The Assembly

Chapter 5 · Roadmap

Three fields, three edges

Each pair of fields has a transfer that would extend one with the other's tools. Each transfer requires a theorem that doesn't exist yet. The triangle below labels the edges by what's missing.

The theorem program

Three theorems would close the triangle. Computational attempts (March 2026) produced one structural finding, one closed-form formula, and one achievability bound. None is a finished theorem. All three advanced the problem.

Theorem 1: R-D bounds for journey observables

Transfers: codec R-D optimization → TVG theory.

Achievability (partial result): Timestamp coarsening into bins of width Δ gives rate below lossless with controlled journey distortion. Navlakha et al. (2008) did this for static graphs under MDL. Bento & Ioannidis (2020) give an information-theoretic framework for lossy link-stream compression but without journey-aware distortion.

Converse (partial): The interval transfer tensor (foremost arrival times between all pairs within a checkpoint interval) is i.i.d. across intervals, giving R ≥ K · R_seg(D). Optimal spacing L* ≈ log(n)/log(np). But journeys couple intervals, so the tensor-level converse does not directly yield end-to-end guarantees. The distortion definition also needs repair (multiplicative vs. additive slack).

Dead end: Serializing the temporal graph as a string and applying lossy LZ fails. Hamming distortion is agnostic to event criticality, so worst-case journey distortion is trivial regardless of serialization order.

Theorem 2: Tropical sheaves (strongest finding)

Transfers: TVG formalism → sheaf cohomology.

Finding: The naive tropicalization of sheaf MFMC breaks on event graphs. On a 4-vertex example with tropical stalks, H⁰ gives earliest-arrival potentials, but max-flow > min-cut because tropical flows are potentials, not conserved commodities. No subtraction in the tropical semiring means the coboundary cannot be written in standard form.

This is a structural obstruction, not an artifact. Existing sheaf MFMC (Krishnan 2014, Ghrist & Hiraoka) works over vector spaces where conservation makes sense. The tropical semiring breaks conservation. The open question is now precise: what cut notion replaces crossing-edge minima when the coefficient semiring has no additive inverses?

Theorem 3: Checkpoint spacing (reframed)

Transfers: sheaf cohomology → codec engineering.

Surprise: Checkpoints increase dim H¹, not decrease it. Each checkpoint adds d independent dimensions of obstruction. The right quantity is not dim H¹ but ‖r*‖ under a drift model.

Closed form: For a linear chain with weight matrix W and per-hop drift N(0, σ²I): E[‖r*‖²] = σ² · Σᵢ (1 − μᵢⁿ)/(1 − μᵢ), where μᵢ = eigenvalues of WᵀW. Optimal checkpoint spacing with cost λ: L* ≈ √(2λ/(σ²d)). The relevant spectral quantity is which singular values of W are near or above 1, not the condition number.

Status: The mathematics (Green's function on a path with transport and drift) is standard (Hansen & Ghrist 2019). The codec interpretation (recovering the GOP heuristic, correcting the condition-number intuition) is new but not sufficient for a standalone paper. The open question is generalization to non-chain dependency graphs.

What this is and what it isn't

This is a Rosetta Stone, not a proof. Three fields study the same directed dependency structure with different operations. The three theorems above are the assembly instructions: well-posed questions with existing tools on each side, nobody working the seam.

The vocabulary barrier is the obstacle, not the mathematics. The codec engineer's I-frame reset is the topologist's clamped sub-complex that zeros the harmonic component. The TVG theorist's chain fragility is the sheaf theorist's exactness failure. These connections have not been drawn in the literature because the three communities don't share venues.

Source material: Temporal Compression (blog post)