} : s \equiv r_i \pmod{p_i} \right\},
\end{equation}
which requires $\lceil T/64\rceil$ 64-bit words and is stored in a supported fixed-width array (\texttt{Bits<W>}). This parity reduction is the relation $h(x)=2\omega(x)+2$ in Ziller and Morack's notation \cite[Corollary~1.2]{ZillerMorack2016}.

For a search at an arbitrary even $G$, a covering of these $T$ positions is an unrestricted witness for $h(x)\ge G$. Its chosen residues may also cover one or both nominal endpoints. Extending the covered interval until both adjacent integers are coprime can therefore produce an actual endpoint gap larger than the tested $G$. An endpoint-safe covering, by contrast, realizes precisely that gap.

\subsection{Top-Down Descending Search and the Endpoint-Pruning Theorem}

A natural question in the design of exact algorithms for $h(x)$ is whether candidate gap sizes should be explored \emph{bottom-up}---incrementally lengthening covered intervals from $T = 1, 2, \dots$ until extension fails---or \emph{top-down} from an initial upper bound $G_0 > h(x)$. In a 2020 preprint, Ziller \cite{Ziller2020} introduced the notion of \emph{restricted coverings}---coverings of an interval of length $L$ that are forbidden from covering the two adjacent boundary integers $0$ and $L + 1$---and noted that any covering realizing the maximal length $h(x) - 1$ must be a restricted covering. However, in a bottom-up search where $h(x)$ is not yet bounded from above, proving that a candidate gap $G$ has no restricted coverings does not by itself prove $h(x) < G$, because an unrestricted covering of $G$ might simply be the restriction of a valid covering of some larger gap $G' > G$.

By contrast, testing candidate even gaps in strictly \emph{descending} order from an initial upper bound $G_0$ down to $h(x)$ makes Ziller's restricted-covering boundary condition sound and complete at \emph{every} step below $G_0$. First, at the starting gap $G_0$ (with $T_0 = (G_0 - 2)/2$ odd spots), the solver establishes $h(x) \le G_0 - 2$ (which, as we show in Corollary~\ref{cor:stride2_bootstrap}, can itself be bootstrapped in two stages without testing all unrestricted boards). Once $h(x) \le G$ is known, every subsequent descending step tests a gap $G$ for which $G + 2$ is \emph{already proven impossible}:

\begin{theorem}[Descending Endpoint Pruning; cf.\ Ziller \cite{Ziller2020}]
\label{thm:endpoint}
Suppose that no valid covering of $T + 1$ consecutive odd spots exists for the odd primes $\{p_2, \dots, p_x\}$ (i.e., $h(x) \le 2T + 2 = G$, so gap $G + 2$ has no covering). Then any valid covering $(r_2, \dots, r_x)$ of $T$ consecutive odd spots $\{0, 1, \dots, T - 1\}$ (corresponding to gap $G$) must be a restricted covering on the odd board, satisfying
\begin{equation}
 r_i \not\equiv p_i - 1 \pmod{p_i} \qquad \text{and} \qquad r_i \not\equiv T \pmod{p_i} \qquad \text{for all } i \in \{2, \dots, x\}.
\end{equation}
\end{theorem}

\begin{proof}
Suppose toward a contradiction that $(r_2, \dots, r_x)$ covers $\{0, 1, \dots, T - 1\}$ and that for some prime $p_j$, either $r_j \equiv -1 \equiv p_j - 1 \pmod{p_j}$ or $r_j \equiv T \pmod{p_j}$.
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item If $r_j \equiv T \pmod{p_j}$, then prime $p_j$ also covers spot $s = T$. Thus $(r_2, \dots, r_x)$ covers the entire $(T + 1)$-spot interval $\{0, 1, \dots, T\}$, contradicting the hypothesis that no covering of length $T + 1$ (gap $G + 2$) exists.
 \item If $r_j \equiv -1 \pmod{p_j}$, then prime $p_j$ covers spot $s = -1$. Shifting all residues by $+1$, the shifted tuple $r'_i = (r_i + 1) \bmod p_i$ covers $\{0, 1, \dots, T\}$ of length $T + 1$, again a contradiction.
\end{itemize}
Therefore, neither $s = -1$ nor $s = T$ can be covered by any prime $p_i$.
\end{proof}

\begin{example}[Worked Example of Endpoint Pruning on Small Primes]
\label{ex:endpoint_small_primes}
Consider $h(54)$ at candidate gap $G = 858$, where $T = (858 - 2)/2 = 428$ odd spots $\{0, 1, \dots, 427\}$ are tested after $G + 2 = 860$ ($T_0 = 429$) has been excluded by an unrestricted check. By Theorem~\ref{thm:endpoint}, neither $s = -1$ nor $s = 428$ can be covered:
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item For $p_2 = 3$: the forbidden endpoint residues are $p_2 - 1 = 2$ and $428 \bmod 3 = 2$. Thus $r_2 \in \{0, 1\}$. (And at gaps where $T \not\equiv 2 \pmod 3$, such as $T = 429$ or $T = 427$, the forbidden residues modulo $3$ are $\{2, 0\}$ or $\{2, 1\}$, leaving \textbf{only one legal residue class modulo $3$}.)
 \item For $p_3 = 5$: the forbidden residues are $5 - 1 = 4$ and $428 \bmod 5 = 3$, eliminating $40\%$ of the domain of $p_3 = 5$ before the search begins.
 \item For $p_4 = 7$: the forbidden residues are $7 - 1 = 6$ and $428 \bmod 7 = 1$, eliminating $2/7 \approx 28.6\%$ of the domain of $p_4 = 7$.
\end{itemize}
\end{example}

\takeaway{Selective Cutoff (\texttt{ENDPOINT\_PRIME\_CUTOFF}) and Distinction from Prior Work}{While Theorem~\ref{thm:endpoint} rests on the same boundary-non-extension principle as Ziller's restricted coverings \cite{Ziller2020}, our framework extends its algorithmic utility in three ways: (1) in the reference orchestrator, Theorem~\ref{thm:endpoint} is parameterized by a selective cutoff \texttt{ENDPOINT\_PRIME\_CUTOFF} (default $109$, or \texttt{None} for all primes), which applies the endpoint restrictions to primes $p_i < 109$ while preserving the unrestricted singleton-bank assignment flexibility of large primes $p_i \ge 109$ at the leaves of the search tree (verified post-hoc by \texttt{endpoint\_safe}); (2) Corollary~\ref{cor:stride2_bootstrap} (Two-Stage Stride-$2$ Bootstrap) uses sub-window sliding on a canonical hitting set $\mathcal{H}$ to unlock Theorem~\ref{thm:endpoint} inside an initial gap $G_0$ without requiring a prior refutation of $G_0 + 2$; and (3) Propositions~\ref{prop:monotonicity_x}--\ref{prop:tier5} apply the endpoint invariant constructively to $p_2 = 3$ and exterior residue sets to establish the five-tier lower-bound hierarchy.}

\section{Anchor Decomposition and the Initial-Bound Bootstrap}
\label{sec:canonical}

\subsection{Small-Prime ``Dogpile'' Anchors, Endpoint Filtering, and Coverage-First Ordering}

To partition the search space into independent, embarrassingly parallel tasks across CPU cores, the solver selects a prefix of $1\le d\le\min(4,x-1)$ small odd primes $D = (p_2, \dots, p_{d+1})$---called the \emph{dogpile primes}---and fixes a common anchor index $a$ where their chosen progressions intersect:
\begin{equation}
 r_i = a \bmod p_i \qquad \text{for each } p_i \in D.
\end{equation}
By the Chinese Remainder Theorem, as $a$ ranges over one period $P_D = \prod_{p \in D} p$ (e.g., $P_D = 3 \cdot 5 \cdot 7 = 105$ for $d = 3$, or $P_D = 3 \cdot 5 \cdot 7 \cdot 11 = 1{,}155$ for $d = 4$), this parameterizes all tuples of small-prime residues whose joint intersection point falls at offset $a$. Notice that $a$ does not need to lie inside $[0, T - 1]$ when $T < P_D$; virtual anchors $a \in [0, P_D - 1]$ represent small-prime configurations whose joint dogpile point lies outside the window $[0, T - 1]$ while their individual progressions still cover dense subsets of $[0, T - 1]$.

First, \texttt{anchor\_indices} removes global mirror-reflection symmetry ($s \mapsto T - 1 - s$) by keying each candidate anchor by
\begin{equation}
 \operatorname{MirrorKey}(a) = \min\big(a \bmod P_D,\; (T - 1 - a) \bmod P_D\big).
\end{equation}
On descending endpoint-pruned searches, \texttt{endpoint\_safe} filters the anchor list using the selected endpoint cutoff. When all primes in $D$ are below that cutoff, as with the default $109$, an anchor survives precisely when $a\bmod p\notin\{p-1,T\bmod p\}$ for every $p\in D$. The number of surviving phases and the number of reflection classes are, respectively,
\begin{equation}
\begin{aligned}
 N_D(T)&=\prod_{p\in D}\left(p-2+\mathbf1_{\{T\equiv-1\pmod p\}}\right),\\
 A_D(T)&=\frac{N_D(T)+\mathbf1_{\{\gcd(T+1,P_D)=1\}}}{2}.
\end{aligned}
\label{eq:anchor_count}
\end{equation}
The first formula follows from CRT. Reflection preserves the surviving set and has a unique fixed phase, satisfying $2a\equiv T-1\pmod{P_D}$. This phase is endpoint-safe exactly when $T\not\equiv-1\pmod p$ for every $p\in D$, which gives the second formula. Under this noncoincidence condition:
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item For $d = 3$ ($D = \{3, 5, 7\}$, period $105$), exactly $1 \times 3 \times 5 = 15$ phases survive under this noncoincidence condition (an \textbf{$85.7\%$ reduction}, which mirror symmetry reduces to \textbf{$8$ anchors}).
 \item For $d = 4$ ($D = \{3, 5, 7, 11\}$, period $1{,}155$), exactly $1 \times 3 \times 5 \times 9 = 135$ phases survive under the same condition (an \textbf{$88.3\%$ reduction}, reduced by mirror symmetry to \textbf{$68$ anchors}).
\end{itemize}

Finally, \texttt{sort\_anchors\_by\_coverage} orders the surviving candidate anchors by \textbf{maximum dogpile coverage first}---that is, in ascending order of the number of remaining uncovered spots $\operatorname{popcount}(u(a))$, tie-broken by a sub-primorial distance to the board center $c=(T-1)/2$. Put $Q=\prod_{i=2}^{\min(d,3)+1}p_i$, so $Q=105$ for $d=3$ or $4$. Then
\begin{equation}
 \operatorname{SortKey}(a) = \Big( \operatorname{popcount}(u(a)),\; \min\big(d_Q(a, c),\, d_Q(T - 1 - a, c)\big),\; |a - c|,\; a \Big),
\end{equation}
where $d_Q(x,c)=\min_{k\in\mathbb{Z}}|x+Qk-c|$. When $d = 4$ ($P_D = 1{,}155$) and $T < 1{,}155$, ordering anchors solely by raw distance $|a - c|$ in $[0, 1{,}154]$ penalizes virtual anchors $a \in [T, 1{,}154]$ whose $(3, 5, 7)$ intersection lies near the board center while its residue modulo $11$ gives additional coverage of the positions left by $\{3,5,7\}$. The coverage gain depends on the board and the chosen residues. Additional coverage reduces the initial number of uncovered positions, but may also reduce the remaining primes\textquotesingle{} capacities. Thus the change in slack need not be exactly one per additional covered position. Coverage-first ordering is a heuristic for finding witnesses early. For unsatisfiable gaps it also attempts to start expensive anchors early; survivor count is only a proxy for task duration, so this is not a guaranteed longest-processing-time schedule.

\subsection{Eliminating Canonical Board Duplicates on Unrestricted Checks}

On \emph{unrestricted} searches (such as the unrestricted stage of the initial upper-bound verification at gap $G_0$), endpoint pruning is inactive, and multiple distinct anchor phases $a_1 \neq a_2 \pmod{P_D}$ frequently leave the \emph{exact same pattern of uncovered odd spots} on $[0, T - 1]$ up to a shift or mirror reflection.

Specifically, let $u(a) \in \{0, 1\}^T$ denote the survivor bitmask of uncovered odd spots remaining after removing the dogpile primes at phase $a$:
\begin{equation}
 u(a) = (2^T - 1) \mathbin{\&} \sim \left( \big|_{p \in D} M(p, a \bmod p) \right).
\end{equation}
If two anchors $a_1, a_2$ produce survivor masks $u(a_1)$ and $u(a_2)$ that differ only by a translation shift (leading/trailing covered zeros at the edges of the window) or a left-right mirror reversal, then the residual covering problems for the remaining primes $\{p_{d+2}, \dots, p_x\}$ are \emph{strictly isomorphic}: any set of arithmetic progressions modulo $\{p_{d+2}, \dots, p_x\}$ that covers the active span of $u(a_1)$ can be shifted or reflected to cover $u(a_2)$, and vice versa.

For the empty survivor board define $\operatorname{Canon}(a)=(0,0)$; that residual problem is already solved. For nonempty boards, to eliminate every such duplicate before invoking the native C++ solver, the helper function \texttt{canonical\_unrestricted\_board} computes a canonical invariant $\operatorname{Canon}(a)$ in three bitwise steps:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{Translation Normalization:} Strip all trailing zeros (covered spots at the left end of the board) by shifting $u(a)$ right by its lowest set bit index $\operatorname{ctz}(u(a))$:
 \begin{equation}
 u_{\text{norm}}(a) = u(a) \gg \operatorname{ctz}(u(a)).
 \end{equation}
 \item \textbf{Active Span Extraction:} Compute the bit-length $L(a) = \lfloor \log_2 u_{\text{norm}}(a) \rfloor + 1$, which is one plus the distance from the first uncovered odd spot to the last uncovered odd spot.
 \item \textbf{Reflection Canonicalization:} Reverse the $L(a)$ bits of $u_{\text{norm}}(a)$ to obtain the mirror-reflected survivor mask $u_{\text{rev}}(a)$, and define the canonical board key
 \begin{equation}
 \operatorname{Canon}(a) = \Big( L(a),\; \min\big(u_{\text{norm}}(a),\, u_{\text{rev}}(a)\big) \Big).
 \end{equation}
\end{enumerate}

\begin{proposition}[Soundness of Canonical Survivor-Board Deduplication]
\label{prop:canon}
For any two dogpile anchors $a_1, a_2$, if $\operatorname{Canon}(a_1) = \operatorname{Canon}(a_2)$, then the residual board $u(a_1)$ admits an unrestricted covering by $\{p_{d+2}, \dots, p_x\}$ if and only if $u(a_2)$ admits an unrestricted covering by $\{p_{d+2}, \dots, p_x\}$.
\end{proposition}

\begin{proof}
If either survivor board is empty, equality of the keys makes both empty and the assertion is immediate. Otherwise let $c_1 = \operatorname{ctz}(u(a_1))$ and $c_2 = \operatorname{ctz}(u(a_2))$, and let $S_k = \{s \in \{0, \dots, T - 1\} : \text{bit } s \text{ of } u(a_k) \text{ is } 1\}$ ($k = 1, 2$). Since $\operatorname{Canon}(a_1) = \operatorname{Canon}(a_2)$, both boards have the same active span $L = L(a_1) = L(a_2)$ and satisfy either $u_{\text{norm}}(a_1) = u_{\text{norm}}(a_2)$ or $u_{\text{norm}}(a_1) = u_{\text{rev}}(a_2)$. In the translation case $u_{\text{norm}}(a_1) = u_{\text{norm}}(a_2)$, we have $S_2 = \{s + (c_2 - c_1) : s \in S_1\}$, so shifting each prime's residue by $r'_i = (r_i + c_2 - c_1) \bmod p_i$ maps any unrestricted covering of $S_1$ bijectively to an unrestricted covering of $S_2$. In the reflection case $u_{\text{norm}}(a_1) = u_{\text{rev}}(a_2)$, we have $S_2 = \{c_1 + c_2 + L - 1 - s : s \in S_1\}$, so reflecting each prime's residue by $r'_i = (c_1 + c_2 + L - 1 - r_i) \bmod p_i$ maps any unrestricted covering of $S_1$ bijectively to an unrestricted covering of $S_2$.
\end{proof}

\begin{remark}[Scope of canonical equivalence]
Proposition~\ref{prop:canon} concerns unrestricted coverability. Translating the survivor set need not preserve legality at the fixed endpoints $-1$ and $T$, or preserve whether a particular covering extends beyond them. Endpoint-pruned searches therefore retain the eligible mirror-unique phases, and exhaustive non-extension checks must replay or correctly transform every merged sibling anchor. An endpoint-unsafe phase does not by itself eliminate its entire canonical class.
\end{remark}

\begin{example}[A translated survivor board]
Take $D=\{3\}$ and $T=3$. Anchor $a_1=0$ leaves the survivor set $\{1,2\}$, while anchor $a_2=2$ leaves $\{0,1\}$. Both have normalized mask $11_2$, span $2$, and canonical key $(2,3)$. Subtracting one from each remaining prime's residue maps a covering of the first survivor set to a covering of the second. The original anchor residues themselves need not be translates within the fixed board.
\end{example}

