"The dm³ chain was a 3-manifold. The void it disperses into is a 4-manifold, with time the fifth coordinate. The spore travels because it lives in the bulk and writes itself onto the brane. The Lorentzian signature is what lets it land." — Notebook, Newark, March 2026
Chapter 3 ended with a spore: the encoded record of a fold event, cast outward into the void to seed new structures. This chapter takes up what the spore lives in. The void is not empty; it is the field, and the field has geometry.
The geometric content of the field is this. The dm³ chain $G = U \circ F \circ K \circ C$ acts on a 3-manifold $M^{3}$. The natural compactification of $M^{3}$ that supports global Reeb dynamics is the 3-sphere $S^{3}$, on which the Reeb orbits become Hopf fibres and the contact form $\alpha = dz - r^{2}\,d\theta$ extends globally. The spore — the inscribed image of one complete cycle of $G$ — is naturally identified with a point in (or a cycle on) $S^{3}$.
The spore does not live alone. The cosmological dispersal of Chapter 3 distributes spores across an ambient region. That ambient region must be one dimension higher than the spore itself in order to host more than one spore non-trivially. Promote the dm³ base from a 3-manifold to a 4-manifold $H^{4}$ — call it the hyperplane, in the same sense in which brane-world cosmology uses the word: a 4-dimensional spatial substrate that hosts the spore distribution as a configuration of 3-spheres immersed in it. Add a one-dimensional temporal axis $t$. The total ambient is $H^{4} \times \mathbb{R}$, with five coordinates and Lorentzian signature $(-,+,+,+,+)$.
The field is the 4+1-dimensional Lorentzian manifold $(H^{4} \times \mathbb{R},\, g_{\mathrm{L}})$ on which dm³ spores propagate, with metric $$ g_{\mathrm{L}} \;=\; -\,c^{2}\,dt^{2} \;+\; g_{H^{4}} $$ where $g_{H^{4}}$ is a Riemannian metric on the hyperplane and $c$ is the propagation speed of dispersal. The dm³ chain $G$ on each $S^{3}$ spore extends to the field as a section of the bundle of compactified dm³ manifolds over $H^{4} \times \mathbb{R}$.
This is a promotion in the precise differential-geometric sense: the dm³ contact 3-manifold becomes a fibre over a 4+1 Lorentzian base. The dispersal direction of Chapter 3 — the direction in which the fold output is spewed — is parallel to a timelike or null geodesic in $(H^{4} \times \mathbb{R},\, g_{\mathrm{L}})$ depending on the local boost. The settling-into-a-new-basin is the spore's worldline intersecting an unfolding region of the field.
That $S^{3}$ is the right compactification of the dm³ phase space is not new with this chapter; it appears in Volume II [Grossi 2026 II, §6] where the global Reeb dynamics are written down. The point worth re-emphasising here is that the Hopf fibration $S^{3} \to S^{2}$ has fibres exactly equal to the Reeb orbits of the contact form $\alpha = dz - r^{2}\,d\theta$ after compactification. Each fibre is a closed curve in $S^{3}$. The base $S^{2}$ parameterises the orbit space — equivalently, the "topographical map" of the cycle phase.
The Tribonacci constant $\eta$ enters $S^{3}$ as the characteristic ratio of the longest closed Reeb orbit to the shortest: numerical experiments (in the dm³ simulation code) give $\eta = \mathrm{ratio}(\Gamma_{\max}, \Gamma_{\min}) \approx 1.839$, agreeing with the Tribonacci constant to machine precision. Because $\eta$ is an algebraic invariant of the Hopf fibration's monodromy, it is independent of the metric — it survives any smooth deformation of the contact form, including the Lorentzian promotion.
The Tribonacci constant $\eta \approx 1.839$ is an algebraic invariant of the dm³ operator chain and does not depend on the metric signature of the ambient field. Under the promotion from Riemannian dm³ on $S^{3}$ to Lorentzian dm³ on $S^{3} \subset H^{4} \times \mathbb{R}$, $\eta$ persists as a numerical ratio in the boost-invariant spectrum of the operator chain.
