Architecture Document

Coherence Interface for Multi-Input Composition

Purpose

Build a coherence interface that makes “multi-input composition” usable without demanding strict syntactic associativity/interchange equations everywhere. Instead, composition is expressed as structured data + commuting comparison maps that guarantee the usual laws (associativity, unitality, whiskering/exchange) up to specified equivalence, and in a way that scales to bicategories / virtual equipments / Segal-style models.

This is an interface architecture: it defines what data you must provide, what the system guarantees, and how you validate coherence.

1) Design goals

G1 — Multi-input first-class

Support composition of many inputs at once (strings, grids, pastings), not only binary composition.

G2 — Coherence by structure, not by brute equations

Replace fragile “prove 200 lemmas that diagrams commute” with a small number of universal commuting squares / naturality laws.

G3 — Segal-like recoverability

Composition should be recoverable from gluing: multi-composition is determined by consistent local composites (Segal condition idea).

G4 — Interchange/whiskering compatibility as a core invariant

Horizontal/vertical interactions (2-cells, whiskering, exchange) must be enforced at the interface, not assumed.

G5 — Implementation neutrality

The architecture should apply whether your backend is:

strict 2-categories,
bicategories,
virtual double categories / equipments,
∞-categorical “spaces of composites”.

2) Non-goals (explicit)

Not a full axiomatization of virtual equipments.
Not a proof assistant formalization.
Not a commitment to “physical geometry.” “Geometric” means “diagrammatic/homotopical structure of composites.”

3) Core idea: “Composition = a contract”

A multi-composition system isn’t defined by an operator “∘” alone. It is defined by a contract:

What counts as a composable configuration (the “shape”).
What data you return when you compose (objects/1-cells/2-cells/multimorphisms).
What comparison maps exist between different ways of composing.
Which diagrams must commute (coherence laws).
Which equalities are strict vs which are witnessed by invertible 2-cells / equivalences.

This contract is the coherence interface.

4) Conceptual model: shapes, cells, and multimorphism spaces

4.1 Shapes (indexing objects)

A shape is a finite compositional pattern, e.g.:

linear string of composable arrows,
tree (multicategory-like),
grid/pasting diagram (2-dimensional),
“equipment” style: objects + proarrows + 2-cells.

We treat shape as a first-class index because coherence depends on shape.

Shape API (conceptual):

Boundary(S) → input/output boundary data (objects, endpoints).
Subshapes(S) → canonical decompositions (Segal cover).
Refinements(S) → ways to subdivide S into smaller pieces.
Gluing(S; pieces) → reassemble pieces into S.

4.2 Cells (what you compose)

We assume at least:

0-cells: objects
1-cells: arrows/proarrows/heteromorphisms
2-cells: transformations/squares

In a virtual equipment setting, “horizontal” and “vertical” directions may differ; the interface abstracts that.

4.3 Multimorphisms / composite spaces

For each shape (S) with boundary (∂S), there is a space (or set) of realizations
[
\mathrm{Comp}(S; ∂S)
]
that represents “all coherent ways to fill the shape with cells.”

In a strict setting: (\mathrm{Comp}) may be a singleton (unique composite).
In a bicategorical/∞ setting: (\mathrm{Comp}) can have multiple points and higher paths (coherence data).

This is the Segal pivot: composition is not a function; it is a controlled space of choices.

5) The Coherence Interface (CI): required components

The coherence interface is a package of five subsystems:

CI-1: Composition constructor (multi-input)

For each shape (S) and boundary data, provide:

a way to construct a composite filler or a point in (\mathrm{Comp}(S))
optionally with chosen “normal form” composite

Output types:

strict mode: a composite cell
weak mode: a composite + a witness that it’s equivalent to any other “legal” composite
∞ mode: a point in (\mathrm{Comp}(S)) with path data

CI-2: Segal decomposition maps (gluing ↔ products)

For canonical decompositions (S \simeq S_1 \cup \dots \cup S_n), provide a comparison:
[
\mathrm{Comp}(S) \longrightarrow \mathrm{Comp}(S_1)\times \cdots \times \mathrm{Comp}(S_n)
]
and an assembly map in the other direction (possibly up to equivalence).

Contract requirement: these maps exhibit “composition by gluing.”

This is where you cash out “Segal-like”: global composites are determined by consistent local composites.

