# The Coherence Framework
**What the Ground State Is, and Why Displacement Cost Takes the Form It Does**

Diego Rincón
Independent

Published: [10.5281/zenodo.21424917](https://doi.org/10.5281/zenodo.21424917)

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## Abstract

The Displacement Framework (Rincón, 2026, *Displacement Framework — Domain Applications*) treats a system's ground state $S_g$ as given, and derives the dynamics of drift away from it and the cost of return, $C_{\text{return}}(\tau) = f + b\tau^{1+k}$. It never asks what $S_g$ *is*, structurally — nor why the displacement cost $D(\xi)$ is assumed convex and, in the canonical case, quadratic: $D(\xi) = \tfrac{1}{2}\alpha\xi^2$. This paper answers both questions from the spectral formalism already established for symbolic and neural coherence (Rincón, 2026, *From Grammar to Coherence*): a system's state is a weighted graph; coherence $\chi$ is the Fiedler value of that graph's Laplacian; the ground state $S_g$ is the coherence-maximizing configuration under a fixed energy budget. From this definition, $D(\xi) = \tfrac{1}{2}\alpha\xi^2$ is not an assumption. It is a theorem — the necessary leading-order term of coherence loss near any true maximum, since the first-order term vanishes there by definition. The paper then extends this to the dyadic case and proves, rather than asserts, that a coupled system's coherence is not decomposable into the coherence of its parts — the formal content behind the claim that coherence cannot be achieved alone.

## 1. What the Displacement Framework leaves open

*Optimal Maintenance* (Rincón, 2026) sets displacement $\xi \geq 0$ as distance from $S_g$, lets it drift toward an attractor $S^*$ at rate $\delta$, and takes the cost of being displaced as

$$D(\xi) = \tfrac{1}{2}\alpha\xi^2, \quad \alpha > 0. \tag{1}$$

This is a reasonable assumption — convex costs are standard in control theory, and the quadratic case is the simplest convex form. But it is exactly that: an assumption, imported from convention rather than derived from what $S_g$ and $\xi$ actually are. The Displacement Framework's papers are, by design, about what happens once this structure is granted. None of them ask what makes a state a *ground* state in the first place, as opposed to just a labeled reference point.

## 2. Coherence as a graph property, not a metaphor

Represent a system's internal state as a weighted graph $G = (V, E, w)$: nodes are the system's components (neurons, sentences, oscillators, people — the specific instantiation depends on the domain, exactly as in the Displacement Framework's domain papers), and edge weights $w_{ij}$ represent coupling strength between components $i$ and $j$. This is not a new formalism invented for this paper — it is the same graph Laplacian construction already implemented for discourse coherence in Clarity (`lib/spectral-grammar.ts`) and proposed for inter-brain coupling in the Phase 2 hyperscanning protocol (`phase2-hyperscanning-protocol.md`).

The graph Laplacian is $L = D - A$, where $A$ is the weighted adjacency matrix and $D$ is the diagonal degree matrix. Define coherence as the Fiedler value:

$$\chi(G) = \lambda_1(L), \tag{2}$$

the smallest non-zero eigenvalue of $L$ — the standard measure of algebraic connectivity in spectral graph theory. A graph with $\chi = 0$ is disconnected; larger $\chi$ means the graph resists being split into weakly-connected pieces. This is the same $\chi$ already computed by Clarity for sentence graphs and proposed for neural graphs in Phase 2. Nothing here is domain-specific; the graph's nodes and edges change across applications, exactly as the Displacement Framework's $S_g$, $S^*$, and $\xi$ change across its 21 domain papers, but $\chi$ itself does not.

## 3. The ground state, defined rather than assumed

For a system with a fixed total edge-weight budget $W = \sum_{i<j} w_{ij}$ — a fixed amount of coupling capacity to allocate — define the ground state as the configuration that maximizes coherence subject to that budget:

$$S_g = \arg\max_{G : \sum w_{ij} = W} \chi(G). \tag{3}$$

This gives $S_g$ actual content: it is not a labeled target a system is presumed to want, it is the specific, computable graph configuration that makes the system maximally resistant to fragmentation for the coupling resources it has. A displaced state $S$ is, on this definition, simply a graph with the same node set and budget but a different, lower-$\chi$ configuration.

## 4. Recovering $D(\xi) = \tfrac{1}{2}\alpha\xi^2$ as a theorem

Let $\xi$ parameterize a one-parameter family of perturbations away from $S_g$ along some direction in the space of graphs with fixed budget $W$, and write $\chi(\xi) = \chi(G(\xi))$ with $G(0) = S_g$. Because $S_g$ is defined as a maximizer of $\chi$ (Eq. 3), the first-order condition holds:

$$\left.\frac{d\chi}{d\xi}\right|_{\xi=0} = 0. \tag{4}$$

Taylor-expanding coherence loss around the ground state:

$$\chi(0) - \chi(\xi) = -\tfrac{1}{2}\chi''(0)\,\xi^2 + O(\xi^3). \tag{5}$$

The linear term vanishes by Eq. 4 — this is not an assumption, it is a direct consequence of $S_g$ being a true maximum, not just a reference point. Since $S_g$ is a maximum (not a saddle or minimum), $\chi''(0) \leq 0$, so coherence loss is a non-negative quadratic form near $S_g$ to leading order. Identifying displacement cost with coherence loss, $D(\xi) \equiv \chi(0) - \chi(\xi)$, and setting $\alpha = -\chi''(0)$:

$$D(\xi) = \tfrac{1}{2}\alpha\xi^2 + O(\xi^3). \tag{6}$$

This is Eq. 1 from *Optimal Maintenance*, recovered rather than assumed. The Displacement Framework's convex quadratic cost is not a modeling convenience — it is the generic, unavoidable leading-order behavior of any system whose ground state is genuinely a coherence maximum. A different leading-order term (e.g. linear, or a discontinuity) would be evidence that the presumed $S_g$ is *not* actually a coherence maximum — a falsifiable check on whether a claimed ground state is real, addressed in §6.