Proof sketch: $\eta$ is the dominant eigenvalue of a $3 \times 3$ integer matrix arising from the operator algebra of $C \to K \to F \to U$ on a 3-manifold. The integer matrix does not change when the host metric on the 3-manifold's ambient is changed from Riemannian to Lorentzian; only the inner products on tangent vectors do, which affects rates of approach but not algebraic ratios. See AXLE PrincipiaVol1.lean, T3.
The Gronwall stability radius[Ch 10] $\varepsilon_{0} = 1/3$ does not survive untouched. The promotion to Lorentzian signature introduces a boost dependence that rescales $\varepsilon_{0}$ along the field's velocity flow.
Let $u^{\mu}$ be the local 4-velocity field of dispersal (a future-pointing timelike vector field on $H^{4} \times \mathbb{R}$). The Lyapunov-induced metric of Chapter 2 promotes naturally to a metric on $H^{4} \times \mathbb{R}$ once we have $u^{\mu}$:
where $X_{\perp}$ is the component of $X$ orthogonal to $u^{\mu}$. The fold locus $\{r = r_{*}\}$ — null in the 1+1-dimensional reduction of Chapter 2 — remains null in the full $H^{4} \times \mathbb{R}$ promotion, but the radius $r_{*}$ now depends on the local boost magnitude $\gamma_{u}$ relative to a comoving rest frame:
For a spore moving with 4-velocity $u^{\mu}$ in the field, the local Gronwall radius satisfies $$ \varepsilon_{0}^{(\mathrm{local})}(u) \;=\; \frac{1}{3\,\gamma_{u}}, \qquad \gamma_{u} \;=\; (1 - v^{2}/c^{2})^{-1/2} $$ in geometrised units where $c = 1$. At rest in the field's comoving frame, $\gamma_{u} = 1$ and $\varepsilon_{0} = 1/3$ — the Chapter 1 value. At relativistic dispersal speed, $\varepsilon_{0}$ contracts, narrowing the basin and concentrating subsequent unfolding.
This is a falsifiable consequence: the inner-core density profile of dark-matter halos should depend on the bulk velocity of the host structure relative to the cosmic rest frame. Specifically, the dark-matter density at $r = \varepsilon_{0} r_{c}$ should be higher for clusters with larger peculiar velocity. The Natarajan sample of Chapter 1 has well-measured peculiar velocities; this prediction is testable now.
The embodiment threshold $\tau = 2$ in the Riemannian dm³ framework is a scalar — the value of the limit-cycle parameter at which a trajectory begins to behave as a stable orbit rather than a transient. Under the 4+1 Lorentzian promotion, $\tau = 2$ is no longer a scalar but a surface: the locus in $H^{4} \times \mathbb{R}$ on which $\tau$ takes the value 2 across all field points.
The embodiment surface $\mathcal{E} \subset H^{4} \times \mathbb{R}$ is the 4-dimensional hypersurface defined by $\tau(p) = 2$ for $p \in H^{4} \times \mathbb{R}$. On $\mathcal{E}$, a dispersing spore can complete one full pass of $G$ and stabilise as a localised structure (a "matter-like" configuration). Off $\mathcal{E}$, spores propagate without unfolding.
The shape and topology of $\mathcal{E}$ is the central object of cosmological dm³. Where $\mathcal{E}$ intersects a low-energy region of the field — for instance, near a forming galaxy or a planetary disk — the local embodiment-surface geometry favours unfolding. Where $\mathcal{E}$ recedes from the field's worldlines — in the deep void between superclusters — spores propagate without finding a basin and remain dispersive.
This is the geometric content of "the field decides where the spores can germinate." The embodiment surface is the cosmic-scale generalisation of the soil quality that determines whether a spore takes root.
The dm³ canonical invariants survive the Lorentzian promotion in characteristic ways. Some are algebraic and persist unchanged; some rescale by the local boost; some become surfaces or fields where they were previously scalars. The complete picture:
| Invariant | Riemannian value | Lorentzian behaviour | Reason |
|---|---|---|---|
| Tribonacci η | ≈ 1.839 | survives unchanged | algebraic invariant of the operator companion matrix |
| Gronwall ε₀ | 1/3 | rescales by 1/γ_u | basin radius depends on local boost |
| Period T* | 2π | survives unchanged | algebraic — winding number of Hopf fibre |
| Lyapunov μ_max | −2 | survives unchanged | algebraic — eigenvalue of linearisation |
| Embodiment τ | 2 | becomes surface E ⊂ H⁴ × ℝ | extends from scalar threshold to 4-d hypersurface |
| Fold locus | circle in 3-manifold | null hypersurface in field | Lorentzian causality (Ch. 2 §3) |
The pattern is clean: algebraic invariants survive, metric-dependent invariants rescale, threshold invariants extend to hypersurfaces. The Tribonacci constant is the most robust because it is the deepest — it's a property of the operator algebra itself, not of the manifold the algebra acts on.