CI-3: Whiskering (action of 1-cells on 2-cells / multimorphisms)

Provide left/right actions that extend composition when you attach extra 1-cells on boundaries:

left whisker: precompose
right whisker: postcompose

Whiskering must be functorial in the appropriate sense.

CI-4: Exchange/interchange law (the commuting square)

This is the backbone visible in your diagram: two ways to combine:

vertical composition of 2-cells and
horizontal composition/whiskering
must agree via a specified commuting diagram (strictly or up to coherent equivalence).

This is not optional. This is the exact place multi-input systems fail if you don’t force it.

CI-5: Natural transformations between presentations (change-of-coordinates)

Whenever you have two valid “presentation choices” (different decompositions, different cut points, different normal forms), provide:

a canonical comparison map (ψ)
and require a commuting diagram relating them

This prevents “coherence drift” where different normal forms silently diverge.

6) Coherence laws (what must hold)

You want a small list that generates everything.

L1 — Segal recoverability (local-to-global determinacy)

For each canonical cover, the decomposition map is an equivalence (or has specified universal property). Intuition:

A composite over the whole shape is the same data as compatible composites over the pieces.

L2 — Associativity as a coherence diagram

Associativity becomes: two different parenthesizations correspond to two decompositions of the same shape. The interface requires that the comparison between them is:

identity (strict),
or an invertible 2-cell (bicategorical),
or a contractible choice in (\mathrm{Comp}(S)) (∞).

L3 — Units as degeneracies

Units are not extra axioms glued on; they appear as degenerate shapes (empty string, identity edge). Degeneracy maps must satisfy simplicial identities (or their weak versions).

L4 — Whiskering functoriality

Whiskering respects composition:

whisker after composing equals composing after whiskering (in the relevant direction)
strict or up to coherent equivalence

L5 — Exchange / whiskering interchange (the key commuting square)

If you have 2-cells (\alpha, \beta) and composable 1-cells around them, then:

“compose vertically then whisker” equals “whisker then compose vertically,”
and similarly with the other direction.

This is what the last line of your image calls the whiskering exchange law.

L6 — Naturality of comparison maps

All (ψ)-maps (change of presentation) must be natural with respect to whiskering and Segal gluing. This stops “presentation-dependent semantics.”

7) Concrete specialization: 2-categories (sanity baseline)

In a strict 2-category (A):

shapes are pasting diagrams
(\mathrm{Comp}(S)) is essentially a singleton (unique composite)
whiskering is strict
interchange holds as a strict equation

Your CI reduces to:

multi-pasting defines a unique composite 2-cell
all the comparison maps are identities
the “commuting squares” become actual commutative diagrams in hom-sets

This is the baseline test: the CI must collapse to ordinary 2-category coherence.

8) Concrete specialization: bicategories / virtual equipments (why CI matters)

In bicategories/equipments:

associativity is not strict: you have associators
units not strict: unitors
composition of proarrows may be defined only up to equivalence
there are multiple legal composites; you need comparison data

CI tells you exactly what to store:

composites + witnesses
gluing equivalences
exchange coherence between “vertical” and “horizontal” compositions

You’re no longer proving coherence from scratch; you’re declaring the coherence interface and then validating it once.

9) Validation strategy (how you know CI is satisfied)

You validate the interface at three layers:

V1 — Local coherence checks (small shapes)

Check:

triangles (unit laws)
pentagons (associativity coherence)
basic interchange square (whiskering exchange)
naturality squares for (ψ)

V2 — Segal cover completeness

For each shape class, ensure your canonical decompositions generate all parenthesizations/refinements you care about, and that gluing comparisons are coherent under refinement of refinements.

V3 — Contractibility / uniqueness-of-composite (if desired)

In “∞/Segal” mode, the strongest form is:

(\mathrm{Comp}(S)) is contractible for admissible shapes (unique composite up to unique higher homotopy).
That gives you “coherence by homotopy” rather than by rewriting.

10) Operational patterns (how the CI is used)

Pattern P1 — Compose by refinement, then forget refinement

To compose a complex shape:

choose a refinement into elementary shapes
compose each elementary piece
glue via Segal maps
erase the refinement by transporting along (ψ)-comparisons

This is the scalable “no brittle parenthesization wars” pattern.

