GVSS-09 — Multi-Provider Visual Capability Portfolios, Fallback Geometry, and Federated Reachability

多生成器視覺能力投資組合、Fallback 幾何與聯邦可達域：覆蓋次模性、失效相關、切換成本與穩健多重覆蓋

Series: Global Visual Space & Generative Navigation — Paper 09
Bridge: GVSS × frozen Reflexive Representation Theory (RRT)
Author: Neo.K / EveMissLab
Version: v0.1
Date: 2026-08-17
Status: Formal provider-portfolio paper. Federated reachable-set union, coverage monotonicity and submodularity, marginal reachability gain, reachability redundancy, worst-case outage multiplicity, union-redundancy/robustness-redundancy separation, switching-cost practical reachability, two-provider correlated-failure diversification, Fréchet reliability bounds, pairwise-correlation insufficiency, critical-region weighted coverage, cardinality-constrained greedy coverage, and portfolio Pareto statements are proved under the stated hypotheses. Multi-model T2I routing, edge/cloud routing, mixture-of-experts image generation, model serving, ensemble diversity, and submodular coverage optimization are prior research and are not claimed as GVSS inventions. No strong novelty claim is made.

Keywords: multi-provider image generation, text-to-image routing, provider portfolio, visual reachability, fallback, robust coverage, correlated failure, submodular coverage, model routing, capability diversification, GVSS

Abstract

GVSS-08 treated provider/model identity as uncertain and introduced:

nominal control;
robust control;
recalibration;
fallback;
quarantine;
stop.

It maintained a joint belief:

\boxed{ \beta_t(F,\nu) = P( F_t=F, \nu_t=\nu \mid H_t ). }

GVSS-09 changes the question.

Instead of asking:

Which one provider is currently active?

it asks:

What capability is created by maintaining a portfolio of providers, models, editing systems, and fallback runtimes simultaneously?

Let the available provider set be:

\boxed{ \mathcal P = \{ 1,\ldots,N \}. }

Provider $\nu$ has a declared visual reachable set:

\boxed{ \mathcal R_\nu \subseteq \Omega_\Sigma. }

If providers operate in different native visual specifications:

\Omega_{\Sigma_\nu},

let a declared semantics-preserving or explicitly approximate normalization map be:

\boxed{ \tau_\nu: \Omega_{\Sigma_\nu} \to \Omega_\Sigma. }

Then all portfolio geometry is computed on:

\boxed{ \widetilde{\mathcal R}_\nu = \tau_\nu( \mathcal R_\nu ). }

For readability, the tilde is dropped below.

Under zero switching/activation cost and a policy allowed to select any provider, the raw portfolio reachable domain is:

\boxed{ \mathcal R(S) = \bigcup_{\nu\in S} \mathcal R_\nu }

for:

S\subseteq\mathcal P.

This union is an upper bound on practical finite-budget reachability.

It is not a claim that switching is free.

Coverage geometry

Let:

\mu

be a finite nonnegative measure on the visual state space.

It can represent:

raw state-count measure in a finite benchmark;
task-frequency weighting;
semantic-region importance;
project-critical visual mass;
empirical benchmark distribution.

Define portfolio coverage:

\boxed{ f(S) = \mu \left( \bigcup_{\nu\in S} \mathcal R_\nu \right). }

Then:

\boxed{ f(\varnothing)=0, }

\boxed{ A\subseteq B \Longrightarrow f(A)\le f(B), }

and:

\boxed{ f(A\cup\{\nu\})-f(A) \ge f(B\cup\{\nu\})-f(B) }

whenever:

A\subseteq B.

Therefore portfolio visual coverage is a monotone submodular set function.

The marginal reachability gain of provider $\nu$ is exactly:

\boxed{ \Delta_\mu( \nu\mid S ) = \mu \left( \mathcal R_\nu \setminus \bigcup_{j\in S} \mathcal R_j \right). }

As the portfolio grows, the new visual mass contributed by an additional provider can only decrease.

This gives a formal diminishing-returns law for provider accumulation.

A provider is reachability-redundant relative to $S$ if:

\boxed{ \Delta_\mu( \nu\mid S )=0. }

For ordinary set cardinality/counting measure, this is equivalent to:

\boxed{ \mathcal R_\nu \subseteq \bigcup_{j\in S} \mathcal R_j. }

For a general measure, redundancy is only up to $\mu$ -null differences.

Redundancy is not robustness redundancy

Raw union coverage ignores provider failure.

Let coverage multiplicity of image/state $I$ be:

\boxed{ m_S(I) = \sum_{\nu\in S} \mathbf 1 \{ I\in\mathcal R_\nu \}. }

Suppose the runtime must remain able to reach an image after any $r$ providers in the portfolio become unavailable.

Define worst-case $r$ -outage reachable set:

\boxed{ \mathcal R_{\mathrm{rob}}^{(r)}(S) = \bigcap_{ F\subseteq S,\, |F|\le r } \bigcup_{ \nu\in S\setminus F } \mathcal R_\nu. }

Then:

\boxed{ \mathcal R_{\mathrm{rob}}^{(r)}(S) = \{ I: m_S(I)\ge r+1 \}. }

Thus worst-case robust reachability is exactly a multi-coverage condition.

This immediately yields an important no-go.

Take two providers:

\mathcal R_1 = \mathcal R_2 = A.

Provider 2 contributes zero raw union coverage:

\boxed{ \Delta(2\mid\{1\})=0. }

But:

\mathcal R_{\mathrm{rob}}^{(1)}(\{1\}) = \varnothing,

while:

\boxed{ \mathcal R_{\mathrm{rob}}^{(1)}(\{1,2\}) = A. }

Therefore:

\boxed{ \text{reachability redundancy} \not\Rightarrow \text{robustness redundancy}. }

A provider can be visually redundant and operationally indispensable as a fallback.

Switching and calibration cost

Let provider $\nu$ have activation/calibration/switch cost:

\boxed{ s_\nu\ge0. }

Let:

g_\nu(I)

be minimum provider-specific generation/search cost required to reach visual state $I$ .

