# Borrowed Gain Geometry — current formal specification

**Status: proposed Flow Hijacked synthesis and testable hypothesis. No empirical relational operator or resolvent was estimated in this work. Conceptual priority has not been established.**

The candidate contribution is not merely that people affect each other. It is a precise question:

> Under which conditions does relational context change the feasible, safe and agentic dynamics of response, beyond changing the external input, baseline state, noise or observation channel?

### Coupled nonlinear formulation

For person \(i\), let \(x_i\) represent faster state variables, \(z_i\) slower learned/adaptive variables, \(m_i\) neuromodulatory variables, and \(r_{ij}\) relational history/context:

\[
\dot x_i = F_i(x_i,z_i,m_i) + C_i(x_i,x_j,r_{ij},u_i) + \eta_i.
\]

A slow subsystem may be written

\[
\dot z_i = \epsilon_i G_i(x_i,z_i,x_j,r_{ij}),
\]

but \(\epsilon_i\ll1\) is a modeling assumption to test, not an automatic fact about attachment. Neuromodulatory actions may span multiple timescales.

### Local dynamics and observations

Around an operating point or trajectory:

\[
\delta\dot x = J_r(t)\delta x + B_r(t)\delta u,\qquad
\delta y = C_r(t)\delta x.
\]

A relationship can change operating points, input channels, coupling, parameters or the measured variables. A difference in observed response does not by itself establish a change in the intrinsic within-person operator.

For an approximately stationary, locally stable approximation \(A_r\):

\[
R_r(z)=(zI-A_r)^{-1}, \qquad
H_r(i\omega)=C_r(i\omega I-A_r)^{-1}B_r.
\]

The resolvent and the measured input-output transfer function are not identical. Both require specified variables, units, norms and feasible input directions.

### Finite-time amplification

\[
G_r(T)=\max_{0\le t\le T}\|e^{A_rt}\|.
\]

This is a norm-amplification measure; a squared-energy convention uses its square. Non-normality, defined relative to an inner product, does not by itself guarantee transient growth. A large resolvent norm can also reflect slowly decaying normal modes. “High” therefore needs a timescale- and spectrum-matched reference rather than an invented clinical threshold.

When the context changes materially during observation, use the state-transition propagator

\[
\partial_t\Phi(t,s)=J_r(t)\Phi(t,s),
\]

not an unqualified frozen resolvent. Delays, nonlinear excursions and constraints may require augmented models or nonlinear reachability analysis.

### An analytical counterexample, not clinical data

Let

\[
A_k=\begin{pmatrix}-1&k\\0&-2\end{pmatrix}.
\]

All values of \(k\) leave eigenvalues \(-1,-2\). For \(k\ne0\), the matrix is non-normal. Its numerical abscissa is

\[
\omega(A_k)=\frac{-3+\sqrt{1+k^2}}2.
\]

Euclidean norm growth in some initial direction is possible when \(k^2>8\). Thus small nonzero \(k\) gives a non-normal system without such growth. For \(x(0)=e_2\),

\[
x_1(t)=k(e^{-t}-e^{-2t}),\qquad x_2(t)=e^{-2t}.
\]

At \(k=8\) and \(t=\ln2\), the norm exceeds 2 even though the system ultimately decays. This is one direction at one time, not the maximum gain.

Conversely, \(A=\operatorname{diag}(-0.01,-1)\) is normal and stable, but \(\|R(0)\|=100\). A high resolvent does not uniquely identify non-normality.

The workbook's ToyModel and the included script calculate these distinctions. Their variables and time units are dimensionless and are not fitted to people.

### What would count as support for BGG?

Compare preregistered, identifiable models:

**M0:** input-only differences with shared dynamics.  
**M1:** starting state, baseline, noise or observation differences.  
**M2:** state-dependent gain or operating-point differences.  
**M3:** context-dependent operator or coupling changes.  
**M4:** time-varying/delayed coupled dynamics.

Use counterbalanced benign tasks, verified inputs, shared-stimulus controls, neutral versus trusted versus recorded/symbolic support, multimodal outcomes and parameter-recovery checks. Added complexity must improve out-of-sample predictions and survive plausible confounds.

Useful outcomes include function and the ability to choose, speak, decline, disengage and re-engage—not simply lower arousal. Later voluntary follow-up can assess retention and generalization without removing clinically necessary support.

The operator-specific hypothesis is weakened if simpler models predict equally well, if parameters cannot be identified, or if effects do not replicate. Such a result would not invalidate the practical value of supportive relationships.

### Safety and prior art

Relational regulation theory, Social Baseline Theory, social allostasis, interpersonal inference and network-control research already address important parts of this terrain. The proposed contribution is a discriminating integration, not a claim that dyadic regulation was newly discovered.

A benefit to one person must not be obtained through coercion or unbounded cost to another. Never use this framework to diagnose, rank relationships, mandate touch/contact, alter medication, deny care, impose surveillance or remove autonomy.

## Current package status note

The original methods file was written during acquisition and contains an older held publication-gate statement. For the current Assimilation Package, research status is governed by the later **Numerical Locked** workbook: 360 included sources, numerical gate passed, final EN/HE lectures complete. The mathematical/falsification specification above remains the intended BGG formal boundary.