FPRD-IC-O01 · open problem

Can any formula beat the product of twos?

The integer complexity $\lVert n\rVert$is the least number of occurrences of 1 in an expression for$n$ using only addition, multiplication, and parentheses. Since

one always has $\lVert2^n\rVert\le2n$. Selfridge's question is whether equality always holds. It remains open; see the current general estimates of Konyagin and Oganesyan and the stability framework of Altman and Arias de Reyna.

FPRD-IC-D01 · restricted grammar

A three-addition fork that preserves factored cost

The active family has the form

where every displayed operand is$(2,3)$-smooth. The paid atoms 1, 2, and 3 cost 1, 2, and 3 ones. Multiplication is free once those atoms have been built, but all three displayed additions are real operations; there is no reuse, subtraction, or division. The exact displayed cost retains the special charge for a unit operand.

Common factors of $(B,C)$ and$(D,E)$ can be transferred to$A$ without increasing cost. A common exterior power of two can likewise be transferred to$M$. A potentially cheaper expression may therefore be assumed primitive with

The fork is called proper when both inner sums are nonsmooth. Smooth inner sums can be rebuilt at no greater cost and reduce to the already classified two-addition family.

FPRD-IC-T01 · normalization theorem

Only three final valuations survive

For a coprime smooth pair, the 2-adic valuation of its sum is 0, 1, or 2. Equality of the two summands' 2-adic valuations in a sum equal to a power of two first gives$f\le4$. A complete periodic analysis of the cases $f=3,4$leaves six residue rows, all contradicted modulo 9, 27, 64, 81, or 271. Hence every normalized primitive proper fork satisfies

There is also a purely elementary mod-eight obstruction. For odd$x$, put$\chi(x)=+1$ on residues 1 and 3 and $\chi(x)=-1$ on residues 5 and 7. If $P_0,Q_0$ are the odd parts of the two nonsmooth inner sums, then

Thus equal nonsmooth factors and every same-character family are impossible before any size estimate is used.

FPRD-IC-T02 · closed infinite branch

The cyclotomic ladder has no nonsmooth rung

One $f=0$ branch reduces to

Exact valuation and character constraints reduce the ladder to five fixed exponential equations and one periodic family. The fixed equations fail modulo 163, 37, 7, 757, and 271. In the last family, a complete residue certificate modulo 7681 forces its parameter to be odd, while a complete certificate modulo 8641 forces it to be even. Therefore no canonical nonsmooth solution remains for $r\ge2$.

The boundary reductions inherit one use of Bennett's theorem on Pillai-type equations. The final six modular contradictions are independently checkable finite quotients, not bounded searches in the exponents.

FPRD-IC-T03 · closed odd branch

The odd-exponent branch has one harmless control

After the cyclotomic boundary, the two nonsmooth canonical shapes at odd $N$ can be classified completely. Their unique solution is

The two relevant factored presentations cost 20 and 21 ones, respectively $2N+2$ and$2N+3$. Neither threatens the conjectured $2N$ minimum. The proof combines multiplicative-order certificates modulo 7, 19, 27, and 73 with Zsigmondy's primitive-divisor theorem on one boundary. Consequently the complete odd-$N$ portion of this normalized branch satisfies the desired cost inequality.

FPRD-IC-T04 · intermediate even-layer reduction

A zero-exponent identity first appeared to lift remotely

In the first even layer, $a=1$, the remaining exceptional equation is

It is a lift of the genuine boundary identity$15\cdot17+1=2^8$. A complete cover modulo 5, 7, 13, 17, 19, 37, and 41 leaves exactly

Expanding the equation gives the exact identity

The resulting simple Hensel root modulo$3^{140}$ proves that any positive solution must satisfy

This enormous lower bound was not a contradiction. It is retained because it records the exact input to the next theorem; the lift is no longer an active survivor.

FPRD-IC-T05 · closed exceptional lift

The first Hensel digit and nine residues close x=4

Put $N=8+3240z$. Dividing the inherited valuation identity by$3^5$ and reducing its first Hensel digit modulo three gives

Write $v=576r$ and$y=144s$. Modulo 19441,

Hence each power of three has only three possible residues. The nine possible left sides of the exceptional equation are

None is 15812. Therefore the entire positive$x=4$ lift is empty. Primality of 19441 is not required; the displayed period identities suffice. Moreover, every hypothetical point on this branch would already have cost at most $2N-1$, so this is a genuine restricted counterexample family that has been ruled out, not merely a formal Diophantine curiosity.