Under a one-provider-per-final-output policy and total budget $B$ , define:

\boxed{ \mathcal R_\nu(B) = \{ I\in\mathcal R_\nu: s_\nu+g_\nu(I)\le B \}. }

The practical portfolio reachable set is:

\boxed{ \mathcal R_{\mathrm{pr}}(S,B) = \bigcup_{\nu\in S} \mathcal R_\nu(B). }

Therefore:

\boxed{ \mathcal R_{\mathrm{pr}}(S,B) \subseteq \mathcal R(S). }

The zero-cost union is only an upper bound.

Increasing budget cannot reduce:

\mathcal R_{\mathrm{pr}}(S,B)

under nested feasibility.

Switching and calibration debt can therefore make a nominally large provider portfolio practically smaller than its union geometry suggests.

Correlated provider failure

Provider diversity must be measured through failure behavior, not names.

For a fixed critical target region/task, let:

X_\nu = \mathbf 1 \{ \text{provider }\nu\text{ fails} \}.

For two providers define:

q_1=P(X_1=1),

q_2=P(X_2=1),

and joint failure:

\boxed{ q_{12} = P( X_1=1, X_2=1 ). }

The portfolio succeeds when not both fail:

\boxed{ P( \text{portfolio success} ) = 1-q_{12}. }

The best single provider succeeds with probability:

1-\min(q_1,q_2).

Therefore diversification gain is:

\boxed{ G_{\mathrm{div}} = \min(q_1,q_2)-q_{12}. }

This gain is zero exactly when the better provider's failures are fully contained in the other provider's failure events:

q_{12} = \min(q_1,q_2).

Under independent failures:

q_{12}=q_1q_2.

But independence is a modeling assumption, not a consequence of provider/model-name diversity.

Fréchet bounds give:

\boxed{ \max( 0, q_1+q_2-1 ) \le q_{12} \le \min(q_1,q_2). }

Hence portfolio success satisfies:

\boxed{ 1-\min(q_1,q_2) \le P(\mathrm{success}) \le 1-\max( 0, q_1+q_2-1 ). }

The same marginal failure rates can therefore support very different portfolio reliability depending on dependence.

Pairwise correlation is insufficient for three or more providers

A provider portfolio can have higher-order shared blind spots not captured by pairwise correlations.

Consider three binary provider failure indicators:

X_1,X_2,X_3.

Model A — independent fair failures

Uniform distribution over all:

8

binary triples.

Then:

P(X_i=1)=1/2,

P(X_i=X_j=1)=1/4,

and:

\boxed{ P(X_1=X_2=X_3=1)=1/8. }

Model B — even-parity law

Uniform distribution over:

000, 011, 101, 110.

Again:

P(X_i=1)=1/2,

P(X_i=X_j=1)=1/4,

so all pairwise correlations are zero just as in the independent model.

But:

\boxed{ P(X_1=X_2=X_3=1)=0. }

Thus identical marginal failure rates and identical pairwise correlation matrix do not determine all-provider failure probability.

Therefore:

\boxed{ \text{pairwise failure diversity} \not\Rightarrow \text{known portfolio tail reliability}. }

Higher-order failure structure matters.

Critical-region weighting

Raw reachable-set size can also mislead.

Let:

w(I)\ge0

be a visual/task importance density.

Define:

\boxed{ \mu_w(A) = \int_A w(I)\,d\mu(I). }

A small stable provider can have lower unweighted coverage but larger critical weighted coverage.

For example, let the visual benchmark have ten equally sized regions.

Provider A reaches nine low-importance regions.

Provider B reaches only one region.

If that one region carries weight:

100

while each of A's regions carries weight:

1,

then:

|\mathcal R_A|>|\mathcal R_B|,

but:

\boxed{ \mu_w(\mathcal R_B) > \mu_w(\mathcal R_A). }

Thus:

\boxed{ \text{larger raw reachable set} \not\Rightarrow \text{larger project value}. }

Fallback portfolios should be evaluated on the visual regions that matter.

Greedy provider selection

Because:

f(S) = \mu( \cup_{\nu\in S}\mathcal R_\nu )

is monotone submodular, the classical greedy algorithm for a cardinality budget $k$ :

repeatedly add the provider with largest current marginal reachability gain,

satisfies:

\boxed{ f(S_{\mathrm{greedy}}) \ge \left[ 1- \left( 1-\frac1k \right)^k \right] f(S^\star) \ge (1-1/e) f(S^\star) }

where $S^\star$ is the optimal size- $k$ portfolio.

This is the classical maximum-coverage/submodular-greedy guarantee, not a GVSS novelty.

It supplies a practical baseline for provider portfolio construction when the objective is pure measured coverage.

Once calibration debt, switching cost, failure correlation, and robustness constraints enter, pure greedy coverage is no longer sufficient.

Routing literature

Provider portfolios are becoming operational in current T2I systems.

Cost-Aware Routing for Efficient Text-to-Image Generation explicitly routes prompts among nine pre-trained T2I generation functions/models according to prompt complexity and cost-quality tradeoffs, reporting higher average quality than any single model under its evaluation.

Adaptive edge-cloud T2I routing routes prompts between edge and cloud image models to trade quality against cloud cost.

OctoT2I uses stateful multi-round routing across T2I tools with a self-evolving capability knowledge base.

HADIS studies adaptive serving/cascading of diffusion models.

These systems are direct precedent for provider routing.

GVSS-09 does not claim multi-model routing as new.

Its contribution is to add visual reachable-set portfolio geometry and robust multi-coverage to the routing problem.

Central conclusion

The relevant provider portfolio is not the list of model names.

