% Copyright 2026 Ricardo Pinho. CC BY 4.0. \documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,array} \usepackage[expansion=false]{microtype} \usepackage[hyphens]{url} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{2em} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathbb E} \newcommand{\supp}{\operatorname{supp}} \title[Derivative controls for relevant variables]{Derivative controls and Boolean completion\protect\\in the relevant-variable recurrence} \author{Ricardo Pinho} \address{Independent researcher} \thanks{ORCID: \href{https://orcid.org/0009-0006-2744-2599}{0009-0006-2744-2599}. Licensed under CC BY 4.0.} \date{2 November 2026; version 3; unrefereed preprint} \subjclass[2020]{06E30, 68Q25, 68R05} \keywords{Boolean functions, relevant variables, local degree, ternary derivatives, Reed--Muller codes} \hypersetup{pdftitle={Derivative controls and Boolean completion in the relevant-variable recurrence},pdfauthor={Ricardo Pinho}} \begin{document} \begin{abstract} We give the unrefereed bound $C_{44}<2.0417086280$ for the number of relevant variables of a Boolean function of real degree $d$, $r(f)\le C_{44}2^d$. The argument bounds the number of coordinates controlling a ternary derivative, exploits its dictator branches and Boolean completion, and propagates local-degree caps. The small-degree inputs are $R_4=22$, $R_5\le50$, and $R_6\le109$. The quartic proof uses an exact calculation on Rasoolzadeh's published cubic catalogue; the remaining finite arithmetic is rational. The lower bound remains $3/2$. There has been no independent human review or formalization. \end{abstract} \maketitle \section{Result, provenance, and scope} For sign Boolean functions on uniform independent sign inputs, let $r(f)$ be the number of relevant coordinates and $\deg f$ the real multilinear Fourier degree. Put $R_d=\max_{\deg f\le d}r(f)$ and $C_{44}=\sup_{d\ge1}R_d/2^d$. \begin{theorem}\label{main} The argument and exact finite checks below give \[ R_4=22,\qquad46\le R_5\le50,\qquad94\le R_6\le109. \] \[ \frac32\le C_{44}\le \frac{80880411492166173705437336107}{39614081257132168796771975168} <2.0417086280. \] \end{theorem} This is a large move for an elementary argument. Please read it with suspicion and scrutinize the proof, especially the classification bridge, the completion inequalities, and the finite case coverage. The weighted restriction method, including its potential recurrence, is due to Chiarelli, Hatami and Saks (CHS) \cite{chs}; Wellens developed sharper recurrences and bounds \cite{wellens}. Wellens's published bound is $4.394$; BakeLab/\'Epi's subsequent unrefereed value is $4.3654674227$, using exact polynomial certificates through degree $48$ \cite{epi}. The author's earlier unrefereed values were $4.3576549226$, $4.1225323803$ (version 2 of this note), $3.9819073803$, and $2.1974348818$. The current claim is filed in Tao-table PR~294 \cite{pr}; it is not a claim of refereed acceptance. The contribution relative to version 2 is the derivative-control argument, the cubic energy envelope and quartic equality exclusions, and the use of Boolean completion to sharpen the subsequent caps. The proof below follows the frozen PROOF files \cite{base,packet}; it uses their \texttt{simple} mode. The earlier $52$-case quartic/quintic control classification is compatible with the result but is unnecessary for this