Iterated Copy Lemma in Entropic Inequalities

• Iterated Copy Lemma is a technique that applies repeated copy steps to extend polymatroid entropy profiles while preserving marginals and imposing conditional independence.
• It converts copy constraints into linear equalities and inequalities, facilitating the derivation of non-Shannon information inequalities through computational methods.
• Leveraging canonical isomorphisms and symmetry, the method separates the Shannon outer cone from the almost-entropic region, impacting applications in network coding and secret sharing.

Searching arXiv for the cited papers to ground the article in current metadata and ensure accurate citation. The iterated Copy Lemma is the repeated application of the Copy Lemma to a polymatroid or entropy profile and, crucially, to the successive extensions created by earlier copy steps. In entropy-region theory it serves both as an entropic existence principle and as a computational device: each copy step preserves designated marginals, enforces a canonical isomorphism between original and copied coordinates, and imposes conditional independence over a chosen over-set. These constraints hold for entropic, almost-entropic, and linear objects, but not for arbitrary polymatroids, which is why the method is used to separate the Shannon outer cone $\Gamma_N$ from the almost-entropic region $\overline{\Gamma_N^*}$ and to derive non-Shannon information inequalities (Csirmaz, 15 Sep 2025).

1. Historical emergence and conceptual role

The Copy Lemma was distilled from the original Zhang–Yeung construction that produced the first non-Shannon inequality. In that lineage, the method is also called the copy/pasting lemma or copy trick, and in proof-system language its corresponding inference rule is called Rule ZY. The later phrase iterated Copy Lemma denotes the closure of this idea under repetition: instead of making a single auxiliary copy and then importing previously known inequalities, one repeatedly performs the copy construction itself on the enlarged system (Kaced, 2013).

Within the modern study of the entropic region, this role is both conceptual and computational. Conceptually, the Copy Lemma gives an operation that is guaranteed for entropic and almost-entropic objects but not for all polymatroids; such operations are precisely what can witness the gap between $\Gamma_N$ and $\overline{\Gamma_N^*}$. Computationally, every copy step becomes a finite system of linear equalities and inequalities in entropy space, so consequences can be searched by linear programming and polyhedral vertex enumeration. The iterated form is the main extension of this viewpoint: iteration can simulate the effect of using previously discovered non-Shannon inequalities while staying within one uniform linear-constraint framework (Csirmaz, 15 Sep 2025).

A proof-theoretic formulation makes the same point in different language. A proof system consists of a pool of inequalities and an inference rule; one repeatedly selects an inequality from the convex closure of the pool, applies the rule, and adds the conclusion back to the pool. Repeated use of Rule ZY is therefore the formal counterpart of iterated copying. In this sense, the iterated Copy Lemma is not merely a combinatorial gadget for constructing auxiliary variables; it is also a composable inference mechanism for non-Shannon derivations (Kaced, 2013).

2. Single-step copy constructions

A single copy step is formulated on a base set partitioned as $N=ED$, where $E$ and $D$ are non-empty and disjoint. For a subset $A\subseteq E$, let $A'$ denote a disjoint copy of $A$. The permutation $\overline{\Gamma_N^*}$0 on $\overline{\Gamma_N^*}$1 that swaps corresponding elements of $\overline{\Gamma_N^*}$2 and $\overline{\Gamma_N^*}$3 and fixes $\overline{\Gamma_N^*}$4 pointwise is the canonical permutation or canonical map. When $\overline{\Gamma_N^*}$5, the construction is a full copy and the canonical map on $\overline{\Gamma_N^*}$6 is denoted $\overline{\Gamma_N^*}$7 (Csirmaz, 15 Sep 2025).