It is:

\boxed{ \left( \{ \mathcal R_\nu \}, \{ s_\nu \}, \text{failure joint law}, \text{calibration state}, \text{provider provenance} \right). }

The canonical GVSS-09 principle is:

\boxed{ \textbf{ A provider adds value in three different ways: it can add new visual territory, add an independent route to existing territory, or reduce the cost/risk of reaching critical territory. These are not the same notion of diversity. } }

1. Position in the GVSS sequence

GVSS-03 introduced one generator's bounded reachable domain.

GVSS-04 made the search regime adaptive.

GVSS-05 diagnosed why a visual trajectory failed.

GVSS-06 selected corrective/diagnostic actions.

GVSS-07 learned provider-specific diagnostic models.

GVSS-08 controlled robustly under provider/model drift.

GVSS-09 studies multiple providers as one federated visual capability portfolio.

2. Provider definition

Definition GVSS09-D1

A provider is a versioned generation/action subsystem:

\boxed{ \nu = ( G, P, \Pi, E, \Sigma_\nu, \mathsf{Prov} ). }

It can include:

base T2I model;
editing model;
local diffusion stack;
cloud image API;
LoRA/control bundle;
repair agent;
deterministic graphics backend.

Provider identity is versioned.

3. Common visual comparison space

Different providers can emit:

different resolution;
different color spaces;
different modalities;
layered/project artifacts.

A common reachability union is meaningful only after a comparison map is stated.

4. Normalization map

Definition GVSS09-D2

\boxed{ \tau_\nu: \Omega_{\Sigma_\nu} \to \Omega_\Sigma. }

If $\tau_\nu$ is lossy, the loss/defect must be recorded.

Portfolio reachability is always relative to the chosen common space.

5. Provider reachable set

\boxed{ \mathcal R_\nu \subseteq \Omega_\Sigma. }

This can mean:

exact finite support;
effective support;
budgeted empirical reachability;
accepted reachable region.

The semantics must be fixed before comparing providers.

6. Raw portfolio reachability

Definition GVSS09-D3

For:

S\subseteq\mathcal P,

\boxed{ \mathcal R(S) = \bigcup_{\nu\in S} \mathcal R_\nu. }

This assumes provider selection is allowed.

7. GVSS09-T1 — Federated union reachability

Theorem GVSS09-T1

Under zero switching/activation cost and a runtime allowed to select any provider in $S$ , the set of states reachable by one provider execution is exactly:

\boxed{ \mathcal R(S) = \bigcup_{\nu\in S} \mathcal R_\nu. }

Proof

Every output is produced by some selected provider and therefore lies in its reachable set.

Conversely, every state in some provider's reachable set is available by selecting that provider.

$\square$

For multi-stage cross-provider compositions, the reachable closure can be larger; GVSS09-T1 concerns one-provider terminal generation.

8. Multi-stage provider composition

If the runtime can:

\text{generate with }\nu_1 \to \text{edit with }\nu_2,

portfolio reachability should be defined using the closure under admissible provider kernels.

This can exceed the simple union.

GVSS-09 uses the union as the base portfolio geometry and records composition as future work.

9. Visual coverage measure

Let:

\boxed{ \mu }

be a finite nonnegative measure on $\Omega_\Sigma$ .

In finite benchmarks:

\mu(A)=|A|

is allowed.

10. Coverage set function

Definition GVSS09-D4

\boxed{ f(S) = \mu( \mathcal R(S) ). }

11. GVSS09-T2 — Coverage monotonicity

Theorem GVSS09-T2

If:

A\subseteq B,

then:

\boxed{ f(A)\le f(B). }

Proof

\cup_{\nu\in A}\mathcal R_\nu \subseteq \cup_{\nu\in B}\mathcal R_\nu.

Apply measure monotonicity.

$\square$

12. Marginal provider gain

Definition GVSS09-D5

\boxed{ \Delta_\mu( \nu\mid S ) = f( S\cup\{\nu\} ) - f(S). }

13. GVSS09-T3 — Marginal reachability formula

Theorem GVSS09-T3

\boxed{ \Delta_\mu( \nu\mid S ) = \mu \left( \mathcal R_\nu \setminus \bigcup_{j\in S} \mathcal R_j \right). }

Proof

For measurable sets:

\mu(A\cup B)-\mu(A) = \mu(B\setminus A).

Set:

A=\cup_{j\in S}\mathcal R_j,

B=\mathcal R_\nu.

$\square$

14. GVSS09-T4 — Visual coverage submodularity

Theorem GVSS09-T4

If:

A\subseteq B

and:

\nu\notin B,

then:

\boxed{ \Delta_\mu( \nu\mid A ) \ge \Delta_\mu( \nu\mid B ). }

Proof

Because:

\cup_{j\in A}\mathcal R_j \subseteq \cup_{j\in B}\mathcal R_j,

we have:

\mathcal R_\nu \setminus \cup_{j\in A}\mathcal R_j \supseteq \mathcal R_\nu \setminus \cup_{j\in B}\mathcal R_j.

Apply measure monotonicity.

$\square$

This is the diminishing-return law.

15. Reachability redundancy

Definition GVSS09-D6

Provider $\nu$ is $\mu$ -reachability-redundant relative to $S$ when:

\boxed{ \Delta_\mu(\nu\mid S)=0. }

16. GVSS09-T5 — Reachability redundancy characterization

Theorem GVSS09-T5

Provider $\nu$ is $\mu$ -redundant relative to $S$ iff:

\boxed{ \mu \left( \mathcal R_\nu \setminus \cup_{j\in S} \mathcal R_j \right) = 0. }

For counting measure on a finite benchmark this becomes:

\boxed{ \mathcal R_\nu \subseteq \cup_{j\in S} \mathcal R_j. }

Proof

Immediate from GVSS09-T3.

$\square$

17. Raw redundancy does not imply uselessness

A provider can add:

robustness;
lower latency;
lower cost;
better calibration;
better project-critical reliability;

without adding new union coverage.

Therefore redundancy must always name the criterion.

18. Coverage multiplicity

Definition GVSS09-D7

\boxed{ m_S(I) = \sum_{\nu\in S} \mathbf 1 \{ I\in\mathcal R_\nu \}. }

Multiplicity counts independent provider routes in set geometry.