\subsection{Sub-Window Sliding, Monotonic Anchor Saturation, and Strided Upper-Bound Certificates}
\label{sec:sliding}

Whereas Theorem~\ref{thm:endpoint} exploits the impossibility of covering $G + 2$ to prune boundary residues at $G$, a complementary structural phenomenon governs what happens as we step \emph{below} $h(x)$ in unrestricted mode. Because any valid covering of a board of length $T$ contains shifted coverings of every shorter board $T' = T - m$ ($m \ge 1$) as contiguous sub-windows, a known covering generates a consecutive arc of solvable dogpile phases as the gap size decreases by steps of $2$, until that arc fills the period:

\begin{theorem}[Sub-Window Sliding and Monotonic Anchor Saturation]
\label{thm:sliding}
Let $D = (p_2, \dots, p_{d+1})$ be the prefix of $d$ dogpile primes with period $P_D = \prod_{p \in D} p$. Suppose an even gap $G = 2T + 2$ admits at least one valid covering $(r_2, \dots, r_x)$ of the $T$ odd interior spots $\{0, 1, \dots, T - 1\}$, and let $a_0 \in \mathbb{Z} / P_D \mathbb{Z}$ denote its unique Chinese Remainder Theorem dogpile phase satisfying $a_0 \equiv r_i \pmod{p_i}$ for all $p_i \in D$. Then for every integer $m \ge 0$ with $T' = T - m \ge 1$ (corresponding to the smaller even gap $G' = G - 2m = 2T' + 2$):
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{(Contiguous Phase Arc from Sub-Window Sliding):} Each of the $m + 1$ contiguous sub-windows of length $T'$ inside $\{0, 1, \dots, T - 1\}$,
 \begin{equation}
 W_\delta = \{\delta,\; \delta + 1,\; \dots,\; \delta + T' - 1\} \subseteq \{0, 1, \dots, T - 1\}, \qquad \delta \in \{0, 1, \dots, m\},
 \end{equation}
 induces a valid \emph{unrestricted} covering $(r'_2(\delta), \dots, r'_x(\delta))$ of $\{0, 1, \dots, T' - 1\}$ with shifted residues $r'_i(\delta) = (r_i - \delta) \bmod p_i$ and dogpile phase
 \begin{equation}
 a(\delta) \equiv a_0 - \delta \pmod{P_D}, \qquad \delta \in \{0, 1, \dots, m\}.
 \end{equation}
 Consequently, a single covering at gap $G$ guarantees that at least $\min(m + 1, P_D)$ consecutive dogpile phases $\{a_0, a_0 - 1, \dots, a_0 - m\} \pmod{P_D}$ (together with their mirror-reflected arc $\{(T' - 1 - a_0 + \delta) \bmod P_D : 0 \le \delta \le m\}$) admit valid unrestricted coverings at gap $G - 2m$, yielding at least $\lceil \min(m + 1, P_D) / 2 \rceil$ solvable mirror-unique phase classes modulo $P_D$.
 \item \textbf{(Full Anchor Saturation Threshold):} Whenever $m \ge P_D - 1$---in particular, applying the statement at $G=h(x)$, at every even gap $G'\ge4$ satisfying
 \begin{equation}
 G' \le h(x) - 2(P_D - 1),
 \end{equation}
 the solvable phase arc wraps completely around $\mathbb{Z} / P_D \mathbb{Z}$, so \textbf{every dogpile phase $a \in \{0, 1, \dots, P_D - 1\}$ ($100\%$ of all anchors)} is guaranteed to possess a valid unrestricted covering of gap $G'$.
\end{enumerate}
\end{theorem}

\begin{proof}
Since $(r_2, \dots, r_x)$ covers $\{0, 1, \dots, T - 1\}$ and $W_\delta = \{\delta, \dots, \delta + T' - 1\} \subseteq \{0, 1, \dots, T - 1\}$ for every $0 \le \delta \le m = T - T'$, every integer $s \in W_\delta$ satisfies $s \equiv r_i \pmod{p_i}$ for at least one odd prime $p_i \in \{p_2, \dots, p_x\}$. Re-indexing $W_\delta$ to local coordinates $s' = s - \delta \in \{0, 1, \dots, T' - 1\}$, we have
\begin{equation}
 s \equiv r_i \pmod{p_i} \iff s' + \delta \equiv r_i \pmod{p_i} \iff s' \equiv r_i - \delta \equiv r'_i(\delta) \pmod{p_i}.
\end{equation}
Thus $(r'_2(\delta), \dots, r'_x(\delta))$ covers every $s' \in \{0, 1, \dots, T' - 1\}$. For each dogpile prime $p_i \in D$, since $r_i \equiv a_0 \pmod{p_i}$, the shifted residue satisfies $r'_i(\delta) \equiv a_0 - \delta \pmod{p_i}$, which by the Chinese Remainder Theorem corresponds to the unique dogpile phase $a(\delta) \equiv a_0 - \delta \pmod{P_D}$. As $\delta$ ranges over $\{0, 1, \dots, m\}$, $a(\delta)$ traverses $\min(m + 1, P_D)$ distinct consecutive residues modulo $P_D$. Under the reflection $s' \mapsto T' - 1 - s'$, each phase $a(\delta)$ maps to $(T' - 1 - a(\delta)) \bmod P_D$, so at most two phases in the arc can share the same mirror key $\operatorname{MirrorKey}(a) = \min(a \bmod P_D, (T' - 1 - a) \bmod P_D)$, giving at least $\lceil \min(m + 1, P_D) / 2 \rceil$ distinct mirror-unique phase classes. Finally, when $m \ge P_D - 1$ at $G = h(x)$, we have $\min(m + 1, P_D) = P_D$, exhausting all of $\mathbb{Z} / P_D \mathbb{Z}$.
\end{proof}

\begin{remark}[Why Sub-Window Solutions Are Unrestricted Rather Than Endpoint-Safe]
Notice that for any shift $\delta \in \{0, \dots, m\}$ with $m \ge 1$, at least one of the two adjacent boundary spots $s' = -1$ ($s = \delta - 1$) or $s' = T'$ ($s = \delta + T'$) of $W_\delta$ lies \emph{inside} the covered parent board $\{0, 1, \dots, T - 1\}$ and is therefore covered by $(r'_2(\delta), \dots, r'_x(\delta))$. Thus, the shifted sub-window coverings guaranteed by Theorem~\ref{thm:sliding} are \emph{unrestricted} coverings of $G - 2m$ that violate the restricted-covering endpoint condition $r'_i \not\equiv -1, T' \pmod{p_i}$ of Theorem~\ref{thm:endpoint}. Sub-window sliding therefore does not establish monotonicity of endpoint-safe existence, whereas it does establish monotonicity of unrestricted coverability. In a descending search, endpoint restrictions are justified by the already established exclusion of a larger gap. With a selective cutoff, only the selected primes' endpoint hits are removed from the search domains.
\end{remark}

\begin{example}[Monotonic Anchor Growth Below $h(50) = 762$]
\label{ex:h50_sliding}
Consider $x = 50$, where $h(50) = 762$ ($T = 380$ odd interior spots):
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item At gap $G = 764$ ($T = 381$): $0$ dogpile phases have coverings.
 \item At gap $G = 762 = h(50)$ ($T = 380$, $m = 0$): at least $1$ dogpile phase $a_0 \pmod{P_D}$ (along with its mirror reflection $(379 - a_0) \bmod P_D$) has a valid covering.
 \item At gap $G' = 760$ ($T' = 379$, $m = 1$): sliding the $379$-spot window inside the $380$-spot covering at offsets $\delta \in \{0, 1\}$ immediately yields $2$ consecutive solvable dogpile phases $\{a_0, a_0 - 1\} \pmod{P_D}$ (and their mirror partners).
 \item At gap $G' = 758$ ($T' = 378$, $m = 2$): sliding the $378$-spot window at offsets $\delta \in \{0, 1, 2\}$ yields $3$ consecutive solvable dogpile phases $\{a_0, a_0 - 1, a_0 - 2\} \pmod{P_D}$.
 \item Continuing downward to $G' = 762 - 2m$, at least $\min(m + 1, P_D)$ consecutive phases have unrestricted coverings, reaching $100\%$ saturation of all dogpile anchors at $G' = 762 - 2(P_D - 1)$ (namely $G' = 758$ for $d = 1, P_D = 3$; $G' = 734$ for $d = 2, P_D = 15$; and $G' = 554$ for $d = 3, P_D = 105$).
\end{itemize}
\end{example}

Taking the contrapositive of Theorem~\ref{thm:sliding} yields a general strided-anchor upper-bound certificate using only a fraction of the dogpile phases:

\begin{corollary}[Strided-Anchor and Single-Anchor Upper-Bound Certificates]
\label{cor:strided_bound}
Let $D = (p_2, \dots, p_{d+1})$ have period $P_D = \prod_{p \in D} p$, fix a stride $s \in \{1, 2, \dots, P_D\}$, and let $\mathcal{C}_s \subseteq \mathbb{Z} / P_D \mathbb{Z}$ be any subset of dogpile phases that intersects every cyclic arc of $s$ consecutive residues modulo $P_D$ (for instance, $\mathcal{C}_s = \{0, s, 2s, \dots, \lfloor (P_D - 1)/s \rfloor s\}$ of cardinality $\lceil P_D / s \rceil$).
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{(Strided-Anchor Upper Bound):} If no phase $a \in \mathcal{C}_s$ admits an unrestricted covering at an even gap $G'\ge4$, then no covering exists at gap $G = G' + 2(s - 1)$, certifying the rigorous upper bound
 \begin{equation}
 h(x) \le G' + 2s - 4.
 \end{equation}
 \item \textbf{(Single-Anchor Upper Bound, $s = P_D$):} In particular, choosing $s = P_D$ ($|\mathcal{C}_{P_D}| = 1$), proving that \textbf{even a single dogpile anchor $a^* \in \mathbb{Z} / P_D \mathbb{Z}$ has no unrestricted covering at gap $G'$} is sufficient to prove
 \begin{equation}
 h(x) \le G' + 2P_D - 4
 \end{equation}
 without evaluating any of the remaining $P_D - 1$ dogpile phases. For $d = 2$ ($P_D = 15$), refuting a single anchor at $G'$ certifies $h(x) \le G' + 26$; for $d = 3$ ($P_D = 105$), refuting a single anchor at $G'$ certifies $h(x) \le G' + 206$.
\end{enumerate}
\end{corollary}

\begin{proof}
Suppose toward a contradiction that gap $G = G' + 2(s - 1)$ admits a covering $(r_2, \dots, r_x)$ with dogpile phase $a_0 \in \mathbb{Z} / P_D \mathbb{Z}$. Applying Theorem~\ref{thm:sliding}(1) with $m = s - 1$, every phase in the cyclic arc $I = \{a_0, a_0 - 1, \dots, a_0 - (s - 1)\} \pmod{P_D}$ of length $s$ admits a valid unrestricted covering at gap $G - 2m = G'$. Because $\mathcal{C}_s$ intersects every cyclic arc of length $s$ in $\mathbb{Z} / P_D \mathbb{Z}$, there exists at least one phase $a \in \mathcal{C}_s \cap I$ that admits an unrestricted covering at $G'$, contradicting the hypothesis that all phases in $\mathcal{C}_s$ are unsatisfiable at $G'$. Therefore no covering exists at $G = G' + 2(s - 1)$, and since $h(x)$ is an even integer strictly less than $G$, we conclude $h(x) \le G - 2 = G' + 2s - 4$. Setting $s = P_D$ yields part~(2).
\end{proof}

Finally, combining Corollary~\ref{cor:strided_bound} ($s = 2$) with Theorem~\ref{thm:endpoint} inside the same gap $G_{\text{test}}$ yields the \emph{Two-Stage Stride-$2$ Bootstrap}, which provides a two-stage test for an initial or single gap without requiring a prior upper bound:

\begin{corollary}[Two-Stage Stride-$2$ Bootstrap for Initial and Single-Gap Verification]
\label{cor:stride2_bootstrap}
Let $G_{\text{test}} = 2T + 2$ be any even gap at least $4$ tested without a prior upper bound, let $\mathcal{K} = \{\operatorname{Canon}(a) : a \in \mathbb{Z} / P_D \mathbb{Z}\}$ be the set of canonical unrestricted boards at $G_{\text{test}}$, and let $\mathcal{H} \subseteq \mathcal{K}$ be a vertex cover (including every vertex with a self-loop) of the stride-$2$ adjacency graph with edge set
\begin{equation}
 E_2 = \Big\{ \big\{\operatorname{Canon}(a_0),\; \operatorname{Canon}((a_0 - 1) \bmod P_D)\big\} : a_0 \in \mathbb{Z} / P_D \mathbb{Z} \Big\}.
\end{equation}
Consider the two-stage search at $G_{\text{test}}$:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{(Stage 1A --- Unrestricted Stride-$2$ Hitting Set):} Search every canonical board $k \in \mathcal{H}$ in unrestricted mode at $G_{\text{test}}$ (replaying any merged sibling anchors $a \in [a_{\text{rep}}]_{\operatorname{Canon}}$ of solved boards when enumerating all solutions).
 \item \textbf{(Stage 1B --- Endpoint-Pruned Remainder):} After an exhaustive Stage~1A search establishes that no covering extends to $G_{\text{test}} + 2$ (where an immediate extension or a one-prime reassignment is detected if some prime covers $s = -1$ or $s = T$, or has $0$ exclusive spots in $\{0, \dots, T - 1\}$ so it can be reassigned to $s = T$), search only the remaining mirror-unique anchors $a$ with $\operatorname{Canon}(a) \notin \mathcal{H}$ that satisfy the selected small-prime endpoint restrictions, with those restrictions enabled in the residual search (Theorem~\ref{thm:endpoint}). Discard the remaining phases that violate a selected endpoint restriction.
\end{enumerate}
Then:
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{(Unsatisfiability Certificate):} Gap $G_{\text{test}}$ admits no valid covering (proving $h(x) \le G_{\text{test}} - 2$) if and only if both Stage~1A and Stage~1B find no covering.
 \item \textbf{(Exhaustive Non-Extension Certificate):} If Stage~1A exhaustively enumerates its coverings at $G_{\text{test}} = G$, including all merged-anchor siblings, and finds at least one covering but no extension to $G + 2$, then $G + 2$ has no valid covering ($h(x) \le G$), and an exhaustive Stage~1B search recovers the remaining valid coverings of $G$, up to global reflection. Enumerating full residue tuples additionally requires expanding any compressed leaf representations.
\end{itemize}
\end{corollary}

\begin{proof}
Suppose $G_{\text{test}} + 2$ admits a valid covering at some dogpile phase $a_0 \in \mathbb{Z} / P_D \mathbb{Z}$. By Theorem~\ref{thm:sliding}(1) with $m = 1$, its $\delta = 0$ and $\delta = 1$ sub-windows are valid unrestricted coverings of $G_{\text{test}}$ at phases $a_0$ and $(a_0 - 1) \bmod P_D$ that extend to $G_{\text{test}} + 2$ (covering $s = T$ and $s = -1$, respectively). Because $\mathcal{H}$ intersects every edge in $E_2$, at least one of $\operatorname{Canon}(a_0)$ or $\operatorname{Canon}((a_0 - 1) \bmod P_D)$ belongs to $\mathcal{H}$, so Stage~1A (with canonical sibling replay) is guaranteed to recover a covering of $G_{\text{test}}$ that extends to $G_{\text{test}} + 2$. Contrapositively, if Stage~1A finds no covering that extends to $G_{\text{test}} + 2$ (including when Stage~1A finds no covering at all), then $G_{\text{test}} + 2$ has no valid covering. By Theorem~\ref{thm:endpoint}, every valid covering of $G_{\text{test}}$ must therefore be endpoint-safe at $G_{\text{test}}$, so its dogpile anchor $a$ is endpoint-safe and is either already searched in Stage~1A ($\operatorname{Canon}(a) \in \mathcal{H}$) or searched with the selected endpoint restrictions in Stage~1B ($\operatorname{Canon}(a) \notin \mathcal{H}$).
\end{proof}

\begin{remark}[Enumeration required for non-extension]
The non-extension conclusion requires every relevant unrestricted residue assignment in the Stage~1A classes to be represented in the endpoint tests, including assignments at merged sibling phases. Distinct assignments can give the same interior coverage and different endpoint behavior. A compressed collection of leaf or graph solutions is sufficient only if its expansion or an equivalent argument accounts for all such behavior. Finding one endpoint-safe witness, or testing only one singleton matching, does not meet this hypothesis. The reported exact values use completed higher-gap exclusions and do not rely on the single-gap enumeration route.
\end{remark}

\begin{remark}[Unified Multi-Core Pipeline for Stage~1A and Stage~1B]
Stage~1B may be computed speculatively, but its refutations are usable only after Stage~1A establishes the required non-extension. A valid covering from either stage remains a lower-bound witness. Because the anchor sets are disjoint, the orchestrator queues all Stage~1A jobs (\texttt{endpoint\_pruning = False}) followed immediately in the same work deque by all Stage~1B jobs (\texttt{endpoint\_pruning = True}). Worker processes can begin Stage~1B anchors while other workers finish Stage~1A, without a mid-gap synchronization barrier. With the default adaptive splitting, subtree donation is requested when the pending queue is empty and active work no longer fills the admitted worker pool. The admitted pool may be smaller than \texttt{NUM\_WORKERS} because of the memory budget.
\end{remark}

