---
title: 'Bi-Constructible: Geometry & Bisheaf Theory'
url: https://www.emergentmind.com/topics/bi-constructible
type: topic
---

# Bi-Constructible: Geometry & Bisheaf Theory

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 [2508.00030]. 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 [1812.05593]. 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 $P_{17}$ [2508.00030]. 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 [1812.05593].

| 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 $F_0=3$, $F_1=5$, $F_2=17$, $F_3=257$, $F_4=65537$ [2508.00030]. Since the only nontrivial Fermat primes below $100$ are $5$ and $17$, 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 $P_5$ and $P_{17}$ [2508.00030]. For each pair of mixing angles $\theta_{ij},\theta_{jk}$, one considers a right-triangle $A_1A_2A_3$ inscribed in two adjacent sectors of a regular $n$-gon. The hypotenuse $A_1A_3$ is one side of the $n$-gon, of length $R$; the acute angle at $A_2$ is half the exterior angle,
$$
\alpha=\epsilon/2,\qquad \epsilon=360^\circ/n,
$$
and the two catheti are identified with the two mixing angles.

The parametrization is given sectorwise, for quarks $q$ or leptons $\ell$, by
$$
\theta_{12}=R_1\cos\alpha_1,\qquad \theta_{23}=R_1\sin\alpha_1,
$$
for the $(12\text{--}23)$ pair, and
$$
\theta_{23}=R_2\cos\alpha_2,\qquad \theta_{13}=R_2\sin\alpha_2,
$$
for the $(23\text{--}13)$ pair. It suffices to study $i=1,2$ since $\tan\alpha_1\cdot\tan\alpha_2=\tan\alpha_3$. The normalized angles are defined by taking unit hypotenuse:
$$
\theta_{12(n1)}=\theta_{12}/R_1=\cos\alpha_1,\qquad \theta_{23(n1)}=\sin\alpha_1,
$$
$$
\theta_{23(n2)}=\cos\alpha_2,\qquad \theta_{13(n2)}=\sin\alpha_2.
$$
This construction turns the angle data into a geometric pattern controlled by the seeds $\alpha_{P5}$ and $\alpha_C$.

## 3. Pentagon and heptadecagon realizations of lepton and quark mixing

For leptons, the pentagon scheme $P5$ is assigned to $(\theta_{12}^\ell,\theta_{23}^\ell)$. Since the pentagon exterior angle is $\epsilon_5=360^\circ/5=72^\circ$, the mixing seed is $\alpha_{P5}=54^\circ$. With the empirical choice $R_1^\ell\approx 1$, one sets $R_1^\ell=1$ and obtains
$$
\theta_{12}^{\ell(P5)}=\cos 54^\circ=\sqrt{1-(\phi/2)^2}\approx 33.6776^\circ,
$$
$$
\theta_{23}^{\ell(P5)}=\sin 54^\circ=\phi/2\approx 46.3533^\circ,
$$
where $\phi=(1+\sqrt{5})/2$ is the golden ratio [2508.00030].

The same source also introduces a dual “golden” scheme $G$, based on the quasi-equalities
$$
\frac12\arccos(1/\phi^2)\simeq \cos 54^\circ,\qquad \frac12\operatorname{arccosh}(\phi^2)\simeq \sin 54^\circ.
$$
This yields
$$
\cos(2\theta_{12}^{\ell(G)})=1/\phi^2\quad\Rightarrow\quad \theta_{12}^{\ell(G)}\approx 33.7722^\circ,
$$
$$
\cosh(2\theta_{23}^{\ell(G)})=\phi^2\quad\Rightarrow\quad \theta_{23}^{\ell(G)}\approx 46.3214^\circ.
$$
Both schemes are reported to agree with data within $1\sigma$.

The reactor angle is then tied to the heptadecagon. Since $\epsilon_{17}=360^\circ/17$, the seed is
$$
\alpha_C=\epsilon_{17}/2=180^\circ/17\approx 10.5882^\circ.
$$
The prediction is
$$
\theta_{13}^{\ell(P5)}=\theta_{23}^{\ell(P5)}\tan\alpha_C
=\sin54^\circ\cdot\tan(180^\circ/17)\approx 8.66^\circ,
$$
which is compared with $\theta_{13}^{\ell(\exp)}=8.56^\circ$. The associated radius parameter is
$$
R_2^{\ell(P5)}=\sin54^\circ/\cos\alpha_C\approx 0.823\simeq R_2^{\ell(\exp)}.
$$

