The Interaction Surplus Framework (Papers A–E)
A Unique Scalar Law for Cross-Domain Interaction, with Open-System Order Dynamics
Author: Michael Laurence Curzi (36N9 GENETICS LLC) · License: MIT (Attribution Required)
Operational encoding: modules/vovina_interaction_surplus.py · Empirically verified in BEHAVIOR.md
Abstract
We present the Interaction Surplus Framework, a five-part theory (Papers A–E) that
derives a unique scalar functional measuring the "surplus" generated when two
representations interact across a block-decomposed inner-product space, and an
open-system dynamics that turns a stream of such surplus into accumulated
order. The framework is not fitted; under four axioms it is the only admissible
law. It is wired into XERO as the stability ceiling against which every module is
benchmarked. All theorems below are machine-verified (verify_axioms) and all
dynamical claims are reproduced by the live behavioral probe.
1. Setup (Paper A)
Let V = H₁ ⊕ H₂ ⊕ … ⊕ H_N be a block-decomposed inner-product space (the default
N = 22 mirrors the 22 modules / Hebrew letters / Tree-of-Life paths, but the
theory holds for any N ≥ 2). For unit vectors x, y, define the geometric
interaction parameter
u(x,y) = 1 − (x·y)² = ‖x ∧ y‖² = sin²θ ∈ [0, 1].
u = 0 means collinear (no interaction); u = 1 means orthogonal (maximal).
2. The axioms and the uniqueness theorem
| Axiom | Statement |
|---|---|
| S1 Geometric dependence | F(x,y) = f(u(x,y)) for some scalar f |
| S2 Zero at zero | f(0) = 0 |
| S3 Affine effective count | g(u) = e^{f(u)} is affine in u |
| S4 Normalization | f(1) = ln N |
Theorem 2.1 (Uniqueness). The only functional satisfying S1–S4 is
f(u) = ln(1 + (N−1)·u), g(u) = e^{f(u)} = 1 + (N−1)·u.
g(u) is the effective count: the number of blocks effectively participating
in the interaction. f is its logarithm — an entropy/order measure.
3. Derived properties (Theorems 3.1–3.4)
f'(u) = (N−1) / (1 + (N−1)u) > 0 ⇒ strict monotonicity (T3.1)
f(u) > 0 for u > 0 ⇒ positivity (T3.2)
f''(u) = −(N−1)² / (1 + (N−1)u)² < 0 ⇒ strict concavity (T3.3)
Sharp Lipschitz bound: |f(u₁) − f(u₂)| ≤ (N−1)·|u₁ − u₂|. The constant N−1
is attained at u = 0 and is used throughout XERO as the stability ceiling —
no module may exhibit a marginal surplus slope exceeding N−1.
Empirical check (N=22): f(0)=0, f(1)=ln 22=3.091, g(1)=22, Lipschitz =21,
measured max slope 17.5 ≤ 21, all sampled f''<0. ✔
4. Two-source decomposition (Theorem 4.1)
Any interaction splits into a crossing (widening) and a divergence (deepening) part:
u = u_cross + u_div = β² + α²·sin²γ
max(f(u_cross), f(u_div)) ≤ f(u) ≤ f(u_cross) + f(u_div).
The widening_ratio = f_cross /(f_cross+f_div) tells us how much of an interaction
is breadth (new cross-domain coverage) versus depth (refinement within a
domain) — a directly actionable diagnostic for routing and curriculum.
5. Information-theoretic bridge & the falsifiable test (Papers C–D)
Because f = ln g, the surplus is a log-effective-count. Paper C issues a
falsifiable prediction: a transformer's empirical effective dimensionality,
measured by the participation ratio
PR = (Σ wᵢ)² / Σ wᵢ²
on its attention/mixture weights, should equal g(u) = 1 + (N−1)u for the block
geometry's interaction parameter u. This makes the framework testable on real
models (flagship_prediction(u, N) returns the predicted observables).
6. Open-system order dynamics (Paper E)
A stream of surplus is converted into accumulated order Q by the discrete law
Q_{t+1} = (1 − δ)·Q_t + η·F(U_t) − C_t
with retention-loss δ ∈ (0,1], conversion efficiency η ≥ 0, surplus input F,
and dissipation cost C.
| Result | Statement |
|---|---|
| Growth (Cor. 3.2) | Q grows iff η·F > δ·Q + C |
| Min surplus (Prop. 4.1) | F_min = (δQ + C + r)/η for target growth r |
| Steady state (Thm. 3.5) | Q∞ = (η·F − C)/δ (0 if ηF < C) |
6.1 Empirical validation (from the behavioral probe)
With constant input such that ηF − C = ln N = 3.091, the system converges to
Q∞ = ln(N)/δ with geometric relaxation factor (1 − δ). Measured:
| δ | predicted Q∞ = 3.091/δ |
measured Q∞ |
steps to 99% (ln 0.01 / ln(1−δ)) |
measured |
|---|---|---|---|---|
| 0.05 | 61.82 | 61.82 | 90 | 90 |
| 0.10 | 30.91 | 30.91 | 44 | 44 |
| 0.25 | 12.36 | 12.36 | 18 | 17 |
| 0.50 | 6.18 | 6.18 | 9 | 7 |
The match is exact for the steady state and within rounding for relaxation — a clean confirmation of Theorem 3.5.
7. The 27/33 fractal binding
Only 27/33 ≈ 0.818 of the available surplus is committed to action; 6/33 ≈ 0.182 is held in reserve for the self-witness protocol. Thus the operational
budget is f(u)·27/33 and the reserve is f(u)·6/33 — coupling the surplus law
to XERO's self-verification quorum.
8. Role in XERO
- Stability ceiling. Every module is benchmarked against the Lipschitz bound
N−1; exceeding it flags an unstable interaction kernel. - Self-proof.
verify_axioms()is XERO's machine proof that its interaction kernel is the unique surplus functional, not an approximation. - Self-evolution engine. The open-system law is the mechanism by which XERO accumulates order from surplus-bearing interactions and decays toward a designable steady state — the quantitative backbone of its autonomy.
Companion documents: WHITEPAPER_01_CAPABILITIES_AUDIT.md,
WHITEPAPER_02_ARCHITECTURE.md, BEHAVIOR.md.