With this notation, a polymatroid $\overline{\Gamma_N^*}$8 is an $\overline{\Gamma_N^*}$9-copy of $\Gamma_N$0 over $\Gamma_N$1 if three conditions hold. First, $\Gamma_N$2 is an extension of $\Gamma_N$3, so the old marginals on the original variables are preserved exactly. Second, the canonical map $\Gamma_N$4 gives an isomorphism between $\Gamma_N$5 and $\Gamma_N$6, so the new variables look, over $\Gamma_N$7, exactly like the original variables $\Gamma_N$8. Third, $\Gamma_N$9, meaning that $\overline{\Gamma_N^*}$0 and $\overline{\Gamma_N^*}$1 are conditionally independent over $\overline{\Gamma_N^*}$2. Using the first two conditions, this may be written equivalently as

$\overline{\Gamma_N^*}$3

The existence statement is the Copy Lemma itself: if $\overline{\Gamma_N^*}$4 is entropic, almost entropic, or linear, then it has an entropic, almost-entropic, or linear $\overline{\Gamma_N^*}$5-copy over $\overline{\Gamma_N^*}$6, respectively (Csirmaz, 15 Sep 2025).

For entropic polymatroids the construction is explicit. If $\overline{\Gamma_N^*}$7 is the entropy profile of $\overline{\Gamma_N^*}$8 and $\overline{\Gamma_N^*}$9, then a full copy $N=ED$0 on $N=ED$1 is defined by

$N=ED$2

where $N=ED$3, $N=ED$4, $N=ED$5, and $N=ED$6 is the canonical map on $N=ED$7. Conditioned on $N=ED$8, the new copy $N=ED$9 is distributed like $E$0 but is conditionally independent of the original $E$1 (Csirmaz, 15 Sep 2025).

An equivalent probabilistic statement appears in the classical random-variable formulation: for jointly distributed random variables $E$2, there exists a fourth random variable $E$3 such that $E$4 and $E$5 have the same distribution, and $E$6 is independent of $E$7 given $E$8. In information-theoretic form, the second condition is $E$9. The same formulation extends to tuples. In notation used for direct LP applications, one writes

$D$0

meaning that $D$1 and $D$2; the tuple version remains valid when $D$3 are tuples rather than single variables [(Kaced, 2013); (Gürpınar et al., 2019)].

A structurally important feature is that not every coordinate of $D$4 is fixed by the definition. Values on sets containing both original and copied variables are constrained only indirectly through polymatroidality together with the copy conditions. This controlled freedom is one source of the method’s strength (Csirmaz, 15 Sep 2025).

3. Iteration as a compositional construction

The iterated Copy Lemma begins from the observation that each copy extension is again entropic or almost entropic, so the Copy Lemma remains available at every subsequent stage. An iteration is specified by a sequence of copy instructions

$D$5

where at step $D$6, the copied set $D$7 and over-set $D$8 are subsets of the base of the polymatroid produced at step $D$9. Each step introduces fresh auxiliary variables $A\subseteq E$0, disjoint from all previously existing variables (Csirmaz, 15 Sep 2025).

At step $A\subseteq E$1, the new polymatroid $A\subseteq E$2 extends $A\subseteq E$3, satisfies the canonical-copy isomorphism on $A\subseteq E$4, and imposes

$A\subseteq E$5

where $A\subseteq E$6 is the complement of $A\subseteq E$7 in the current base. Equivalently,

$A\subseteq E$8

Thus every iteration contributes three kinds of constraints: marginal preservation on the previous base, canonical isomorphism on subsets of $A\subseteq E$9, and conditional independence of the new copy from the rest over $A'$0. The conditional-independence statement propagates to smaller copied subsets: from $A'$1, one obtains $A'$2 for every $A'$3 and $A'$4 (Csirmaz, 15 Sep 2025).

The distinction between partial and full copies is technically useful. Every partial $A'$5-copy can be realized as a restriction of a full copy: the $A'$6-copy of a minor is a minor of a full copy. In principle, this means that iterated full copies suffice; in practice, partial copies are often preferred because they keep the ambient dimension smaller (Csirmaz, 15 Sep 2025).

A proof-system formulation captures the same compositional logic. In System ZY, one repeatedly applies Rule ZY; in System ZY+b, one also performs balancing at each step. This turns repeated copy-based reasoning into a derivation system in which conclusions from earlier stages become premises for later ones. The phrase iterated Copy Lemma does not appear explicitly in that proof-system presentation, but repeated use of Rule ZY is exactly the proof-theoretic content of iterated copying (Kaced, 2013).

