Bi-Constructible: Geometry & Bisheaf Theory

• Bi-Constructible is a dual-use concept where, in particle phenomenology, it models weak and flavour mixing via the geometry of constructible polygons, and in topology, it defines bisheaf constructibility on triangulated spaces.
• The geometric formulation uses the pentagon and heptadecagon to derive accurate mixing angles and electroweak couplings, linking empirical data to mathematical constants like the golden ratio and Fermat primes.
• In bisheaf theory, bi-constructibility ensures that both the sheaf and cosheaf maps are locally constant with respect to a canonical stratification, enabling efficient computation of persistence modules.

Searching arXiv for the specified papers to ground the article and citations. “Bi-Constructible” appears in two distinct technical senses in the supplied arXiv literature. In particle phenomenology, it denotes a semi-empirical pattern in which weak and flavour mixing are described through Euclidean geometry of the regular pentagon and heptadecagon, with implications for the Weinberg angle and electroweak couplings (Ciborowski, 30 Jul 2025). In sheaf-theoretic topology, “bi-constructibility” denotes constructibility of a bisheaf—simultaneous local constancy conditions for a sheaf, a cosheaf, and their compatible cap-product maps—together with the canonical stratification on which that bisheaf remains constructible (Nanda et al., 2018). The two usages are terminologically adjacent but mathematically independent in the supplied literature.

1. Two technical uses of the term

The supplied literature associates the term with two different composite structures. One is geometric and phenomenological: weak, quark, and lepton mixing angles are modeled by right-triangles whose catheti align with sides or half-exterior angles of regular constructible polygons, specifically the pentagon $P_5$ and heptadecagon $P_{17}$ (Ciborowski, 30 Jul 2025). The other is categorical and topological: a bisheaf is a triple consisting of a sheaf, a cosheaf, and compatible maps from sheaf stalks to cosheaf costalks, and bi-constructibility means local constancy with respect to a triangulation (Nanda et al., 2018).

Usage	Domain	Core content
Bi-Constructible	Weak and flavour mixing	Pentagon–heptadecagon geometric pattern for quark, lepton, and electroweak mixing
bi-constructibility	Bisheaf theory	Constructibility of a bisheaf on a triangulated space

A potential source of confusion is that the shared prefix “bi-” refers to different structures. In the mixing paper it points to a two-polygon scheme based on the only nontrivial Fermat primes below $100$, namely $5$ and $17$. In the bisheaf paper it points to the joint presence of a sheaf and a cosheaf, together with compatibility morphisms.

2. Geometric Bi-Constructible hypothesis in weak and flavour mixing

The phenomenological usage is built around Wantzel’s theorem: a regular $n$-gon is compass-and-straightedge constructible if and only if

$n=2^k\cdot p_1\cdot p_2\cdots p_m,$

where the $p_i$ are distinct Fermat primes $F_0=3$, $F_1=5$, $P_{17}$0, $P_{17}$1, $P_{17}$2 (Ciborowski, 30 Jul 2025). Since the only nontrivial Fermat primes below $P_{17}$3 are $P_{17}$4 and $P_{17}$5, the regular pentagon and heptadecagon are taken as the simplest “new” constructible polygons after the triangle, square, and hexagon.

The Bi-Constructible hypothesis is stated as follows: at lowest order, both weak (electroweak) mixing and flavour mixing can be described by right-triangles whose catheti align with sides or half-exterior angles of $P_{17}$6 and $P_{17}$7 (Ciborowski, 30 Jul 2025). For each pair of mixing angles $P_{17}$8, one considers a right-triangle $P_{17}$9 inscribed in two adjacent sectors of a regular $100$0-gon. The hypotenuse $100$1 is one side of the $100$2-gon, of length $100$3; the acute angle at $100$4 is half the exterior angle,

$100$5

and the two catheti are identified with the two mixing angles.

The parametrization is given sectorwise, for quarks $100$6 or leptons $100$7, by

$100$8

for the $100$9 pair, and

$5$0

for the $5$1 pair. It suffices to study $5$2 since $5$3. The normalized angles are defined by taking unit hypotenuse:

$5$4

$5$5

This construction turns the angle data into a geometric pattern controlled by the seeds $5$6 and $5$7.

3. Pentagon and heptadecagon realizations of lepton and quark mixing

For leptons, the pentagon scheme $5$8 is assigned to $5$9. Since the pentagon exterior angle is $17$0, the mixing seed is $17$1. With the empirical choice $17$2, one sets $17$3 and obtains

$17$4

$17$5

where $17$6 is the golden ratio (Ciborowski, 30 Jul 2025).

