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Compatibility Matrix

Reference artifact — instantiates Controlled Demixing and Domain Formation

A pairwise register of which constituents may share a domain and which must be kept apart, each verdict tied to the antagonism condition and the evidence behind it.

A Compatibility Matrix is the up-front, curated artifact that answers one question for every pair of constituents in a mixed system: may these two share a region, or must they be held in separate domains? It lays the constituents along both axes and fills each cell with a governance verdict — allowed, conditional, discouraged, or prohibited — and, critically, attaches to each verdict the condition under which the antagonism bites and a pointer to the evidence for it. Its defining move is that it is qualitative and human-curated, not computed: where a phase diagram derives coexistence from thermodynamics, the matrix records considered preferences and rules, so it can encode judgments ("never co-locate these, per regulation") that no free-energy model would produce. It is the design-time backbone the rest of the separation is built on.

Example

A specialty-chemicals warehouse builds a compatibility matrix over the ~40 hazard classes it stores: oxidizers, flammable organics, strong acids, bases, water-reactives, cyanides, peroxides. The classes are the rows and columns — the constituent inventory. Each cell gets a verdict: co-store freely / co-store with secondary containment / segregate by distance / never together. A prohibited cell doesn't just say "no"; it names the antagonism condition (oxidizer + flammable → ignition on contact; acid + cyanide → toxic HCN release) and links the evidence (the safety data sheet, a regulatory segregation table, a past incident class).

That artifact then drives the physical layout. The racking plan, the aisle assignments, and the pick paths are all read off the matrix rather than re-argued each time. When a new product arrives, the receiving clerk doesn't reason about chemistry — they look up its class row and place it where every cell says it may safely go. The matrix has turned tacit "don't shelve those together" lore into an auditable object anyone can query.

How it works

Every pair gets a cell; the verdict comes from a small fixed vocabulary so the whole grid reads consistently. What makes it this mechanism rather than a generic table is the discipline attached to each cell: a condition (the concentration, temperature, or presence-of-water under which the pairing flips from safe to hostile) and an evidence pointer (why the verdict is what it is). Conditional cells carry their mitigation inline — the barrier, distance, or additive that downgrades a prohibition to a permission. The matrix is deliberately symmetric or explicitly directional, and it is versioned, because verdicts age as conditions and evidence change.

Tuning parameters

  • Verdict granularity — a binary allowed/forbidden grid versus a four- or five-level scale. Finer scales capture "conditionally fine" pairings but multiply the judgment calls and the upkeep.
  • Conditioning depth — whether cells are context-free or annotated with the condition that flips them. Context-rich cells are truer to reality but heavier to build and maintain.
  • Constituent grain — broad classes versus individual species. Coarse classes keep the grid compact but hide intra-class exceptions; fine grain is exhaustive but explodes combinatorially.
  • Evidence bar — how strong the proof required to write a prohibition (a regulation, one incident, a suspicion). A high bar avoids over-cautious false prohibitions but is slow to add warranted caution.
  • Symmetry stance — treating A-with-B as identical to B-with-A, or allowing directional entries where one constituent contaminates the other but not the reverse.

When it helps, and when it misleads

Its strength is making tacit separation knowledge explicit, auditable, and fast to query, and forcing every "keep these apart" into the open together with why and under what condition. It is the shared reference a layout, a routing rule, or a formulation roster can all be built from without re-litigating the chemistry.

Its structural blind spot is that it is pairwise by construction, so it cannot see higher-order interactions — three constituents each individually compatible can still be jointly hostile, and no cell will warn you.[1] Verdicts also freeze while the world moves: a cell that is true at low concentration is wrong at high, and an un-versioned matrix quietly rots. The classic misuse is filling it in after a layout or recipe is already chosen, to rationalize the decision rather than to design it. The discipline that keeps it honest is to date and source every cell, flag the conditions that flip each conditional verdict, and review on a cadence rather than treating the grid as permanent truth.

How it implements the components

  • interaction_preference_map — the matrix is the map of pairwise like/unlike affinity, rendered as governance verdicts rather than raw energies.
  • constituent_and_composition_inventory — its rows and columns enumerate the constituents (and, where tracked, their proportions) that are in play.
  • functional_separation_objective — each "hold apart" or "may combine" verdict encodes what the separation is meant to achieve or prevent, cell by cell.

It does not measure how sensitive those preferences are to changing conditions — that is Interaction-Parameter Sweep — nor does it locate the actual coexistence boundary in state space, which is Phase-Diagram Mapping's job. The matrix records preferences; it does not compute where or when separation happens.

Notes

The matrix is a design-time artifact — it says what should be held apart, not whether a running batch is drifting toward a forbidden mixing. That live question belongs to Phase-Boundary Monitor, which watches the process against boundaries the matrix helped define. Keeping the static register and the live monitor separate lets each be revised without disturbing the other.

References

[1] The non-additivity of interactions — that pairwise compatibility need not compose to three-way compatibility — is a recognized limit of interaction matrices in community ecology and in drug-combination screening, where triplets can be antagonistic even when every pair is benign. A pairwise grid is structurally blind to it. ↩