Branches and the Discriminant

model derivation

The most satisfying concrete grounding in this note set: for an algebraic factor, the abstract latent space $[[c]]$ of Definition 1 is literally the set of solution branches, and the discriminant is where four different things fail simultaneously.

Sources: original to this vault (design and analysis; no single paper).

Theory (CT-ML wiki): Bayesian Inversion · Statistical Game · Open Model

The projection is a branched cover

Fix a polarity. The projection

$\pi :V_{\mathbb{R}}(r_{\Theta })⟶{\mathbb{R}}^p,(x_o,x_u)\mapsto x_o$

is, away from a proper closed subset, a covering map of degree $D$: every $x_o$ in a connected component of the complement has exactly $D$ preimages, varying smoothly.

$\boxed{[[c]]={\pi }^{-1}(x_o)=\{1,\dots ,D\}\text{, the branch index}}$

So an algebraic factor’s forward kernel really does have the type of Definition 1,

$c:X⇝[[c]]\times Y$

and the latent variable has a meaning: which solution you are on. Elbow up or elbow down. Which of the ten essential matrices. This is not a bookkeeping device; it is the physically meaningful hidden state.

Everything the vault says abstractly about $[[c]]$ now has a concrete reading:

abstract (Open Models and Latent Channels, Composition of Open Models)	algebraic reading
latent space $[[c]]$	the set of branches over $x_o$
reveal — a free retyping	tell the consumer which root was taken
marginalising $[[c]]$	sum/average over all real roots
composition files $Y$ into the latent	the intermediate branch choice is remembered
a pure model, $[[c]]\cong 1$	$D=1$: the relation is a function in this polarity

That last row is worth pausing on. A factor is a function exactly when its branch count is one. Explicit learning is the special case $D\equiv 1$, and the entire “implicit” story is about what happens when $D>1$. Implicit Learners’s table row “may be multi-valued” is the statement $D>1$; “or have no solution” is $D=0$.

The discriminant

$D$ is not constant. It changes across the discriminant locus

$\Delta =\{x_o:\exists x_u,r_{\Theta }(x_o,x_u)=0\text{and}\mathrm{rank}{J}_u(x_o,x_u)<q\}$

— the set of observed values over which two branches collide. For a square system, $\Delta =\{x_o:\mathrm{Res}(\cdot )=0\}$, the vanishing of a resultant, hence itself an algebraic hypersurface.

Four failures, one locus

At $\Delta$, all of the following happen at the same points, and they are the same phenomenon seen from four sides:

1. The implicit function theorem fails. $J_u$ is singular, so $\partial x_u/\partial x_o=-J_u^{-1}{J}_o$ is undefined and the adjoint solve of Backpropagation by the Implicit Function Theorem is ill-conditioned. Gradients blow up like $1/\mathrm{dist}(x_o,\Delta )$.
2. Inference is discontinuous. Two real roots merge and become a complex-conjugate pair. The map $x_o\mapsto \{\text{real roots}\}$ is continuous only on the complement of $\Delta$.
3. The entropy jumps. With a uniform belief over branches, ${\mathbf{H}}^c=\log D$, which is a step function of $x_o$. The statistical game’s regulariser is genuinely discontinuous here.
4. The Sampson precision blows up. $(J{J}^{\top }{)}^{-1}$ of Algebraic versus Geometric Distance is singular exactly on $\Delta$ — the noise model says “infinite uncertainty along the collapsing direction”, which is correct and useless.

This is intrinsic, not a defect

Take the circle: $y=\pm \sqrt{1-x^2}$ is genuinely undefined for $|x|>1$ and genuinely two-valued for $|x|<1$, and $dy/dx\to \infty$ at $x=\pm 1$. No parametrisation, solver or regulariser removes this, because it is a property of the relation, not of the representation. Any implicit learner that permits multi-valued relations inherits it.

The honest engineering response is to detect it — $\kappa (J_u)$ is a cheap, exact proximity indicator to $\Delta$ — and to report it, rather than to return a confident gradient that is numerically meaningless.

The branch belief

The natural output of an algebraic inversion is therefore not a point but a belief over a finite set:

$c_{\pi }^{\prime }(x_o)={\sum }_{j=1}^D{w}_j{\delta }_{x_u^{(j)}},w_j\propto \pi \left(x_u^{(j)}\right)$

a SampleBelief supported on the real roots, weighted by the prior. Two readings:

• Selection as a prox. Taking $\arg {\max }_j{w}_j$ is $\arg {\min }_{x_u:r=0}-\log \pi (x_u)$ — a proximal step onto the variety. That is exactly the structure of the diffusion family in ImplicitREDDiff, with the hard constraint $r=0$ in place of a soft energy. The algebraic and diffusion families differ in how they enforce the constraint, not in what they compute.
• Entropy from branch count. ${\mathbf{H}}^c(\pi ,x_o)=-{\sum }_j{w}_j\log w_j\le \log D$. The abstract “entropy or regularizer” of Definition 20 is here a perfectly ordinary discrete entropy over a set you can enumerate.

Monodromy — the branches are not independently labelled

Transporting $x_o$ around a loop in ${\mathbb{R}}^p\setminus \Delta$ can permute the branches. The resulting monodromy group $\subseteq S_D$ measures how globally inconsistent any branch labelling is: if the monodromy is transitive, there is no continuous global choice of branch, and the “elbow up” branch is only locally well defined.

Two consequences:

• Any implementation that caches “which branch we took last time” is making a local choice that cannot be made global. Message passing around a cycle in the factor graph can return to a different branch than it started on — a genuine and under-appreciated failure mode for equilibrium-style iteration.
• Monodromy is also a tool: monodromy loops are the cheapest known way to find many solutions of a polynomial system from one, and are how modern solvers bootstrap.

Related: Inference as Root Finding, Open Models and Latent Channels, Factors are Parameterized Statistical Games, The Algebraic Factor as a Statistical Game