Cohesion Dynamics — Paper B v0.9 - Continuum Physics in Cohesive Phases

Cohesion Dynamics — Paper B (v0.9)

Continuum Physics in Cohesive Phases

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1. Introduction

This paper develops the continuum limit of Cohesion Dynamics (CD) in the special case where the substrate is in a cohesive phase, as defined in Paper A (Substrate Mechanics). A cohesive phase ensures:

• finite height on all finite regions,
• locally bounded mismatch propagation, and
• stable refinement of mismatch-minimising configurations.

Under these conditions, CD admits a continuum description consisting of:

1. a limit field $\phi(x)$,
2. a hyperbolic partial differential equation governing its evolution in closure time, and
3. an effective Lorentzian causal structure derived from the PDE’s principal symbol.

This paper makes no claims regarding gravity, Einstein equations, quantum structure, or cosmology. Its goal is strictly structural:

In any cohesive phase satisfying assumptions (B1) and (M2), CD necessarily yields second-order hyperbolic dynamics with a Lorentzian cone structure.

The paper is organised as follows:

• Sections 2–4 construct the continuum field from refinement and control its regularity.
• Section 5 derives the universal mismatch Lagrangian from assumption M2.
• Section 6 derives the second-order closure recurrence.
• Sections 7–9 take the continuum limit and obtain the metric.
• Section 10 introduces the operational time field.
• Section 11 analyses divergence surfaces and horizon formation.

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2. Refinement and Coarse Fields

Let $\mathcal{R}_\ell$ be a refinement of the substrate into regions of diameter $O(\ell)$. Each region $R$ carries a symbol configuration $X(R)$ in a finite alphabet $\Sigma$.

2.1 Embedding assumption (B1)

A cohesive phase admits a compatible embedding

$$x_\ell : \mathcal{R}_\ell \to \mathbb{R}^D,$$

with:

1. bounded distortion: $$c_1 \ell \le |x_\ell(R)-x_\ell(R')| \le c_2 \ell$$ for adjacent $R,R'$;
2. diameters satisfy $\operatorname{diam}(x_\ell(R)) = O(\ell)$;
3. refinement compatibility: diameters $\to 0$ as $\ell \to 0$.

Dimensionality (imported from v0.6)

Paper F and Paper A justify that the substrate’s mismatch-tolerance structure admits exactly three independent modes of spatial variation. These modes generate coordinate charts via least-distortion embedding. Compatibility of neighbouring charts ensures a 3-dimensional manifold structure (Gromov–Hausdorff convergence). Thus $D=3$ is not chosen—it is the dimension of the substrate’s tolerance degrees of freedom.

2.2 Coarse field definition

Choose an injective encoding $v : \Sigma \to \mathbb{R}^N$. Define the coarse field:

$$\phi_\ell(x(R)) := v(X(R)),$$

extended piecewise-constantly across $x(R)$.
The resulting maps $\Phi_\ell : \Omega \to \mathbb{R}^N$ approximate the continuum field.

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3. Regularity and Compactness of Coarse Fields

A cohesive phase enforces bounded mismatch and finite height. Combined with M2 (Section 5), this yields:

3.1 Uniform Lipschitz bound (v0.6 reasoning)

For adjacent $R,R'$:

$$\|\phi_\ell(R)-\phi_\ell(R')\|^2 \le C_0 M_\ell(R;X) = O(\ell^2),$$

so

$$\|\phi_\ell(R)-\phi_\ell(R')\| \le C\ell.$$

Thus coarse fields are uniformly Lipschitz at scale $\ell$.

3.2 Equicontinuity

Bounded degree and bounded distortion imply neighbouring regions map to distances $O(\ell)$. Combined with Lipschitz bounds, the interpolants $\Phi_\ell$ are equicontinuous.

3.3 Compactness (Arzelà–Ascoli)

Since:

• $\Omega$ is compact,
• $\Phi_\ell$ are bounded and equicontinuous,

the family has a uniformly convergent subsequence:

$$\Phi_{\ell_j} \to \phi : \Omega \to \mathbb{R}^N.$$

This $\phi$ is Lipschitz and therefore weakly differentiable almost everywhere (Rademacher).

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4. Mismatch Universality Class M2

A substrate mismatch function belongs to M2 when, under refinement:

1. it is local and symmetric;
2. first-order (linear gradient) terms vanish due to symmetry;
3. the leading variation under small deviations is quadratic in differences;
4. higher-order terms scale as $O(\ell^3)$ or higher and vanish in the limit.

4.1 Quadratic scaling (v0.6 transplant)

Because $\|\phi_\ell(R')-\phi_\ell(R)\| = O(\ell)$:

$$\|\phi_\ell(R')-\phi_\ell(R)\|^2 = O(\ell^2).$$

Summed over $O(1)$ neighbours, mismatch contributions scale as $\ell^2$.
Rescaled by region volume $\ell^D$, the continuum density becomes:

$$A(\phi) + B(\phi) \|\nabla \phi\|^2.$$

Higher derivatives are suppressed as $\ell^{k}$ with $k\ge 3$.

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5. Effective Spatial Lagrangian

Collecting the M2-scaled mismatch over refined regions yields:

$$\mathcal{L}_{\text{eff}}(\phi,\nabla\phi) = A(\phi) + B(\phi) \|\nabla\phi\|^2.$$

No linear gradient term (symmetry).
No higher-order term (scaling).

