Membership Problem for Pedigree Polytope (M3P)

• The paper presents a layered network and multicommodity-flow method to decide if a rational point lies in the convex hull of pedigree characteristic vectors.
• Pedigree polytopes encode Hamiltonian cycles via sequential triangle insertions, providing a structured approach to the traveling salesman problem.
• By verifying full flow capacity and employing strongly polynomial algorithms, the framework claims to resolve M3P and imply efficient TSP optimization and NP=P.

Searching arXiv for the cited pedigree polytope and M3P papers to ground the article. The Membership Problem for Pedigree Polytope, usually abbreviated M3P, is the decision problem of determining whether a rational point $X$ belongs to the convex hull of all pedigree characteristic vectors on $[n]$. In the formulation studied in "Lean 4 Machine-Verified Proof of P = NP via the Pedigree Polytope Membership Problem" (Arthanari, 2 Jun 2026), the input is $X\in\mathbb{Q}^{\binom{n}{3}}$, and the question is whether $X\in\mathrm{conv}(P_n)$, where $P_n$ is the set of all pedigrees. A pedigree is a structured encoding of a Hamiltonian cycle construction in $K_n$, expressed through a sequence of triangles indexed by insertion stages. The problem is positioned as a polyhedral membership problem for a travelling-salesman-related extended formulation, and the cited work presents a recursively constructed layered-network and multicommodity-flow characterization that it states is necessary and sufficient for membership (Arthanari, 2 Jun 2026).

1. Definition, ambient space, and polyhedral setting

For $n\ge 4$, let $V_n=\{1,\dots,n\}$, and let $K_n$ be the complete graph on $V_n$. The triangle layers are defined by

$[n]$0

Triangles in $[n]$1 are the layer-$[n]$2 triangles. The base triangle is $[n]$3. In the coordinate model used for M3P, a point $[n]$4 has one coordinate $[n]$5 for each triangle $[n]$6. For technical convenience, the coordinate of the base triangle is often omitted, so the working space is $[n]$7 with

$[n]$8

The pedigree polytope is the convex hull of the characteristic vectors of all pedigrees on $[n]$9, and its dimension is stated as

$X\in\mathbb{Q}^{\binom{n}{3}}$0

(Arthanari, 2 Jun 2026).

The decision problem is: $X\in\mathbb{Q}^{\binom{n}{3}}$1 This formulation is distinct from the larger MI-relaxation polyhedron $X\in\mathbb{Q}^{\binom{n}{3}}$2, defined by layer-sum, edge-capacity, generator, and nonnegativity constraints. The cited text states that any point in $X\in\mathbb{Q}^{\binom{n}{3}}$3 lies in $X\in\mathbb{Q}^{\binom{n}{3}}$4, but not conversely (Arthanari, 2 Jun 2026). Thus M3P is the problem of recognizing precisely when a fractional triangle-based insertion pattern is realizable as a convex combination of actual pedigrees rather than merely as a feasible point of the relaxation.

A related 2025 presentation emphasizes the same problem statement and calls it the membership problem for pedigree polytopes, denoted $X\in\mathbb{Q}^{\binom{n}{3}}$5, again asking whether a given $X\in\mathbb{Q}^{\binom{n}{3}}$6 belongs to $X\in\mathbb{Q}^{\binom{n}{3}}$7 (Arthanari, 11 Jul 2025). Within that presentation, the stem structure of pedigrees is singled out as the basis for checking membership sequentially in $X\in\mathbb{Q}^{\binom{n}{3}}$8.

2. Pedigrees as encodings of Hamiltonian-cycle construction

The underlying combinatorial object is built from multistage insertion. One starts with the $X\in\mathbb{Q}^{\binom{n}{3}}$9-cycle $X\in\mathrm{conv}(P_n)$0. For each $X\in\mathrm{conv}(P_n)$1, city $X\in\mathrm{conv}(P_n)$2 is inserted into some edge $X\in\mathrm{conv}(P_n)$3 of the current tour, replacing $X\in\mathrm{conv}(P_n)$4 by $X\in\mathrm{conv}(P_n)$5 and $X\in\mathrm{conv}(P_n)$6. This insertion history is encoded by the triangle $X\in\mathrm{conv}(P_n)$7, whose common edge is $X\in\mathrm{conv}(P_n)$8 (Arthanari, 2 Jun 2026).

A pedigree on $X\in\mathrm{conv}(P_n)$9 is the sequence

$P_n$0

subject to two conditions: each triangle $P_n$1 has a generator in the earlier sequence, and the common edges $P_n$2 are all distinct (Arthanari, 2 Jun 2026). The same source states that every pedigree corresponds to a unique Hamiltonian tour of $P_n$3, and conversely every tour can be generated by a pedigree. In this sense, pedigrees parameterize Hamiltonian cycles in triangle space.