4. Polyhedral and linear-programming formulations

The computational power of the iterated Copy Lemma comes from translating entropic existence statements into linear constraints. After fixing an iterated copy construction, one introduces LP variables for the entropy values of all nonempty subsets of the final base. These split into main variables $A'$7, indexed by subsets of the original base, and auxiliary variables $A'$8, indexed by subsets involving variables created during the iteration. Each copy step contributes linear equalities expressing the canonical-map isomorphism and the conditional-independence consequences; Shannon inequalities are then imposed on the final base. When one is interested only in balanced inequalities, it is enough to use the basic submodularity inequalities (#B2), not the monotonicity inequalities (#B1) (Csirmaz, 15 Sep 2025).

After simplification, each constraint has the form

$A'$9

Collecting these into matrices $A$0 yields the feasible region

$A$1

By Farkas’ lemma, the coefficients $A$2 of all inequalities $A$3 implied by the iterated copy construction are exactly those in the cone

$A$4

The extremal rays of $A$5 are therefore the minimal implied inequalities. Computing them is a polyhedral vertex-enumeration problem; double-description and vector-optimization style algorithms are explicitly cited for this purpose. This framework has been used to generate “several hundreds of new four-variable Shannon inequalities” (Csirmaz, 15 Sep 2025).

A central practical point is that iteration is usually preferable to adding previously known non-Shannon inequalities explicitly. A $A$6-iterated copy has the same consequences as a first copy step supplemented by all linear inequalities on the first extension that guarantee the remaining $A$7 steps; conversely, adding those extra inequalities can be mimicked by further copy steps. The two methods are theoretically equivalent, but iteration avoids the need to solve a larger preliminary enumeration problem merely to discover which auxiliary inequalities should be added (Csirmaz, 15 Sep 2025).

A complementary operational formulation uses direct LPs with copy constraints rather than explicit enumeration of derived inequalities. In that approach, one enlarges the variable set by adding copy variables, imposes Shannon-type inequalities on the enlarged system, adds the copy equalities and conditional-independence constraints, and optimizes the target linear functional directly. This realizes one-step, two-step, or multi-step copy constructions without explicitly writing down the non-Shannon inequalities they imply (Gürpınar et al., 2019).

5. Canonical applications and explicit constructions

The archetypal single-step application is the exclusion of the Vámos polymatroid $A$8 from the almost-entropic region. On $A$9,

$\overline{\Gamma_N^*}$00

with

$\overline{\Gamma_N^*}$01

$\overline{\Gamma_N^*}$02

and Ingleton value

$\overline{\Gamma_N^*}$03

Assuming $\overline{\Gamma_N^*}$04 were almost entropic, one takes a $\overline{\Gamma_N^*}$05-copy over $\overline{\Gamma_N^*}$06, introducing $\overline{\Gamma_N^*}$07, and then applies the five-variable Shannon inequality

$\overline{\Gamma_N^*}$08

Substituting $\overline{\Gamma_N^*}$09, the first term is $\overline{\Gamma_N^*}$10 and the other four are $\overline{\Gamma_N^*}$11, yielding a contradiction. This is the basic pattern by which one copy step plus Shannon inequalities yields a genuinely non-Shannon conclusion (Csirmaz, 15 Sep 2025).

The same mechanism produces the Zhang–Yeung inequality. Starting from an almost-entropic $\overline{\Gamma_N^*}$12 on $\overline{\Gamma_N^*}$13, take a $\overline{\Gamma_N^*}$14-copy over $\overline{\Gamma_N^*}$15, so $\overline{\Gamma_N^*}$16 satisfies

$\overline{\Gamma_N^*}$17

and, by isomorphism,

$\overline{\Gamma_N^*}$18

Applying the same five-variable inequality with $\overline{\Gamma_N^*}$19 gives

$\overline{\Gamma_N^*}$20

This is the Zhang–Yeung inequality in the notation of the modern treatment (Csirmaz, 15 Sep 2025).