It does not imply independent stochastic failure.

19. Worst-case outage model

Suppose any:

r

providers may become unavailable.

No probability distribution is assumed.

20. Robust reachable set

Definition GVSS09-D8

\boxed{ \mathcal R_{\mathrm{rob}}^{(r)}(S) = \bigcap_{ F\subseteq S,\, |F|\le r } \bigcup_{ \nu\in S\setminus F } \mathcal R_\nu. }

21. GVSS09-T6 — Robust reachability equals multi-coverage

Theorem GVSS09-T6

\boxed{ \mathcal R_{\mathrm{rob}}^{(r)}(S) = \{ I: m_S(I)\ge r+1 \}. }

Proof

If:

m_S(I)\le r,

remove every provider that reaches $I$ .

At most $r$ removals make $I$ unreachable.

Conversely, if:

m_S(I)\ge r+1,

removing at most $r$ providers leaves at least one provider reaching $I$ .

$\square$

22. Single-outage robust coverage

For:

r=1,

\boxed{ \mathcal R_{\mathrm{rob}}^{(1)}(S) = \{ I: m_S(I)\ge2 \}. }

A state needs at least two provider routes.

23. GVSS09-N1 — Union redundancy is not robustness redundancy

Take:

\mathcal R_1 = \mathcal R_2 = A.

Then:

\Delta(2\mid\{1\})=0.

But:

\mathcal R_{\mathrm{rob}}^{(1)}(\{1\}) = \varnothing,

whereas:

\boxed{ \mathcal R_{\mathrm{rob}}^{(1)}(\{1,2\}) = A. }

Thus a provider can be raw-coverage redundant and robustly essential.

$\square$

24. Robust marginal gain

Adding provider $\nu$ increases $r$ -outage robust coverage exactly on points that:

lie in $\mathcal R_\nu$ ;
previously had multiplicity exactly $r$ .

25. GVSS09-C1 — Robust marginal-gain formula

For counting/measure-compatible multiplicity sets:

\boxed{ \Delta_{\mathrm{rob}}^{(r)} ( \nu\mid S ) = \mu \left( \{ I\in\mathcal R_\nu: m_S(I)=r \} \right). }

Proof

A new provider increments multiplicity by one on $\mathcal R_\nu$ .

Only points moving from $r$ to $r+1$ cross the robust threshold.

$\square$

26. Provider switching cost

Let:

s_\nu

contain:

activation;
API setup;
calibration;
prompt translation;
format conversion;
provider rebind;
provenance synchronization.

27. Generation cost

Let:

g_\nu(I)

be minimal per-state cost.

It may be infinite when $I$ is unreachable.

28. Budgeted provider reachable set

Definition GVSS09-D9

\boxed{ \mathcal R_\nu(B) = \{ I: s_\nu+g_\nu(I)\le B \}. }

29. Practical portfolio reachability

Definition GVSS09-D10

\boxed{ \mathcal R_{\mathrm{pr}}(S,B) = \bigcup_{\nu\in S} \mathcal R_\nu(B). }

30. GVSS09-T7 — Practical reachability is bounded by raw union

Theorem GVSS09-T7

\boxed{ \mathcal R_{\mathrm{pr}}(S,B) \subseteq \mathcal R(S). }

If:

B_1\le B_2,

then:

\boxed{ \mathcal R_{\mathrm{pr}}(S,B_1) \subseteq \mathcal R_{\mathrm{pr}}(S,B_2). }

Proof

The budgeted provider sets are subsets of provider reachable sets and are nested in $B$ .

Union preserves both inclusions.

$\square$

31. Zero-cost union no-go

\boxed{ \text{portfolio union reachability} \not\Rightarrow \text{finite-budget practical reachability}. }

An expensive provider can contribute enormous theoretical territory that is unusable under the current task budget.

32. Calibration debt

Let:

k_\nu

be the cost of maintaining a trusted provider model:

evaluator calibration;
diagnostic-model calibration;
version monitoring;
test fixtures;
human audit.

Portfolio fixed cost:

\boxed{ K(S) = \sum_{\nu\in S} k_\nu }

in a simple additive model.

33. Pure-coverage dominated provider

If provider $\nu$ is reachability redundant and has positive calibration cost, then it cannot improve a pure objective:

f(S)-\lambda K(S)

for:

\lambda>0,

unless it provides value through another unmodeled coordinate such as robustness or latency.

34. GVSS09-T8 — Redundant-provider penalty under pure coverage-cost objective

Theorem GVSS09-T8

Suppose:

\Delta_\mu(\nu\mid S)=0

and:

k_\nu>0.

For:

J(S) = f(S)-\lambda K(S), \qquad \lambda>0,

\boxed{ J(S\cup\{\nu\}) < J(S). }

Proof

Coverage gain is zero and fixed cost rises by $k_\nu$ .

$\square$

This theorem does not include robustness gain.

35. Provider failure event

For task/region $\theta$ , define:

X_\nu(\theta) = 1

when provider $\nu$ fails to deliver an accepted result within the declared budget.

Failure is therefore task and budget relative.

36. Two-provider reliability

Let:

q_i=P(X_i=1).

Let:

q_{12}=P(X_1=X_2=1).

37. GVSS09-T9 — Two-provider portfolio success law

Theorem GVSS09-T9

If the portfolio succeeds whenever at least one provider succeeds:

\boxed{ P( \mathrm{success} ) = 1-q_{12}. }

The gain over the better individual provider is:

\boxed{ G_{\mathrm{div}} = \min(q_1,q_2)-q_{12}. }

Proof

Portfolio failure is the intersection of both provider failure events.

The best single-provider success is:

1-\min(q_1,q_2).

Subtract.

$\square$

38. Independent-failure specialization

If provider failures are independent:

\boxed{ q_{12}=q_1q_2. }

Then:

P(\mathrm{success}) = 1-q_1q_2.

Do not use this formula without an independence argument.