\subsection{Operational proof modes}
\label{sec:proof_modes}
The orchestrator supports four uses of the preceding reductions. An initial test is an attempt to establish an upper bound, not a supplied proof that the chosen gap is large enough.
\begin{enumerate}
 \item \textbf{Two-gap verification.} Given a candidate $G$, exclude $G+2$ and find one covering at $G$. Use \texttt{RUN\_MODE="exact"}, \texttt{MANUAL\_UPPER\_GAP}$=G+2$, and \texttt{COLLECT\_ALL\_SOLUTIONS=False}. The exclusion uses the bootstrap; the witness search then uses endpoint pruning.
 \item \textbf{Single-gap non-extension verification.} Exhaustively enumerate the required unrestricted coverings at $G$ and verify non-extension, with replay of merged anchor siblings. Use \texttt{MANUAL\_UPPER\_GAP}$=G$ and \texttt{COLLECT\_ALL\_SOLUTIONS=True}. The upper-bound conclusion requires complete enumeration and the extension checks of Corollary~\ref{cor:stride2_bootstrap}; finding a single endpoint-safe covering is insufficient.
 \item \textbf{Descending discovery.} Start at an even trial gap $G_0$, exclude it, and descend in steps of two until a covering is found. Each completed exclusion justifies endpoint pruning on the next step. If the initial gap is too small, the run must report that the initial upper bound was not established.
 \item \textbf{Upper-bound computation.} With \texttt{RUN\_MODE="bound"}, exclude an even goal gap $G_{\mathrm{goal}}$ to prove $h(x)\le G_{\mathrm{goal}}-2$. Set \texttt{MANUAL\_UPPER\_GAP\_GOAL}$=G_{\mathrm{goal}}$ and choose \texttt{BOUND\_METHOD="direct"} or \texttt{"descending"}. A direct test uses the bootstrap at the goal; a descending test retains a completed upper-bound chain.
\end{enumerate}
For the reported candidate values, the two-gap verification targets are
\[
\begin{array}{c|rrrrrr}
x&65&66&67&68&69&70\\
G+2&1136&1162&1194&1218&1250&1276
\end{array}
\] The index-$71$ run uses descending discovery from $G_0=1304$, excluding $1304$, $1302$, and $1300$ before finding a covering at $1298$. The reported high-index upper bounds use $(x,G_{\mathrm{goal}})=(100,2800)$, $(150,6900)$, and $(200,12000)$ in direct mode. Interruptions, resource limits, incomplete enumerations, and cancelled subtrees do not count as refutations. The retained source snapshots and metadata record each run's settings; the solver's editable defaults are not a production-run record.

\section{The Native Engine and Capacity Bounds}
\label{sec:watched}

The initial dogpile anchor fixes $d$ odd primes, leaving $x-1-d$ unassigned odd primes. Any additional sub-anchor assignments are removed before the remaining prime array is passed to the native C++17 solver \texttt{Engine<W>::dfs}, specialized to a supported word count $W\ge\lceil T/64\rceil$. The supplied build supports $W\in\{1,\ldots,16,20,24,32,48,64,128,256\}$.

In the implementation descriptions of Sections~\ref{sec:watched}--\ref{sec:mrv}, $n$ is the length of this remaining array, $j$ is a local array index, and $p_j$ denotes the prime in that slot. This local indexing is distinct from the global prime indices used elsewhere in the article.

\subsection{Prime Capacity, Combinatorial Slack, and Active vs.\ Singleton Primes}

At any node of the branch-and-bound search tree, let $u \in \{0, 1\}^T$ denote the bitmask of odd interior spots that have \emph{not yet been covered} by the primes assigned at depths $0, \dots, \text{idx} - 1$, so that $\operatorname{popcount}(u)$ is the exact number of uncovered spots remaining on the board.

Legal domains are assumed nonempty; an empty domain rejects the node. For each unassigned prime $p_j$ ($j \ge \text{idx}$) and each legal residue class $r \in \mathcal{A}(p_j)$, the bit-parallel intersection $u \mathbin{\&} M(p_j, r)$ represents the subset of \emph{currently uncovered} spots that prime $p_j$ would newly cover if assigned residue $r$.

\begin{definition}[Live Prime Capacity and Combinatorial Slack]
\label{def:slack}
Let $u \in \{0, 1\}^T$ be the current uncovered bitboard at depth $\text{idx}$.
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item The \emph{live capacity} of an unassigned prime $p_j$ ($j \ge \text{idx}$) on $u$ is the maximum number of currently uncovered spots that $p_j$ can cover under a single legal residue choice:
 \begin{equation}
 c_j(u) = \max_{r \in \mathcal{A}(p_j)} \operatorname{popcount}\big(u \mathbin{\&} M(p_j, r)\big).
 \end{equation}
 Primes are partitioned into \emph{active multi-hit primes} ($\text{idx} \le j < n_{\text{active}}$, $p_j < T$), whose current capacity may be at least $2$, and \emph{singleton reserve primes} ($n_{\text{active}} \le j < n$, $p_j \ge T$, plus eligible active primes whose capacity on $u$ has dropped to $1$), which can cover at most $1$ spot of $u$ each and are accumulated into a \emph{singleton bank} of size $B_{\text{bank}}$ for leaf bipartite matching.
 \item In this definition $B_{\text{bank}}$ counts usable capacity-one primes; capacity-zero primes contribute zero, and bank primes are excluded from the active sum so that no capacity is counted twice. Endpoint-restricted singleton assignments still require a legal injective matching. The \emph{total remaining capacity} is the sum of the maximum capacities of all remaining unassigned primes:
 \begin{equation}
 C(u) = B_{\text{bank}} + \sum_{j = \text{idx}}^{n_{\text{active}} - 1} c_j(u).
 \end{equation}
 \item The \emph{combinatorial slack} $\sigma(u)$ of the node is the surplus between the total remaining capacity $C(u)$ and the number of uncovered spots $\operatorname{popcount}(u)$:
 \begin{equation}
 \sigma(u) = C(u) - \operatorname{popcount}(u) = \left( B_{\text{bank}} + \sum_{j = \text{idx}}^{n_{\text{active}} - 1} c_j(u) \right) - \operatorname{popcount}(u).
 \end{equation}
\end{enumerate}
\end{definition}

\paragraph{Loss accounting.}
Let $J$ be the set of all remaining primes, including the singleton bank. For any choice of legal rows, put $A_j=u\cap M(p_j,r_j)$ and define
\begin{equation}
 D=\sum_j(c_j-|A_j|),\qquad
 O=\sum_j|A_j|-\left|\bigcup_j A_j\right|.
\end{equation}
Here $D$ counts capacity drops and $O$ counts duplicate coverage with multiplicity. For a completion, $\bigcup_jA_j=u$, so $D+O=C(u)-|u|=\sigma(u)$. Thus negative slack proves impossibility; zero slack forces every prime to attain capacity with no overlap. Positive slack is the total loss budget. If a singleton bank is represented by an upper bound rather than exact legal capacities, the resulting slack is conservative and leaf feasibility still requires matching.

For $S\subseteq J$, define the assignment-dependent loss
\[
 L(S)=\sum_{j\in S}c_j(u)-\left|\bigcup_{j\in S}A_j\right|.
\]
Writing $U_S=\bigcup_{j\in S}A_j$, disjoint prime sets $S,R$ satisfy the exact identity
\[
 L(S\cup R)=L(S)+L(R)+|U_S\cap U_R|.
\]
This proves superadditivity and identifies the extra cost of overlaps between motifs. In a completion $L(J)=\sigma(u)$. Loss certificates and forced masks refer to this fixed board and its current domains; deductions after a change of board or domains require valid updated capacities and invariants.

\paragraph{Singleton feasibility.}
When only singleton primes remain, join each uncovered position to every remaining prime that can legally cover it. Completion is equivalent to a matching assigning a distinct prime to every uncovered position. A matching gives the required residues; conversely, each prime in a singleton completion covers at most one position. The count $B_{\mathrm{bank}}\ge|u|$ is necessary but can be insufficient: with three holes $a,b,c$ and three primes whose eligible hole sets are $\{a\},\{a\},\{b,c\}$, total capacity is three, yet $b$ and $c$ cannot both be assigned. This is why restricted leaf assignments require matching rather than a capacity count alone.

For example, if $|u|=38$, active capacities sum to $25$, and $15$ usable singleton primes remain, then $C=40$ and $\sigma=2$. A capacity-four prime cannot choose a one-hit row, which would consume three units. Similarly, three disjoint conflicts each requiring a loss of one prove failure without branching.

Equipped with Definition~\ref{def:slack}, we can now formalize the relationship between the window-sliding verification modes and the exact stepwise slack identity linking gap $G$ and gap $G + 2$:

\begin{proposition}[Verification and the stepwise slack identity]
\label{prop:method1_opt}
Write $\operatorname{Sat}(G)$ for existence of an unrestricted covering of the odd board corresponding to even $G\ge4$. Then
\[
 h(x)=G\quad\Longleftrightarrow\quad
 \operatorname{Sat}(G)\ \text{and}\ \neg\operatorname{Sat}(G+2).
\]
To compare slack on boards of lengths $T$ and $T+1$, fix the same partial assignment, the same remaining primes, and the same legal residue domains on both boards. If the extra position $T$ is uncovered, then
\begin{equation}
 \sigma(u_{T+1})=\sigma(u_T)-1+\Delta C,
 \qquad \Delta C=\sum_j\big(c_j(u_{T+1})-c_j(u_T)\big)\ge0.
\end{equation}
Thus slack decreases by one when $\Delta C=0$. There is no unconditional monotonicity of slack with board length.
\end{proposition}
\begin{proof}
The first assertion follows by restricting a longer covering to shorter intervals. For the second, the number of uncovered positions increases by one and total capacity increases by $\Delta C$. Subtracting proves the identity. Identical domains are essential: gap-dependent endpoint exclusions can change when $T$ changes.
\end{proof}
This identity explains one possible local advantage of refuting $G+2$ rather than enumerating coverings of $G$. It does not imply a general runtime comparison between Methods~1 and~2.

\subsection{Generalizing SAT's Two-Watched-Literals to Maximum-Cardinality Domains}
\label{sec:2wl}

In modern Conflict-Driven Clause Learning (CDCL) SAT solvers \cite{Moskewicz2001}, the Two-Watched-Literal (2WL) scheme maintains pointers to two non-false literals in each clause so that unit propagation never inspects a clause unless one of its watched literals becomes false.

In our covering solver, a prime $p_j$ is not a boolean disjunction; instead, it has a domain of residue rows $\{M(p_j, r)\}$ sorted in descending order of their static hit count, and $p_j$ becomes a \emph{full-capacity unit} (meaning it has only one way to achieve its full capacity $c_j(u)$) if and only if \emph{exactly one} residue row satisfies $\operatorname{popcount}(u \mathbin{\&} M(p_j, r)) = c_j(u)$.

We generalize 2WL to \emph{Two-Watched-Capacity-Witnesses}:
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{Ungrouped Primes ($c_j \ge 3$):} Each prime maintains a primary witness index $\text{w\_idx}$ (along with its cached bitmask $\texttt{witness}$) and a secondary witness index $\text{w2} = \texttt{wit2\_idx}$.
 \item \textbf{Grouped Pair Primes ($c_j = 2$):} Primes whose maximum capacity is $2$ store compact shift-and-mask pair groups (\texttt{Group<W>}) and maintain two watched bit-pairs: $(\texttt{wit\_b0}, \texttt{wit\_b1})$ and $(\texttt{wit2\_b0}, \texttt{wit2\_b1})$.
\end{itemize}

The following invariants avoid repeated row scans within the current node across the capacity loop, Pass A1, Pass A2, and MRV selection. The node-local prefix and multi-witness flags are reset for each new node:

\begin{enumerate}\setlength{\itemsep}{3pt}\setlength{\parskip}{2pt}
 \item \textbf{In-Place Secondary Witness Replacement (Two-Watched-Literal Diversity):} For a grouped prime in \texttt{check\_prime\_unit}, when the primary watched pair $(b_0, b_1)$ is still alive on $u$ (\texttt{w1\_ok == true}) and only the secondary watched pair is dead (\texttt{w2\_ok == false}), \texttt{find\_grouped\_up\_to\_2(..., w1\_ok=true)} scans the pair groups for a single live pair distinct from $(b_0, b_1)$ and updates $(\texttt{wit2\_b0}, \texttt{wit2\_b1})$ in place without resetting $(b_0, b_1)$ back to the lexicographically first pair of the table.
 \item \textbf{Dead-Prefix Tracking (\texttt{cap\_prefix\_dead}):} When the Capacity Loop scans $d.\texttt{rows}$ from index $0$ and discovers a new maximum-capacity witness at index $\text{best\_idx}$, every row $0 \le \text{ri} < \text{best\_idx}$ has just been proven on the current $u$ to have strictly fewer than $c_j(u)$ hits. The engine sets $\text{w2} = \text{best\_idx} + 1$ and records $\texttt{cap\_prefix\_dead} \mathrel{|=} (1 \ll j)$, allowing \texttt{check\_prime\_unit} in Pass A1 to skip the wrap-around scan over $0 \dots \text{w2} - 1$.
 \item \textbf{Multi-Witness Bitmask (\texttt{cap\_two\_wit}) and Live-Hit Protection ($h \ge c_j$):} Whenever the Capacity Loop or Pass A1 proves that prime $j < 64$ has at least two full-capacity witnesses on $u$, it sets bit $j$ in $\texttt{cap\_two\_wit}$. When $\sigma_{\text{eff}} \ge 1$ ($\text{mh} = c_j - \sigma_{\text{eff}} < c_j$), \texttt{eval\_mrv} scans rows with live hit count $h \ge \text{mh}$; guarding secondary witness updates with the live count check $h \ge c_j$ (rather than the static count $\texttt{rows[ri].hits} \ge c_j$) guarantees that a valid $c_j$-hit secondary witness is never overwritten by a $(c_j - 1)$-hit row.
\end{enumerate}

\subsection{Dynamic Row Compaction (\texorpdfstring{\texttt{reb\_lvl}}{reb\_lvl}) with 64-Bit Scalar Key Sorting}

As the DFS descends through the highest-capacity primes at depths $\text{idx} \in \{1, 2, 3, 4, 5\}$, many initial bits in $u$ may be cleared. When the rebuild guards permit it, these shallow checkpoints (\texttt{reb\_lvl}) compact each ungrouped prime's row table into a depth-indexed scratch buffer $\texttt{sub\_rows}[\text{reb\_lvl}]$, discarding all rows whose live hit count on $u$ has fallen below $2$ and re-sorting the surviving rows in descending order of live hits $h$.

This optimization is enabled only for $W\le16$, with additional tests on remaining active primes, branch count, and slack. It is not the large-board path used for the index-$100$, $150$, and $200$ exclusions.

Rather than running an in-place shell sort over full $\texttt{Row<W>}$ structs, \texttt{reb\_lvl} packs each surviving row into an 8-byte scalar key
\begin{equation}
 \text{key}(\text{ri}) = \big((4096 - h) \ll 32\big) \mathbin{|} (r \ll 16) \mathbin{|} \text{ri},
\end{equation}
tracks a boolean flag \texttt{need\_sort} that remains \texttt{false} whenever keys are already monotonically non-decreasing, and bypasses sorting entirely whenever $\texttt{srows}[0].\texttt{hits} \le 2$ or when only the primary witness row has $h > 2$. This reduces the amount of data moved during sorting; the benefit depends on board size and row distribution.

\section{Conflict Motifs and Conditional Propagation}
\label{sec:conflict_graph}

The central deduction engine inside \texttt{Engine<W>::dfs} is a multi-pass bit-parallel pipeline that constructs a \emph{conflict hypergraph} over the active primes at each node.