The same source also introduces a dual “golden” scheme $17$7, based on the quasi-equalities

$17$8

This yields

$17$9

$n$0

Both schemes are reported to agree with data within $n$1.

The reactor angle is then tied to the heptadecagon. Since $n$2, the seed is

$n$3

The prediction is

$n$4

which is compared with $n$5. The associated radius parameter is

$n$6

For quarks, the Cabibbo angle is generated from the lepton seed by the empirical relation $n$7. The chosen “para-golden” fraction is

$n$8

so that

$n$9

reported to lie within $n=2^k\cdot p_1\cdot p_2\cdots p_m,$0 of the measured $n=2^k\cdot p_1\cdot p_2\cdots p_m,$1. The second quark angle is then

$n=2^k\cdot p_1\cdot p_2\cdots p_m,$2

to be compared with $n=2^k\cdot p_1\cdot p_2\cdots p_m,$3. For the smallest quark angle, the empirical ratio $n=2^k\cdot p_1\cdot p_2\cdots p_m,$4 motivates the choice $n=2^k\cdot p_1\cdot p_2\cdots p_m,$5, giving

$n=2^k\cdot p_1\cdot p_2\cdots p_m,$6

compared with $n=2^k\cdot p_1\cdot p_2\cdots p_m,$7.

The same framework also notes that all three quark angles can be written, $n=2^k\cdot p_1\cdot p_2\cdots p_m,$8-approximately, as fractions of $n=2^k\cdot p_1\cdot p_2\cdots p_m,$9:

$p_i$0

4. Weak–quark–lepton complementarity and electroweak couplings

A central claim of the geometric framework is the existence of concise Weak–Quark–Lepton Complementarity relations. The first is

$p_i$1

with reported accuracy $p_i$2, leading to

$p_i$3

where $p_i$4 (Ciborowski, 30 Jul 2025). A second relation identifies the normalized Cabibbo angle with the normalized lepton $p_i$5 angle:

$p_i$6

The Weinberg angle is then inserted through the empirical ratios

$p_i$7

which imply

$p_i$8

to be compared with the on-shell value $p_i$9. The resulting WQLC sum rule is

$F_0=3$0

equivalently

$F_0=3$1

The electroweak couplings are then expressed phenomenologically in terms of $F_0=3$2. Starting from $F_0=3$3 and $F_0=3$4, one finds

$F_0=3$5

where $F_0=3$6. Imposing the phenomenological condition $F_0=3$7 gives

$F_0=3$8

Using

$F_0=3$9

the framework reports

$F_1=5$0

at the per-mille level. The electric charge is

$F_1=5$1

and the fine-structure constant becomes

$F_1=5$2

The language of the source is explicitly semi-empirical: it presents “semi-empirical evidence,” derives “concise” complementarity relations, and suggests a “semi-empirical unification pattern of weak and flavour mixing.” A natural implication is that the construction is offered as a phenomenological organization of observed angles and couplings rather than as a derivation from a microscopic dynamical model.

5. Bi-constructibility of bisheaves on triangulated spaces

In the topological usage, the relevant object is a bisheaf on a compact topological space $F_1=5$3 equipped with a finite simplicial triangulation $F_1=5$4. Writing $F_1=5$5 for the poset of simplices ordered by the face relation $F_1=5$6, and $F_1=5$7 for the category of abelian groups, a sheaf on $F_1=5$8 is taken to be a covariant functor

$F_1=5$9

while a cosheaf is a functor

$P_{17}$00

(Nanda et al., 2018).

A bisheaf $P_{17}$01 consists of such a sheaf and cosheaf together with homomorphisms

$P_{17}$02

for each simplex $P_{17}$03, subject to commutativity of the square

$P_{17}$04

Equivalently, restriction in the sheaf followed by capping down coincides with capping followed by extension in the cosheaf.

Constructibility is then defined relative to the chosen triangulation. The bisheaf is constructible, or locally constant, with respect to $P_{17}$05 if for every face relation $P_{17}$06, both the sheaf restriction map $P_{17}$07 and the cosheaf extension map $P_{17}$08 are isomorphisms in $P_{17}$09. In sheaf-theoretic language, this means that $P_{17}$10 is locally constant on each star of $P_{17}$11, and similarly for $P_{17}$12.

The associated notion of a $P_{17}$13-stratification is a filtration by simplicial subcomplexes

$P_{17}$14

whose strata satisfy three axioms: dimension, frontier, and constructibility. A $P_{17}$15-stratum must contain at least one $P_{17}$16-simplex and no simplex of higher dimension; the frontier relation must define a graded partial order; and if $P_{17}$17 lie in the same connected stratum, then both $P_{17}$18 and $P_{17}$19 must be isomorphisms.