FPRD-IC-T06 · intermediate large-x reduction

The large-x gates make every surviving point cost-threatening

Two families remain at the first even layer. Their 2-adic gates are

Exact compatible lifts at the least surviving precisions give

These are unbounded consequences of the gates, not search cutoffs. The defining equations also imply the desired strict cost saving automatically at such sizes. Thus every remaining equation solution would be an actual counterexample inside the normalized grammar; the problem is now existence rather than cost accounting.

There is a first exact reduction in II2. If$y<v$ and$y\ge5$, then$v_3(2^N-13)=y+1$ forces$N\equiv332\pmod{486}$. Combining that class with the periods modulo 19441 eliminates 17 of the 28 inherited classes for$x\pmod{540}$. The surviving necessary classes are

These eleven classes were only an intermediate quotient. The next theorem closes all of them, the other seventeen classes, the small values $y=2,3,4$, and both relative orders of $v,y$.

FPRD-IC-T07 · complete type-II2 closure

The entire first-layer type-II2 family is empty

Consider the remaining large-$x$ equation

under the inherited type-II2 conditions. Every allowed$x$ is 1 modulo 3, while$N\equiv8\pmod{18}$. A complete reduction modulo 7 forces $3\mid y$, so $y=2,4$ are impossible.

If $y=3$, the exact 3-adic valuation fixes $N\equiv8\pmod9$, and the 2-adic gate forces $v\equiv0\pmod4$. Modulo 73, the nine possible left sides are

whereas the right side is 37. Thus$y=3$ is impossible as well.

For $y\ge5$, expanding gives

Both terms are divisible by $3^6$, independently of whether $y<v$,$y=v$, or$y>v$. Hence$N\equiv332\pmod{486}$. Exact finite quotients reduce the 28 allowed$x\pmod{540}$ classes as

Together with the earlier $x=4$theorem, this closes the complete $a=1$type-II2 branch. The next theorem closes type-II1; exterior layers $a\ge2$ remain open.

FPRD-IC-T08 · complete first-layer closure

Two small moduli close the last a=1 family

The sole remaining equation was

with odd $v$,$N\equiv16\pmod{18}$, and sixteen inherited $x$classes modulo 270, all 2 modulo 6. Modulo 7 the complete quotient is

The required residue 2 occurs only at$(v,y)\equiv(3,0)\pmod6$. Hence $3^v\equiv3^y\equiv1\pmod{13}$. The left side modulo 13 is then in$\{0,7\}$, while the allowed powers $2^N$ are$\{3,10\}$. The sets are disjoint.

Therefore type-II1 is empty. With the earlier canonical closures, the complete $a=1$layer of the active branch is closed. The unused 2-adic gate independently raises the hypothetical lower bound to$v\ge137494267053$, but the modulus-13 contradiction makes that bound only a control.

FPRD-IC-B01 · next-layer method boundary

The coarse a=2 frontier defeats coprime-modulus covers

The first serious next-layer survivor is

Modulo 7 and 13 first force the coarse classes

The equation also has a formal rational boundary at$(v,y,N)=(-1,0,6)$. For any finite collection of moduli coprime to 6, take these three exponents modulo the corresponding multiplicative orders. The formal boundary then satisfies every chosen coarse congruence simultaneously. This blocks a direct repetition of the first-layer prime-cover method. It does not block the sharper 3-adic refinement used next.

FPRD-IC-T09 · first a=2 closure

A 3-adic lift breaks the formal boundary

Subtracting the boundary value 64 gives

On the coarse classes, the bracket has 3-adic valuation one. LTE therefore gives $v_3(N-6)=2$, or equivalently

This excludes the formal boundary from the admissible progression. Finally,$\operatorname{ord}_{271}(2)=135$and $\operatorname{ord}_{271}(3)=30$. The allowed exponent classes produce 25 distinct left residues and 10 right residues modulo 271, and those sets are disjoint. Hence the first serious $a=2$type-I2 family has no solution.