mode. The old block-sensitivity certificates are also unnecessary for this endpoint. The historical block-sensitivity route also credits D. Koshta, I. Stepanovs, and Alexander Richard Towell; see \cite{base}. The lower construction is CHS's, also observed by Shinkar and Tal. Minimum-support structure is due to Valyuzhenich; the binary low-weight spectrum to Berlekamp--Sloane and Kasami--Tokura; the lattice gap argument to Nebe, building on Barnes--Wall and Brou\'e--Enguehard. Related ternary energy arguments occur in Potapov \cite{potapov}. No priority or optimality is asserted. The bounded searches recorded in the packet's PRIOR\_ART files and refreshed on 2 November 2026 located no superseding applicable claim; authenticated global code search and private discussions were outside coverage. \section{Potential and derivative controls} Write $f=\sum_S\widehat f(S)x_S$, $x_S=\prod_{i\in S}x_i$. For relevant $i$ let $d_i(f)=\max\{|S|:i\in S,\widehat f(S)\ne0\}$ and put $\D(f)=\sum_i2^{-d_i(f)}$, $W_d=\max_{\deg f\le d}\D(f)$. A nonzero degree-at-most-$k$ polynomial has support density at least $2^{-k}$: write $P=x_jA+B$ with $A\ne0$ of degree at most $k-1$; on the support of $A$, at least one of $B+A,B-A$ is nonzero. Induction, starting with constants, proves the assertion. Thus the ternary derivative $A_i=(f_{i=1}-f_{i=-1})/2$ has influence $p_i=\E A_i^2\ge2^{1-d_i}$. Fourier orthogonality gives \begin{equation}\label{influence} I(f)=\sum_i p_i=\sum_S|S|\widehat f(S)^2\le\deg f, \qquad \D(f)\le I(f)/2. \end{equation} In particular $R_d\le d2^{d-1}$, so both maxima exist after removing dummy coordinates and relabeling. We use the CHS identity $W_d=R_d/2^d$ \cite[Corollary 2.2]{chs}. Here is its extremal argument. In a potential maximizer duplicate all coordinates $L$ of local degree below $d$, sharing the other coordinates $Y$, and form $h=(1+z)f(L,Y)/2+(1-z)f(L',Y)/2$. Degree-$d$ terms use only $Y$ and cancel in the selector difference, so $\deg h\le d$. Each duplicated coordinate increases its local degree by one; its two copies retain its total weight. Shared coordinates retain local degree $d$, and the relevant selector adds positive weight. Hence $L$ was empty. The reverse inequality follows by taking a count maximizer. A count maximizer separately has all local degrees $d$: replace a bit coordinate of smaller local degree by the AND of two fresh bits. Degree stays at most $d$, every old dependence survives, and the count increases. For any fixed $H$ and any chosen subset $Z$ of coordinates outside $H$, \begin{equation}\label{restrict} \sum_{i\in Z}2^{-d_i(f)}\le\E_\alpha\D_Z(f_\alpha). \end{equation} Here absent coordinates contribute zero. For a single restriction, if $i$ survives both branches its local degrees cannot increase; if it survives only one, its derivative factors as $(1\pm x_j)/2$ times the surviving branch derivative, and its local degree falls by exactly one. In either case its mean weight cannot decrease. Iterate this statement for each $i\in Z$, then sum. In particular $\D(f)\le\sum_{j\in H}2^{-d_j(f)}+\E\D(f_\alpha)$. Let $A$ be ternary of degree at most $k\ge1$, support density $p$, and relevant count $h$. Each nonzero $\Delta_jA=(A_{j=1}-A_{j=-1})/2$ has nonzero magnitude at least $1/2$. The support bound and orthogonality give \begin{equation}\label{energy} \E(\Delta_j A)^2\ge2^{-k-1},\qquad