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6. Closure Dynamics and the Universal Second-Order Recurrence

Let $X_n$ be the substrate configuration after the $n$-th global closure.
Define the coarse field:

$$\phi^{(n)}(x) := \lim_{\ell\to 0} \phi^{(n)}_\ell(x).$$

6.1 Why closure cannot be first-order (v0.6)

Relaxation erases all subcycle history. Closure depends only on mismatch gradients between $X_n$ and its relaxed neighbours. Thus closure cannot depend on $X_{n-1}$ except through its effect on $X_n$.

6.2 Why closure must depend on two prior fields (v0.6)

Variation of mismatch across successive closures yields:

$$\delta M(X_{n+1},X_n) = -\delta M(X_n,X_{n-1}) + \Delta_{\text{spatial}}\phi_n.$$

Rearranging gives the discrete second derivative in closure index:

$$\phi^{(n+1)} - 2\phi^{(n)} + \phi^{(n-1)} = \ell^2 \nabla\cdot\!\big(B(\phi^{(n)})\nabla\phi^{(n)}\big) + O(\ell^3).$$

6.3 Universal recurrence

Thus:

$$\phi^{(n+1)} - 2\phi^{(n)} + \phi^{(n-1)} = \ell^2 \nabla\cdot\!\big(B(\phi^{(n)})\nabla\phi^{(n)}\big) + O(\ell^3).$$

This is the time-discrete ancestor of a hyperbolic PDE.

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7. Continuum Limit and PDE

Choose $t_n = n\Delta t$ with $\Delta t = O(\ell)$. Pass to the limit:

7.1 Time second derivative

$$\frac{\phi^{(n+1)}-2\phi^{(n)}+\phi^{(n-1)}}{(\Delta t)^2} \to \partial_t^2 \phi.$$

7.2 Spatial term

Define

$$c(x)^2 = \lim_{\ell\to0} \frac{\ell^2}{(\Delta t)^2}.$$

Then

$$\partial_t^2 \phi = \nabla\cdot\!\big(c(x)^2 B(\phi)\nabla\phi\big).$$

7.3 Weak formulation

Since $\phi$ is Lipschitz, all derivatives are defined in the weak sense, satisfying:

$$\int \phi\,\partial_t^2 u = \int c^2 B\,\nabla\phi \cdot \nabla u$$

for all test functions $u$.

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8. Causal Cones

Linearising around $\phi_0$:

$$\omega^2 = c(x)^2 B(\phi_0) |k|^2.$$

Thus the principal symbol is:

$$P(\omega,k) = -\omega^2 + c^2 B |k|^2.$$

This defines cones:

$$|\Delta x| \le v_{\max}(x) |\Delta t|, \qquad v_{\max}(x)=c(x)\sqrt{B(\phi)}.$$

Propagation is finite; influence is restricted to these cones.

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9. Emergent Lorentzian Metric

Hyperbolic cone fields determine a metric up to scale:

$$g_{\mu\nu} v^\mu v^\nu = 0 \iff |\Delta x| = v_{\max}(x)|\Delta t|.$$

Choose the representative:

$$g^{00}=-1, \qquad g^{ij}=c^2 B \,\delta^{ij}.$$

Then

$$g_{00}=-1, \qquad g_{ij}=(c^2 B)^{-1}\delta_{ij}.$$

This metric is effective, encoding the propagation structure and nothing beyond that.

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10. Operational Time and the Cycle-Duration Field

Closure defines a global monotone ordering parameter $n$.
Different regions may require varying relaxation depth, producing a cycle-duration field:

$$\theta(x)>0, \qquad t = \theta(x)\,n.$$

Reparameterising time yields:

$$g_{\tau\tau} = -\theta(x)^2.$$

$\theta(x)$ is refinement-stable in cohesive phases (v0.6 reasoning) because variations in mismatch-minimisation complexity converge under refinement.

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11. Divergence Surfaces and Horizons

In Paper A, a region is divergent when $H(R)=\infty$.
Let $S$ be the set where divergence persists under refinement.

11.1 Refinement breakdown (v0.6)

If $H=\infty$, then:

• $\phi_\ell$ is not equicontinuous near $S$,
• coefficients $c^2 B$ degenerate or diverge,
• cones collapse.

Thus the continuum limit cannot extend across $S$.

11.2 PDE breakdown

Hyperbolicity fails when:

• $c^2 B \to 0$, or
• gradients blow up, or
• $\theta(x)\to\infty$.

11.3 Horizon definition

A surface $S$ is a CD horizon when:

$$\theta(x)\to\infty \quad\text{or}\quad c^2 B \to 0.$$

This is equivalent to failure of refinement convergence and failure of PDE extendibility.

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12. Summary

Given a cohesive substrate and assumptions (B1) and M2:

1. refinement yields a Lipschitz continuum field;
2. mismatch scaling yields an effective quadratic Lagrangian;
3. closure dynamics impose a second-order recurrence;
4. the continuum limit yields a hyperbolic PDE;
5. cones define an emergent Lorentzian metric;
6. cycle duration determines temporal weighting;
7. divergence surfaces mark horizon boundaries.

This is the minimal structural continuum framework of CD.
Future papers develop gravitational dynamics, matter coupling, and interior models for divergent regions.

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