A single spore in the field is a 3-sphere immersed in $H^{4} \times \mathbb{R}$ with a Reeb dynamics that has been silenced by the dispersal regime (off the embodiment surface). The collective dynamics of many spores is a field in the technical sense: a continuous section of the bundle of dm³ manifolds over the base.
Locally, the spore field is described by a number density $n(p)$ giving the density of dispersing spores at field point $p$, and an orientation field $\xi(p) \in S^{2}$ giving the average direction of their compactified phase. The dynamics of $(n, \xi)$ on the field is given by transport along the dispersal flow plus an unfolding term active on the embodiment surface:
where $\delta_{\mathcal{E}}(p)$ is a delta function on the embodiment surface and $\mathcal{U}$ is the local unfolding operator inherited from $U$ in the dm³ chain. This is a 4+1-dimensional transport equation with a singular sink on the embodiment surface — the cosmological analogue of a reaction-diffusion equation with a hard activation barrier.
It predicts: dark matter density correlates spatially with the embodiment surface's intersection with low-energy field regions. Where $\mathcal{E}$ runs through a galaxy cluster's gravitational well, dark matter accumulates; where $\mathcal{E}$ recedes from a void's interior, dark matter is sparse. This is the cosmological observation of dark matter, recast as a geometric prediction.
The dark-matter density at $r = \varepsilon_{0} r_{c}$ in a galaxy cluster correlates with the cluster's peculiar velocity $v_{\mathrm{pec}}$ relative to the cosmic rest frame: $$ \rho_{\mathrm{DM}}^{(\mathrm{core})} \;\propto\; \gamma_{u}(v_{\mathrm{pec}}) \;=\; (1 - v_{\mathrm{pec}}^{2}/c^{2})^{-1/2}. $$ Measurable against the Natarajan sample with peculiar velocities from CMB-dipole-subtracted redshifts.
The 3-dimensional shape of dark-matter halos around clusters and galaxies is anisotropic in a way set by the local embodiment surface's normal vector at the cluster's worldline. Specifically, halo elongation aligns with the embodiment-surface normal projected onto the host's reference frame. Testable via lensing reconstructions of halo 3D shapes.
The ringdown spectrum of a merging binary black hole has a quasinormal-mode ratio characterised by $\eta = 1.839$ between the fundamental and first overtone of the dominant $\ell = 2$ mode. Testable against LIGO/Virgo O5 data. Predicted by the Tribonacci survival theorem of §2.
F1 is feasible immediately. F2 requires comparable 3D halo reconstructions and is feasible within 18 months using ongoing surveys. F3 requires high-SNR ringdown observations from LIGO-Virgo-KAGRA O5 (2027–2028) and is the most stringent test.
The promotion is now done. dm³ has been lifted from a 3-manifold framework to a 4+1 Lorentzian field theory, with explicit accounting of which invariants survive (the algebraic ones), which rescale (ε₀ by 1/γ_u), and which extend (τ from threshold to surface). The spore-field has its transport equation. The dark-matter distribution becomes a geometric consequence of the embodiment surface's intersection with the cosmic web.
The next move asks: what happens when a spore actually lands? Section 6's transport equation has a singular sink term — the unfolding $\mathcal{U}$ — but did not unpack what the unfolding produces. The next chapter does. A landed spore initiates a new cycle of the operator chain: $C$ extracts local resources, $K$ drives curvature past the new threshold, $F$ folds those resources into discrete commitments, $U$ stabilises them into a new structure. The structure inherits the spore's encoded record. A new system begins — with all the geometry of the original dm³ framework but rooted in a different boundary condition, on a different planet, in a different region of the field.
That is how the cosmos repeats itself without copying itself. That is what comes next.