FPRD-IC-T10 · complete second-layer closure

Moduli 73 and 577 close the last a=2 family

The remaining equation was

with positive $y$ and odd$v$. The second exact gate is

Modulo 27 first gives $N\equiv6\pmod{18}$. The complete modulus-73 quotient has six rows; the primary gate removes three and leaves

These classes also give$v_3(2^N-37)=\min(v,y)+2$and $N\equiv96\pmod{162}$.

For the final quotient, modulo 577 one has$3=2^{105}$ and$\operatorname{ord}_{577}(2)=144$. Set $t=x+105y$. The three surviving classes require $t\equiv1,4,7\pmod9$. Solving the remaining equation for $2^t$gives the complete logarithm table

A dash means the quotient is not a power of two modulo 577. The seven existing logarithms occupy only$\{2,3,5,6\}\pmod9$, disjoint from the required classes. Therefore this family is empty. Since the preceding results exhaust the other canonical forms, the complete $a=2$ layer of the active branch is closed.

FPRD-IC-T11 · third-layer entry theorem

The complete a=3 entry table is empty through x=20

The third exterior layer has the four canonical equations

Modulo 81 gives the exact progressions$N\equiv18\pmod{54}$ when$P\equiv1\pmod3$, and$N\equiv36\pmod{54}$ when$P\equiv2\pmod3$. The two$x=1$ families fall modulo 13 and 73. Every entry with $x\ge3$obeys the exact second gate

For the remaining low-bit roots, none of the possible values of$P$ vanishes modulo 73. Since $2^N\equiv1\pmod{73}$, the complete quotient forces

Thus $3\mid x$. Exact gate-residue covers at the remaining values give

Each cell lists a complete successive cover, not a bounded exponent search. At $x=18$, the I2 gate quotient contracts$262144\to8192\to512\to0$; each II quotient contracts$65536\to2048\to0$. Consequently every survivor satisfies

and the first 2-adic gate forces$v\ge132267$ in I2,$v\ge457519$ in II1, and$v\ge736154$ in II2. These values exceed the exact cost-saving thresholds, so every remaining solution would genuinely beat$2N$ within this grammar. The next theorem closes all of these remote classes.

FPRD-IC-T12 · complete third layer

The remote a=3 tail is empty

Let $p=262657$. It is prime, and its relevant orders are

Using the inherited classes for $x,y,v,N$, the complete quotient modulo $p$checks 623,808 residue triples for each surviving negative-factor shape. It leaves only

Six rows fail modulo 7. For the other two, modulo 13 the possible left residues are $\{0,4,7,11\}$and $\{3,5,6,8\}$, while the right side lies in $\{1,12\}$. Hence no remote row survives, and the complete$a=3$ positive-type-I layer is empty.

FPRD-IC-T13 · uniform exterior obstruction

Two primes eliminate every layer a at least four

An odd $N$ would force$a=0$. In the even branch,$a=1+v_3(N)$. Thus$a\ge4$ implies$27\mid N$. Modulo 73 and 262657 the right side is therefore one. Every canonical negative factor is nonzero at both primes, so each prime forces the positive factor $Q$ to vanish.

The exponent moduli in the two columns are$(9,12)$ and$(27,14592)$. Projecting the second column to the first gives no common row for either form of$Q$. This excludes all$a\ge4$ at once; no exterior-layer induction remains.

FPRD-IC-T14 · branch synthesis

The complete g=0 positive-type-I branch is closed

The cyclotomic and odd-exponent theorems handle$a=0$. The complete first, second, and third-layer theorems handle$a=1,2,3$, and the paired-prime obstruction handles every $a\ge4$. Therefore no normalized primitive proper fork in this entire positive-type-I branch has displayed cost below$2N$. This completes one infinite branch of the three-addition grammar, not the grammar itself.

FPRD-IC-T15 · complementary third-layer closure

The positive-type-II a=3 layer is empty

For the complementary positive factor, the remaining low-layer equation is

At $a=3$, the exact 2-adic gate quotient modulo $2^{11}$contains 524,288 admissible states. Solution-preserving filters give the complete chain

Each filter deliberately allows independent exponent lifts, so it only enlarges the genuine solution set. The empty intersection therefore proves that the complete complementary$a=3$ layer has no solution.