h2^{-k-1}\le\sum_j\E(\Delta_j A)^2 =\sum_S|S|\widehat A(S)^2\le kp. \end{equation} Take a count maximizer $f$, $N=R_d$, $d\ge2$, and choose $i$ of minimum influence. Set $k=d-1$, $M=2^k$, $\lambda=Mp_i$, and let $H$ be the relevant coordinates of $A_i$, $h=|H|$. Then \begin{equation}\label{basic} 1\le\lambda\le dM/N,\qquad h\le\lfloor2k\lambda\rfloor. \end{equation} Write $f=x_iA_i(H)+B(H,Y)$. On active assignments $A_i=\pm1$, Booleanity forces $B=0$ identically in $Y$; the restriction is a signed dictator. Moreover $H$ hits every degree-$d$ monomial: a degree-$d$ coefficient of $B$ missing $H$ survives restriction unchanged, contradicting an active assignment, and a top monomial involving $i$ meets $H$ because $d\ge2$. Thus every restriction has degree at most $d-1$. For $c\ge W_{d-1}$, $c\ge1/2$, the potential inequality gives \begin{equation}\label{oldbudget} N\le2^dc+h+(1-2c)\lambda. \end{equation} We record the exact possible-weight relaxation. Integer-valued functions have integer bit coefficients by finite differences: $a_T=\sum_{U\subseteq T}(-1)^{|T|-|U|}A(1_U)$. Apply this to $(1-f)/2$ and substitute bit $=(1-x)/2$: the sign Fourier coefficients of $f$, hence of $A_i$, lie in $M^{-1}\mathbb Z$. Write them $z_S/M$. Since $\sum z_S=MA_i(\boldsymbol1)$ and $M$ is even, $\sum z_S^2$ is even, giving $\lambda\in(2/M)\mathbb Z$. Reducing the integer bit coefficients of $A_i$ itself modulo two gives its support indicator in $\mathrm{RM}(k,n)$, without squaring it. The published below-twice-minimum weight theorem \cite{kt,carlet} says that $\lambda<2$ must be $2-2^{-j}$, $j\ge0$; dummy inputs ensure $n\ge k+2$. The energy grid excludes $j\ge k$. We therefore allow \begin{equation}\label{weights} \{2-2^{-j}:0\le j3$ and $\sum w_S^2=2^{20}$; determine local degrees from nonzero masks. If $a=\sum_{i:d_i>0}2^{3-d_i}$ and $T=\sum_{|S|=3}w_S^2$, the check for \eqref{envelope} is precisely $(a-8)2^{20}\le2T$. Direct integer character sums on every row pass. The equality data are \[ \begin{array}{c|ccc} \text{catalogue line}&E_3&\D&\text{relevant count, common local degree}\\\hline 757&0&1&4,2\\ 716&1/2&9/8&9,3\\ 700&1&5/4&10,3 \end{array} \] The data and direct checker are pinned in \cite{packet}, \path{r4-exact/sources/F10T6.txt} and \path{r4-exact/verify_catalogue.cpp}. The data SHA-256 is \begin{center}\scriptsize\texttt{d0ca4026c3bfa61f7c40945e4627feecd9264d7854114806c533d64e06836ba0}.\end{center} This is a finite computational proof of the envelope and equality table; completeness of the catalogue remains the published classification premise. Choose a quartic count maximizer. If no derivative has influence $1/8$, \eqref{weights} gives $N\le\lfloor64/3\rfloor=21$. Otherwise choose one with influence $1/8$. Valyuzhenich's minimum-support classification \cite[Theorem 2(2)]{val} describes its controls as disjoint point factors and blocks of size $m\ge2$ supported on the two constant strings. Even blocks have opposite signs, odd blocks equal signs; their degree cost is $m-1$, and a point factor has cost one. Thus $r+\sum(m-1)=3$, $h=r+\sum m\le6$. For $h\le4$, \eqref{oldbudget} gives $N/16\le45/32$, so $N\le22$. The only remaining types are a point and two pairs, a pair and a triple, and three pairs. Let $Q_\alpha(Y)$ be the homogeneous cubic part after fixing $H$. Its coefficients are affine in $H$ and vanish on the active