\subsection{Unit Primes and the Conflict Graph \texorpdfstring{$\mathcal{G}(u)$}{G(u)}}

Suppose prime $p_j$ has a unique residue row $r_j$ achieving its full capacity $c_j(u)$ on $u$. Define its \emph{live unit mask}
\begin{equation}
 M_j = u \mathbin{\&} M(p_j, r_j), \qquad \text{with } \operatorname{popcount}(M_j) = c_j(u).
\end{equation}
If prime $p_j$ does \emph{not} choose residue $r_j$, then $p_j$ covers at most $c_j(u) - 1$ spots of $u$, consuming at least $1$ unit of the global slack $\sigma(u)$. Now suppose two such unit primes $p_a$ and $p_b$ have overlapping live unit masks:
\begin{equation}
 M_a \mathbin{\&} M_b \neq 0.
\end{equation}
Even if both $p_a$ and $p_b$ choose their full-capacity rows $r_a$ and $r_b$, any spot $s \in M_a \cap M_b$ is covered \emph{twice}, so the union $M_a \cup M_b$ covers at most $c_a(u) + c_b(u) - |M_a \cap M_b| \le c_a(u) + c_b(u) - 1$ spots of $u$. Thus:

\begin{lemma}[Fundamental Conflict-Pair Inequality]
\label{lem:conflict}
Let $p_a, p_b$ be distinct active primes with unique $c_a(u)$-hit and $c_b(u)$-hit live unit masks $M_a, M_b$ on $u$. If $M_a \cap M_b \neq \emptyset$, then in \textbf{every} valid covering of $u$, the pair $\{p_a, p_b\}$ loses at least $1$ unit of effective capacity (either because $p_a$ or $p_b$ chooses a row below its own capacity, or because both choose $M_a, M_b$ and overlap on $M_a \cap M_b$). Consequently, $M_a$ and $M_b$ cannot both be realized disjointly on $u$.
\end{lemma}

By packing vertex-disjoint conflict structures and counting their mandatory slack consumption $n_{\text{cp}}$, the solver tightens the remaining slack to $\sigma_{\text{rem}} = \sigma(u) - n_{\text{cp}}$. Whenever $n_{\text{cp}} > \sigma(u)$, the mandatory wasted hits exceed the node's entire slack budget $\sigma(u)$, so the node is immediately unsatisfiable and pruned.

\subsection{Conflict motifs and their loss bounds}

Rather than restricting the conflict graph to disjoint edges ($K_2$), our engine packs a collection of conflict motifs across four sequential passes, caching the unit bitmasks $\texttt{pb\_ma}[p]$ and $\texttt{pb\_mb}[p]$ in place so subsequent passes never recompute \texttt{get\_unit\_mask}:

\begin{table}[!htbp]
\centering
\small
\caption{Conflict structures used by the proposed deduction pipeline. Loss bounds require the hypotheses of Proposition~\ref{prop:motifs_soundness} and disjoint sets of represented primes.}
\label{tab:motifs}
\begin{tabularx}{\textwidth}{@{}l c X@{}}
\toprule
Structure & Loss & Required condition\\
\midrule
Edge & $\ge1$ & Two unique full-capacity rows intersect.\\
Claw & $\ge1$ & A unit center intersects unit leaves; a later disjoint edge may increase the bound.\\
Triangle & $\ge2$ & Three unique full-capacity rows intersect pairwise.\\
Path or $4$-cycle & $\ge2$ & The structure contains two vertex-disjoint conflict edges.\\
Implication chain & $\ge0$ & Zero loss forces a unique row avoiding the accumulated forced mask.\\
Clause pair & $\ge1$ & Every full-capacity row of a new prime intersects a forced mask.\\
Clause upgrade & $\ge2$ & The three participants satisfy Proposition~\ref{prop:motifs_soundness}(5).\\
Watched-pair upgrade & $\ge1$ & Every live maximum-capacity pair conflicts with a zero-loss forced mask.\\
\bottomrule
\end{tabularx}
\end{table}
\FloatBarrier

The following arguments establish loss bounds under explicit disjointness and live-capacity hypotheses. Each implementation path must preserve these invariants; the regression checks and run audits described in Section~\ref{sec:new_results} complement these arguments but do not constitute a formal proof of the implementation.

\begin{proposition}[Soundness of Higher Conflict Motifs: $K_3$, $P_4$, Claws $K_{1,k}$, $C_4$, and Clause-Pairs]
\label{prop:motifs_soundness}
Use the assignment-dependent loss $L(S)$ defined in Section~\ref{sec:watched}. For disjoint sets of represented primes, the following motif bounds can be added by superadditivity:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{(Triangle $K_3$, $\Delta n_{\text{cp}} = +2$):} If three unit primes $S = \{a, b, k\}$ have pairwise intersecting live unit masks ($M_a \cap M_b \neq \emptyset$, $M_a \cap M_k \neq \emptyset$, $M_b \cap M_k \neq \emptyset$), then the total loss is at least two, whether the assignment sacrifices full-capacity rows or accepts overlaps; thus $L(\{a, b, k\}) \ge 2$.
 \item \textbf{(4-Path $P_4$ Split, $\Delta n_{\text{cp}} = +2$):} If four unit primes $S = \{u_a, a, b, u_b\}$ satisfy $M_{u_a} \cap M_a \neq \emptyset$ and $M_b \cap M_{u_b} \neq \emptyset$, partitioning $S$ into the two vertex-disjoint edges $S_1 = \{u_a, a\}$ and $S_2 = \{b, u_b\}$ guarantees $L(S) \ge L(S_1) + L(S_2) \ge 1 + 1 = 2$ by Lemma~\ref{lem:conflict}.
 \item \textbf{(Claw $K_{1,k}$ and Re-Splitting, certified cost $1$ or $2$):} If a center unit prime $j$ intersects pairwise-disjoint leaf units $l_1, \dots, l_k$ ($k \ge 2$), then $L(\{j, l_1, \dots, l_k\}) \ge 1$ since $M_j \cap M_{l_1} \neq \emptyset$. Moreover, if any subsequent unit $u'$ intersects a stored leaf $l_m$ ($m \ge 2$), provided $u'$ is distinct from the claw's primes, the enlarged structure contains the two disjoint edges $\{j,l_1\}$ and $\{l_m,u'\}$, with total certified cost $2$. Replacing an already charged cost-one claw adds only one to $n_{\mathrm{cp}}$; it does not add a further two.
 \item \textbf{(Forced Implication Chain and 1-vs-2 Clause-Pair, $\Delta n_{\text{cp}} = +1$):} Let $f \in u_f$ be an active unit slot with accumulated forced mask $\texttt{uf\_m}[f]$ (meaning that $0$ slack loss on the primes represented by slot $f$ requires covering all bits of $\texttt{uf\_m}[f]$ disjointly). Suppose a multi-witness prime $j \notin f$ has live capacity $c_j(u)$:
 \begin{itemize}\setlength{\itemsep}{1pt}\setlength{\parskip}{0pt}
  \item If all $c_j(u)$-hit rows of $j$ except a single surviving row $M_{j,\text{free}}$ intersect $\texttt{uf\_m}[f]$, then achieving $0$ total slack loss on $f \cup \{j\}$ forces slot $f$ to be realized \emph{and} forces prime $j$ to choose $M_{j,\text{free}}$; hence $\texttt{uf\_m}[f]$ validly expands to $\texttt{uf\_m}[f] \cup M_{j,\text{free}}$ at cost $\Delta n_{\text{cp}} = 0$.
  \item If \emph{every} $c_j(u)$-hit row of $j$ intersects $\texttt{uf\_m}[f]$ ($\text{n\_free} = 0$), then $f \cup \{j\}$ cannot achieve $0$ slack loss: either $f$ loses $\ge 1$ hit, or $j$ chooses a row of size $< c_j(u)$ (losing $\ge 1$ hit), or $j$ chooses a $c_j(u)$-hit row that overlaps $\texttt{uf\_m}[f]$ (losing $\ge 1$ hit to collision), certifying $L(f \cup \{j\}) \ge 1$.
 \end{itemize}
 \item \textbf{(Cycles and clause upgrades):} A $4$-cycle of unit primes contains two vertex-disjoint edges, so its loss is at least $2$. For the typed clause upgrade used by the implementation, let $a$ and $k$ be unit primes with $M_a\cap M_k\ne\varnothing$, and suppose every full-capacity row of a third prime $b$ intersects both $M_a$ and $M_k$. Then $L(\{a,b,k\})\ge2$. The same argument applies to disjoint aggregate slots when their zero-loss assignments force the stated masks.
\end{enumerate}
\end{proposition}

\begin{proof}
For (1), let $t\in\{0,1,2,3\}$ count the primes in $S=\{a,b,k\}$ that choose their unique full-capacity rows. Each of the other $3-t$ primes loses at least one hit relative to its own capacity. If $t\le1$, these drops already give $L(S)\ge2$. If $t=2$, the third prime loses one hit and the two chosen unit masks overlap, giving another loss. If $t=3$, the union of the three pairwise-intersecting masks has size at most $c_a(u)+c_b(u)+c_k(u)-2$: the first two masks overlap, and the third intersects their union. Thus $L(S)\ge2$ in every case. Parts (2)--(4) follow from the displayed inequalities and the zero-loss invariant for each slot. For the clause upgrade in (5), if both unit primes use full-capacity rows, they lose one hit through overlap; a non-full row of $b$ loses another, while a full row of $b$ overlaps each unit row and gives total overlap loss at least two. If exactly one unit prime drops capacity, that drop costs one and a full row of $b$ overlaps the other; a non-full row of $b$ instead incurs a second capacity drop. If both units drop, their drops already cost two. For aggregate slots use the same cases, with any failure of the zero-loss mask condition costing at least one. Slots must represent disjoint sets of primes; counting the same prime in two independently charged motifs is not permitted.
\end{proof}

\paragraph{Virtual intersection slots in Pass A2.}
A packed conflict pair also supplies a conditional coverage mask:

\begin{proposition}[Soundness of Virtual Intersection Slots $I_p = M_a \mathbin{\&} M_b$]
\label{prop:virtual_slot}
Let $p = (a, b)$ be a packed $K_2$ conflict pair of unit primes with live unit masks $M_a, M_b$ and intersection $I_p = M_a \mathbin{\&} M_b \neq 0$, contributing $+1$ to $n_{\text{cp}}$. If a third prime $j \notin \{a, b\}$ has every one of its $c_j(u)$-hit rows intersecting $I_p$, then the triple $\{a, b, j\}$ loses at least $2$ units of slack in every valid completion ($L(\{a, b, j\}) \ge 2$, certifying $\Delta n_{\text{cp}} = +1$ beyond the $+1$ already counted for $p$).
\end{proposition}

\begin{proof}
Suppose toward a contradiction that $L(\{a, b, j\}) \le 1$. Since $L(\{a, b\}) \ge 1$ by Lemma~\ref{lem:conflict}, achieving $L(\{a, b, j\}) = 1$ requires that $\{a, b\}$ loses \emph{at most} $1$ unit of slack and that $j$ loses $0$ units of slack. For $\{a, b\}$ to lose at most $1$ unit of slack, at least one of $a$ or $b$ must choose its full-capacity unit mask $M_a$ or $M_b$ (otherwise both lose $\ge 1$ hit each, giving $L(\{a, b\}) \ge 2$). Because $I_p = M_a \cap M_b \subseteq M_a$ and $I_p \subseteq M_b$, every spot in $I_p$ is covered by $\{a, b\}$ in \emph{both} cases. Meanwhile, for $j$ to lose $0$ slack, $j$ must choose one of its $c_j(u)$-hit rows $M_j$ disjointly from the spots covered by $\{a, b\}$. However, by hypothesis every $c_j(u)$-hit row of $j$ intersects $I_p$, creating at least $1$ additional overlap hit with $I_p \subseteq M_a \cap M_b$ and forcing $L(\{a, b, j\}) \ge 1 + 1 = 2$, a contradiction.
\end{proof}

\paragraph{Fast-Path Filtering in Pass A2.}
Because Pass A2 inspects all multi-witness primes against the union mask $s_{1,\text{forced}}$, three exact fast paths keep its overhead minimal:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{\texttt{count\_row}-First Filtering:} Static rows with $\texttt{hits} \ge c$ may have lost bits on $u$. Testing \texttt{u.count\_row(rows[ri].mask) == c} before testing whether \texttt{s1\_forced} intersects \texttt{rows[ri].mask} avoids intersection tests on rows whose live count is below $c$.
 \item \textbf{\texttt{w2\_known\_c} Bypass:} Whenever bit $j$ of \texttt{cap\_two\_wit} is set from Pass A1, the secondary witness $w_2$ is already guaranteed to have $c$ live hits on $u$, skipping the redundant call to \texttt{u.count\_row} on $w_2$ in Pass A2.
 \item \textbf{Single-slot \texttt{n\_uf == 1} fast path:} When $n_{\text{uf}} = 1$, $s_{1,\text{forced}}$ consists of the single unit slot $\texttt{uf\_m}[0]$. Every row in \texttt{full\_ris} was added precisely because it intersected $s_{1,\text{forced}} = \texttt{uf\_m}[0]$, so the solver sets $\texttt{common} = 1, \texttt{union\_m} = 1$ immediately without executing $O(n_{\text{full}} \cdot W)$ word-intersection loops.
\end{enumerate}

\subsection{Slack-Saturated Dilemma Shaving (\texorpdfstring{\texttt{while (n\_cp == slack)}}{while (n\_cp == slack)})}
\label{sec:shaving}

When the conflict graph saturates the slack budget ($n_{\text{cp}} = \sigma$), all $\sigma$ units of the node's wasted-hit budget are already consumed by the packed conflict structures in $\text{CP}$:

\begin{proposition}[Soundness of Slack-Saturated Dilemma Shaving]
\label{prop:shaving_sound}
Suppose the packed vertex-disjoint conflict motifs $\text{CP} = \{S_1, \dots, S_m\}$ on board $u$ have total certified slack loss $n_{\text{cp}} = \sum_{r=1}^m \ell_r = \sigma(u)$, where $L(S_r) \ge \ell_r$ in every valid completion. Then in every valid covering extending $u$:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item Every motif $S_r \in \text{CP}$ must incur \emph{exactly} its certified slack loss $L(S_r) = \ell_r$, and the set of all primes outside $\text{CP}$ must incur $0$ total slack loss ($L(\text{Complement}(\text{CP})) = 0$). In particular, any prime $j \notin \text{CP}$ must choose a $c_j(u)$-hit row disjoint from all other chosen rows, so if $j \notin \text{CP}$ has a unique surviving $c_j(u)$-hit row, that row is unconditionally forced.
 \item For any 2-prime $K_2$ conflict pair $p = (a, b) \in \text{CP}$ ($\ell_p = 1$), since $L(\{a, b\}) = 1$, at least one of $a$ or $b$ must choose its unique full-capacity mask $M_a$ (World~A) or $M_b$ (World~B). Consequently, if assuming World~A (or World~B) leads to a contradiction under forced unit propagation (\texttt{test\_world\_fast}), the complementary world is mandatory; and if both World~A and World~B force a prime $j$ to select the same residue row $r$, then $(p_j, r)$ is unconditionally forced at $u$.
\end{enumerate}
\end{proposition}

\begin{proof}
Let $J$ be the set of remaining primes and $R=J\setminus\bigcup_r S_r$. In any completion, the loss identity and superadditivity on disjoint prime sets give $\sigma(u)=L(J)\ge\sum_{r=1}^m L(S_r)+L(R)\ge\sum_{r=1}^m\ell_r+L(R)=\sigma(u)+L(R)$. Because $L(S_r) \ge \ell_r$ and $L(R) \ge 0$, equality must hold in every term: $L(R) = 0$ and $L(S_r) = \ell_r$ for each $r$. For a $K_2$ pair $p = (a, b)$ with $\ell_p = 1$, if neither $a$ nor $b$ chose its unique full-capacity unit mask, then $a$ and $b$ would each cover at most $c_a(u) - 1$ and $c_b(u) - 1$ spots, giving $L(\{a, b\}) \ge 2 > \ell_p = 1$, a contradiction. Thus every valid completion of $u$ belongs to World~A ($M_a$ chosen) or World~B ($M_b$ chosen), so any consequence that holds in all non-contradictory worlds of $\{A, B\}$ holds in every valid completion of $u$.
\end{proof}

\section{Dynamic Prime Selection and Conflict-Aware Branching}
\label{sec:mrv}

Dynamic prime selection chooses the next unassigned prime from the current uncovered board instead of fixing the entire prime order in advance. A directly relevant precedent is the Greedy Permutation Algorithm (GPA) of Ziller and Morack \cite[Section~2.4, Algorithm~6]{ZillerMorack2016}: at each recursive stage, it selects a prime and residue class attaining the greatest number of newly covered positions. Their live residue counts also support the capacity rejection criterion discussed in Section~\ref{sec:watched}. We credit this coverage-based selection principle to that earlier Jacobsthal algorithm; dynamic prime ordering itself is not claimed as a new contribution here.

The present implementation combines live coverage capacities with a conflict-aware Minimum Remaining Values (MRV) heuristic (\texttt{eval\_mrv}). If capacity bounding, conflict deductions, and dilemma shaving neither reject the node nor force a move, a maximum-capacity candidate provides the initial pivot. The solver may then prefer an eligible prime with fewer surviving residue branches, using effective slack and the filters described below. Thus it can select a lower-capacity prime when its branching domain is sufficiently constrained. For the selected prime, surviving multi-hit residues are tried in decreasing live coverage, with deferred singleton handling retained when allowed. This is a particular combination of selection, ordering, and pruning rules rather than an identification of the implementation with GPA. Reordering alone preserves completeness; deleting a branch requires a separate valid exclusion argument.