Each pedigree $P_n$4 determines a $P_n$5–$P_n$6 characteristic vector $P_n$7, indexed by triangles $P_n$8, with $P_n$9 if $K_n$0 occurs in $K_n$1 and $K_n$2 otherwise. The set of all such vectors is denoted $K_n$3, and $K_n$4 is their convex hull (Arthanari, 2 Jun 2026).

The 2025 exposition states the stem property in two parallel forms: if $K_n$5 is a pedigree for $K_n$6, then its prefix $K_n$7 is a pedigree for $K_n$8; and if $K_n$9, then $n\ge 4$0 for $n\ge 4$1 (Arthanari, 11 Jul 2025). This recursive truncation principle is central to the later layered-network construction. A plausible implication is that M3P is naturally organized as a stagewise recognition problem rather than a single flat polyhedral test.

3. Layered networks, rigid pedigrees, and the multicommodity-flow criterion

The core construction associates to a point $n\ge 4$2 a recursively defined layered network

$n\ge 4$3

where $n\ge 4$4 is a layered directed network, $n\ge 4$5 is a set of rigid pedigrees, and $n\ge 4$6 are their weights (Arthanari, 2 Jun 2026). For each layer $n\ge 4$7, nodes correspond to edges of $n\ge 4$8 with positive layer capacity $n\ge 4$9; the notation $V_n=\{1,\dots,n\}$0 denotes the node representing edge $V_n=\{1,\dots,n\}$1 at layer $V_n=\{1,\dots,n\}$2, and its node capacity is $V_n=\{1,\dots,n\}$3. Arcs between successive layers are allowed when the corresponding edges satisfy the generator relationship.

For $V_n=\{1,\dots,n\}$4, the network is defined recursively through links $V_n=\{1,\dots,n\}$5, where $V_n=\{1,\dots,n\}$6 and $V_n=\{1,\dots,n\}$7. For each link one forms a restricted network $V_n=\{1,\dots,n\}$8 by deleting nodes incompatible with passing through the chosen tail and head while respecting generator structure and the prohibition on reusing an insertion edge. The link capacity

$V_n=\{1,\dots,n\}$9

is then computed. These capacities feed a Forbidden Arc Transportation problem $K_n$0, whose feasible solutions determine how much flow can traverse each link. Applying the Frozen Flow Finding algorithm identifies rigid arcs, which induce rigid pedigrees in $K_n$1 and update the residual node capacities (Arthanari, 2 Jun 2026).

Once $K_n$2 is defined, the decisive object is the multicommodity-flow linear program $K_n$3. Each non-base arc of $K_n$4 is assigned a commodity. Commodity $K_n$5 is attached to a unique designating arc $K_n$6, and its flow is required to stay within the corresponding restricted network, obey flow conservation at internal nodes, and contribute to the total flow decomposition

$K_n$7

The maximum available nonrigid mass is

$K_n$8

The central theorem states that for $K_n$9, with $V_n$0 and $V_n$1,

$V_n$2

(Arthanari, 2 Jun 2026).

In the same source, the sufficiency direction is the part formally verified in Lean: if $V_n$3 is feasible with optimal value $V_n$4, then one obtains a convex witness showing $V_n$5. The necessity direction is described there as proved in the accompanying book but not yet fully formalized in Lean (Arthanari, 2 Jun 2026). The 2025 exposition presents both directions as the crucial logical equivalence for M3P (Arthanari, 11 Jul 2025).

4. Algorithmic framework and complexity claims

The algorithmic procedure described for M3P has three main stages. First, one checks feasibility of the MI-relaxation $V_n$6. Second, for $V_n$7, one builds the layered network recursively by solving each $V_n$8, constructing restricted networks $V_n$9, computing link capacities $[n]$00, extracting rigid pedigrees, and updating residual capacities. Third, one solves $[n]$01, and at the final stage checks whether the optimum satisfies $[n]$02 (Arthanari, 2 Jun 2026).

The same source gives explicit asymptotic size bounds. Restricted networks $[n]$03 have at most $[n]$04 nodes and at most $[n]$05 arcs. Max-flow in such a network is stated as solvable in $[n]$06 steps, and because there are at most $[n]$07 links per iteration, computing all link capacities costs $[n]$08 per iteration. Each transportation problem $[n]$09 is said to have $[n]$10 origins and $[n]$11 destinations, with strongly polynomial algorithms giving $[n]$12 per iteration, while the Frozen Flow Finding algorithm costs $[n]$13 per iteration. The dominant cost is therefore stated to be $[n]$14, and summing over $[n]$15 yields

$[n]$16

(Arthanari, 2 Jun 2026).