6. Canonical stratification, localization, and algorithmic construction

The bisheaf paper identifies a canonical $P_{17}$20-stratification, defined as the unique minimal element in the refinement order: every other $P_{17}$21-stratification refines it (Nanda et al., 2018). Its main theorem states existence and uniqueness of such a filtration

$P_{17}$22

and further states that its $P_{17}$23-dimensional strata are exactly the isomorphism classes of $P_{17}$24-simplices in a suitable localization of the face poset of $P_{17}$25.

The construction proceeds inductively for $P_{17}$26 by defining a subcomplex $P_{17}$27 and a set

$P_{17}$28

of face relations closed under composition. The complement $P_{17}$29 is the union of those simplices of dimension $P_{17}$30 that become isomorphic in the localized category $P_{17}$31, while $P_{17}$32 records face relations whose sheaf and cosheaf maps are invertible not merely on the pair itself but on the entire open star of the lower simplex.

Concretely, one starts from

$P_{17}$33

Having defined $P_{17}$34, one removes from $P_{17}$35 those simplices of dimension $P_{17}$36 that become isomorphic to some higher-dimensional simplex in the localization, obtaining $P_{17}$37. One then enlarges $P_{17}$38 to $P_{17}$39 by closing under stars of invertible relations. The construction preserves the frontier and constructibility axioms and yields the coarsest valid stratification.

The procedure is entirely combinatorial once $P_{17}$40 and $P_{17}$41 are presented simplicially. Each step requires testing which restriction and extension maps are isomorphisms and then computing connected components of the resulting isomorphism graph. If $P_{17}$42 is the number of simplices of $P_{17}$43, the complexity is stated as at worst

$P_{17}$44

once matrix invertibility tests have been preprocessed. An equivalent nerve-theoretic description views $P_{17}$45 as a functor on the nerve of the covering by open stars and localizes that nerve at the quasi-isomorphisms.

A toy example is given for the closed $P_{17}$46-simplex with vertices $P_{17}$47 and edge $P_{17}$48. If $P_{17}$49 and $P_{17}$50 are constant of rank $P_{17}$51 except that $P_{17}$52 is zero while $P_{17}$53 is the identity, then $P_{17}$54 contains only $P_{17}$55. In the localized category, $P_{17}$56 becomes isomorphic to $P_{17}$57, yielding one $P_{17}$58-stratum $P_{17}$59, while $P_{17}$60 forms its own $P_{17}$61-stratum. The canonical stratification is therefore

$P_{17}$62

7. Applications, significance, and terminological caution

The bisheaf-theoretic notion is motivated by homological stability of fibers. For a tame map

$P_{17}$63

into a real-analytic, triangulated manifold $P_{17}$64 of dimension $P_{17}$65, one defines

$P_{17}$66

$P_{17}$67

and lets $P_{17}$68 be cap-product with the pullback of a generator of $P_{17}$69 (Nanda et al., 2018). This produces a bisheaf constructible with respect to a chosen triangulation. Its canonical stratification identifies the coarsest partition of $P_{17}$70 on which the “persistent local system” of subquotients

$P_{17}$71

is constant. Restricting to a single stratum permits one to compute the associated classical persistence modules once per stratum rather than once per simplex, which the source presents as central both to theoretical clarity and to computational efficiency.

In the phenomenological usage, the significance is different. The pentagon–heptadecagon construction is presented as a unified geometric picture in which quark, lepton, and electroweak mixing angles arise from right-triangles built on $P_{17}$72 and $P_{17}$73, the golden ratio appears naturally from $P_{17}$74, the heptadecagon contributes nested radicals expressible in terms of $P_{17}$75 up to $P_{17}$76-level, and the Standard Model couplings $P_{17}$77, $P_{17}$78, and $P_{17}$79 admit elegant expressions in $P_{17}$80 alone (Ciborowski, 30 Jul 2025). The paper further suggests a low-energy unification pattern summarized schematically as geometry of Fermat-prime polygons $P_{17}$81 mixing and coupling constants $P_{17}$82 anthropically viable values.

The principal misconception to avoid is terminological conflation. In the supplied literature, “Bi-Constructible” in particle physics does not refer to bisheaves, canonical stratifications, or local constancy; conversely, “bi-constructibility” in topology does not refer to constructible polygons, Fermat primes, or flavour mixing. The shared label masks two separate research programs: one semi-empirical and geometric, the other categorical, algorithmic, and sheaf-theoretic.