For quarks, the Cabibbo angle is generated from the lepton seed by the empirical relation $\theta_C/\theta_{12}^{\hat\ell}\simeq 1/\phi^2$. The chosen “para-golden” fraction is
$$
b^{(P5)}=\cos(2\cdot 33.6776^\circ)\approx 0.3850,
$$
so that
$$
\theta_C^{(P5)}=b^{(P5)}\theta_{12}^{\ell(P5)}\approx 0.3850\cdot 33.6776^\circ\approx 12.97^\circ,
$$
reported to lie within $1\sigma$ of the measured $13.004^\circ$. The second quark angle is then
$$
\theta_{23}^{q(P5)}=\theta_C^{(P5)}\tan\alpha_C\approx 12.97^\circ\cdot\tan(10.588^\circ)\approx 2.424^\circ,
$$
to be compared with $2.397^\circ$. For the smallest quark angle, the empirical ratio $\alpha_2^q/\alpha_C\approx \tfrac12$ motivates the choice $\alpha_2^q=\alpha_C/2$, giving
$$
\theta_{13}^{q}=\theta_{23}^{q}\tan(\alpha_C/2)\approx 2.424^\circ\cdot\tan(5.2941^\circ)\approx 0.2247^\circ,
$$
compared with $0.2138^\circ$.

The same framework also notes that all three quark angles can be written, $\epsilon$-approximately, as fractions of $\pi$:
$$
\theta_C\simeq \pi\,b\,\tan\alpha_C,\qquad
\theta_{23}^q\simeq \pi\,b\,\tan^2\alpha_C,\qquad
\theta_{13}^q\simeq \pi\,b\,\tan^2\alpha_C\,\tan(\alpha_C/2).
$$

## 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
$$
\cos\alpha_{P5}+\cos\alpha_C\simeq \pi/2
$$
with reported accuracy $\sim 10^{-5}$, leading to
$$
\theta_{12}^{\ell(P5)}+\theta_{C(n)}=\pi/2,
$$
where $\theta_{C(n)}=\theta_C/R_1^q=\cos\alpha_C$ [2508.00030]. A second relation identifies the normalized Cabibbo angle with the normalized lepton $(23\text{--}13)$ angle:
$$
\theta_{C(n)}=\theta_{23(n2)}^\ell.
$$

The Weinberg angle is then inserted through the empirical ratios
$$
\theta_W/\theta_{C(n)}\simeq \tfrac12,\qquad \theta_W/\theta_{23(n2)}^\ell\simeq \tfrac12,
$$
which imply
$$
\theta_W=\tfrac12\cos\alpha_C\simeq \tfrac12\cos(180^\circ/17)\simeq 28.1601^\circ,
$$
to be compared with the on-shell value $28.1931^\circ$. The resulting WQLC sum rule is
$$
\theta_{12}^{\ell(P5)}+2\theta_W=\pi/2,
$$
equivalently
$$
\sin^2\theta_W=\tfrac12(1-\sin\theta_{12}^\ell).
$$

The electroweak couplings are then expressed phenomenologically in terms of $\phi$. Starting from $\tan\theta_W=g'/g$ and $f_W=\tfrac12$, one finds
$$
\tan\theta_W=\frac{1-t}{1+t},
$$
where $t=\tan\theta_W\approx 0.3027$. Imposing the phenomenological condition $g'+g=1$ gives
$$
g'=\frac{1-t}{2},\qquad g=\frac{1+t}{2}.
$$
Using
$$
M_W=(gv)/2,\qquad M_Z=(v/2)\sqrt{g^2+g'^2},
$$
the framework reports
$$
M_W\approx 80.19\,\mathrm{GeV},\qquad M_Z\approx 90.95\,\mathrm{GeV},
$$
at the per-mille level. The electric charge is
$$
e=\frac{gg'}{\sqrt{g^2+g'^2}}=\frac{1-t^2}{2\sqrt2\,\sqrt{1+t^2}}\approx 0.3074,
$$
and the fine-structure constant becomes
$$
\alpha=\frac{e^2}{4\pi}=\frac{(1-t^2)^2}{32\pi(1+t^2)}\approx 7.533\times10^{-3}\simeq 1/133.
$$

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 $X$ equipped with a finite simplicial triangulation $K$. Writing $\mathrm{Fc}(K)$ for the poset of simplices ordered by the face relation $\sigma\le\tau$, and $\mathrm{Ab}$ for the category of abelian groups, a sheaf on $K$ is taken to be a covariant functor
$$
\mathcal F:\mathrm{Fc}(K)\longrightarrow \mathrm{Ab},
$$
while a cosheaf is a functor
$$
\mathcal G:\mathrm{Fc}(K)^{\mathrm{op}}\longrightarrow \mathrm{Ab}
$$
[1812.05593].

A bisheaf $\mathcal B=(\mathcal F,\mathcal G,\varphi)$ consists of such a sheaf and cosheaf together with homomorphisms
$$
\varphi_\sigma:\mathcal F(\sigma)\to \mathcal G(\sigma)
$$
for each simplex $\sigma$, subject to commutativity of the square
$$
\begin{CD}
\mathcal F(\sigma) @>{\mathcal F(\sigma\le\tau)}>> \mathcal F(\tau)\\
@V{\varphi_\sigma}VV @VV{\varphi_\tau}V\\
\mathcal G(\sigma) @<{\mathcal G(\sigma\le\tau)}<< \mathcal G(\tau).
\end{CD}
$$
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 $K$ if for every face relation $\sigma\le\tau$, both the sheaf restriction map $\mathcal F(\sigma\le\tau)$ and the cosheaf extension map $\mathcal G(\sigma\le\tau)$ are isomorphisms in $\mathrm{Ab}$. In sheaf-theoretic language, this means that $\mathcal F$ is locally constant on each star of $K$, and similarly for $\mathcal G$.