\subsection{Effective Slack per Prime}

Because $n_{\text{cp}}$ units of the node's global slack $\sigma(u)$ are already spoken for by the vertex-disjoint conflict structures in $\text{CP}$, we can bound the maximum number of hits any individual prime $j$ can afford to lose below $c_j(u)$:

\begin{proposition}[Effective slack with a residual loss certificate]
\label{prop:eff_slack}
For a remaining prime $j$, let $q_j$ be a certified lower bound on the total loss of the other remaining primes, after $j$ has been removed. Then every completing row for $j$ satisfies
\begin{equation}
 |u\cap M(p_j,r)|\ge c_j(u)-\big(\sigma(u)-q_j\big).
\end{equation}
In particular, $q_j=n_{\mathrm{cp}}$ is valid outside all packed motifs. Inside a motif of cost $\ell$, removing the whole motif always gives the conservative choice $q_j=n_{\mathrm{cp}}-\ell$. The sharper choice $q_j=n_{\mathrm{cp}}-1$ is valid only if deleting $j$ leaves a certified loss of at least $\ell-1$ within its motif.
\end{proposition}
\begin{proof}
The capacity drop of $j$ is $c_j(u)-|u\cap M(p_j,r)|$. This drop and the loss of the other primes concern disjoint prime sets, so their sum is at most the total slack in a completion. Rearranging gives the result.
\end{proof}
For an edge or a cost-one clause, the deletion condition is immediate. Deleting a vertex from a triangle leaves an edge; deleting a vertex from a pair of disjoint edges leaves at least one edge. In the clause upgrade of Proposition~\ref{prop:motifs_soundness}(5), deletion of any one of the three participants leaves a conflict of cost one. A linked upgrade must be treated together with the pair on which its proof depends, rather than counted as an independent disjoint motif. These residual-loss facts justify the compact implementation formula
\[
 \sigma_{\mathrm{eff}}(j)=\sigma(u)-n_{\mathrm{cp}}
        +\mathbf1_{\{j\text{ belongs to a packed motif}\}}
\]
for structures satisfying the deletion condition. They are an invariant required of any additional motif optimization.

If a candidate row uses all slack remaining after its capacity drop and the residual loss certificate, every other structure must attain its certified minimum and no additional cross-structure overlap is possible. Thus the row must avoid masks forced under those equality conditions. Such masks include zero-loss unit-slot masks and, for an unmodified cost-one pair, its intersection mask from Proposition~\ref{prop:virtual_slot}. The exclusion must not include the selected prime's own dependent slot unless separately justified.

\subsection{Adjacent-Tier Effective-Slack MRV Pivoting When \texorpdfstring{$\text{best\_br} > 2$}{best\_br > 2}}

Inside \texttt{eval\_mrv}, it is important to distinguish between a prime's \emph{live hit capacity} $c_j(u)$ (the number of uncovered spots on $u$ that a single residue covers, which can be $3, 4, 5,$ or more) and its \emph{surviving branch count} $\text{nb}(j)$ (the number of legal residue choices $r \in \{0, \dots, p_j - 1\}$ that cover at least $\text{mh}(j) = c_j(u) - \sigma_{\text{eff}}(j)$ spots without violating conflict-graph constraints, i.e., the branching factor in the DFS tree if $p_j$ is chosen next). The solver tracks the smallest surviving branch count $\text{best\_br} = \min_j \text{nb}(j)$ seen so far, starting from the highest-capacity prime $\text{selected}$ (with capacity $c_{\text{sel}} = \texttt{caps\_ptr}[\text{selected}]$). When global slack $\sigma > 2$, inspecting lower-capacity primes ($c < c_{\text{sel}}$) across $50\text{--}60$ active primes on every node would be expensive if unrestricted. The heuristic therefore inspects an adjacent capacity tier only while the best known branch count exceeds two and the candidate's effective slack is at most one. These thresholds are performance choices; they do not alter the set of residue choices that must be searched for the selected prime.

We therefore govern lower-tier MRV evaluation with the dynamic two-part filter:
\begin{equation}
\begin{aligned}
 \texttt{if (}&\texttt{slack > 2 \&\& c < sel\_cap \&\& jd.row1\_hits >= mh} \\
 &\texttt{\&\& (best\_br <= 2 || eff\_slack > 1 || c < sel\_cap - 1 || jd.grouped)) return;}
\end{aligned}
\end{equation}
Once a candidate with at most two surviving branches is found, the filter skips further lower-tier evaluations. The heuristic is not a proof that a globally minimum branching factor has been found.

\subsection{Pre-Masked Conflict Bitboards (\texorpdfstring{\texttt{u\_no\_s1}}{u\_no\_s1}) and Branchless MRV Counting}

To accelerate \texttt{eval\_mrv} when $\sigma_{\text{eff}} \le 1$, the engine precomputes the conflict-free bitboard
\begin{equation}
 u_{\text{no\_s1}} = u \mathbin{\&} \sim s_{1,\text{forced}}
\end{equation}
once before the MRV loop (using $W$ wordwise Boolean operations, which a supported compiler may implement with \texttt{ANDN}). Inside \texttt{eval\_mrv}, let $u_{\text{mh}} = \text{check\_s1} \mathbin{?} u_{\text{no\_s1}} : u$. We then exploit the following exact identity:

\begin{proposition}[Branchless Minimum-Hit Row Identity]
\label{prop:umh}
Suppose a candidate row $M = \texttt{rows[ri].mask}$ of prime $j$ has static rebuild hit count $\texttt{rows[ri].hits} == \text{mh}$ (or suppose $\text{mh} == c_j(u)$). Since $\operatorname{popcount}(u \mathbin{\&} M) \le \min(\texttt{rows[ri].hits}, c_j(u)) \le \text{mh}$, the row $M$ satisfies both
\begin{equation}
 \operatorname{popcount}(u \mathbin{\&} M) \ge \text{mh} \qquad \text{and} \qquad \big(\text{check\_s1} \implies (u \mathbin{\&} M \mathbin{\&} s_{1,\text{forced}}) = 0\big)
\end{equation}
if and only if
\begin{equation}
 \operatorname{popcount}(u_{\text{mh}} \mathbin{\&} M) == \text{mh}.
\end{equation}
\end{proposition}

\begin{proof}
Since $u$ is a subset of the board at the last shallow rebuild checkpoint (\texttt{reb\_lvl}) and $c_j(u)$ is the maximum live hit count of prime $j$ on $u$, we always have $\operatorname{popcount}(u \mathbin{\&} M) \le \min(\texttt{rows[ri].hits}, c_j(u)) \le \text{mh}$. When $\text{check\_s1}$ is false, $u_{\text{mh}} = u$, so $\operatorname{popcount}(u \mathbin{\&} M) \ge \text{mh} \iff \operatorname{popcount}(u_{\text{mh}} \mathbin{\&} M) == \text{mh}$. When $\text{check\_s1}$ is true, $u_{\text{mh}} = u \mathbin{\&} \sim s_{1,\text{forced}}$ partitions $u \mathbin{\&} M$ into two disjoint bitmasks: $(u_{\text{mh}} \mathbin{\&} M)$ and $(u \mathbin{\&} M \mathbin{\&} s_{1,\text{forced}})$. Their popcounts sum to $\operatorname{popcount}(u \mathbin{\&} M) \le \text{mh}$, so $\operatorname{popcount}(u_{\text{mh}} \mathbin{\&} M) == \text{mh}$ holds if and only if $\operatorname{popcount}(u \mathbin{\&} M) == \text{mh}$ and $\operatorname{popcount}(u \mathbin{\&} M \mathbin{\&} s_{1,\text{forced}}) == 0$.
\end{proof}

Splitting the \texttt{eval\_mrv} row loop into a small loop over rows with $\texttt{rows[ri].hits} > \text{mh}$ (skipped entirely when $\text{mh} == c_j(u)$) and a second loop over $\texttt{rows[ri].hits} \le \text{mh}$ that tests $\operatorname{popcount}(u_{\text{mh}} \mathbin{\&} M) == \text{mh}$ eliminates all \texttt{s1\_forced.intersects} calls on minimum-hit rows. By the same identity, for grouped and $c = 2$ primes when $\text{mh} = 2$, calling \texttt{u\_mh.count\_pair\_bounded} counts only the 2-hit pairs disjoint from $s_{1,\text{forced}}$ without a separate intersection test on each accepted pair.

The implementation uses two local coverage-based ordering rules. Neither asserts that MRV always selects a globally maximum-capacity prime:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{Max-Capacity Tie-Breaking Across Primes:} Inside \texttt{eval\_mrv}, whenever multiple candidate primes tie for the minimum surviving branch count $\text{nb} == \text{best\_br}$, the condition \texttt{tie\_wins = (c > best\_cap || (c == best\_cap \&\& jd.p < best\_p))} selects the prime with the largest live coverage capacity $c_j(u)$.
 \item \textbf{Max-Coverage-First Value Ordering Across Residues:} For the chosen prime $\text{selected}$, sub-branch conflict lookahead (\texttt{prune\_\allowbreak sub\_branch}) first prunes any candidate residue $r$ whose mask collides with more than $h - \text{min\_hits}$ disjoint conflict units. All surviving multi-hit candidate residues ($h = \operatorname{popcount}(u \mathbin{\&} M(p, r)) \ge \max(2, \text{min\_hits})$) are then packed into 64-bit sort keys \texttt{(uint64\_t(4096 - hits) << 32) | r} and sorted via \texttt{insertion\_sort\_scalar} in nonincreasing order of their live uncovered hit count $h$, with residue value breaking ties (the deferred 1-hit singleton option \texttt{-1} is appended last when $\text{min\_hits} \le 1$), so the highest-coverage surviving multi-hit residues are tried first at a branching node.
\end{enumerate}

\section{A Five-Tier Framework for Constructive Lower Bounds}
\label{subsec:four_tier_lower_bounds}
A complete covering supplies a finite lower bound independently of whether its length is maximal. This section organizes ways to strengthen that bound into five tiers: a general modulo-$3$ argument, a pigeonhole capacity bound, the exterior geometry of one certificate, an envelope over several certificates, and replacement of the base by a complete or repaired covering at a higher index. The last two tiers make use of computational information that a formula based only on the prime index cannot capture.

The hierarchy concerns the information and constructions available. Individual methods need not give strictly increasing numerical outputs. If $b_j(y)$ is a bound supplied by tier $j$, define the cumulative bound
\[
 B_{\le j}(y)=\max_{1\le i\le j}b_i(y).
\]
Retaining earlier constructions guarantees $B_{\le1}(y)\le\cdots\le B_{\le5}(y)$; strict improvement is an empirical or separately proved property. A successful construction is certified by its residue vector, so these lower bounds hold independently of any maximality computation.

\subsection{Strict Monotonicity and Modulo-3 Boundary Bridging Across Primorial Indices}
\label{sec:monotonicity_x}

Strict increase of $h(x)$ follows from the classical coprime-product property of Jacobsthal's function; see \cite[Remark~1.1]{ZillerMorack2016}. Combining boundary extension at $s = -1$ and $s = T$ with the Descending Endpoint-Pruning condition for $p_2 = 3$ (Theorem~\ref{thm:endpoint}) yields sharpened single-step and two-step lower bounds:

\begin{proposition}[Strict Monotonicity and Modulo-$3$ Boundary Bridging of $h(x)$]
\label{prop:monotonicity_x}
For every integer $x \ge 1$, $h(x + 1) > h(x)$. Moreover, for every integer $x \ge 2$, let $T_x = (h(x) - 2)/2$ denote the extremal odd-board length at index $x$. Then:
\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{(Single-Prime Increment):} Every single-prime step satisfies
 \begin{equation}
 h(x + 1) \ge \begin{cases}
 h(x) + 4, & \text{if } T_x \not\equiv 1 \pmod 3 \quad \big(\text{i.e., } h(x) \not\equiv 4 \pmod 6\big), \\[2pt]
 h(x) + 2, & \text{if } T_x \equiv 1 \pmod 3 \quad \big(\text{i.e., } h(x) \equiv 4 \pmod 6\big).
 \end{cases}
 \end{equation}
 \item \textbf{(Two-Prime Modulo-$3$ Bridge):} Every two-prime step bridges at least one boundary to the next odd spot covered by $p_2 = 3$, guaranteeing
 \begin{equation}
 h(x + 2) \ge \begin{cases}
 h(x) + 8, & \text{if } T_x \equiv 0 \pmod 3 \quad \big(\text{i.e., } h(x) \equiv 2 \pmod 6\big), \\[2pt]
 h(x) + 6, & \text{if } T_x \equiv 1, 2 \pmod 3 \quad \big(\text{i.e., } h(x) \equiv 0, 4 \pmod 6\big).
 \end{cases}
 \end{equation}
 \item \textbf{(Multi-Step Growth):} Consequently, for all integers $y > x \ge 2$,
 \begin{equation}
 h(y) \ge h(x) + 6 \left\lfloor \frac{y - x}{2} \right\rfloor + 2\big((y - x) \bmod 2\big) \ge h(x) + 3(y - x) - 1 > h(x).
 \end{equation}
\end{enumerate}
\end{proposition}

\begin{proof}
For $x = 1$, $h(1) = 2 < h(2) = 4$. Now fix $x \ge 2$, put $G = h(x) = 2T_x + 2$, and let $(r_2, \dots, r_x)$ be any valid covering of the $T_x$ consecutive odd interior spots $\{0, 1, \dots, T_x - 1\}$ by $\{p_2, \dots, p_x\}$. Because $h(x) = G$, no covering of $T_x + 1$ consecutive odd spots exists for $\{p_2, \dots, p_x\}$; hence by Theorem~\ref{thm:endpoint} applied to $p_2 = 3$, neither endpoint $s_{\text{left}} = -1$ nor $s_{\text{right}} = T_x$ is covered by $p_2 = 3$:
\begin{equation}
 r_2 \not\equiv -1 \equiv 2 \pmod 3 \qquad \text{and} \qquad r_2 \not\equiv T_x \pmod 3.
\end{equation}
Thus $r_2 \in \{0, 1\}$, and among the three consecutive odd spots $\{-3, -2, -1\}$ on the left (corresponding to $n_0 - 4, n_0 - 2, n_0$) and $\{T_x, T_x + 1, T_x + 2\}$ on the right (corresponding to $n_0 + G, n_0 + G + 2, n_0 + G + 4$), $p_2 = 3$ already covers either $s = -2$ (if $r_2 = 1$) or $s = -3$ (if $r_2 = 0$), and already covers either $s = T_x + 1$ (if $r_2 \equiv T_x + 1 \pmod 3$) or $s = T_x + 2$ (if $r_2 \equiv T_x + 2 \pmod 3$):
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \emph{Case 1 ($T_x \equiv 0 \pmod 3$):} Since $r_2 \notin \{2, 0\}$, we must have $r_2 = 1$. Then $p_2 = 3$ already covers \emph{both} $s = -2 \equiv 1 \pmod 3$ and $s = T_x + 1 \equiv 1 \pmod 3$. Assigning $p_{x+1}$ to cover $s = T_x$ extends the covered odd board to $\{0, \dots, T_x + 1\}$ of length $T_x + 2$ ($h(x + 1) \ge h(x) + 4$), and assigning $p_{x+2}$ to cover $s = -1$ simultaneously bridges the left end to $\{-2, -1, 0, \dots, T_x + 1\}$ of length $T_x + 4$, proving $h(x + 2) \ge 2(T_x + 4) + 2 = h(x) + 8$.
 \item \emph{Case 2 ($T_x \equiv 2 \pmod 3$):} Here $r_2 \in \{0, 1\}$. If $r_2 = 0$, $p_2 = 3$ covers $s = T_x + 1 \equiv 0 \pmod 3$; if $r_2 = 1$, $p_2 = 3$ covers $s = -2 \equiv 1 \pmod 3$. In either subcase, one new prime $p_{x+1}$ covering the adjacent boundary spot ($s = T_x$ or $s = -1$) bridges across $2$ new odd spots ($h(x + 1) \ge h(x) + 4$), and a second new prime $p_{x+2}$ covering $s = -1$ or $s = T_x$ adds at least $1$ further odd spot, giving $T_{x+2} \ge T_x + 3$ ($h(x + 2) \ge h(x) + 6$).
 \item \emph{Case 3 ($T_x \equiv 1 \pmod 3$):} Since $r_2 \notin \{2, 1\}$, we must have $r_2 = 0$, so $p_2 = 3$ already covers $s = -3$ and $s = T_x + 2 \equiv 0 \pmod 3$. One new prime $p_{x+1}$ covering $s = T_x$ gives $T_{x+1} \ge T_x + 1$ ($h(x + 1) \ge h(x) + 2$), while two new primes $p_{x+1}, p_{x+2}$ assigned to cover $s = T_x$ and $s = T_x + 1$ bridge to $s = T_x + 2$ (covered by $p_2 = 3$), covering $\{0, \dots, T_x + 2\}$ of length $T_x + 3$ and proving $h(x + 2) \ge 2(T_x + 3) + 2 = h(x) + 6$.
\end{itemize}
Applying $h(m + 2) \ge h(m) + 6$ inductively $\lfloor (y - x)/2 \rfloor$ times and $h(y) \ge h(y - 1) + 2$ on any remaining odd step yields part~(3).
\end{proof}


\subsection{Five tiers of information and construction}
\subsubsection{Tier 1: modulo-3 boundary bridging}
The increments of Proposition~\ref{prop:monotonicity_x} also propagate lower bounds without knowing an exact value.
\begin{corollary}[Propagation from any even lower bound]
\label{cor:lower_bound_propagation}
If $x\ge2$ and $G\ge4$ is even with $h(x)\ge G$, then
\begin{align*}
 h(x+1)&\ge G+\begin{cases}2,&G\equiv4\pmod6,\\4,&G\not\equiv4\pmod6,\end{cases}\\[3pt]
 h(x+2)&\ge G+\begin{cases}8,&G\equiv2\pmod6,\\6,&G\not\equiv2\pmod6.\end{cases}
\end{align*}
Consequently, for $y>x$,
\[
 h(y)\ge G+6\left\lfloor\frac{y-x}{2}\right\rfloor
                 +2\big((y-x)\bmod2\big).
\]
\end{corollary}
\begin{proof}
If $h(x)=G$, apply Proposition~\ref{prop:monotonicity_x}. Otherwise evenness gives $h(x)\ge G+2$, and the unconditional increments in that proposition give $h(x+1)\ge G+4$ and $h(x+2)\ge G+8$, which imply both displayed cases. Iteration proves the last inequality.
\end{proof}
These are inequalities for the function. To implement boundary bridging on a particular nonmaximal covering, first extend its fixed residues to uncovered boundaries and then apply the corresponding modulo-$3$ construction.