39. Perfect common-mode failure

If:

X_1=X_2

almost surely and:

q_1=q_2=q,

then:

q_{12}=q,

and:

\boxed{ P(\mathrm{portfolio\ success}) = 1-q. }

No reliability diversification is obtained.

Two brands can behave like one failure mode.

40. GVSS09-T10 — Fréchet reliability bounds

Theorem GVSS09-T10

For two provider failure events:

\boxed{ \max( 0, q_1+q_2-1 ) \le q_{12} \le \min(q_1,q_2). }

Therefore:

\boxed{ 1-\min(q_1,q_2) \le P(\mathrm{portfolio\ success}) \le 1-\max( 0, q_1+q_2-1 ). }

Proof

Standard Fréchet bounds on intersection probability.

$\square$

Marginals alone do not determine diversification value.

41. Pairwise failure correlation

One can estimate:

\operatorname{Corr}(X_i,X_j).

This is useful but incomplete for larger portfolios.

42. GVSS09-N2 — Pairwise correlations do not determine three-provider tail failure

Counterexample

Distribution A

$X_1,X_2,X_3$ independent Bernoulli $(1/2)$ .

Then:

P(X_i=1)=1/2,

P(X_i=X_j=1)=1/4,

and all pairwise correlations are zero.

Triple failure:

\boxed{ P(111)=1/8. }

Distribution B

Uniform on:

000, 011, 101, 110.

Again:

P(X_i=1)=1/2,

P(X_i=X_j=1)=1/4,

and all pairwise correlations are zero.

But:

\boxed{ P(111)=0. }

Thus identical marginals and pairwise correlations can have different all-provider failure probabilities.

$\square$

Higher-order dependence matters.

43. Brand diversity no-go

Provider names, model families, or company labels do not mathematically determine failure dependence.

Two independently branded services may:

use related base models;
share safety filters;
share evaluators;
share cloud infrastructure;
share training distributions;
fail on the same compositional prompts.

Therefore:

\boxed{ \text{brand diversity} \not\Rightarrow \text{failure diversity}. }

44. Failure-mode diversity

A more meaningful portfolio diagnostic records failure vectors over benchmark/task regions.

For provider $\nu$ :

\boxed{ x_\nu = ( X_\nu(\theta_1), \ldots, X_\nu(\theta_m) ). }

Portfolio diversity can be studied from joint error structure.

45. Capability diversity

Reachability diversity instead records which visual regions each provider covers.

Failure diversity and capability diversity are related but distinct.

A provider can cover the same region but fail independently.

A provider can cover unique regions but share common-mode outages.

46. Critical visual measure

Let:

w: \Omega_\Sigma \to [0,\infty).

Define:

\boxed{ \mu_w(A) = \int_A w\,d\mu. }

47. Critical-region coverage

\boxed{ f_w(S) = \mu_w( \mathcal R(S) ). }

All coverage/submodularity results still hold because $\mu_w$ is a measure.

48. GVSS09-N3 — Raw coverage ranking can reverse under critical weighting

Counterexample

Let visual benchmark regions be:

A_1,\ldots,A_{10}

with equal raw measure.

Provider $P_A$ covers:

A_1,\ldots,A_9.

Provider $P_B$ covers only:

A_{10}.

Set importance:

w(A_i)=1 \quad (i\le9),

w(A_{10})=100.

Then raw coverage:

9>1,

but weighted coverage:

\boxed{ 100>9. }

Thus a small provider can dominate on the critical region.

$\square$

49. Fallback provider geometry

A fallback provider may be valuable precisely because it covers:

identity-critical portraits;
text rendering;
local offline operation;
safe deterministic rendering;
specific style regimes;

that the main provider handles unreliably.

Fallback value is weighted by importance and failure complementarity.

50. Small stable fallback can dominate

A lower-capability provider can dominate a larger provider on a project-specific fallback objective if:

its critical-region coverage is higher;
its calibration debt is lower;
its common-mode failure with the primary is lower;
its switching cost is manageable.

"Capability" is objective relative.

51. Prompt-conditioned provider reachability

Reachability can depend on prompt/task class:

\boxed{ \mathcal R_\nu(\theta). }

A provider router maps:

\theta \to \nu.

GVSS-09 portfolio geometry can therefore be conditioned on task distributions.

52. Current cost-aware T2I routing

Cost-Aware Routing for Efficient Text-to-Image Generation routes prompts among multiple pre-trained T2I functions according to prompt complexity and computation cost.

The reported nine-model router demonstrates that a portfolio can outperform uniform commitment to any single model on an average quality-cost objective.

GVSS interprets the router as selecting among portfolio reachable/cost regions.

53. Edge-cloud routing

Adaptive T2I edge-cloud routing explicitly decides whether requests should use lightweight edge or expensive cloud models.

This is a direct provider-cost/fallback problem.

54. OctoT2I

OctoT2I performs stateful multi-round routing across image-generation tools and maintains a self-evolving knowledge base about tool capability.

This is close to GVSS provider capability learning and routing.

GVSS-09 contributes set/robustness geometry rather than the general idea of routing.

55. HADIS

HADIS studies adaptive diffusion-model serving, routing prompts based on expected difficulty and resource use.

This reinforces the fact that provider selection is a cost-sensitive serving problem.

56. Mixture-of-experts relation

RAPHAEL and ERNIE-ViLG 2.0 use expert routing internally inside a single generative architecture.

GVSS-09 concerns an external portfolio of versioned providers.

The mathematical intuition of specialization is related, but the governance/cost/failure semantics differ.

57. Cardinality-constrained provider selection

Suppose the portfolio may contain at most:

k

providers.

Objective:

\max_{|S|\le k}f(S).

58. Greedy selection

Start:

S_0=\varnothing.

At step $t$ , add provider with maximum:

\Delta_\mu(\nu\mid S_t).