The associated notion of a $\mathcal B$-stratification is a filtration by simplicial subcomplexes
$$
\emptyset=K_{-1}\subset K_0\subset\cdots\subset K_m=K,
$$
whose strata satisfy three axioms: dimension, frontier, and constructibility. A $d$-stratum must contain at least one $d$-simplex and no simplex of higher dimension; the frontier relation must define a graded partial order; and if $\sigma\le\tau$ lie in the same connected stratum, then both $\mathcal F(\sigma\le\tau)$ and $\mathcal G(\sigma\le\tau)$ must be isomorphisms.

## 6. Canonical stratification, localization, and algorithmic construction

The bisheaf paper identifies a canonical $\mathcal B$-stratification, defined as the unique minimal element in the refinement order: every other $\mathcal B$-stratification refines it [1812.05593]. Its main theorem states existence and uniqueness of such a filtration
$$
\emptyset=K_{-1}\subset K_0\subset\cdots\subset K_m=K,
$$
and further states that its $d$-dimensional strata are exactly the isomorphism classes of $d$-simplices in a suitable localization of the face poset of $K$.

The construction proceeds inductively for $d=m,m-1,\dots,0$ by defining a subcomplex $K_{d-1}\subset K_d$ and a set
$$
W_d\subset \{(\sigma\le\tau)\mid \sigma,\tau\in K_d\}
$$
of face relations closed under composition. The complement $K_d\setminus K_{d-1}$ is the union of those simplices of dimension $d$ that become isomorphic in the localized category $\mathrm{Fc}(K)[W_d^{-1}]$, while $W_d$ 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
$$
K_m=K,\qquad
W_m=\{(\sigma\le\tau)\mid \mathcal F(\sigma\le\tau),\mathcal G(\sigma\le\tau)\text{ are iso's}\}.
$$
Having defined $(K_d,W_d)$, one removes from $K_d$ those simplices of dimension $d$ that become isomorphic to some higher-dimensional simplex in the localization, obtaining $K_{d-1}$. One then enlarges $W_d$ to $W_{d-1}$ 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 $\mathcal F$ and $\mathcal G$ 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 $n$ is the number of simplices of $K$, the complexity is stated as at worst
$$
O(m\,n+n\log n)
$$
once matrix invertibility tests have been preprocessed. An equivalent nerve-theoretic description views $\mathcal B$ 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 $1$-simplex with vertices $v_0,v_1$ and edge $e$. If $\mathcal F$ and $\mathcal G$ are constant of rank $1$ except that $\mathcal G(v_0\le e)$ is zero while $\mathcal F(v_0\le e)$ is the identity, then $W_1$ contains only $v_1\le e$. In the localized category, $v_1$ becomes isomorphic to $e$, yielding one $1$-stratum $\{v_1,e\}$, while $v_0$ forms its own $0$-stratum. The canonical stratification is therefore
$$
K_{-1}=\emptyset,\qquad K_0=\{v_0\},\qquad K_1=K.
$$

## 7. Applications, significance, and terminological caution

The bisheaf-theoretic notion is motivated by homological stability of fibers. For a tame map
$$
f:X\to M
$$
into a real-analytic, triangulated manifold $M$ of dimension $m$, one defines
$$
\mathcal F(\sigma)=H^{\mathrm{BM}}_{\ast+m}(f^{-1}(\mathrm{star}(\sigma))),
$$
$$
\mathcal G(\sigma)=H_\ast(f^{-1}(\mathrm{star}(\sigma))),
$$
and lets $\varphi_\sigma$ be cap-product with the pullback of a generator of $H_c^m(\mathrm{star}(\sigma))$ [1812.05593]. This produces a bisheaf constructible with respect to a chosen triangulation. Its canonical stratification identifies the coarsest partition of $M$ on which the “persistent local system” of subquotients
$$
\mathrm{im}\bigl(\varphi_\sigma:\mathcal F(\sigma)\to\mathcal G(\sigma)\bigr)
$$
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 $n=5$ and $n=17$, the golden ratio appears naturally from $P_5$, the heptadecagon contributes nested radicals expressible in terms of $\phi$ up to $\epsilon$-level, and the Standard Model couplings $g$, $g'$, and $\alpha_{\mathrm{em}}$ admit elegant expressions in $\phi$ alone [2508.00030]. The paper further suggests a low-energy unification pattern summarized schematically as geometry of Fermat-prime polygons $\to$ mixing and coupling constants $\to$ 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.

Source: https://www.emergentmind.com/topics/bi-constructible