\subsubsection{Tier 2: pigeonhole capacity and period averaging}
Fix $y>x\ge2$ and define integers $K_j$ backward by
\begin{equation}
 K_y=0,\qquad
 K_{j-1}=K_j+1+\left\lfloor\frac{K_j}{p_j-1}\right\rfloor
 \quad(j=y,y-1,\ldots,x+1).\label{eq:pigeonhole}
\end{equation}
Every set of at most $K_x$ uncovered positions can be covered by one residue class for each of $p_{x+1},\ldots,p_y$.
Indeed, among $U$ positions some residue modulo $p_j$ contains at least $\lceil U/p_j\rceil$ positions. The largest $U$ satisfying $U-\lceil U/p_j\rceil\le K_j$ is exactly $K_{j-1}$ in~\eqref{eq:pigeonhole}; induction proves the assertion. Ascending prime order is a specified construction, not a claim of global optimality.

A period-average statement must be kept separate from the geometry of a particular anchor. Put $Q_x=P_x/2$ and $\rho_x=\varphi(Q_x)/Q_x$. For any fixed choice of one residue per odd prime through $p_x$, CRT gives exactly $\varphi(Q_x)$ uncovered odd positions per period $Q_x$. The following finite consequence is valid without any assumption about the distribution near a chosen covering.
\begin{corollary}[Period-average extension bound]
\label{cor:totient_bridge}
With $K_x$ as in~\eqref{eq:pigeonhole},
\begin{equation}
 h(y)\ge 2\left\lceil\frac{K_x+1}{\rho_x}\right\rceil.
\end{equation}
\end{corollary}
\begin{proof}
Let $L=\lceil(K_x+1)/\rho_x\rceil-1$. Averaging the number of uncovered positions over all $Q_x$ translates of an $L$-position interval gives $L\rho_x<K_x+1$. The counts are integers, so at least one translate contains at most $K_x$ holes. Cover these holes using~\eqref{eq:pigeonhole}, and apply parity compression.
\end{proof}
This bound need not extend the given anchor and must not be added to its gap length. Here $1/\rho_x$ is the mean spacing in odd-board indices, while $P_x/\varphi(P_x)=2/\rho_x$ is the mean spacing in ordinary integers. Neither is a guaranteed spacing between particular exterior holes. The bound uses integrality of the hole count and strengthens the weaker estimate $2\lfloor K_x/\rho_x\rfloor+2$.

\subsubsection{Tier 3: bidirectional extension of one certificate}
Let $\mathcal C=(r_2,\ldots,r_x)$ cover $[0,T-1]$. For nonnegative integers $L,R$, form
\begin{equation}
 H_{L,R}=\{s\in[-L,T+R-1]: s\not\equiv r_i\pmod{p_i}
 \text{ for every }2\le i\le x\}.
\end{equation}
If $|H_{L,R}|\le y-x$, assigning one distinct new prime to each hole gives
\begin{equation}
 h(y)\ge 2(T+L+R)+2.\label{eq:exterior}
\end{equation}
More generally, $|H_{L,R}|\le K_x$ is sufficient by~\eqref{eq:pigeonhole}, and an explicit residue assignment covering $H_{L,R}$ is sufficient even when that inequality fails. The search may assign new primes in ascending order to residues covering the greatest number of remaining holes. This greedy procedure is a witness search: success proves~\eqref{eq:exterior}, but failure does not disprove a covering. Translating by $L$ gives residues $(r_i+L)\bmod p_i$ on a board starting at zero.

\subsubsection{Tier 4: multi-certificate envelopes}
For a finite collection $\mathcal M(x)$ of valid certificates, let $B(\mathcal C,y)$ be any bound obtained by the preceding construction. Then
\begin{equation}
 h(y)\ge \max_{\mathcal C\in\mathcal M(x)}B(\mathcal C,y).
\end{equation}
Different certificates may maximize the expression at different indices. Completeness of the collection is unnecessary for the inequality. A claim that a collection contains \emph{all} optimal coverings, however, requires exhaustive enumeration and a precise equivalence convention. Section~\ref{sec:catalog_envelope} verifies the supplied $62$-entry family and its constructive uses. The envelope construction is valid for any supplied family; completeness is a separate computational assertion.

\subsubsection{Tier 5: relocation to complete or repaired coverings}
\label{sec:tier5}
The first four tiers can retain the same base residues for a long range of target indices. Tier~5 changes that base: a bounded search at a larger index $x_1$ can produce useful residues even if it does not determine $h(x_1)$. There are two distinct ways to obtain the new starting certificate.

\paragraph{Tier 5A: complete, possibly nonmaximal coverings.}
If the search covers every position of a board, its residue tuple immediately supplies a lower bound and can be extended by Tiers~3 and~4. There is no need to prove that the board is longest possible. The resulting even lower bound can also be propagated by Corollary~\ref{cor:lower_bound_propagation}. For a direct geometric extension of the saved tuple, first establish its actual uncovered boundaries. Increasing the number of fixed primes alone does not guarantee an improved bound.

\paragraph{Tier 5B: repair of an incomplete covering.}
If $k$ holes remain, $k$ additional primes always suffice by assigning one to each hole. Distinct-prime pairings can reduce that cost: every disjoint pair covered by its own new prime saves one prime. The pairing must use distinct holes and distinct prime labels. Once all holes are covered, the repaired tuple becomes an ordinary complete certificate for later extensions.

\paragraph{Relation to earlier covering constructions.}
Leaving a small residual set after partial sieving and completing the covering with unused primes is classical in the Erd\H{o}s--Rankin approach to large prime gaps; see the account in Ford, Green, Konyagin, and Tao \cite[Section~1]{FGKT2016}. Maier and Pomerance \cite[Sections~2 and~4]{MaierPomerance1990} used a single remaining prime to cover two surviving positions congruent modulo that prime, selecting disjoint pairs with distinct prime labels. Pintz \cite{Pintz1997} strengthened the associated matching result; this development is also summarized in \cite[Section~1]{FGKT2016}. A later explicit singleton-completion step, assigning each survivor to a distinct additional prime, appears in Ford, Green, Konyagin, Maynard, and Tao \cite[Section~3]{FGKMT2018}.

Method~5B implements these principles at a fixed finite target index: it searches for a covering with a bounded deficit, repairs the holes using a prescribed reserve of primes, and combines repair with exterior extension to obtain explicit lower-bound certificates. The contribution here is this computational workflow and the reported constructions. We make no priority claim for partial-covering completion, reserve-prime repair, or distinct-prime pair matching, or for the first implementation of their combination. The following elementary proposition records the finite prime-prefix condition needed for these certificates.
\begin{proposition}[Tier 5A/5B: complete and deficit-patched stepping]
\label{prop:tier5}
Let residues for the odd primes through $p_{x_1}$ cover all but $k$ positions of $[0,T-1]$, and set $G=2T+2$.
For Tier~5A, if $k=0$, then $h(x_1)\ge G$. For Tier~5B, if $0\le d\le\lfloor k/2\rfloor$ disjoint pairs of holes have differences divisible by $d$ distinct primes among
$p_{x_1+1},\ldots,p_{x_1+k-d}$, then
\begin{equation}
 h(x_1+k-d)\ge G.
\end{equation}
The completed assignment is a valid base for further certificate extensions.
\end{proposition}
\begin{proof}
Assign the $d$ distinct new primes to their respective pairs. Each pair has a common residue modulo its assigned prime. Assign the remaining $k-2d$ new primes injectively to the remaining $k-2d$ holes. All $T$ positions are then covered. The case $k=0$ follows directly from parity compression.
\end{proof}
Only an actual matching with distinct holes and distinct prime labels may be counted; the existence of many eligible pairs alone does not give the saving $d$. A higher-index base supplies more fixed residues, but is not guaranteed to improve every bound. Retaining the maximum over old and new certificates prevents a decrease.

The prime-index restriction in Proposition~\ref{prop:tier5} is essential: saving $d$ assignments does not prove a bound at index $x_1+k-d$ if a used prime lies beyond that prefix. The implemented hunt fixes the target index as $H+K$, where $K$ is the configured number of reserve primes, even when fewer than $K$ holes remain. It can use reserve primes freed by pair repair to cover exterior holes. Initially reserving these primes for singleton repair restricts the construction search; an unsuccessful hunt, including exhaustion of that restricted search, supplies no upper bound for $h(H+K)$.


\subsection{Verified envelopes from the 62-covering catalog}
\label{sec:catalog_envelope}
The supplied catalog contains $62$ residue vectors at index $65$, each covering the $566$-position odd board and hence proving $h(65)\ge1134$. Independent checks verify every vector and its reflected copy, the CRT starting points, both coprime endpoints, and the accompanying position-by-position coverage data. The vectors are distinct modulo reflection $s\mapsto565-s$. The printed index-$65$ certificate agrees with catalog entry $62$ up to reflection.

These checks establish the validity and distinctness of the supplied family, not that it contains every optimal covering. The envelope inequality requires only a valid family, so catalog completeness does not affect the constructions below. Catalog IDs refer to the CSV's \texttt{solution\_index} field.

For the singleton construction, list a certificate's exterior holes as
\[
 \ell_1>\ell_2>\cdots\quad(\ell_1=-1),\qquad
 r_1<r_2<\cdots\quad(r_1=T).
\]
For $m=y-65$ added primes, assigning $a$ of them to left holes and $m-a$ to right holes covers the interval from $\ell_{a+1}+1$ through $r_{m-a+1}-1$. Counting only the assigned holes, the best bound over these left/right splits is
\begin{equation}
 B_{\rm single}(\mathcal C,y)
 =2\max_{0\le a\le m}\big(r_{m-a+1}-\ell_{a+1}-1\big)+2.
\end{equation}
This is an optimization over splits of the hole budget, not an assumption of uniform exterior spacing. Taking the maximum over the $62$ entries gives the singleton envelope in Table~\ref{tab:catalog_envelope}. The pair column records explicit achieved bounds; it is not a claim that every possible multi-hit assignment has been optimized.

\begin{table}[!htbp]
\centering
\small
\caption{Verified lower bounds constructed from the supplied index-$65$ catalog. The single-certificate column uses entry $62$. The last column gives one entry attaining the displayed achieved bound; there can be ties.}
\label{tab:catalog_envelope}
\begin{tabular}{rrrrrr}
\toprule
$y$ & $1134+2m$ & Entry $62$ & Singleton envelope & With pairs & Example ID\\
\midrule
66 & 1136 & 1138 & 1144 & 1144 & 5\\
70 & 1144 & 1168 & 1176 & 1176 & 2\\
72 & 1148 & 1178 & 1190 & 1190 & 58\\
73 & 1150 & 1184 & 1198 & 1198 & 60\\
75 & 1154 & 1206 & 1206 & 1206 & 5\\
76 & 1156 & 1210 & 1224 & 1224 & 59\\
80 & 1164 & 1248 & 1250 & 1250 & 36\\
82 & 1168 & 1258 & 1296 & 1296 & 59\\
85 & 1174 & 1280 & 1314 & 1314 & 59\\
90 & 1184 & 1314 & 1340 & 1350 & 59\\
95 & 1194 & 1366 & 1396 & 1402 & 60\\
100 & 1204 & 1420 & 1450 & 1482 & 60\\
\bottomrule
\end{tabular}
\end{table}
\FloatBarrier

\paragraph{Changes in the leading certificate.}
The singleton computation reproduces the original pattern: entry $5$ is among the leaders for indices $66$--$71$; entry $58$ is the unique leader at $72$; entry $60$ leads at $73$--$74$; and entry $59$ leads at $76$--$77$ and throughout $81$--$90$. Entry $60$ leads throughout $91$--$100$. Different exterior hole patterns therefore matter at different horizons. The ancillary summary lists all ties rather than selecting one winner silently.

\paragraph{An explicit multi-hit example at index $100$.}
For catalog entry $60$, extending the base interval $[0,565]$ to $[-22,717]$ leaves exactly $39$ holes. Four pairs can be covered using four distinct new primes:
\[
\begin{aligned}
-2&\equiv704\pmod{353}, &-8&\equiv626\pmod{317},\\
-13&\equiv661\pmod{337}, &-1&\equiv697\pmod{349}.
\end{aligned}
\]
Here each pair's difference is twice its displayed prime. Assign the other $31$ primes among $p_{66},\ldots,p_{100}$ injectively to the remaining $31$ holes. The resulting $740$-position covering proves
\[
 h(100)\ge2\cdot740+2=1482.
\]
The same independent construction verifies $h(90)\ge1350$ from entry $59$ and $h(95)\ge1402$ from entry $60$. Entry $62$ separately gives $h(100)\ge1426$ with pairs, reproducing the reported $56$-unit improvement of the catalog example over that single-certificate example. These are comparisons between achieved constructions, not assertions of global optimality.

The full CSV, a catalog verifier, a deterministic extension-construction script, and complete vectors for $35$ singleton-envelope witnesses and four pair examples are included in \texttt{anc/catalog/}. The extension checks use only Python's standard library and do not call the native solver.

\subsection{Verified extensions from the index-70 certificate}
\label{sec:h70_extensions}
Table~\ref{tab:h70_lower_bounds} lists selected lower bounds obtained from the $h(70)$ certificate in Table~\ref{tab:certificates}. Complete residue vectors are supplied in \texttt{anc/certificates.json}; \texttt{anc/verify\_certificates.py} checks their coverage without using the native search engine. These are finite constructive bounds, not estimates from a density or an asymptotic formula. The index-$71$ entry illustrates extension of this fixed base; the separate descending computation establishes the stronger exact value $h(71)=1298$.
\begin{table}[!htbp]
\centering
\caption{Selected certificate-verified extensions of the printed $h(70)$ covering. Each entry asserts $h(y)\ge B$. The classical finite bound $2p_{y-1}$ is included for comparison; $B$ need not improve it.}
\label{tab:h70_lower_bounds}
\begin{tabular}{rrr@{\hspace{1cm}}rrr}
\toprule
$y$ & $B$ & $2p_{y-1}$ & $y$ & $B$ & $2p_{y-1}$\\
\midrule
71 & 1284 & 698 & 170 & 2286 & 2018\\
72 & 1288 & 706 & 200 & 2664 & 2434\\
73 & 1296 & 718 & 300 & 3970 & 3958\\
75 & 1310 & 746 & 400 & 5316 & 5462\\
84 & 1376 & 862 & 500 & 6694 & 7118\\
95 & 1472 & 982 & 1070 & 15212 & 17162\\
100 & 1510 & 1046 & 10070 & 169420 & 211058\\
\bottomrule
\end{tabular}
\end{table}
\FloatBarrier
The rows at $y=400,500,1070,10070$ illustrate verified fixed-base extensions, rather than improvements on known bounds. In particular, the classical construction already gives $h(1070)\ge17162$ and $h(10070)\ge211058$. This comparison shows why retaining one base indefinitely can be less effective than changing the base or using another construction.

For $y=100$, a particularly short verification is available. Retain the first $70$ primes' residues and extend the original board $[0,635]$ to $[-107,646]$. Exactly $30$ positions of this $754$-position interval remain uncovered. Assigning the $30$ primes $p_{71},\ldots,p_{100}$ injectively to those holes proves $h(100)\ge2\cdot754+2=1510$; no multi-hit assumption is needed for this witness.

\Needspace{18\baselineskip}
\section{Computational Results and Reproducibility}
\label{sec:experiments}

\subsection{Verified witnesses and exhaustive computations}
\label{sec:new_results}

\begin{theorem}[Computational results with explicit lower-bound witnesses]
\label{thm:main_comp}
The exhaustive computations and the independently verified witnesses give
\begin{equation}
 \begin{aligned}
 h(65)&=1134,& h(66)&=1160,& h(67)&=1192,\\
 h(68)&=1216,& h(69)&=1248,& h(70)&=1274,\\
 h(71)&=1298.
 \end{aligned}
\end{equation}
Separately verified integer certificates and direct exclusions establish
\[
 \begin{aligned}
 2054&\le h(100)\le2798,\\
 3558&\le h(150)\le6898,\\
 5172&\le h(200)\le11998.
 \end{aligned}
\]
\end{theorem}
\begin{proof}[Computational proof and verification scope]
The witnesses in Table~\ref{tab:certificates} establish the lower bounds at indices $65$--$71$. The Method~5B certificates described below establish the three lower bounds by direct integer arithmetic. The Method~1 runs exclude gaps $1136,1162,1194,1218,1250,1276$ at indices $65,66,67,68,69,70$, respectively, and then find coverings two units shorter. The index-$71$ descending run excludes gap $1304$ using the bootstrap, then gaps $1302$ and $1300$ using endpoint pruning justified by the preceding exclusions, and finds a covering at $1298$. The direct checks exclude gaps $2800$, $6900$, and $12000$ at indices $100$, $150$, and $200$, respectively, proving the displayed upper bounds. The bootstrap and descending reductions are those of Section~\ref{sec:canonical}.