59. GVSS09-T11 — Classical greedy coverage guarantee

Theorem GVSS09-T11

For monotone submodular coverage $f$ and a cardinality constraint $k$ :

\boxed{ f(S_k) \ge \left[ 1- \left( 1-\frac1k \right)^k \right] f(S^\star) \ge (1-1/e)f(S^\star). }

Proof sketch

Let optimal size- $k$ set be $S^\star$ .

At greedy step $t$ , by submodularity the sum of marginal gains of elements of $S^\star\setminus S_t$ is at least:

f(S^\star)-f(S_t).

At most $k$ elements contribute, so the largest available marginal is at least:

\frac1k [ f(S^\star)-f(S_t) ].

Thus residual gap contracts:

f(S^\star)-f(S_{t+1}) \le \left( 1-\frac1k \right) [ f(S^\star)-f(S_t) ].

Iterate $k$ times.

$\square$

This is classical submodular maximum coverage theory.

60. Coverage-only greedy is incomplete

The greedy theorem does not account for:

provider activation cost;
calibration cost;
latency;
robust multiplicity;
correlation;
version drift;
provenance requirements.

GVSS portfolio selection is multiobjective.

61. Cost-aware marginal score

A practical heuristic can use:

\boxed{ \frac{ \Delta_\mu(\nu\mid S) }{ s_\nu+k_\nu } }

or a multiobjective Pareto rule.

This ratio has no universal optimality claim.

62. Robust provider selection objective

For outage budget $r$ :

\boxed{ f_{\mathrm{rob}}^{(r)}(S) = \mu( \mathcal R_{\mathrm{rob}}^{(r)}(S) ). }

Unlike ordinary coverage, its combinatorial properties should be analyzed separately.

GVSS-09 does not claim it is submodular in all regimes.

63. Robust coverage can reward duplicate providers

Ordinary coverage penalizes exact duplicates with zero marginal gain.

Robust multi-coverage can reward duplication until multiplicity reaches the required threshold.

This is precisely why one objective cannot represent both territory expansion and failover.

64. Provider portfolio state

Define:

\boxed{ \mathsf{PF}_t = ( S_t, \{ \nu, \mathcal R_\nu, s_\nu, k_\nu, Q^\nu, T^\nu \}_{\nu\in S_t}, \mathcal D_{\mathrm{fail}}, \mathsf{Prov}_t ). }

Here:

\mathcal D_{\mathrm{fail}}

stores joint failure statistics/models.

65. Portfolio provenance

Every provider capability estimate should record:

provider name;
exact model/version;
endpoint/backend;
evaluator version;
reachability benchmark;
budget;
calibration date;
failure traces;
normalization map;
source artifacts.

Do not merge provider identities in the capability database.

66. Provider version change

A version change creates:

\nu \to \nu'.

Reachable set, cost, failure correlations, and calibration can all change.

Treat it as a new portfolio asset until transfer is validated.

67. Portfolio drift

The portfolio itself is dynamic:

providers appear;
providers disappear;
prices change;
APIs change;
models drift;
local hardware becomes available.

GVSS-09 is a snapshot geometry.

A later routing paper should treat continual portfolio evolution.

68. Failure correlation under version updates

Correlation estimates are version-specific.

A provider update can:

reduce common failure;
introduce a shared safety filter;
change inference infrastructure;
alter correlations without large marginal accuracy changes.

Re-estimate joint failures after behaviorally relevant updates.

69. Infrastructure common-mode failure

Two independent models hosted on the same service can fail together due to:

outage;
authentication;
billing;
rate limits;
region failure.

Thus model-level diversity and infrastructure-level diversity are distinct.

70. Evaluator common-mode failure

If every provider output is accepted/rejected by one shared evaluator, evaluator failure can collapse the entire portfolio.

Provider diversification does not protect against a centralized judge.

This connects back to GVSS-08 quarantine.

71. GVSS09-N4 — Provider diversification does not diversify a shared evaluator

If provider outputs are independent but every final decision is deterministically controlled by the same failed evaluator, the decision system can fail on all providers simultaneously.

Thus:

\boxed{ \text{provider diversity} \not\Rightarrow \text{observer diversity}. }

72. Multi-observer portfolio

A truly resilient visual runtime can diversify both:

generators;
evaluators.

This creates a two-layer portfolio problem.

GVSS-09 does not fully solve the joint portfolio.

73. Portfolio routing

Given task $\theta$ , provider routing chooses:

\boxed{ \nu_t = \pi_{\mathrm{route}} ( \theta, b_t, B_t, \mathsf{PF}_t ). }

The router can use:

predicted quality;
region reachability;
cost;
latency;
failure probability;
switching cost;
calibration confidence.

74. Routing cannot create new provider support

GVSS09-T12 — Routing union upper bound

Any router restricted to provider set $S$ and one-provider terminal generation can only output:

\boxed{ I\in \mathcal R(S). }

Proof

The router ultimately selects one provider in $S$ .

Apply GVSS09-T1.

$\square$

Routing improves allocation/search over the portfolio.

It does not create visual support absent from every provider.

75. Stateful multi-round routing

A stateful router can:

try provider A;
inspect failure;
switch to B;
repair with C.

This can enter compositional reachable closure beyond simple union.

OctoT2I-like agentic routing motivates this extension.

76. Federated visual reachability graph

Represent providers as nodes.

Edges represent feasible output transfer:

\nu_i \to \nu_j

when output of $i$ can serve as input/control/reference to $j$ .

The compositional reachable set becomes graph-path dependent.

This is a natural future GVSS direction.

77. Switching graph

Each directed provider switch has cost:

\boxed{ c_{ij}. }

Provider routing becomes a shortest-path / stochastic-control problem on the provider graph.

GVSS-09 keeps only first-order switching costs.

78. Provider cold-start calibration

A newly added provider has uncertain:

capability;
failure correlations;
evaluator compatibility.

Its nominal reachable set may be large while trusted reachable set is initially small.

Calibration debt should discount immediate portfolio value.

79. Trusted reachability

Define calibration confidence:

\kappa_\nu(I)\in[0,1].