set; it is independent of $i$. Write $e=\E_\alpha\|Q_\alpha\|_2^2$. Apply \eqref{envelope} on inactive assignments and use potential $1/2$ on the active fraction $1/8$. The restriction inequality yields \begin{equation}\label{quarticbudget} N\le h+15+4e. \end{equation} All vector norms below are in the Hilbert space of cubic polynomials in $Y$. For a point and two pairs, normalize the active conditions to $u=1,s=t,v=w$. Affine vanishing gives $Q=(1-u)P_0/2+(s-t)P_1/2+(v-w)P_2/2$. On the full-mismatch slice, averaging $\|Q\|^2\le1$ over independent signs gives $\sum\|P_j\|^2\le1$; the full average is half that sum. Thus $e\le1/2$ and \eqref{quarticbudget} gives $N\le22$. For a pair and a triple, normalize to $s=t,u=v=w$. Then $Q=(s-t)P/2+uC_1+vC_2+wC_3$, with $C_1+C_2+C_3=0$. On nonconstant triple assignments and an unequal pair, averaging signs gives $\|P\|^2+4\|C_j\|^2\le1$ for each $j$. Hence $e=\|P\|^2/2+\sum\|C_j\|^2\le3/4-\|P\|^2/4\le3/4$. This gives $N\le23$. Equality forces $P=0$, $\|C_j\|^2=1/4$. Each $-2C_j$ is then sign Boolean by Parseval on its restriction, but their sum is zero, impossible for three signs. Thus $N\le22$. For three pairs, $Q=\sum_{j=1}^3(s_j-t_j)P_j/2$. Full-mismatch averaging gives $\sum e_j\le1$, $e_j=\|P_j\|^2$, and $e=\sum e_j/2\le1/2$; again $N\le23$. If $N=23$, all inequalities are equalities. All eight full-mismatch norms equal one, so expanding them gives $\langle P_j,P_l\rangle=0$ for $j\ne l$. On a mismatch set $K$ the cubic energy is $\sum_{j\in K}e_j$. The equality table on singleton mismatches forces $e_j\in\{0,1/2,1\}$ and $\sum e_j=1$. If one $e_j=1$, then on $s=-t$ the entire function is $sP_j(Y)$. We use the following pair argument. If a degree-at-most-$d$ Boolean $F$ satisfies $F(s,-s,Z)=s g(Z)$, write $F=U+sV+tW+stT$. Then $U=T$, $V-W=g$, $\deg U\le d-2$, and $h(s,Z)=sF(s,s,Z)=V+W+2sU$ is Boolean of degree at most $d-1$. Every relevant $Z$ occurs in $g$ or $h$, so $r(F)\le2+2R_{d-1}$. Here this is $22$, a contradiction. If two $e_j=1/2$, each singleton mismatch is an equality cubic with nine variables all of local degree three; hence each $P_j$ uses nine $Y$ coordinates. Since $|Y|=23-6-1=16$, they share some $y$. On either single mismatch (probability $1/4$ each), $y$ has local degree three; on a double mismatch it survives for at least half the signs, since its derivatives in $P_1+P_2$ and $P_1-P_2$ cannot both vanish. Its expected weight is therefore at least $(1/4+1/4+1/8)/8=5/64>1/16$, contradicting equality in the coordinatewise restriction inequality. This proves $R_4\le22$. For attainment and later lower bounds use the CHS/Shinkar--Tal family \begin{equation}\label{lower} \Xi_1=x,\qquad \Xi_d=\frac{s+t}{2}\Xi_{d-1}(X)+\frac{s-t}{2}\Xi_{d-1}(Y) \end{equation} on disjoint inputs. Exactly one summand is active. All Fourier masks have size $d$, every variable remains relevant, and $L_1=1,L_d=2L_{d-1}+2$, whence $L_d=3\cdot2^{d-1}-2$. This proves $R_4=22$, the lower counts $46,94$, and $C_{44}\ge3/2$. \section{Equality and the integer energy gap} Two refinements of \eqref{energy} are needed. First, for $k\ge3$, \begin{equation}\label{strictcontrol} \lambda\notin\mathbb Z\quad\Longrightarrow\quad h\le\lceil2k\lambda\rceil-1. \end{equation} Indeed equality in \eqref{energy} makes $A$ homogeneous of degree $k$ and each partial minimum-supported with nonzero magnitude $1/2$. In