assembly-economics/models/CrossBorder.md

# Cross-border coupling in GADM

**Status:** v0.1 (2026-10-13), ASSEMBLY-WP-0011-T01 (channel inventory) and
T02 (coupling design). Implementation: T03.

## 1. Why

The calibrated model (`models/Calibration.md` v0.4) puts domination at or
above the floor in every jurisdiction (US 0.68, EU 0.72, AR 0.75), and the
largest contributions are measured. Three firms take 88 % of frontier-model
spend, one search engine has 85–94 %, and two mobile operating systems divide
the market. Outside the US, all of these operators are foreign. A closed
model reads them as domestic, so the EU, already at rights-first levers,
cannot lower its domination (BR-007). The coupled model must say whether, and
through which channel, levers can act on operators set abroad.

## 2. Channel inventory (T01)

Foreign share f = share of the channel held by operators headquartered in
another jurisdiction, as seen from the host.

| # | Channel | Model quantities | Evidence (2026) | f for EU / AR (US) | Indicator | Priority |
|---|---|---|---|---|---|---|
| C1 | Discovery and membrane operators | Pd, Pmem; `disc`, `m` | Google 85–94 % search share; iOS + Android ≈ 100 % of mobile OS (StatCounter, Sept 2026) | ≈ 1.0 / ≈ 1.0 (0) | StatCounter shares × operator home | **1** |
| C2 | Lineage import | Plin; `L`, `lin_open` | top-3 US developers 88 % of enterprise model spend (Menlo, Dec 2025); open-weight lineages from US, China and EU (Mistral) | ≈ 0.9 / ≈ 1.0 (≈ 0.1) | P4 + origin of open-weight downloads (Hugging Face) | **1** |
| C3 | Compute and infrastructure | Pi; `std` | Amazon, Microsoft, Google 63 % of world cloud (Synergy Q2 2026); EU providers a small share of their home market | ≈ 0.75 (prior) / ≈ 0.9 (0) | P1 by region | **2** |
| C4 | Investment and capital | `aO` inflow, K_Ω | US information-processing investment $1.92 trn SAAR (L5, 2026 Q2); frontier capex concentrated in the US | adoption follows the frontier set abroad | L5 per jurisdiction; FDI in data centres | **2** |
| C5 | Regulatory spillover ("Brussels effect") | foreign part of Pd, Pmem, `expl` | DMA obligations (choice screens, interoperability) and AI Act transparency (since Aug 2026) bind global operators | host rules act on operators abroad | L7 and operator compliance changes | **2** |
| C6 | Trade in organism output | demand `dem`, Y* | digital services exports | — | WDI ICT services exports | 3 |
| C7 | Talent and founders | H, F | AI talent migration (AI Index) | — | AI talent flows | 3 |
| C8 | Sentiment and media | Σ | sentiment trends move together across countries (Pew, Ipsos) | — | L6 across jurisdictions | 3 |

Minimal set for T03: **C1, C2, C3** (where domination comes from) and **C5**
(the only lever a host has on foreign operators), plus **C4** for adoption.
C6–C8 come later, when a reading depends on them.

## 3. Coupling structure (T02)

### 3.1 Split channels

For each host j and channel c ∈ {Pd, Pmem, Plin, Pi}, the channel level is
a mix:

    P_c^j = (1 − f_c^j) · D_c^j + f_c^j · O_c^{h(c)}

- `D_c^j` is the **domestic part**. Its dynamics are the current GADM
  channel equation under the host's own levers.
- `O_c^h` is the **operator part**. It is the channel level of the
  operators' home jurisdiction h (for C1–C3 today: the US), under the home's
  levers, and **modified by the host's market-access rules** (C5):

      dO_c^{h→j} = (home channel dynamics)
                   − a_c^j · O_c^{h→j}        (host access rule, e.g. DMA)

  `a_c^j` ≥ 0 is the host's market-access lever. One host's rule lowers only
  its own operator part, unless the operator applies it globally. That is the
  spillover share `σ_c`: a fraction `σ_c` of every host's rule also acts in the
  home market and in all other hosts.

- The foreign share `f_c^j` is a parameter set from data (§ 2). In later
  versions it can become a state that moves with relative quality and access
  rules.