A trusted coverage measure can weight:

\boxed{ f_{\mathrm{trusted}}(S) = \int \max_{\nu\in S} \kappa_\nu(I) \mathbf 1\{ I\in\mathcal R_\nu \} \,d\mu(I). }

This is an engineering proposal.

No submodularity claim is made here without additional assumptions.

80. Fallback readiness

A fallback provider that has never been recently tested is not a reliable fallback.

Maintain:

health check;
authentication;
latency;
version;
calibration;
sample fixture pass.

Fallback readiness is a state, not merely a configured endpoint.

81. Provider health probe

A health probe is a diagnostic action over provider availability/capability.

Its value can be analyzed with GVSS-06 value-of-diagnosis.

82. Portfolio criticality

For provider $\nu$ , define critical unique weighted coverage:

\boxed{ C_\nu(S) = \mu_w \left( \mathcal R_\nu \setminus \bigcup_{j\in S\setminus\{\nu\}} \mathcal R_j \right). }

High $C_\nu$ means removing the provider loses important unique territory.

83. Robust criticality

A provider can have zero unique coverage but large robust criticality because it supplies the second route needed for outage tolerance.

Track both.

84. Portfolio frontier vector

Define:

\boxed{ \mathbf C_{\mathrm{portfolio}} = ( -f_w, -f_{\mathrm{rob}}^{(r)}, C_{\mathrm{switch}}, C_{\mathrm{cal}}, C_{\mathrm{latency}}, R_{\mathrm{failure}}, D_{\mathrm{corr}}, D_{\mathrm{prov}} ). }

Lower is better after sign convention.

85. GVSS09-T13 — Portfolio Pareto necessity

Theorem GVSS09-T13

Every optimum of a scalar objective strictly increasing in all declared portfolio costs/risks and strictly decreasing in all declared coverage/reliability benefits lies on the portfolio Pareto frontier.

Proof

Standard dominance argument.

$\square$

86. Portfolio objectives can conflict

A provider can:

add unique style territory;
have expensive calibration;
be highly correlated with main provider;
be excellent on one critical region;
have slow latency.

No universal provider ranking exists.

87. Critical fallback example

Main provider:

huge global coverage;
high identity failures.

Fallback provider:

narrow portrait domain;
extremely stable identity.

For a character-production project, the narrow provider can be more valuable as fallback despite smaller raw state coverage.

88. Correlated blind-spot example

Two frontier models trained on similar web-scale distributions can share:

counting errors;
spatial relation errors;
typography failures.

A small structurally different renderer may offer more failure diversity on a critical region.

This is a hypothesis to test empirically, not inferred from architecture labels.

89. Capability audit

Provider portfolio audit should report:

unique reachability;
overlap;
robust multiplicity;
critical-region coverage;
failure correlation;
higher-order joint failures;
activation/switch cost;
calibration debt;
evaluator compatibility;
provenance.

90. Current routing benchmark implication

Cost-aware routing papers usually optimize expected quality/cost over prompt distributions.

GVSS adds another evaluation axis:

Does the model pool actually span complementary reachable/failure regions?

A router cannot exploit diversity that has not been measured.

91. Provider benchmark matrix

Rows:

prompt/task regions.

Columns:

providers.

Cells can store:

\boxed{ ( \text{success}, \text{quality}, \text{cost}, \text{latency}, \text{calibration confidence} ). }

This is the empirical approximation of the capability portfolio.

92. Failure tensor

For higher-order dependence, store failure samples per task episode rather than only a pairwise correlation matrix.

This allows estimation of:

pairwise intersections;
triple failures;
conditional failures;
common-mode clusters.

93. Portfolio sample complexity

High-order failure estimation becomes data intensive as provider count grows.

GVSS-09 does not solve this statistical problem.

Sparse factor/common-cause models are possible future approaches.

94. Common-cause latent variable

A practical dependence model can introduce:

Z_{\mathrm{common}}

such as:

prompt difficulty;
evaluator failure;
service outage;
shared safety filter.

Conditional on $Z$ , provider failures may become less correlated.

This is an engineering/statistical modeling option.

95. Worst-case versus probabilistic robustness

Worst-case $r$ -outage robustness:

m_S(I)\ge r+1

does not need failure probabilities.

Probabilistic reliability uses the joint failure law.

These are different robustness notions.

96. Deterministic multiplicity can be conservative

Two provider routes can exist but both rely on one common cloud service.

Set multiplicity says two routes.

Infrastructure failure analysis may say one common cause.

Therefore provider-route independence must be modeled separately.

97. Provider decomposition

A provider can itself be represented as dependency tuple:

\boxed{ \nu = ( \text{model}, \text{service}, \text{evaluator}, \text{auth}, \text{network}, \text{hardware} ). }

Portfolio resilience should ultimately reason over dependency components.

This connects GVSS to RRT-19 trust/federation graphs.

98. Federated capability versus federated trust

A provider portfolio federates capability.

RRT-19 federates proof/trust.

A mature system needs both:

can the provider reach the target?
can we trust the capability/evaluation claim?

99. Portfolio provenance principle

Never collapse:

Q^{\nu_1}, Q^{\nu_2}, \mathcal R_{\nu_1}, \mathcal R_{\nu_2}

into one anonymous "ensemble model" if future routing/failure diagnosis requires provider attribution.

Preserve source identity through every aggregate statistic.

100. What is classical / neighboring

GVSS-09 does not claim as inventions:

union coverage;
maximum coverage;
submodularity;
greedy $1-1/e$ coverage approximation;
ensemble diversity;
correlated-failure analysis;
Fréchet probability bounds;
mixture-of-experts routing;
cost-aware model routing;
edge/cloud routing;
provider portfolios.