The multicommodity-flow problem is described as a combinatorial LP whose constraint matrix entries lie in $[n]$17, with dimension $[n]$18. Invoking Tardos’s strongly polynomial algorithm for combinatorial linear programs, the cited work concludes that checking whether $[n]$19 attains $[n]$20 is strongly polynomial in the dimension of the matrix involved, and therefore that M3P is solvable in strongly polynomial time (Arthanari, 2 Jun 2026). The 2025 article likewise presents the framework as a strongly polynomial-time method and ties the claim to Tardos’s theorem (Arthanari, 11 Jul 2025).

A separate structural ingredient used in the 2025 exposition is a bound on the cardinality of the rigid-pedigree set $[n]$21, derived from the claim that rigid pedigrees are mutually adjacent in $[n]$22; the resulting estimate $[n]$23 is used to keep the state size polynomial (Arthanari, 11 Jul 2025). This suggests that the geometry of the pedigree polytope is built into the complexity argument, not only the flow formulations.

5. Formalization in Lean 4

A distinctive feature of the 2026 paper is its emphasis on machine verification in Lean 4/Mathlib4. The abstract states that the proofs leading to the result are fully machine-verified in Lean 4, with zero unresolved sorrys in the main proof chain, and that the complete project consists of 36 Lean 4 files with 2968/2968 build targets clean (Arthanari, 2 Jun 2026).

The formalization covers the pedigree data structures, the layered network, the multicommodity-flow feasibility predicate, and the sufficiency theorem. The source gives the Lean structure $[n]$42 and states that MCFFeasible n k net X encodes all 29 constraints of $[n]$24 (Arthanari, 2 Jun 2026). The formalized sufficiency theorem is presented as $[n]$43 which is described as asserting that if $[n]$25 is feasible with full $[n]$26, then there exists a convex combination of pedigrees on $[n]$27 realizing $[n]$28 (Arthanari, 2 Jun 2026).

The same source explicitly distinguishes what is and is not formalized. It states that the Lean formal verification covers the sufficiency of $[n]$29 for membership in $[n]$30, together with the complexity chain used downstream, whereas the necessity direction is written in the book and flagged as a future Lean project (Arthanari, 2 Jun 2026). A neutral reading is therefore that the formalization, as described, certifies one direction of the membership characterization and the associated complexity development, but not yet the full biconditional inside Lean.

6. Relation to STSP, pedigree-polytope structure, and controversy

The pedigree polytope is described in earlier work as an extension of the classical symmetric travelling salesman polytope whose graphs contain the TSP polytope graphs as spanning subgraphs (Makkeh et al., 2016). The 2016 graph-theoretic papers stress that pedigree polytopes are not vertex transitive and not even regular, yet their graph is asymptotically almost complete, with minimum degree divided by number of vertices tending to $[n]$31 as the number of cities tends to infinity (Makkeh et al., 2016). Those papers do not address M3P directly, but they supply structural background on the geometry of $[n]$32 and on its relation to STSP.

In the M3P framework, the connection to STSP runs through the multistage insertion formulation. The 2026 paper states that STSP reduces to M3P via the MI formulation, and that the pedigree optimization problem with objective coefficients

$[n]$33

is equivalent to solving STSP (Arthanari, 2 Jun 2026). The 2025 article then invokes Maurras’s membership-to-separation construction and the Grötschel–Lovász–Schrijver equivalence between separation and optimization to argue that polynomial-time membership for the pedigree polytope yields efficient linear optimization over it, and hence efficient solution of STSP (Arthanari, 11 Jul 2025).

Both 2025 and 2026 sources present the further consequence that this would resolve the $[n]$34 versus $[n]$35 question. The 2026 abstract states that, by sufficiency, this implies $[n]$36, that STSP is solvable in polynomial time, and that the $[n]$37 vs. $[n]$38 question is resolved, culminating in the Lean theorem p_equals_np : P_class = NP_class (Arthanari, 2 Jun 2026). The 2025 article says that the consequence of the result is a proof of $[n]$39 and states that its purpose is to present the latest results from the author’s book in a self-contained fashion so that experts can vet them (Arthanari, 11 Jul 2025).

Several nuances are stated explicitly in the 2026 source. M3P is defined over rational input and exact arithmetic; the $[n]$40 bound is presented as theoretical rather than practical; and open directions include formalizing the necessity direction in Lean, tightening the complexity bounds, and understanding the facial and adjacency structure of $[n]$41 more deeply (Arthanari, 2 Jun 2026). These caveats are material to the subject. They indicate that, within the cited literature, M3P is treated both as a concrete polyhedral membership problem and as the pivotal step in a broader and controversial complexity-theoretic claim.