### 3.2 Lineage import (C2)

The host's effective lineage levers are the import-weighted mix:

    L_eff^j        = (1 − f_lin^j) · L^j        + f_lin^j · L^h
    lin_open_eff^j = (1 − f_lin^j) · lin_open^j + f_lin^j · lin_open^h

A rights-first host with open domestic lineages, which uses mostly imported
closed lineages, therefore has low effective openness. That is the
mechanism BR-007 suspects.

### 3.3 Adoption and frontier (C4)

Organism learning in the host draws on the frontier, not only on its own
stock:

    learning term:  ell · max(K_Ω^j, κ · K_Ω^frontier) · dem · Q

with access κ ∈ [0, 1] (κ = 0 recovers the closed model). The frontier is the
largest K_Ω among the coupled jurisdictions.

### 3.4 Numerical scheme

Synchronous step: at each time step, all jurisdictions compute their
derivatives from the **previous** state of all others, then all update
together. The step size stays at dt = 0.1 years.

### 3.5 Required properties (tests for T03)

1. **Uncoupled = closed.** With all f = 0, κ = 0 and a = 0, each jurisdiction
   reproduces `gadm04.simulate` exactly.
2. **Symmetry.** Identical jurisdictions coupled symmetrically stay identical
   to each other and to one uncoupled jurisdiction.
3. **Operator consistency.** With f = 1 and a = 0, a host's channel equals
   the home channel.
4. **Lever reach.** A host rule a_c^j > 0 lowers the host's channel and
   leaves the home channel unchanged when σ = 0.

### 3.6 Two scenario modes

- **Global:** all jurisdictions (US, EU, AR, and a J4 reference bloc) are
  simulated together with this coupling.
- **Local in context:** one host is simulated in full. The operator parts
  `O_c^h(t)` and the frontier `K_Ω^frontier(t)` are given paths (T04 context
  scenarios), taken from a global run or set by hand.

The comparison between the two modes (T05) shows when the context
approximation fails.

### 3.7 Calibration hook

Hosts start from `results/calibrated_starts.json`. Split the measured
channel level into parts so that P_c^j stays at its calibrated value at t 0:
`D_c^j(0) = O_c^h(0) = P_c^j(0)`. The coupling then changes the *dynamics*,
not the measured starting point.

## 4. First results: the BR-007 test (T05, first part)

`simulations/gadm02/world.py` implements § 3 (tests `test_world.py`: closed
equivalence, symmetry, operator tracking, lever reach, spillover, frontier
access). `cross_border.py` runs US (operators' home), EU and AR (hosts) from
Calibration v0.4 (`results/cross_border.md`). Foreign shares are from § 2;
frontier access κ = 0.8 is a prior.

| Lever (coupled model) | EU P at t 100 | US | AR |
|---|---|---|---|
| current levers | 0.82 (floor broken) | 0.82 | 0.83 |
| DMA as observed (a fitted ≈ 0, at most 0.006 per year) | 0.82 | 0.82 | 0.83 |
| strong access rule, a = 0.15 per year on foreign Pd and Pmem | **0.68** | 0.82 | 0.83 |
| strong access rule, applied globally by operators (σ 0.5) | 0.68 | 0.75 | 0.76 |
| US moves to rights-first (J1) levers | **0.63** | 0.60 | 0.66 |
| US J1 + strong EU access rule | **0.42** | 0.60 | 0.66 |

**Calibrating the access rule.** The DMA has applied since March 2024.
Google's share fell 6.5 points in Germany, 3.0 in France and 3.1 in the US
(no DMA), comparing March 2023–February 2024 with October 2025–September 2026
(StatCounter). The EU-specific excess is about −1.7 points (0 to −3.4). The
coupled model without any rule already produces −2.6 points, because the
EU's operator part converges toward the operator's home level. So the
observed decline is consistent with **no measurable DMA effect on search
concentration**, and the data allow at most a ≈ 0.006 per year. That is about
1/25 of the strong rule.

