Local Coherence, Global Quotients, and Task‑Relative Identification

“A recently announced counterexample to the Jacobian conjecture demonstrates a structural gap: local coherence does not guarantee global injectivity. The SIOS distinguishability framework generalises this insight — and shows why it matters for AI representations, policies, and world-models.”

Scope

This note introduces a conceptual framework rather than a new mathematical theory. All mathematical objects used—equivalence relations, quotient spaces, fibre cardinality, geometric degree, differential rank and singular values—are standard.

The contribution of SIOS is to organise these notions into a unified framework for analysing:

• local coherence,
• global distinguishability,
• and task‑relative admissibility

across mathematical and computational systems.

Abstract

Local differential coherence does not guarantee global injectivity. The recently announced counterexample to the Jacobian conjecture demonstrates this sharply: a polynomial map $F:{\mathbb{C}}^3\to {\mathbb{C}}^3$ with constant nonzero Jacobian determinant is everywhere locally biholomorphic yet globally non‑injective.

This note introduces a SIOS framework for analysing such local–global decouplings. The framework is not a new mathematical theory; it is a geometric organisation of existing notions—local differential behaviour, global identification structure, and task‑relative admissibility—into a unified object:

$$\boxed{{\displaystyle \mathfrak{S}(F)=({\mathcal{D}}_F,{\mathrm{\Delta }}_F,{\mathcal{Q}}_F)}}$$

The Jacobian counterexample serves as a canonical illustration of a map that is locally coherent yet globally non‑injective. The SIOS framework generalises this insight to mathematical maps, AI representations, policies, world‑models and evaluators.

1. Introduction

Many systems—mathematical, computational, or cognitive—exhibit a structural gap between local coherence and global distinguishability. A map may behave perfectly in every infinitesimal neighbourhood while still failing to preserve distinctions that matter globally.

The Jacobian counterexample is a particularly clean mathematical instance of this broader phenomenon, where the failure appears as loss of global injectivity:

$$\det JF\equiv c\mathrm{\ne }0\text{centernot}\Rightarrow F\text{ injective}\mathrm{.}$$

SIOS does not modify this theorem. Instead, SIOS provides a geometric framework for interpreting such phenomena across domains.

2. Exact Identification: The Induced Equivalence Relation

For any map $F:X\to Y$, the correct nonlinear notion of “identified states” is the induced equivalence relation:

$$x{\sim }_F{x}^{\mathrm{\prime }}⟺F(x)=F(x^{\mathrm{\prime }})\mathrm{.}$$

This partitions $X$ into fibres. The quotient $X\mathrm{/}{\sim }_F$ is the exact global identification structure of the map.

Task‑relative admissibility

Let ${\sim }_{\mathrm{a}\mathrm{d}\mathrm{m}}$ be the equivalence relation representing distinctions the task permits to be erased.

Exact quotient legitimacy is:

$${\mathcal{Q}}_F^{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}=\mathbf{1}[{\sim }_F\subseteq {\sim }_{\mathrm{a}\mathrm{d}\mathrm{m}}]\mathrm{.}$$

Ideal abstraction is:

$${\sim }_F={\sim }_{\mathrm{a}\mathrm{d}\mathrm{m}}\mathrm{.}$$

3. Approximate Identification: Tolerance Relations

In many continuous or high‑dimensional operational settings, exact equality is too restrictive to serve as the sole collision criterion.

Define the tolerance relation:

$$x{\approx }_{F,\epsilon }{x}^{\mathrm{\prime }}⟺d_Y(F(x),F(x^{\mathrm{\prime }}))\le \epsilon \mathrm{.}$$

This relation is reflexive and symmetric but not transitive. If an equivalence is required, use the transitive closure:

$${\sim }_{F,\epsilon }^{\ast }=\mathrm{EqClosure}({\approx }_{F,\epsilon })\mathrm{.}$$

Operational evaluation typically uses ${\approx }_{F,\epsilon }$ directly.

4. Local Differential Coherence

Local coherence is represented abstractly by a differential‑coherence profile. When $X$ and $Y$ are equipped with specified Riemannian or Hermitian metrics, one possible profile is:

$${\mathcal{D}}_F(x)=(\mathrm{rank}d{F}_x,{\sigma }_{\min }(d{F}_x),{\kappa }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}(d{F}_x))\mathrm{.}$$

• Rank is intrinsic.
• Singular values and conditioning are metric‑relative.
• For maps $X^m\to Y^n$ with $m\le n$, full column rank characterises immersion‑type local coherence.
• For equal‑dimensional maps, full rank gives local invertibility.