FPRD-IC-T16 · uniform high-x obstruction

Every remaining low-layer solution has x at most ten

The exact gates are

Independent prime filters leave false high-$x$ states because they may choose different exponent lifts. Coupling the same$x,N$ residues at 17, 193, 257, and 12289 leaves only

Here the coordinates after the shape are$(v,y,x,N)$. Each row has 81 simultaneous lifts to periods$(1536,1536,18432,18432)$. None satisfies the equation modulo all of 7, 13, 73, 97, and 577. Hence every solution at$a=1,2,3$ has$x\le10$.

FPRD-IC-T17 · exact residual quotient

The complementary low layers reduce to twenty-one lift classes

A coupled exact quotient leaves only$x\in\{1,3,4,5,6,8\}$ and the following rows. The columns $v,y,N$are residues modulo 1536, 1536, and 18432.

These are necessary lift classes, not solutions. A zero residue also allows positive exponents such as$y=1536$. A bounded search through $v,y\le384$ finds ten control presentations, representing four factorizations, all with displayed cost at least$2N+4$. The table was the exact finite input to the later second-gate classification below; its nonzero lifts are no longer open.

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FPRD-IC-T18 · shared modular lift

Five rows lock to two boundary factorizations

Five rows in the residual table share$a=2,\ x=6,\ N\equiv12\pmod{18432}$. Applying simultaneous exponent periods at the twelve primes

leaves exactly one lift of each row. Every solution therefore has

The base residues encode only$9\cdot7\cdot65+1=2^{12}$ and$9\cdot5\cdot91+1=2^{12}$. No single-prime search through five million eliminated any row; the useful obstruction is the shared lift, not an isolated lucky modulus.

This lock remains an exact intermediate theorem. The full valuation gate in FPRD-IC-T21 now excludes every nonboundary lift in all five rows.

FPRD-IC-T19 · certified height barrier

Every non-boundary lift begins beyond 26 quadrillion

Put $k=r+s$. Exact logarithmic normalization turns every non-boundary solution into the one-sided approximation

Rational atanh-series bounds certify the logarithms. An exact integer sweep through $k=737{,}837{,}167$finds 111 broad one-sided candidates. All 111 miss the fourteen row-specific correction limits by at least 0.0020764, against a correction error below $10^{-6}$. Hence every non-boundary solution satisfies

This conditional height theorem remains correct, but the full valuation gate below now excludes every non-boundary lift in the cluster. The logarithmic sweep is therefore superseded as a frontier and should not be extended.

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FPRD-IC-T20 · full 2-adic gate

The residual rows are valuation-rigid or one-dimensional

Put$F(v,y)=3^aP(v)(2^x+3^y)+1$. The earlier gate determined $x$, but equality with a power of two also requires

In eleven rows this valuation is fixed at the displayed base exponent, immediately excluding every nonboundary lift. In each of the other ten rows, solutions modulo$2^e$ form a one-dimensional Hensel graph. With$L_e=3\cdot2^{e-2}$, there are exactly $2^{e-11}$ states modulo $(v,y)=(L_e,L_e)$, and each $v$ residue determines a unique $y$ residue.

The lifting mechanism is elementary. LTE gives$v_2(3^{L_e}-1)=e$; shifting$y$ by$L_e$ therefore toggles the next binary digit, so precisely one of its two lifts works for each next $v$ digit.

FPRD-IC-T21 · complete restricted classification

The positive-type-II even tower has only ten controls

At $e=22$, each nonrigid row has 2,048 Hensel states modulo$L_{22}=3{,}145{,}728$. The prime 65,537 is aligned with both remaining exponent periods:

None of the 20,480 states satisfies the equation modulo 65,537. The 21-row quotient therefore contains exactly ten canonical proper solutions, representing four factorizations:

Their minimum displayed excess is four ones. Combined with the retained $a=3$ and uniform$a\ge4$ exclusions, this classifies the complete positive-type-II tower with$a\ge1$ and proves that it cannot beat $2N$.

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FPRD-IC-T22 · complementary-orientation gate

Quadratic reciprocity leaves six of twelve residue pairings

In the complementary orientation, division by the common final power of two gives

where $R$ is the negative-character odd factor and$S$ the positive-character factor. Canonical smooth-operand sums give$R\bmod24\in\{5,7,13\}$ and$S\bmod24\in\{1,11,17,19\}$. Reducing the equation modulo each factor and taking Jacobi symbols yields

These identities and the equation modulo 24 leave exactly

This is an unbounded arithmetic gate: the six omitted canonical pairings cannot occur at any exponent height.