Iteration enlarges this mechanism by building towers of copies. A concrete three-step example starts from base

$\overline{\Gamma_N^*}$21

First, take a $\overline{\Gamma_N^*}$22-copy over $\overline{\Gamma_N^*}$23, introducing $\overline{\Gamma_N^*}$24. Second, take an $\overline{\Gamma_N^*}$25-copy over $\overline{\Gamma_N^*}$26, introducing $\overline{\Gamma_N^*}$27. Third, take a $\overline{\Gamma_N^*}$28-copy over $\overline{\Gamma_N^*}$29, introducing $\overline{\Gamma_N^*}$30. The compact notation is

$\overline{\Gamma_N^*}$31

Because $\overline{\Gamma_N^*}$32 persists through all stages, the final restricted construction lives on seven points $\overline{\Gamma_N^*}$33. The corresponding iterated full-copy sequence ends on 14 points, and its final full copy contains eight isomorphic instances of the original four-variable polymatroid at the sets $\overline{\Gamma_N^*}$34, where $\overline{\Gamma_N^*}$35 and $\overline{\Gamma_N^*}$36. This proliferation of embedded instances is one stated reason that iteration is powerful (Csirmaz, 15 Sep 2025).

Direct LP applications show how such repeated constructions are used operationally. For the normalized Ingleton score, one introduces four auxiliary variables $\overline{\Gamma_N^*}$37 by three dependent copy steps: $\overline{\Gamma_N^*}$38

$\overline{\Gamma_N^*}$39

$\overline{\Gamma_N^*}$40

On the entropy profile of $\overline{\Gamma_N^*}$41, this gives an LP with $\overline{\Gamma_N^*}$42 coordinates, and the resulting optimization proves

$\overline{\Gamma_N^*}$43

The final exact proof can be reduced to a rational linear combination of 167 equalities and inequalities (Gürpınar et al., 2019).

A second example concerns a Vámos-matroid secret-sharing access structure. Writing

$\overline{\Gamma_N^*}$44

one first takes $\overline{\Gamma_N^*}$45 to be an $\overline{\Gamma_N^*}$46-copy of $\overline{\Gamma_N^*}$47 over $\overline{\Gamma_N^*}$48, and then $\overline{\Gamma_N^*}$49 to be an $\overline{\Gamma_N^*}$50-copy of $\overline{\Gamma_N^*}$51 over the same base. The associated independence constraints are

$\overline{\Gamma_N^*}$52

$\overline{\Gamma_N^*}$53

On 12 scalar variables, this yields an LP over $\overline{\Gamma_N^*}$54 entropy coordinates and proves the lower bound

$\overline{\Gamma_N^*}$55

The exact dual solution involves more than $\overline{\Gamma_N^*}$56 equalities and inequalities (Gürpınar et al., 2019).

6. Balancing, symmetry, relation to other methods, and limitations

Balanced inequalities are fundamental to the proof theory of iterated copying. An information inequality

$\overline{\Gamma_N^*}$57

is balanced for variable $\overline{\Gamma_N^*}$58 if

$\overline{\Gamma_N^*}$59

and balanced if this holds for every variable. The balancing transformation subtracts

$\overline{\Gamma_N^*}$60

producing an equivalent valid balanced inequality. Rule ZY preserves balance for every variable, and the main proof-theoretic comparison shows that repeated Zhang–Yeung reasoning and repeated MMRV reasoning are equivalent modulo balancing. More precisely, the systems ZY+b and R+b have the same provability power (Kaced, 2013).

Balancing also has a geometric role in entropy-region computation. Every valid information inequality $\overline{\Gamma_N^*}$61 decomposes uniquely into a conic combination of the basic monotonicity inequalities $\overline{\Gamma_N^*}$62 and a balanced part $\overline{\Gamma_N^*}$63. A balanced inequality is characterized by

$\overline{\Gamma_N^*}$64

where $\overline{\Gamma_N^*}$65 is the entropic indicator vector of “$\overline{\Gamma_N^*}$66”. Since genuinely new inequalities obtained from Copy Lemma computations can be taken balanced, the dimension of the main-variable space can be reduced by $\overline{\Gamma_N^*}$67, which is significant for enumeration (Csirmaz, 15 Sep 2025).