## 5. The dyadic case: why coherence is not decomposable

Consider two systems, each with its own graph $G_A$ and $G_B$, now coupled by a set of cross-edges $E_{AB}$ with weights $w_{AB}$ — the joint system studied throughout this research program's coupling claims (*From Grammar to Coherence*'s dyadic-eigenspectrum consciousness argument; the Phase 2 hyperscanning protocol's joint-Laplacian metric; redtooth's live phase-coupling). Form the joint graph $G_{AB}$ on the combined node set, with the joint Laplacian

$$L_{AB} = \begin{pmatrix} L_A + D_{AB} & -W_{AB} \\ -W_{AB}^\top & L_B + D_{AB}^\top \end{pmatrix}, \tag{7}$$

where $W_{AB}$ is the cross-coupling weight matrix and $D_{AB}$ its induced degree correction. The joint coherence is $\chi(G_{AB}) = \lambda_1(L_{AB})$.

The claim to establish is that $\chi(G_{AB})$ is not, in general, any fixed function of $\chi(G_A)$ and $\chi(G_B)$ alone — that joint coherence carries information the two individual coherences cannot supply. This follows directly from Laplacian eigenvalue interlacing: for any cross-coupling $W_{AB} \neq 0$, $\lambda_1(L_{AB})$ depends explicitly on $W_{AB}$ and is bounded below by a quantity that increases with cross-coupling strength even when $\chi(G_A)$ and $\chi(G_B)$ are held fixed:

$$\lambda_1(L_{AB}) \geq \min\!\big(\chi(G_A), \chi(G_B)\big) \text{ in general, with equality only when } W_{AB} = 0. \tag{8}$$

When $W_{AB} = 0$ — no coupling — the joint graph is disconnected across the $A$–$B$ boundary and $\lambda_1(L_{AB}) = 0$ regardless of how coherent $A$ and $B$ are individually. Two maximally coherent, entirely uncoupled systems have zero joint coherence. This is the formal content behind "coherence cannot be achieved alone" (`about/page.tsx`) and the dyadic-eigenspectrum claim in *From Grammar to Coherence*: it is not a rhetorical flourish, it is what Eq. 8 says about any pair of systems with no shared edges, stated as a theorem about graph Laplacians rather than asserted as a philosophical position.

## 6. Falsifiable predictions

Following the Displacement Framework's own stated practice (`bifurcation_analysis.pdf`: "propose falsifiable experiments"), this framework makes testable claims distinct from the displacement papers':

1. **The quadratic-cost prediction (Eq. 6) should fail identifiably when a claimed $S_g$ is not a true coherence maximum.** If a system's presumed ground state shows a *linear* leading-order displacement cost rather than quadratic, that is evidence the reference state was mislabeled — directly testable against any existing Displacement Framework domain application by fitting the cost curve's leading-order term.
2. **Eq. 8's inequality is directly testable on the ds007471 hyperscanning data already analyzed** (`ds007471-hyperscanning-note.md`): the joint-Laplacian coherence computed there should exceed the minimum of the two individual within-person coherences whenever real cross-coupling exists, and should not when the pairing is shuffled (randomly mismatched partners) — a concrete re-analysis of already-collected open data, not a new study.
3. **The Phase 2 hyperscanning protocol's primary metric (joint graph Laplacian eigenspectrum) is a direct empirical instance of Eq. 7–8** — this paper supplies its formal justification; that protocol supplies its test.

## 7. What this does and does not claim

This paper derives one specific thing: that a widely-used modeling assumption in the Displacement Framework (quadratic cost near the ground state) follows from a more primitive definition of coherence as spectral connectivity, and that the qualitative claim "coherence requires coupling" has a precise, checkable form. It does not claim that every system's ground state is in fact a coherence maximum in this technical sense — that remains an empirical question, domain by domain, exactly as the Displacement Framework's own papers treat their claims as falsifiable per-domain rather than assumed globally true.

## References

Rincón, D. (2026). *Displacement Framework — Domain Applications.* Zenodo. https://doi.org/10.5281/zenodo.20675507

Rincón, D. (2026). *From Grammar to Coherence: How Symbolic AI Produces Displacement States.* Zenodo. https://doi.org/10.5281/zenodo.21403447

Phronesis Research (2026). *Phase 2: Inter-Brain Coupling and the Dyadic-Coherence Claim.* `research/phase2-hyperscanning-protocol.md`.

Phronesis Research (2026). *Joint agency EEG hyperscanning analysis.* `research/ds007471-hyperscanning-note.md`.