Valyuzhenich's tensor classification a homogeneous minimum polynomial has only two-coordinate factors: point factors and blocks of size at least three have distinct Fourier degrees that cannot cancel in disjoint products. Each partial therefore has Fourier masks the transversals of $k-1$ disjoint pairs. The pure simplicial complex generated by the $k$-masks of $A$ has every vertex link a cross-polytope boundary. Masuda--Panov's published Lemma 4.7 \cite{mp} implies that each connected component is a cross-polytope boundary, on $2k$ vertices. Hence $h$ is a multiple of $2k$, forcing $\lambda=h/(2k)$ integral. This proves \eqref{strictcontrol}; away from equality the integer ceiling is immediate. \begin{lemma}[Integer energy gap]\label{integergap} For a nonzero integer-valued polynomial $Q$ of real degree at most $r$, $\E Q^2=2^{-r}$ or $\E Q^2\ge(3/2)2^{-r}$. \end{lemma} \begin{proof} We give the lattice proof expanded in the packet, crediting Nebe \cite{nebe}, so the recent gap is not an unexplained premise. On $n$ bit coordinates let $b_T$ be the bit-monomial truth vector. The classical Barnes--Wall bases are $2^{\lfloor|T|/2\rfloor}b_T$ for $\Lambda_n$ and $2^{\lceil|T|/2\rceil}b_T$ for $\Delta_n$; their minimum squared norms are $a_n=2^{n-1}$ and $2a_n$ \cite[Theorems 3.1(ii), 3.2]{bw}. We also use the classical pair-preserving minimum-vector transitivity \cite[Theorem II.4]{be} (with the dimension-eight stabilizer convention specified in \cite[Theorem 2.5]{nebe}). The map $S(u,v)=(u+v,u-v)$ on the last-coordinate branches satisfies $S\Lambda_n=\Delta_n$, $S\Delta_n=2\Lambda_n$, and $S^2=2I$. For a monomial not using the last bit, $S$ sends its equal branches to $2b_T(1-b_n)$; for one using it, $S$ sends $(0,b_T)$ to $b_T-2b_nb_T$. Comparing the floor and ceiling exponents proves both containments, and $S^2=2I$ proves equality. Thus the normalized norm spectra agree. The bases likewise give \[ \Lambda_{n+1}=\{(\ell+\delta,\ell):\ell\in\Lambda_n, \ \delta\in\Delta_n\}. \] A minimum-vector coset $\ell+\Delta_n$ has no norm strictly between $a_n$ and $2a_n$. For odd $n$, transitivity permits $\ell=2^{(n-1)/2}e_0$. If $x=\ell+\delta$ has largest common coordinate power $2^j$ with $j\ge(n-1)/2$, norm divisibility proves the claim. Otherwise the odd positions of $x/2^j$ equal those of $\delta/2^j$. The bit-coefficient divisibilities in $\Delta_n$ make this nonzero binary residue have degree at most $2j$. Its support has at least $2^{n-2j}$ positions, so $\|x\|^2\ge2^{2j}2^{n-2j}=2a_n$. The binary support bound follows by branching induction: a zero branch costs one degree, and two nonzero branches add their lower weights. For even $n$ use $(\Delta_n,2\Lambda_n)$ and minimum vector $2^{n/2}e_0$. The corresponding residue degree is at most $2j-1$, giving norm at least $2^{n+1}$; $j=0$ would make the residue zero. Transport by $S$ gives the same coset claim for $(\Lambda_n,\Delta_n)$. Every $\Lambda_n$ norm is divisible by $2^{\lceil n/2\rceil}$ for $n\ge2$: basis-vector norms are $2^n$ or $2^{n-1}$, and twice the inner product for $T,U$ has exponent $1+\lfloor|T|/2\rfloor+\lfloor|U|/2\rfloor+n-|T\cup U| \ge\lceil n/2\rceil$ (split at $|T|+|U|=n$). For $2\le n\le5$ this gives the $3/2$ gap; $n=1$ is the integer lattice. Induct in the displayed recursion, ordering the branches by norm. If the smaller branch is zero, use the