The accompanying audits check completion flags, counters, source and settings hashes, native-build identity, and complete coverage of the initial anchor plan. Witness checks use independently generated primes and integer arithmetic. The large native exclusions were performed by the author and were not independently repeated during review; their upper-bound conclusions depend on correct exhaustive execution of the supplied search, not on standalone machine-checkable refutation certificates.
\end{proof}

Table~\ref{tab:new_values} presents the resulting exact values and bounds alongside the preceding values $h(60)$--$h(64)$ across the three linked OEIS sequences:
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{OEIS A048670:} $h(x) = j(P_x)$, the maximum distance between consecutive integers coprime to $P_x = \prod_{i=1}^x p_i$.
 \item \textbf{OEIS A058989:} $h(x) - 1$, the maximum length of a contiguous block of integers each divisible by at least one prime $p_i \le p_x$.
 \item \textbf{OEIS A072752:} $T = (h(x) - 2)/2$, the maximum number of consecutive odd interior integers divisible by at least one odd prime $p_i \in \{p_2, \dots, p_x\}$.
\end{itemize}

\begin{table}[!htbp]
\centering
\caption{Known values and bounds. The exact rows $60$--$64$ are due to Bozek. Reviewed exclusions and covering witnesses support equality at every index $65$--$71$. The lower bounds at indices $100$, $150$, and $200$ have explicit integer certificates. The three upper bounds are supported by exhaustive exclusions; intervals enclose the corresponding values.}
\label{tab:new_values}
\begin{tabular}{rrrrr}
\toprule
$x$ & $p_x$ & $h(x)$ & $h(x)-1$ & $(h(x)-2)/2$\\
\midrule
60&281&1002&1001&500\\
61&283&1030&1029&514\\
62&293&1058&1057&528\\
63&307&1098&1097&548\\
64&311&1110&1109&554\\
\midrule
65&313&1134&1133&566\\
66&317&1160&1159&579\\
67&331&1192&1191&595\\
68&337&1216&1215&607\\
69&347&1248&1247&623\\
70&349&1274&1273&636\\
71&353&1298&1297&648\\
\midrule
100&541&$[2054,2798]$&$[2053,2797]$&$[1026,1398]$\\
150&863&$[3558,6898]$&$[3557,6897]$&$[1778,3448]$\\
200&1223&$[5172,11998]$&$[5171,11997]$&$[2585,5998]$\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[!htbp]
\centering\small
\caption{Exclusion-run records supporting the stated computational upper bounds. Indices $69$, $150$, and $200$ used the author's home Windows desktop; the other rows used AWS. Times are for the tested gap only, in seconds; subsequent witness searches are excluded. All recorded anchors completed with zero cancelled work. The three index-$71$ rows form a descending chain; the exclusions at $1302$ and $1300$ depend on the preceding upper bounds.}
\label{tab:exclusion_records}
\begin{tabular}{rrrrrr}
\toprule
$x$ & Excluded $G$ & Anchors & Workers & Native nodes & Seconds\\
\midrule
65 & 1136 & 259 & 60 & 21,061,838,753 & 1344.896\\
66 & 1162 & 276 & 60 & 37,652,138,616 & 2585.889\\
67 & 1194 & 344 & 60 & 42,196,903,741 & 2950.694\\
68 & 1218 & 354 & 60 & 122,827,419,889 & 8679.226\\
69 & 1250 & 278 & 24 & 112,909,607,345 & 42336.531\\
70 & 1276 & 293 & 59 & 195,378,532,115 & 13861.350\\
71 & 1304 & 259 & 59 & 239,964,866,900 & 18171.184\\
71 & 1302 & 162 & 59 & 127,429,403,459 & 9707.427\\
71 & 1300 & 90 & 59 & 108,655,045,645 & 8365.597\\
100 & 2800 & 301 & 60 & 242,066,346,403 & 5462.775\\
150 & 6900 & 374 & 24 & 1,981,848,749 & 1267.193\\
200 & 12000 & 372 & 24 & 945,191,148 & 7324.362\\
\bottomrule
\end{tabular}
\end{table}

\FloatBarrier
The index-$69$ run completed in $48208.400$ seconds on the home desktop, including $42336.531$ seconds for the gap-$1250$ exclusion and $5871.644$ seconds for the gap-$1248$ witness search. Its $624$-spot exclusion board has $346$ canonical boards and uses $188$ Stage~1A plus $90$ Stage~1B jobs. The source/setup audit checks every phase and the full initial plan. All six saved coverings and their reflections pass independent arithmetic verification. Together the two-gap records support the exact-value series $65$--$70$.

The index-$71$ descending discovery run completed in $37973.311$ seconds on AWS with $59$ admitted workers, visiting $499{,}221{,}756{,}584$ native nodes. The three exclusions in Table~\ref{tab:exclusion_records} were followed by a $1728.899$-second search yielding a gap-$1298$ covering. At the initial gap $1304$, the $651$-spot board has $319$ canonical boards, and the bootstrap uses $191$ Stage~1A plus $68$ Stage~1B jobs. The subsequent endpoint-pruned exclusions at gaps $1302$ and $1300$ complete all $162$ and $90$ admissible anchor representatives. The audit reconstructs every initial phase key and every descending endpoint-safe phase class. The saved covering and its reflection pass independent CRT, modular-coverage, and integer-gcd checks. These records extend the exact-value series through $h(71)=1298$.

The direct upper-bound searches at indices $100$, $150$, and $200$ excluded gaps $2800$, $6900$, and $12000$, respectively, establishing $h(100)\le2798$, $h(150)\le6898$, and $h(200)\le11998$. The gap exclusions took $5462.775$, $1267.193$, and $7324.362$ seconds, with total run times of $5462.923$, $1267.303$, and $7324.547$ seconds. All $301$, $374$, and $372$ initial anchors completed without cancellation. Memory admission selected $60$ of $64$ requested workers at index $100$ and all $24$ workers at each of indices $150$ and $200$. Table~\ref{tab:exclusion_records} records the corresponding node counts.

The respective odd boards contain $1399$, $3449$, and $5999$ positions and have $413$, $413$, and $400$ canonical survivor boards. Their Stage~1A/1B job counts are $240/61$, $247/127$, and $237/135$. Independent set-based checks reconstruct all $1155$ anchor phases, check every adjacent-phase edge, and verify the full remaining endpoint-safe plan for each run. Source, settings, and native-build hashes link the run records to the supplied implementation; supporting regression-test records and detailed validation provenance are retained in the ancillary evidence.

\paragraph{Method 5B lower bounds at indices 100, 150, and 200.}
The strongest saved checkpoints use $(H,k)=(95,5),(145,5),(195,5)$ on tested gaps $2048$, $3550$, and $5170$, respectively. In each case, the first $H$ primes leave exactly five odd-board positions uncovered. The next five primes fill these holes, and extending to adjacent coprime endpoints gives gaps $2054$, $3558$, and $5172$. The strongest three witnesses have no left extension and right extensions of $6$, $8$, and $2$, respectively. These are constructive lower bounds for $h(H+5)$; they do not determine $h(H)$ or assert maximality at the target index.

All $44$ saved covering checkpoints ($1$, $35$, and $8$ at indices $100$, $150$, and $200$) and their reflections pass independent checks of CRT reconstruction, coprime endpoints, every interior integer, and the recorded residue coverage. Compact integer certificates and a standard-library verifier are supplied as \path{anc/method5b_lower_bound_certificates.json} and \path{anc/verify_method5b_certificates.py}. Source snapshots and regression-test records accompany the witness files. The archives contain saved witnesses but no completed-run summary or populated timing rows, so no elapsed-time or speedup claim is made for these hunts. The first-solution collections are not exhaustive catalogs.

\paragraph{Explicit integer certificates for the strongest lower bounds.}
For each $(n,B)\in\{(100,2054),(150,3558),(200,5172)\}$, let $a_n$ be the integer printed below and $P_n$ the product of the first $n$ primes. The last primes are $p_{100}=541$, $p_{150}=863$, and $p_{200}=1223$. Line breaks within an integer are typographical only; concatenate the digits.

\noindent\begin{minipage}{\linewidth}
\textbf{$a_{100}$, certifying $h(100)\ge2054$:}
\begin{quote}\small\ttfamily
2996194938546716811833273503997122017356587994385090867084276362\\
5985526907744630355438778156557604638472262931198577241443217021\\
2912031793123214054530485578052213099722407564044959635698651820\\
496487498205341191387422933
\end{quote}
\end{minipage}\par\medskip

\noindent\begin{minipage}{\linewidth}
\textbf{$a_{150}$, certifying $h(150)\ge3558$:}
\begin{quote}\small\ttfamily
4324567726126838606236100858732080099290338768084355918536659445\\
5513263656147804980229095827208129889519132397750203809166075042\\
9130702380474588678508562826696400209964858657664338834655770001\\
6511496030607238372854841937926515406556088502143719577542441891\\
5391033493967370978383323332817383143929922495935686302827410665\\
280924133330134192192406671544549220171989
\end{quote}
\end{minipage}\par\medskip

\noindent\begin{minipage}{\linewidth}
\textbf{$a_{200}$, certifying $h(200)\ge5172$:}
\begin{quote}\small\ttfamily
6177466790250688825461723318644337348708909036695717970170700339\\
9455245939352550485339573909494491932260611800658326737648508346\\
1971619059034381232449897827506189919441175723640019427483126753\\
2704719423852275320225042521528671145480089417322404646378265011\\
3948778157955780667083472910294355992057524425223799003015380639\\
6905452265730336686873364693985517240234947506588367747187614777\\
5809873925377917500540975839357027708848183477917865983911067018\\
6537638174433698357539116364651618511031230788140722288706094451
\end{quote}
\end{minipage}\par\medskip

Direct integer arithmetic verifies, for each pair $(n,B)$ above,
\[
 \gcd(a_n,P_n)=\gcd(a_n+B,P_n)=1,
 \qquad \gcd(a_n+t,P_n)>1\quad(1\le t<B).
\]
Thus the endpoints are consecutive integers coprime to $P_n$, separated by $B$, which proves $h(n)\ge B$. No search completeness assumption is needed for these lower bounds. If desired, the residue of each odd prime $p\le p_n$ on the odd board is recovered from the single integer by
\[
 r_p\equiv -(a_n+2)\,2^{-1}\pmod p,\qquad 0\le r_p<p.
\]
The accompanying short verifier generates the first $n$ primes and checks the two endpoints and every interior integer; it does not import the native search engine.

Throughout the historical timing discussion below, $G_x$ denotes the reported final successful gap; maximality is now supported by the reviewed two-gap records for every $65\le x\le70$. Saved witnesses from first-solution runs are not claimed to form exhaustive catalogs.

To make the lower bounds $h(65) \ge 1{,}134$, $h(66) \ge 1{,}160$, $h(67) \ge 1{,}192$, $h(68) \ge 1{,}216$, $h(69) \ge 1{,}248$, $h(70) \ge 1{,}274$, and $h(71) \ge 1{,}298$ specified completely in the text and independently checkable, Table~\ref{tab:certificates} records an explicit Chinese Remainder Theorem starting integer $n_0 + 1 \in [1, P_x]$ and the corresponding odd-board residue vector $(r_2, \dots, r_x)$ for each candidate gap $G_x$. For every $x \in \{65, \dots, 71\}$, one verifies directly that $\gcd(n_0, P_x) = \gcd(n_0 + G_x, P_x) = 1$ and that every integer $m \in [n_0 + 1, n_0 + G_x - 1]$ satisfies $\gcd(m, P_x) > 1$.

\begingroup
\small
\setlength{\tabcolsep}{5pt}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.77\textwidth}@{}}
\caption{Explicit lower-bound certificates. The start is $n_0+1$; the residues are in increasing odd-prime order.}\label{tab:certificates}\\
\toprule
Certified lower bound & CRT starting integer and odd-board residues\\
\midrule
\endfirsthead
\toprule
Certified lower bound & CRT starting integer and odd-board residues (continued)\\
\midrule
\endhead
\bottomrule
\endfoot

$h(65) \ge 1134$ ($T = 566$, $p_{65} = 313$) & \textbf{Start $n_0 + 1$:} \texttt{5206461640133296580289665432701325\allowbreak 5450793639260712485313165369580935\allowbreak 5679835390816055416026726322519795\allowbreak 7092009709663673284440284}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{65})$:} \small\raggedright\arraybackslash $($0, 0, 0, 2, 5, 4, 4, 5, 1, 17, 15, 10, 30, 1, 29, 12, 54, 18, 42, 53, 72, 43, 47, 27, 16, 25, 0, 64, 110, 62, 8, 67, 108, 15, 76, 22, 121, 26, 37, 149, 0, 72, 86, 58, 169, 11, 34, 2, 94, 104, 32, 27, 206, 12, 131, 197, 253, 127, 158, 19, 254, 60, 41, 103$)$ \\
\midrule
$h(66) \ge 1160$ ($T = 579$, $p_{66} = 317$) & \textbf{Start $n_0 + 1$:} \texttt{5869317644368218372673464836061974\allowbreak 7909535522776451798934921213977890\allowbreak 9569829017451419514647343022091255\allowbreak 9041934313827686068723626532}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{66})$:} \small\raggedright\arraybackslash $($1, 1, 3, 6, 11, 8, 2, 14, 3, 23, 20, 39, 30, 0, 12, 39, 56, 53, 33, 24, 35, 75, 48, 77, 5, 22, 4, 15, 52, 62, 92, 18, 107, 48, 123, 84, 9, 160, 132, 115, 15, 166, 1, 25, 179, 134, 34, 183, 44, 74, 153, 68, 99, 225, 36, 69, 113, 95, 252, 29, 27, 140, 260, 200, 210$)$ \\
\midrule
$h(67) \ge 1192$ ($T = 595$, $p_{67} = 331$) & \textbf{Start $n_0 + 1$:} \texttt{3389420191648110027993454481194093\allowbreak 1265759420126566168232951104489730\allowbreak 7007091553697517434365424327062738\allowbreak 6017225161999655330231676515530}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{67})$:} \small\raggedright\arraybackslash $($0, 2, 1, 2, 1, 8, 11, 5, 0, 20, 29, 10, 7, 19, 18, 31, 54, 20, 9, 48, 4, 56, 11, 44, 8, 78, 12, 19, 86, 95, 41, 55, 34, 126, 26, 132, 61, 38, 168, 16, 88, 189, 158, 193, 60, 98, 63, 23, 42, 185, 164, 130, 245, 188, 236, 191, 223, 91, 89, 73, 146, 94, 104, 70, 65, 184$)$ \\
\midrule
$h(68) \ge 1216$ ($T = 607$, $p_{68} = 337$) & \textbf{Start $n_0 + 1$:} \texttt{9569145227297279341370932931881261\allowbreak 5886828820242943939767248098564856\allowbreak 7603555405329170464802223826107397\allowbreak 202135895624954457180105915505234}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{68})$:} \small\raggedright\arraybackslash $($0, 0, 4, 5, 4, 11, 14, 8, 3, 23, 32, 13, 10, 22, 21, 34, 57, 23, 12, 51, 7, 59, 18, 47, 2, 81, 15, 22, 89, 98, 44, 112, 37, 101, 29, 135, 64, 41, 75, 19, 58, 1, 161, 75, 63, 133, 66, 26, 45, 194, 39, 73, 167, 191, 239, 63, 226, 94, 92, 76, 149, 97, 188, 91, 281, 187, 68$)$ \\
\midrule
$h(69) \ge 1248$ ($T = 623$, $p_{69} = 347$) & \textbf{Start $n_0 + 1$:} \texttt{2592499624862335368059936192918898\allowbreak 7938297026575135989563745599772624\allowbreak 0652358579957867550255324103638903\allowbreak 1947873166140529667405788663520750}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{69})$:} \small\raggedright\arraybackslash $($0, 2, 3, 9, 1, 4, 17, 10, 12, 6, 18, 11, 18, 5, 15, 6, 34, 21, 12, 43, 60, 26, 75, 44, 48, 76, 85, 102, 71, 103, 50, 23, 19, 72, 113, 35, 145, 151, 27, 13, 58, 114, 12, 39, 180, 49, 48, 176, 135, 110, 29, 25, 87, 8, 78, 249, 10, 91, 28, 46, 173, 16, 256, 100, 224, 133, 214, 98$)$ \\
\midrule
$h(70) \ge 1274$ ($T = 636$, $p_{70} = 349$) & \textbf{Start $n_0 + 1$:} \texttt{4389848693639099009886917445655692\allowbreak 4212894115872099882423041016745041\allowbreak 9216247685335699753182384371930228\allowbreak 1688777494019035234641746080745314\allowbreak 5124}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{70})$:} \small\raggedright\arraybackslash $($1, 0, 2, 0, 1, 0, 11, 0, 25, 19, 31, 24, 31, 18, 3, 19, 47, 34, 25, 59, 73, 39, 45, 57, 15, 89, 79, 6, 84, 98, 63, 36, 32, 130, 114, 48, 126, 8, 98, 26, 21, 29, 41, 164, 28, 62, 113, 129, 40, 213, 42, 158, 12, 237, 123, 189, 23, 104, 111, 71, 38, 284, 243, 218, 249, 146, 266, 30, 252$)$ \\