5. Global Distinguishability Defect

5.1 Exact defect (finite fibres)

$${\mathrm{\Delta }}_F(y)=\mathrm{\#}{F}^{-1}(y)-1.$$

5.2 Generic defect (algebraic maps)

$${\mathrm{\Delta }}_{\mathrm{g}\mathrm{e}\mathrm{n}}(F)={\mathrm{deg}}_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(F)-1.$$

Recent structural analyses identify geometric degree $3$ for the Jacobian counterexample, giving ${\mathrm{\Delta }}_{\mathrm{g}\mathrm{e}\mathrm{n}}(F)=2$.

5.3 Approximate defect (continuous systems)

Define the harmful collision probability:

This measures how often states that must remain distinguishable are represented as effectively identical.

6. Quotient Legitimacy

6.1 Exact legitimacy

$${\mathcal{Q}}_F^{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}=\mathbf{1}[{\sim }_F\subseteq {\sim }_{\mathrm{a}\mathrm{d}\mathrm{m}}]\mathrm{.}$$

6.2 Distributional legitimacy

Let $(X_1,X_2)\sim \nu$ be a chosen pair distribution.

Warning: Distributional legitimacy is measure‑relative and weaker than exact legitimacy. In continuous settings, exact collisions may have probability zero even when inadmissible fibres exist. Thus:

$${\mathcal{Q}}_F^{\nu }=1\nRightarrow {\mathcal{Q}}_F^{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}=1.$$

Operational evaluation should instead use tolerance‑based harmful‑collision probability.

6.3 Ideal abstraction

$${\sim }_F={\sim }_{\mathrm{a}\mathrm{d}\mathrm{m}}\mathrm{.}$$

7. Intrinsic Geometry and Task‑Relative Interpretation

The intrinsic geometry of a map is described by:

$$({\mathcal{D}}_F,{\mathrm{\Delta }}_F)\mathrm{.}$$

Task‑relative interpretation introduces ${\mathcal{Q}}_F$.

Classification

Local differential coherence	Global defect	Quotient legitimacy	Interpretation
Admissible	(0)	(1)	Locally coherent and globally distinguishable
Admissible	(>0)	(1)	Constraint‑valid abstraction
Admissible	(>0)	(0)	Hidden global aliasing
Inadmissible	Any	Any	Local differential failure or near‑collapse

8. Why Distinguishability Matters

Established notions such as Lyapunov stability, structural stability and numerical stability address dynamical, perturbative or computational robustness. They do not, by themselves, encode whether a map’s induced identifications are permitted by a task‑relative admissibility relation.

SIOS introduces distinguishability and quotient legitimacy as additional analytical axes rather than replacements for existing stability notions.

This allows compression, abstraction, policy equivalence, representation learning, symbolic reasoning and causal modelling to be analysed using one unified language.

9. The Jacobian Counterexample as Illustration

The announced polynomial map has been independently formally verified in Isabelle/HOL to have constant Jacobian determinant $-2$ and an explicit fibre containing three distinct points. Recent structural analysis further identifies its generic geometric degree as $3$.

Thus:

$${\mathrm{\Delta }}_{\mathrm{g}\mathrm{e}\mathrm{n}}(F)=2.$$

Within the SIOS interpretation, this is a canonical example of a locally coherent yet globally aliased mapping.

10. The SIOS Distinguishability Principle

$$\boxed{{\displaystyle \text{Task‑valid distinguishability requires admissible local coherence and constraint‑valid global identification.}}}$$

Exact form:

$$\boxed{{\displaystyle {\mathcal{D}}_F\text{ admissible}\wedge {\sim }_F\subseteq {\sim }_{\mathrm{a}\mathrm{d}\mathrm{m}}\mathrm{.}}}$$

Approximate operational form:

$$\boxed{{\displaystyle {\mathcal{D}}_F\text{ admissible}\wedge {\mathrm{\Delta }}_{F,\epsilon }^{\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}}\le {\delta }_{\max },}}$$

where $\epsilon$, ${\delta }_{\max }$, and the pair distribution $\nu$ are fixed by the task and evaluation protocol.

11. Conclusion

This note establishes the initial SIOS formulation of distinguishability geometry. Future work will test, refine and extend the framework through applications to AI representations, policies, world models and evaluators.