gap for $\Delta_n$; if nonminimal, both branches have norm at least $3a_n/2$; if minimal, the coset claim makes the larger branch minimal or of norm at least $2a_n$. Thus any total norm above $2a_n$ is at least $3a_n$, proving the gap. Finally append dummy inputs so $n\ge\max(r,2)$. Integer bit coefficients put $2^{(r-1)/2}Q$ in $\Lambda_n$ for odd $r$, and $2^{r/2}Q$ in $\Delta_n$ for even $r$. In both cases its squared norm divided by the lattice minimum is $2^r\E Q^2$. The norm gap proves the lemma. \end{proof} For a relevant partial of our ternary $A$, apply the lemma to $Q=2\Delta_jA$ of degree at most $k-1$. With $e_0=2^{-k-1}$, \begin{equation}\label{partialgap} \E(\Delta_jA)^2=e_0\quad\text{or}\quad \E(\Delta_jA)^2\ge3e_0/2. \end{equation} Since total derivative energy is at most $2k\lambda e_0$, the condition $3h>4k\lambda$ forces a minimum partial. Equality in its support and integer-magnitude bounds makes its values $0,\pm1/2$. \section{Boolean completion and the next two degrees} Restrict $H$ and apply \eqref{restrict} only to $Y$. Active branches have no $Y$ dependence; inactive branches have degree at most $d-1$ and potential at most $c$. Since $|Y|=N-h-1$, this gives the improved budget \begin{equation}\label{ordinary} N\le O_d(h,\lambda):=2^dc+h+1-2c\lambda. \end{equation} Now assume $d\ge4$, $\lambda>1$ and $3h>4k\lambda$; choose a minimum partial $j$ from \eqref{partialgap}. Among $j$-edges indexed by $H\setminus\{j\}$, let $q$ be the fraction with one active endpoint and $r$ the fraction with two. Double-active signs agree, since the partial never has magnitude one. Thus $q=4e_0=2^{1-k}$ and $r=p-q/2=(\lambda-1)2^{-k}>0$. An affine function vanishing on the active set therefore has zero $x_j$ coefficient, by subtracting its values on a double-active edge. Set $K=H\setminus\{j\}$. A degree-$d$ monomial of $B$ missing $K$ has $Y$-size $d$ or $d-1$. Its constant or affine control coefficient vanishes on the active set; the preceding edge argument kills its possible $j$ coefficient. An $x_iA$ term missing $K$ uses only $i,j$ and has degree at most two. Hence every restriction to $K$ still has degree at most $d-1$. For a section on $(i,j,Y)$, no active endpoint gives no $i$ dependence and $\D_{j,Y}\le c$; two active endpoints give the dictator $\pm i$ and $\D_{j,Y}=0$. On a one-active edge change signs to write $F=(1+j)i/2+(1-j)g(Y)/2$. If $t=\deg g\ge1$, then $t\le d-2$ because the top $j$-multiple cannot cancel, and $\D_{j,Y}(F)=2^{-t-1}+\D(g)/2$. For constant $g$, it equals $1/4$. With already proved caps $c_t\ge W_t$ put \begin{equation}\label{bdef} b_d=\max\left(\frac12,\max_{1\le t\le d-2} \left(\frac14+2^{-t-1}+\frac{c_t}{2}\right)\right). \end{equation} Apply \eqref{restrict} to $\{j\}\cup Y$, of size $N-h$. The mean section potential is at most $c(1-q-r)+(b_d-1/4)q$. Using $2^dq=4$ gives the second budget \begin{equation}\label{retained} N\le T_d(h,\lambda):=2^dc+h-2c\lambda +4\left(b_d-\frac{c+1/2}{2}\right). \end{equation} At degree five $c=11/8,b_5=15/16$. For $\lambda=1$, $h\le8$ and \eqref{ordinary} gives $201/4$, hence $N\le50$. For $\lambda=3/2$, \eqref{strictcontrol} gives $h\le11$. For $h\le8$ use \eqref{ordinary}, at most $391/8$; for $9\le h\le11$ use \eqref{retained}, at most $407/8$. Both imply $N\le50$. All remaining allowed weights are at least $7/4$, so \eqref{basic} gives $N\le\lfloor80/(7/4)\rfloor=45$. Thus $R_5\le50$ and $c_5=25/16$. At degree six $b_6=31/32$ and the correction in \eqref{retained} is $-1/4$. For $\lambda=1$ the ordinary maximum is $863/8$. For $\lambda=3/2$, $h\le14$; the maxima for $h\le10$ and $11\le h\le14$ are $1701/16$ and $1745/16$. All imply $N\le109$. For $\lambda\ge7/4$, the influence bound gives $N\le\lfloor192/(7/4)\rfloor=109$. Hence $R_6\le109$. \section{Exact propagation and the infinite tail} Start with $c_1=1/2,c_2=1,c_3=5/4,c_4=11/8$; the first two are exact by \eqref{influence} and \eqref{lower}. At each $d\ge5$ use only earlier caps, $c=c_{d-1}$ and \eqref{bdef}. For a candidate excluded count $n$, test the weights \eqref{weights} in $[1,d2^{d-1}/n]$. Let $h_{\max}=\lfloor2k\lambda\rfloor$, demoted by one when $\lambda$ is nonintegral and $2k\lambda$ integral, as required by \eqref{strictcontrol}. At each integer $0\le h\le h_{\max}$ use $O_d$ and, when $\lambda>1,3h>4k\lambda$, also $T_d$. Take the smaller applicable budget and maximize over $h$. Both ranges increase with slope one in $h$, so it suffices to test $h_{\max}$ and $\min(h_{\max},\lfloor4k\lambda/3\rfloor)$. If the maximum is strictly below $n$ for every weight, all $N\ge n$ are excluded by \eqref{basic}; set $c_d=(n-1)/2^d$ and apply the CHS identity. Here is the complete finite cover, avoiding an exponentially long grid. Write $G=2^{d-2}$, $\lambda=z/G$. Below two enumerate \eqref{weights}. At or above two test $z=2G$, each cell start $\lceil aG/(2k)\rceil$ and its next integer, and each start $\lceil3aG/(4k)\rceil$ of a cell for $\lfloor4kz/(3G)\rfloor$. Retain indices in the tested interval. These are all joint floor changes; the extra next point covers a demoted equality endpoint. Between changes each applicable budget has slope $-2c$ in $\lambda$, hence is nonincreasing. No untested interior point can be worse. The exact certificate \path{round8/completion_certificate.json} in \cite{packet} records every exclusion through degree $96$. The two supplied checkers, replayed for this version, agree in both modes. One checks rational next-count exclusions; the other derives integer maxima before comparing with the certificate and scans the full grid through degree $16$. Selected ceiling rows in the simple mode are \[ \begin{array}{c|rrrrrrrr} d&5&6&7&8&9&10&11&12\\\hline 2^dc_d&50&109&233&487&1001&2037&4117&8287 \end{array} \] They verify nondecreasing $c_d\le d/2$ and obtain exactly \[ c_{96}=\frac{80880411492166173705437331353}{39614081257132168796771975168}. \] Feasible scalar points stored with the rows are not Boolean witnesses; only the strict exclusions are needed for this upper bound. To cover all larger degrees, \eqref{lower} gives $R_d\ge2^d$ for $d\ge2$, so its least-influence derivative has $\lambda\le d/2$. Apply the weaker \eqref{oldbudget} with the actual $c=W_{d-1}$ and $h\le2(d-1)\lambda$. Since $1/2\le W_{d-1}\le(d-1)/2$, \[ W_d\le W_{d-1}+\frac{\lambda(2d-1-2W_{d-1})}{2^d} \le W_{d-1}+\frac{d(d-1)}{2^d}. \] Differentiating the geometric series gives $\sum_{j\ge1}(1,j,j^2)2^{-j}=(1,2,6)$; expanding $(H+j)(H+j-1)$ yields \[ \sum_{d>H}\frac{d(d-1)}{2^d}=\frac{H^2+3H+4}{2^H}. \] The nondecreasing finite caps cover every $d\le96$. Telescoping beyond $96$ adds $9508/2^{96}$ to $c_{96}$, exactly the rational in Theorem~\ref{main}. Its comparison with the printed decimal is a strict