FPRD-IC-T23 · exact subbranch exclusion

The double-even complementary branch is empty

If $f=2$ is split as one factor of two from each inner sum, both sums have the form$1+3^b$ with even exponent. Their odd parts satisfy

Opposite character forces one residue of each kind, hence the pair $(5,17)$. That pair is forbidden by FPRD-IC-T22, proving that the entire infinite subbranch has no solution.

The same gate compresses $f=1$to three residue rows. For the two rows whose negative factor is even, divisibility by 5 adds$K\equiv3g\pmod4$. These are the input to the valuation theorem below.

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FPRD-IC-T24 · exact complementary valuation theorem

Tied minima cannot create an exceptional f=1 lift

Let the unique even inner sum be$1+3^b$, with$b$ positive and even, and write the other odd factor as$W=2^h+2^q3^y$. The reduced equation is

After multiplying by two, removing the constant$2^h$, and applying LTE,

Ordinarily the right side has valuation$\min(b,y,g)$. Canonical parity reduces every possible cancellation of its lowest 3-adic digit to six families. Complete exponent periods modulo 7 and 73 have empty joint survivor sets in all six; the only minimum-one exception is checked against its three possible mod-73 targets separately. Therefore every actual solution in the three $f=1$ rows obeys

Thus tied minima produce no exceptional lift graph: the least ternary exponent is always logarithmically small in the output exponent. This controls the branch but does not prove that every$f=1$ row is empty.

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FPRD-IC-T25 · exact complementary valuation theorem

The remaining f=2 split has no exceptional tied-minimum lift

In the split where one inner sum contributes both factors of two, put $V_b=(1+3^b)/4$, where$b\ge3$ is odd, and write the other odd factor as$W=2^h+2^q3^y$. The reduced equation is

Multiplying by four, removing the constant term$2^h$, and applying LTE gives

The four possible residues of $V_b$modulo 24 leave eight placements in the quadratic gate. Canonical parity then reduces every possible cancellation at the least ternary exponent to ten families. Complete periods modulo 7 and 73 eliminate nine. The tenth leaves one class; it has 343 survivors modulo 487 and four modulo 2593, but no pair satisfies the coordinatewise CRT compatibility conditions. The sole minimum-one boundary also has no mod-73 survivor. Therefore every actual solution in this split obeys

Equivalently,$2\cdot3^{\min(b,y,g)-1}\mid K+2-h$. The least ternary exponent is logarithmically small in the output exponent, but the valuation law does not by itself make the split empty.

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FPRD-IC-T26 · exact modular lift classification

The first two complementary minimum layers lock to forty periodic graphs

The two valuation theorems have the uniform form

At $m=1$ and$m=2$, this fixes$K+f-h$ modulo 18. Keeping the exact coordinates attaining the minimum produces 32 canonical strata. A complete sequence of CRT-compatible order-modulus quotients proves that nine are empty and that all states in the other 23 lie on exactly 40 lift graphs. Their common coordinate period is

Centering the graph coordinates modulo$P$ gives seventeen distinct proper presentations on 29 graphs and nine distinct improper rational boundary identities on eleven graphs. The proper roots are harmless: their displayed costs exceed$2N$ by at least six ones. Any other proper solution must move at least one exponent by a full period; the defining equation then gives

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FPRD-IC-T27 · complete complementary-layer classification

The forty lift graphs contain exactly seventeen proper presentations

The next ternary digit closes the gap left by the lift lock. Because the order of two modulo$3^6=729$ is 486 and divides the common graph period, each ternary coordinate is either its small centered value or contributes zero modulo 729. Exhausting these low/high patterns eliminates all eleven graphs rooted at improper rational identities. It also fixes every ternary coordinate on 26 of the 29 control-rooted graphs; a direct 2-adic divisibility check then fixes their binary coordinate.

The three residual graphs reduce to only two equations:

Here $b$ or$g$ is congruent to six modulo $20{,}652{,}025{,}680$. Matveev's explicit lower bound for rational linear forms in logarithms gives$b,g<10^{15}$. Exact rational bounds on $\log_2 3$ leave 117,363 possible lift-index pairs for each equation. Successive exact reductions modulo 29, 43, 59, 83, and 101 leave none.