101. Candidate GVSS-specific synthesis

Subject to broader literature audit, the GVSS-specific synthesis is:

treating external image-generation providers as a visual reachable-set portfolio;
defining marginal provider value as new visual measure outside the existing union;
separating reachability redundancy from outage-robustness redundancy through multi-coverage;
identifying worst-case $r$ -provider outage reachability exactly with provider multiplicity $\ge r+1$ ;
combining switching/calibration cost with practical finite-budget portfolio reachability;
separating capability diversity, failure diversity, and provider-name diversity;
proving pairwise failure correlations are insufficient for higher-order portfolio tail risk;
introducing critical-region weighted visual reachability and robust fallback geometry;
linking current multi-model T2I routing to explicit GVSS capability/reliability geometry.

No strong novelty claim is made in v0.1.

102. What GVSS-09 proves

Under explicit hypotheses, GVSS-09 proves:

zero-cost one-provider terminal portfolio reachability equals the union of provider reachable sets;
measured union coverage is monotone;
provider marginal reachability gain equals the measure of its uncovered region;
visual union coverage is submodular;
provider reachability redundancy is characterized by zero uncovered measure;
worst-case $r$ -provider-outage reachable set equals states with reachability multiplicity at least $r+1$ ;
a provider can be raw-union redundant while increasing outage-robust reachability;
robust marginal coverage is exactly the mass whose multiplicity rises from $r$ to $r+1$ ;
practical finite-budget reachability is a subset of raw union reachability and is monotone in budget;
a purely coverage-redundant positive-cost provider worsens a pure coverage-minus-calibration-cost objective;
two-provider portfolio success depends on joint failure probability, not only marginal failure rates;
Fréchet bounds quantify possible portfolio reliability from marginal failure probabilities;
identical marginals and pairwise correlations do not determine three-provider all-failure probability;
weighted critical-region coverage can reverse raw provider rankings;
classical greedy size- $k$ coverage obtains the standard $1-1/e$ -type guarantee;
provider routing cannot exceed the raw union of provider supports in the one-provider terminal setting;
every scalar portfolio optimum with monotone coordinate preferences lies on the provider Pareto frontier.

103. What GVSS-09 does not prove

It does not prove:

exact reachable sets of proprietary providers;
that provider failures are independent;
that pairwise correlation captures higher-order common-mode failures;
that brand/model-family diversity implies reliability diversity;
that raw reachable-set union is affordable under finite switching budget;
that greedy raw coverage is optimal after costs/robustness enter;
that robust multi-coverage is the only useful failure criterion;
that provider dependencies are independent at infrastructure/evaluator layers;
that a small fallback provider is always better on critical tasks;
that provider portfolio statistics remain stable across version updates;
that multi-stage cross-provider composition equals simple union reachability.

104. Proposed GVSS-10

The next natural paper should move from static provider portfolio geometry to task-conditioned routing and capability attribution.

Proposed title:

\boxed{ \textbf{ GVSS-10 — Task-Conditioned Visual Routing, Capability Attribution, and Portfolio Learning } }

Chinese:

任務條件式視覺路由、能力歸因與生成器投資組合學習

Main questions:

How should a router estimate which provider covers a prompt/visual region?
How should provider capability matrices be learned online?
How should routing explore undercovered providers?
How should switching/cold-start calibration affect routing?
Can provider contribution be attributed after multi-stage edits?
What is routing regret against an oracle provider portfolio?
How should higher-order correlated failure affect routing decisions?

105. References

Qinchan Li, Kenneth Chen, Changyue Su, Wittawat Jitkrittum, Qi Sun, Patsorn Sangkloy, Cost-Aware Routing for Efficient Text-To-Image Generation, arXiv:2506.14753, 2025.
Adaptive Routing of Text-to-Image Generation Requests between Edge and Cloud Models, arXiv:2411.13787.
OctoT2I: A Self-Evolving Agentic Text-to-Image Router, arXiv:2606.01803, 2026.
HADIS: Hybrid Adaptive Diffusion Model Serving for Efficient Text-to-Image Generation, arXiv:2509.00642, 2025.
Zeyue Xue et al., RAPHAEL: Text-to-Image Generation via Large Mixture of Diffusion Paths, arXiv:2305.18295.
Zhida Feng et al., ERNIE-ViLG 2.0: Improving Text-to-Image Diffusion Model with Knowledge-Enhanced Mixture-of-Denoising-Experts, arXiv:2210.15257.
Rafael Rosales, Pablo Munoz, Michael Paulitsch, Exploring Resiliency to Natural Image Corruptions in Deep Learning using Design Diversity, arXiv:2303.09283.
GVSS-01 through GVSS-08, internal series artifacts, 2026.
RRT-20, Reflexive Representation Theory: Unified Closure, Meta-Theorems, Limits, and Research Program, internal series artifact, 2026.

106. Conclusion

GVSS-08 asks how to survive uncertainty about one provider/model regime.

GVSS-09 treats the provider set itself as a portfolio.

Its raw visual territory is:

\boxed{ \mathcal R(S) = \bigcup_{\nu\in S} \mathcal R_\nu. }

Its marginal visual gain is:

\boxed{ \Delta_\mu( \nu\mid S ) = \mu \left( \mathcal R_\nu \setminus \bigcup_{j\in S} \mathcal R_j \right). }

Its worst-case $r$ -provider-outage territory is:

\boxed{ \mathcal R_{\mathrm{rob}}^{(r)}(S) = \{ I: m_S(I)\ge r+1 \}. }

And its probabilistic reliability depends not merely on marginal provider quality but on the joint structure of provider failure.

The same provider can have:

zero unique raw coverage;
high fallback value;
high robust multiplicity value;
low critical-region value;
high calibration debt.

Therefore provider diversity has at least three meanings:

\boxed{ \text{territory diversity} }

\boxed{ \text{failure diversity} }

\boxed{ \text{cost/availability diversity}. }

The canonical GVSS-09 principle is:

\boxed{ \textbf{ Do not ask how many image models are connected. Ask which visual regions have genuinely different routes, which failures are genuinely independent, and what those routes cost to keep calibrated and ready. } }

This establishes provider-portfolio geometry as the next operational layer of the Global Visual Space framework.

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