rational comparison, rounded upward. This completes the argument. \section{Evidence and disclosure} All code, catalogue data and certificates are linked at the single pinned revision in \cite{base,packet}; Hexagon carries this TeX note. The finite calculation supports the written proof and its named published dependencies; it does not independently prove catalogue completeness or the classical lattice group theorem. The packet includes AI reviews of each stage. This version replayed the two scalar checkers and direct cubic character-sum checker; it did not rerun the historical large exhaustive controls or searches. This is an unrefereed preprint. There has been no independent human review and no formalization. The programs check finite arithmetic, not all prose or the cited external theorems. No exact value of $R_5,R_6$, or $C_{44}$ is claimed. The short derivative recursion may already exist; the source search was bounded, and expert scrutiny is requested. The arguments were found and checked by AI agents under Ricardo Pinho's direction. Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared certificates, checking software, and publication materials. No line-by-line human verification is claimed. \begin{thebibliography}{99} \raggedright \bibitem{bw} E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, \emph{J. Austral. Math. Soc.} 1 (1959), 47--63; \href{https://doi.org/10.1017/S1446788700025064}{doi:10.1017/S1446788700025064}. \bibitem{be} M. Brou\'e and M. Enguehard, Une famille infinie de formes quadratiques enti\`eres; leurs groupes d'automorphismes, \emph{Ann. Sci. \'Ecole Norm. 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Rasoolzadeh, Classification of all $t$-resilient Boolean functions with $t+4$ variables, \emph{IACR Trans. Symmetric Cryptol.} 2023(3), 213--226; \href{https://doi.org/10.46586/tosc.v2023.i3.213-226}{doi:10.46586/tosc.v2023.i3.213-226}. \bibitem{val} A. Valyuzhenich, Optimal functions with spectral constraints in hypercubes, \emph{Discrete Math.} 347 (2024), 113816, Theorem 2(2); \url{https://arxiv.org/abs/2303.10995}. \bibitem{wellens} J. Wellens, Relationships between the number of inputs and other complexity measures of Boolean functions, \emph{Discrete Analysis} 2022:20; \url{https://discreteanalysisjournal.com/article/57741}. \bibitem{epi} BakeLab, \emph{\'Epi-results}, exact Wellens certificates, revision \texttt{97fa6a420f417458e11723c8bceb70f1529e447a}; \href{https://github.com/rpinho/math-certificates/tree/36325fade4b73afb787c37ee8cc4c90ae4adf856/c44/reed-muller-caps/baseline}{Retained data at the pinned math-certificates revision}. \bibitem{base} R. Pinho, \emph{Math certificates}, revision \texttt{36325fade4b73afb787c37ee8cc4c90ae4adf856}, derivative-control base; \href{https://github.com/rpinho/math-certificates/tree/36325fade4b73afb787c37ee8cc4c90ae4adf856/c44/reed-muller-caps}{\texttt{c44/reed-muller-caps}}. \bibitem{packet} R. Pinho, \emph{Math certificates}, same revision, quartic proof and completion packet; \href{https://github.com/rpinho/math-certificates/tree/36325fade4b73afb787c37ee8cc4c90ae4adf856/c44/small-degree-classification}{\texttt{c44/small-degree-classification}}. \bibitem{pr} R. Pinho, Tao optimization table, PR 294 (open, unrefereed; read 2 November 2026); \url{https://github.com/teorth/optimizationproblems/pull/294}. \end{thebibliography} \end{document}