\midrule
$h(71) \ge 1298$ ($T = 648$, $p_{71} = 353$) & \textbf{Start $n_0 + 1$:} \texttt{67340696081459809308362916592035\allowbreak 57629340817214005240745103491165\allowbreak 14676106895042618684182285293198\allowbreak 97680255561577500742104434411660\allowbreak 428649537261332}\\*[2pt] & \textbf{Odd-board residues $(r_2, \dots, r_{71})$:} \small\raggedright\arraybackslash $($1, 1, 1, 6, 7, 6, 17, 6, 2, 25, 0, 30, 37, 24, 34, 25, 53, 40, 31, 68, 0, 45, 5, 63, 18, 95, 62, 12, 90, 122, 69, 42, 38, 44, 132, 54, 48, 3, 117, 32, 27, 133, 47, 58, 34, 187, 35, 135, 55, 192, 164, 77, 106, 243, 129, 195, 14, 110, 279, 65, 34, 9, 269, 104, 168, 152, 275, 300, 224, 219$)$ \\

\end{longtable}
\endgroup

\subsection{Reported Search Timings (\texorpdfstring{$h(50)$--$h(70)$}{h(50)--h(70)}), Hardware Environment, and Scaling Projections to \texorpdfstring{$h(80)$}{h(80)}}
\label{sec:scaling}

The production timings in the original manuscript were reported by the author for evaluations executed on the author's personal desktop workstation equipped with an \textbf{Intel Core i7-13700} processor ($16$ physical cores: $8$ Performance cores and $8$ Efficiency cores, $24$ hardware threads) running 64-bit Windows with \texttt{NUM\_WORKERS = 24} and \texttt{ENDPOINT\_PRIME\_CUTOFF = 109}. On this platform, the Python orchestrator dispatches dogpile anchor and sub-anchor tasks across all $24$ hardware threads to the precompiled x86-64 native C++17 shared library (\texttt{jacobsthal\_core.dll}), utilizing hardware \texttt{POPCNT}, \texttt{BMI1/BMI2}, and \texttt{LZCNT} bit-manipulation instructions.

Table~\ref{tab:scaling} and Figure~\ref{fig:scaling} report the measured wall-clock execution times (in hours) on this system across $50 \le x \le 70$, together with an exponential regression trendline projecting Method~1 verification times out to $h(80)$:
\begin{itemize}\setlength{\itemsep}{2pt}\setlength{\parskip}{0pt}
 \item \textbf{Method~1 Verification Timings ($h(50)$--$h(63)$):} For $50 \le x \le 63$, each value $h(x) = G$ was tested via Method~1 (excluding $G + 2$, followed by endpoint-pruned verification of $G$ terminating on the first valid covering). Reported wall-clock search times grow from $0.0039\text{ hours}$ ($14.0\text{ s}$) at $h(50)$ and $0.0011\text{ hours}$ ($4.0\text{ s}$) at $h(51)$ to $0.1684\text{ hours}$ ($10.1\text{ min}$) at $h(60)$ and $0.2009\text{ hours}$ ($12.1\text{ min}$) at $h(63)$.
 \item \textbf{Method~3 Blind Discovery Timings ($h(64)$--$h(70)$):} For $64 \le x \le 70$, where the reported candidates at indices $65$--$70$ had not previously been established, runs were executed via Method~3 (starting at a conservative upper gap $G_0 > h(x)$ and stepping down by $2$ with endpoint pruning until the first covering at $h(x)$ was discovered), requiring $1.3380\text{ hours}$ at $h(64)$, $5.7173\text{ hours}$ at $h(65)$, $4.8865\text{ hours}$ at $h(66)$, $5.6241\text{ hours}$ at $h(67)$, $15.6579\text{ hours}$ at $h(68)$, $22.5168\text{ hours}$ at $h(69)$, and $16.7277\text{ hours}$ at $h(70)$.
 \item \textbf{Exponential Trendline and Projections to $h(80)$:} Fitting an ordinary least-squares linear regression to $\ln T_{\text{M1}}(x)$ over the measured Method~1 series $50 \le x \le 63$ yields the exponential scaling model
 \begin{equation}
 T_{\text{M1}}(x) = 4.791 \times 10^{-12}\, e^{0.4018\,x}\text{ hours} \approx 4.791 \times 10^{-12} \times (1.4945)^x\text{ hours} \qquad (R^2 = 0.9008),
 \end{equation}
 corresponding to a multiplicative growth factor of $1.4945\times$ per additional prime ($+49.5\%$ per prime, or $55.6\times$ per $+10$ primes). Notice in Figure~\ref{fig:scaling} that the Method~3 discovery times at $x = 64$--$70$ lie approximately $1.9\times$--$5.4\times$ above the Method~1 trendline. These runs include additional refutations, but also differ in their starting gaps and, in some cases, anchor settings; the ratio therefore does not isolate a single optimization. In particular, blind discovery must refute several intermediate unsatisfiable gaps $G_0, G_0 - 2, \dots, h(x) + 2$ before reaching $h(x)$, whereas Method~1 tests only $h(x) + 2$ and $h(x)$. Extrapolating the Method~1 trendline to $x = 71$--$80$ projects single-workstation verification times of $\sim 11.7\text{ hours}$ at $h(71)$, $\sim 26.2\text{ hours}$ ($1.1\text{ days}$) at $h(73)$, $\sim 58.5\text{ hours}$ ($2.4\text{ days}$) at $h(75)$, and $\sim 436.5\text{ hours}$ ($18.2\text{ days}$) at $h(80)$. These extrapolations are not runtime guarantees or new function values; no uncertainty interval is inferred from this short timing series.
\end{itemize}

\begin{table}[!htbp]
\centering
\small
\caption{Author-reported wall times (hours) and the fit to the rounded Method~1 data. The fit column is a model evaluation, not another measurement.}
\label{tab:scaling}
\begin{tabular}{rrrr}
\toprule
$x$ & Method 1 & Method 3 & Method 1 fit\\
\midrule
50 & 0.0039 & --- & 0.0025\\
51 & 0.0011 & --- & 0.0038\\
52 & 0.0058 & --- & 0.0057\\
53 & 0.0061 & --- & 0.0085\\
54 & 0.0131 & --- & 0.0127\\
55 & 0.0506 & --- & 0.0190\\
56 & 0.0235 & --- & 0.0283\\
57 & 0.0786 & --- & 0.0423\\
58 & 0.0669 & --- & 0.0633\\
59 & 0.1127 & --- & 0.0945\\
60 & 0.1684 & --- & 0.1413\\
61 & 0.2546 & --- & 0.2112\\
62 & 0.2948 & --- & 0.3156\\
63 & 0.2009 & --- & 0.4716\\
64 & --- & 1.3380 & 0.7048\\
65 & --- & 5.7173 & 1.0533\\
66 & --- & 4.8865 & 1.5742\\
67 & --- & 5.6241 & 2.3527\\
68 & --- & 15.6579 & 3.5160\\
69 & --- & 22.5168 & 5.2547\\
70 & --- & 16.7277 & 7.8531\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[!htbp]
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\caption{Reported search and discovery times (hours, $\log_{10}$ vertical scale) on an Intel Core i7-13700 ($24$ threads) for $h(50)$--$h(63)$ (Method~1) and $h(64)$--$h(70)$ (Method~3), with the fitted Method~1 exponential trendline projected out to $h(80)$.}
\label{fig:scaling}
\end{figure}



\FloatBarrier
\subsection{Per-Gap Runtime Escalation During Top-Down Descent (\texorpdfstring{$h(65)$--$h(70)$}{h(65)--h(70)})}
\label{sec:gap_escalation}

To illustrate how computational cost evolves across individual candidate gaps during a top-down descending search, Figure~\ref{fig:gap_escalation} and Table~\ref{tab:gap_escalation} trace the per-gap execution times recorded in the original timing summaries (the production \texttt{*\_gaps.csv} files were not supplied) for the six Method~3 discovery runs $h(65)$--$h(70)$, plotted as a function of the remaining distance $\Delta G = G - G_x$ above the reported final gap $G_x$ (moving left to right from the initial upper gap $G_0$ down to $\Delta G = 0$ at $G = h(x)$). The trajectories show several patterns, with structural explanations that should be distinguished from controlled performance measurements:

\begin{enumerate}\setlength{\itemsep}{2pt}\setlength{\parskip}{1pt}
 \item \textbf{Observed runtime drop at the first endpoint-pruned gap ($G_0 \to G_0 - 2$):} At the initial gap $G_0$ ($\Delta G_{\max}$, plotted as open squares), the solver executes the initial bootstrap search, which includes unrestricted Stage~1A. Once $G_0$ is refuted, Theorem~\ref{thm:endpoint} (\texttt{ENDPOINT\_PRUNED}) activates on the very next step $G_0 - 2$ ($\Delta G_{\max} - 2$), coinciding with a per-gap runtime reduction of $\mathbf{5.7\times}$ on $h(69)$ ($1{,}583.7\text{ s} \to 277.9\text{ s}$), $\mathbf{7.0\times}$ on $h(66)$ ($1{,}917.3\text{ s} \to 272.8\text{ s}$), $\mathbf{7.1\times}$ on $h(68)$ ($5{,}096.8\text{ s} \to 717.6\text{ s}$), $\mathbf{8.0\times}$ on $h(67)$ ($4{,}983.2\text{ s} \to 626.7\text{ s}$). The tested gap and anchor plan also change, so these ratios do not isolate the effect of endpoint pruning.
 \item \textbf{Increasing cost near the threshold:} Shrinking the odd board can increase slack when live capacities do not fall; Proposition~\ref{prop:method1_opt} explains why this is not automatic. The reported times fluctuate, but some descents become substantially more expensive near the final gap. On $h(68)$, refutation time rises from $717.6$ seconds at $G=1244$ to $12467.8$ seconds at $G=1218$, a factor of $17.4$. On $h(69)$, the first endpoint-pruned test at $G=1298$ takes $277.9$ seconds, while the final successful search at $G=1248$ takes $14755.0$ seconds, a factor of $53.1$. The latter comparison includes a witness search, not just refutations. These observations do not establish exponential growth at every step or a monotone runtime law.
 \item \textbf{A periodic component from endpoint coincidences:} When $T\equiv2\pmod3$, equivalently $G\equiv0\pmod6$, the two forbidden residues modulo $3$ coincide. This leaves two legal residues instead of one (Example~\ref{ex:endpoint_small_primes}). Holding the other prime factors fixed doubles the phase count $N_D(T)$; the reflection correction is given exactly by equation~\eqref{eq:anchor_count}. Coincidences for other dogpile primes and differences in branch difficulty also affect the total cost, so the modulo-$3$ condition alone does not determine the height or location of every runtime peak.
 \item \textbf{Early Termination at the Final Successful Gap ($\Delta G=0$):} At the final successful gap $G = G_x$ ($\Delta G = 0$), the solver terminates as soon as the first valid covering is found, producing a sharp downward drop whenever coverage-first anchor ordering hits a winning anchor early in the queue (e.g., dropping by $6.5\times$ on $h(67)$ from $4{,}335.9\text{ s}$ at $1{,}194$ to $663.2\text{ s}$ at $1{,}192$, and by $3.2\times$ on $h(68)$ from $12{,}467.8\text{ s}$ at $1{,}218$ to $3{,}880.6\text{ s}$ at $1{,}216$).
\end{enumerate}

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 \put(27.5,185.00){\makebox(0,0)[l]{\small $h(70)$: $1280\to1274$, $1.3\times$/a}}
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\end{picture}}
\caption{Per-gap execution time (hours, $\log_{10}$ vertical scale) during top-down Method~3 discovery of $h(65)$--$h(70)$ as a function of distance $\Delta G = G - G_x$ above the reported final gap.}
\label{fig:gap_escalation}
\end{figure}

\begin{table}[!htbp]
\centering
\small
\caption{Author-reported per-gap wall times in seconds. ``Peak failed EP'' excludes the final successful gap. EP denotes endpoint-pruned search.}
\label{tab:gap_escalation}
\begin{tabular}{rrrrrr}
\toprule
$x$ & Tested gap range & Initial & First EP & Peak failed EP & Final\\
\midrule
65 & $1160\to1134$ & 1509.8 & 557.4 & 3213.4 & 5722.5\\
66 & $1186\to1160$ & 1917.3 & 272.8 & 3035.7 & 764.9\\
67 & $1210\to1192$ & 4983.2 & 626.7 & 4335.9 & 663.2\\
68 & $1246\to1216$ & 5096.8 & 717.6 & 12467.8 & 3880.6\\
69 & $1300\to1248$ & 1583.7 & 277.9 & 11082.7 & 14755.0\\
70 & $1280\to1274$ & 32062.3 & 14626.5 & 14626.5 & 3111.8\\
\bottomrule
\end{tabular}
\end{table}

\FloatBarrier
\section{Discussion and limitations}
The results combine two distinct forms of computational evidence. A covering gives a short, independently checkable lower-bound certificate, whereas an upper bound requires complete exclusion of the relevant search space. The endpoint and sliding reductions reduce that space without changing the conclusion when their hypotheses hold. The supplied run records document those exclusions; independently implemented searches or standalone refutation certificates would provide additional assurance.

Method~5B implements the classical completion principle discussed in Section~\ref{sec:tier5} as a finite deficit search with reserve-prime repair and exterior extension. This permits a family of partial solutions to yield different final gaps, which can make the method effective even when a direct search at one prescribed gap encounters a difficult branch order. The present certificates establish feasibility, not a measured general speedup. A performance comparison should use the same hardware, worker budget, time limits, and target index, and include a direct search spread across several candidate gaps. Recording time to each independently verified lower bound, including unsuccessful attempts, would distinguish the benefit of deficit repair from the benefit of exploring several gaps.

\section*{Code and data availability}
The complete reproducibility package contains the Python orchestrator, C++ search engine, build instructions, regression tests, run records, source snapshots, lower-bound certificates, and verification scripts. It is publicly available as GitHub release \texttt{v1.0-paper}:
\begin{center}
\small\url{https://github.com/albenweeks/JacobsthalFunction/releases/tag/v1.0-paper}
\end{center}
Download \path{jacobsthal_Oct292026.zip} from the release's Assets section. This Hexagon upload contains only the manuscript; references throughout the paper to ancillary or supplied files refer to the complete reproducibility package. The package's \path{README.txt} describes the files and reproduction commands, and a SHA256 manifest identifies the files. Platform binaries are omitted, and the build script creates the native library locally.

The evidence includes the two-gap runs at indices $65$--$70$, the descending index-$71$ run, the direct exclusions at indices $100$, $150$, and $200$ (gaps $2800$, $6900$, and $12000$), all $44$ saved Method~5B checkpoints at indices $100$, $150$, and $200$, and the $62$-entry index-$65$ catalog with its verified extensions. Search records retain native binary hashes and installation-check logs. Lower-bound arithmetic verifiers use the Python standard library and do not require a native build. Historical timing figures use the author's earlier rounded summaries and are distinguished from the retained exclusion records. Exhaustive completeness of the first-solution collections or the $62$-entry catalog is not claimed.

\section*{Use of generative AI tools}
Google Gemini assisted with C++ optimization and preparation of an earlier manuscript. OpenAI ChatGPT assisted with development, debugging, and optimization of the solver; preparation of regression tests and certificate-verification scripts; mathematical and editorial review; and drafting and revision of the exposition. These uses included code and text generation. The tools are not authors. Alben Weeks is the sole author and takes full responsibility for the mathematical claims, software, computations, references, and final text.

Independent arithmetic verification in this article means checking certificates with separate integer-arithmetic routines that do not use the native search engine; it does not mean independent human peer review. AI assistance is not evidence for a mathematical claim. The supplied certificates, verification routines, mathematical arguments, and explicitly scoped exhaustive-search records provide the stated support for the results.

\section*{Acknowledgments}
The author thanks Prof.~Max~A.~Alekseyev for independently verifying the six earlier covering certificates at indices $65$--$70$ in Table~\ref{tab:certificates} and for helpful comments on an earlier draft. This acknowledgment concerns those covering witnesses, not an independent exhaustive proof of their maximality or verification of the index-$71$ or later Method~5B certificates.

\clearpage
\begingroup
\small
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\setlength{\itemsep}{2pt}
\setlength{\parskip}{0pt}

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\end{thebibliography}
\endgroup

\vspace{10pt}
\noindent\textit{Email address:} \texttt{weeks.alben@gmail.com}

\end{document}