Symmetry is the other major optimization. The Copy Lemma itself yields copies in which the canonical map is a symmetry, and for full copies imposing canonical symmetry does not strengthen the method because averaging a copy with its canonical image yields a symmetric copy with the same copy properties. For partial copies, by contrast, imposing $\overline{\Gamma_N^*}$68-symmetry can produce stronger consequences: there exist polymatroids having an $\overline{\Gamma_N^*}$69-copy but no $\overline{\Gamma_N^*}$70-symmetric $\overline{\Gamma_N^*}$71-copy. Pre-existing symmetries of the original polymatroid can also be propagated through full-copy steps. In a 14-point iterated full-copy example, the successive canonical maps together with inherited symmetries reduce the total number of LP variables from $\overline{\Gamma_N^*}$72 to 2351, even before conditional-independence constraints are used. One consequence of such a highly symmetric arrangement is the strengthened non-Shannon inequality

$\overline{\Gamma_N^*}$73

Symmetrization is therefore not merely a computational shortcut; it can change the strength of the derived inequalities (Csirmaz, 15 Sep 2025).

The iterated Copy Lemma sits in a hierarchy of related techniques. The Maximum Entropy Method (MEM) includes the Copy Lemma as a special case: when only a single separated 3-partition $\overline{\Gamma_N^*}$74 is imposed, the MEM extension is equivalent to taking an $\overline{\Gamma_N^*}$75-copy over $\overline{\Gamma_N^*}$76. MEM can prove inequalities that are not consequences of a single application of the Copy Lemma, but whether MEM is stronger than the iterated Copy Lemma is unknown. No numerically convenient “iterated MEM” framework comparable to iterated copying is known. The generalized MEM (GMEM), intended to mimic many embedded copies at once via a family of transversals, has a reported limitation: a natural attempt to compress an iterated full-copy configuration into one GMEM instance “failed spectacularly,” yielding no new inequality at all in the worked example. The Ahlswede–Körner comparison is sharper. The generalized Ahlswede–Körner operation

$\overline{\Gamma_N^*}$77

preserves polymatroids and therefore cannot by itself generate non-Shannon inequalities. The stronger MMRV/Ahlswede–Körner variant hides an application of the Copy Lemma; for singleton copying it is equivalent to the Copy Lemma, but overall it is strictly weaker, since it cannot directly prove that the Vámos vector is not almost entropic whereas the Copy Lemma can (Csirmaz, 15 Sep 2025).

There are also explicit preconditions under which a copy step is guaranteed to be useless. If the over-set $\overline{\Gamma_N^*}$78 is modular enough that

$\overline{\Gamma_N^*}$79

then an $\overline{\Gamma_N^*}$80-copy over $\overline{\Gamma_N^*}$81 already exists for every polymatroid, so that step cannot separate almost-entropic from polymatroidal behavior. Likewise, no new information comes from copying over $\overline{\Gamma_N^*}$82 or over $\overline{\Gamma_N^*}$83, and non-flat over-sets may as well be replaced by their closures. These criteria can be checked at intermediate stages of an iterated construction (Csirmaz, 15 Sep 2025).

The method’s limitations are equally explicit. Auxiliary-variable counts grow rapidly, so only “a couple of iterations” are usually tractable unless symmetry is heavily exploited. There is no guarantee that the iterated Copy Lemma captures the whole entropy region, and there are stated indications that it does not. Open problems include systematic numerical exploration on five variables; comparison of iterated copying with MEM and GMEM; and whether the Copy Lemma has consequences for linear polymatroids beyond those obtainable from common-information extraction. The current practical conclusion is that iterating the Copy Lemma remains the best developed computational framework for finding non-Shannon inequalities: repeated copy constraints carve out a polyhedral cone of consequences in the original entropy coordinates, and its extremal rays are the information inequalities one seeks (Csirmaz, 15 Sep 2025).