### Readings

1. **BR-007 is explained.** The EU's domination channels are run by foreign
   operators, so its own rights levers act only on the small domestic part.
   This is not visible in a closed model, which attributes the channels to
   the host.
2. **Two levers reach foreign-run channels:** a host access rule on foreign
   operators, and the home's own levers. The home's levers are the strongest
   single lever for every jurisdiction. A US move to rights-first levers
   brings all three under the floor (US 0.60, EU 0.63, AR 0.66).
3. **The access rule works only far above its observed strength.** As
   practised so far (2024–2026), it has no measurable effect on search
   concentration. A rule about 25 times stronger would bring the EU under the
   floor. This is a quantitative statement about enforcement, not about the
   law's text.
4. **Spillover helps others.** If operators apply the host's rule worldwide
   (σ 0.5), US and AR domination fall too (0.82 → 0.75, 0.83 → 0.76).
5. **Lineage concentration (Plin ≈ 0.9) is reachable only through the home.**
   No host rule in this setup acts on imported lineages. An access rule for
   lineages (interoperability, model portability) is the open design
   question, and the most direct link to break-register entry BR-003.

Caveats: the foreign shares and κ are first estimates. The access rule is a
removal rate on the operator part, not a model of how regulation works. The
legitimacy loop is off.

## 5. Context scenarios and hypothesis re-test (T04, T05)

`world.py` supports **local runs in a context**: `context(rows)` makes a
jurisdiction follow given rows instead of being simulated (test: a local run
against a context taken from a global run reproduces that run to nine
decimals). `world_scenarios.py` (`results/world_scenarios.md`) runs four
contexts globally (US, EU, AR and a J4 reference bloc) with the legitimacy
loop off and on. Calibration v0.5 (Argentina's E measured too, 40 of 40
targets in range).

| Context | Definition |
|---|---|
| X1 US-led concentrated frontier | current levers |
| X2 open-lineage world | US moves to rights-first (J1) levers |
| X3 accelerating frontier | US and J4 with self-enhancement (B4-like) |
| X4 fragmented blocs | hosts' frontier access 0.2, foreign shares halved |

Output is compared **relative to each jurisdiction's own start**. Model
output levels are normalized and not comparable across jurisdictions.

### 5.1 "Europe lags" (H1)

| | loop off | loop on |
|---|---|---|
| EU adoption below US and AR at t 10 | yes, in all four contexts | no, in all four contexts |
| EU − US relative output, years 0–25 | −0.03 to −0.24 (largest when the frontier accelerates) | −0.04 to +0.08 |
| EU − US VFAC at t 100 | +0.03 to +0.05, except open-lineage world −0.02 | +0.01 to +0.02, except open-lineage world −0.02 |

After the BR-009 fix (sentiment anchored at measured levels), the verdicts
stay the same. With the loop on, EU and US adoption at t 10 are now equal in
X1 and X4 (0.27 / 0.26–0.27), and in X2 and X3 the US leads (0.44). The
relative-output differences with the loop on become −0.09 to +0.05.

Reading: Europe lags in adoption **only if sentiment does not matter**. With
the measured sentiment levels (US Σ 0.29, EU 0.54) and the legitimacy loop,
US adoption brakes and the EU no longer lags. Lagging costs output early,
most in an accelerating-frontier world, but the EU ends with higher VFAC than
the US unless the US itself moves to rights-first levers.

### 5.2 "Argentina may thrive" (H2)

Not supported in any context or loop setting. Relative to its start,
Argentina grows less than the EU (−0.02 to −0.07 over years 0–25) and ends
with lower VFAC (−0.04 to −0.06). It starts with the highest domination
(0.75), all of it run from abroad (f = 1), under deregulated levers. That
leaves no domestic lever on domination, and its frontier access does not
compensate. Argentina gains most from the open-lineage world (X2), that is,
from US policy, not from its own.

### 5.3 When the context approximation fails

For host outcomes, the local run in a context equals the global run (max |ΔP|
< 0.001), including with a strong EU access rule and spillover. Hosts do not
feed back on the home in the current coupling: spillover changes the
*home's* path, and the host already applies its own rule. Global runs are
therefore needed for the home's outcomes, for spillover effects on third
jurisdictions, and for frontier leadership changes. Local runs suffice for a
single host's questions.

### 5.4 Limitations

- Foreign shares f are fixed. In reality they would move with access rules
  and domestic alternatives.
- Frontier access κ = 0.8 is a prior.
- The J4 bloc is uncalibrated. It reuses the US start with J4 levers.
- Trade, talent and sentiment contagion (C6–C8) are not modeled.
- No host rule reaches imported lineages (BR-008).

These are handed to ASSEMBLY-WP-0016.