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FPRD-IC-T28 · classical S-unit corollary

The normalized complementary family is finite—but the theorem gives no height cutoff

After multiplying away the fixed factor$2^f$, every complementary presentation has the form

Expansion and division by the right side makes five positive$\{2,3\}$-units sum to one. Their multiplicative group has rank at most six, and positivity makes every solution nondegenerate. The Evertse–Schlickewei–Schmidt theoremtherefore bounds each fixed factor-shape pattern by$\exp(7\cdot30^{15})$solutions. Across the eight patterns under study,

This established theorem bounds a number of solutions, not their exponent heights. It therefore proves finiteness but does not supply the uniform Matveev cutoff needed to compare directly with the growing 3-adic period. That distinction redirects the attack to exact finite quotients in the unopened$f=0$ branch.

FPRD-IC-T29 · exact complementary f=0 classification

The two triple-minimum cancellation strata contain one harmless control

For $RS+3^g=2^K$, with each factor of type $1+2^x3^v$ or$2^x+3^v$, the quadratic gate leaves forty ordered canonical placements. The first ternary digit permits twenty tied-minimum cancellations. Only two can have $v=y=g=m$.

The first is

with $x,K$ even and$m$ odd. Its complete solution tables modulo 7 and 73 have no CRT-compatible pair. The second family has odd $x,m,K$:

Reduction modulo $2^x$ and$v_2(1+3^m)=2$ force$x=3$. With$z=3^m$, completing the square gives

The two positive factor pairs of 65 leave exactly$(m,x,K)=(1,3,7)$:

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FPRD-IC-T30 · complete complementary f=0 cancellation closure

Every exceptional lowest-digit cancellation is now classified

The remaining eighteen strata have exactly two of$v,y,g$ equal to the least ternary exponent $m$. The squarefree quotient

makes seven families empty and reduces the other eleven to 23 complete states modulo$(36,12,12,36)$. Modulo 27, an exponent congruent to one or two modulo 12 can be its exact low value or a high value whose power of three vanishes; every other positive representative is necessarily high. Exhausting the feasible low/high patterns leaves one state:

with $m=1$ exactly. Reduction modulo $2^x$ gives$2^x\mid7+3^g$. Since$4\mid g$, the latter number has exact 2-adic valuation three, so $x=3$. The equation becomes

Its factor pairs $(1,175),(5,35),(7,25)$leave only $(K,g)=(8,4)$. Hence the entire pair-minimum cancellation locus contains the single control

Together with FPRD-IC-T29, this classifies all twenty possible first-digit cancellations. If$2^r$ and$2^u$ are the constant binary terms of the two factors, every other$f=0$ solution therefore obeys

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Evidence and reproducibility

Normalization and thin-branch archive · Cyclotomic closure archive · Odd-branch archive · Even-layer archive · Exceptional-lift closure and large-x archive · Complete type-II2 closure archive · Complete first-layer closure archive · First-layer and first a=2 closure archive · Complete second-layer closure archive · Third-layer entry-table archive · Positive-type-I and high-layer closure archive · a=2, x=6 boundary-lock archive

Sources and dependency boundary

• Sergei Konyagin and Kristina Oganesyan, Upper and lower estimates for integer complexity (2026): current general context for the open problem.
• Harry Altman and Juan Arias de Reyna, Integer complexity: Stability and self-similarity: background on defects and stability.
• Michael A. Bennett, On Some Exponential Equations of S. S. Pillai (2001): imported classification for specified boundary equations.
• Karl Zsigmondy, Zur Theorie der Potenzreste (1892): primitive divisors on one odd-branch boundary.
• Bajpai and Bennett, Effective S-unit Equations Beyond 3 Terms: a methodological comparator. Its five-term theorem does not directly close the full six-term expansion of the fork.
• Jan-Hendrik Evertse, Hans Peter Schlickewei, and Wolfgang Schmidt, Linear equations in variables which lie in a multiplicative group (2002): the imported uniform bound for nondegenerate S-unit equations.
• E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II (2000): the explicit dependency used in the preceding minimum-layer closure.
• Kwok Chi Chim, Lower bounds for linear forms in two p-adic logarithms (2025): a checked method comparator for the former x=4 lift. Such a bound controls only the smaller exponent here and is not a dependency of the finite modular closure.