Pattern P2 — Multi-input composition as “multimorphism spaces”

Instead of a single composite morphism, return (\mathrm{Comp}(S)) or a canonical point with evidence it’s equivalent to others. This is where “robustness” comes from: the system admits multiple realizations but insists their differences are tracked.

11) Deliverables (what an implementation must provide)

A coherence interface implementation must supply:

A shape vocabulary (strings/trees/grids/pastings) and canonical decompositions
A definition of (\mathrm{Comp}(S)) for each shape
Segal maps: restriction to pieces and gluing back
Whiskering actions
Exchange/interchange coherence (the commuting square)
Presentation comparison maps (ψ) and their naturality
A validation suite (minimal diagrams that generate the rest)

12) Common failure modes (what this architecture prevents)

Parenthesization dependence: two ways to compose give different results with no canonical bridge.
Whiskering drift: left/right actions don’t respect vertical composition.
Refinement instability: changing the decomposition changes semantics.
Hidden strictness: pretending things are strict and then getting contradictions later.

The CI makes these failures explicit as violations of named laws rather than mysterious “coherence bugs.”

Physical realization category for SCIT-P

Category choice: QTD-Proc

A physically grounded target that fits your thermodynamics + coherence context (Liu–Segal, TUR, quantum absorption refrigerators) is a 2-category of quantum thermodynamic processes.

Objects (0-cells): open-system interfaces

An object is a triple
[
X := ( \mathcal{H}_X,; H_X,; \mathsf{Ports}_X )
]

(\mathcal{H}_X): finite-dimensional system Hilbert space
(H_X): system Hamiltonian (defines “energy” observable)
(\mathsf{Ports}_X): declared coupling interfaces (e.g., hot/cold/work bath ports, measurement ports, control knobs)

This forces “what counts as heat/work/current” to be well-typed at the boundary.

1-morphisms (1-cells): implementable process fragments

A 1-cell (f: X \to Y) is an implementable protocol fragment with explicit thermodynamic readout structure. Pick one concrete representation (either works):

(A) Channel representation
[
f = (\Phi_f,; \mathsf{Readout}_f,; \mathsf{Cost}_f)
]

(\Phi_f): CPTP map from states on (X) to states on (Y) (possibly on (X) itself if (Y=X))
(\mathsf{Readout}_f): declared observables you can extract (heat increments, current estimators, entropy production estimators, etc.)
(\mathsf{Cost}_f): resource bookkeeping (time window, control bandwidth, energy budget, sampling budget)

(B) Generator representation
[
f = ( \mathcal{L}_f(t),; t\in[0,T_f],; \mathsf{Readout}_f,; \mathsf{Cost}_f )
]

(\mathcal{L}_f(t)): Lindbladian/Redfield-type generator segment (you can support coherences explicitly)
same readout/cost structure

Composition (g\circ f) is just concatenation in time (plus interface matching of ports/readouts). Monoidal product (f\otimes g) is parallel composition (two devices running side-by-side).

2-morphisms (2-cells): calibrated refinements / witnesses

A 2-cell (\alpha: f \Rightarrow g) is not “equality”; it’s a witness that (f) and (g) are physically the same under a declared observational regime.

Concretely, (\alpha) can be:

a refactoring equivalence (different circuit decompositions / different dilation choices)
a coarse-graining map (discarding micro-details while preserving macro readouts)
a calibration transformation (renormalizing measurement interpretation)

But crucially: it carries a quantified error bound (below).

Bake “indistinguishable observables under budgets” into the theory

Budgets

Fix a budget profile
[
B := (T,; \mathcal{S},; \epsilon,; N,; E_{\max},; \delta)
]
where typical components are:

(T): horizon / integration time
(\mathcal{S}): admissible set of initial states (e.g., energy-bounded states, or “prepared states allowed by lab”)
(\epsilon): tolerance
(N): sampling budget / trajectory count (if you estimate currents/noise statistically)
(E_{\max}): energy constraint (if you want energy-constrained distinguishability)
(\delta): resolution thresholds (instrument precision / binning)

Observable family

Choose an observable family (\mathcal{O}_B). For thermodynamic machines this is naturally:

heat currents (J_v) into each bath (v)
current noise / second cumulant (\langle!\langle J_v^2\rangle!\rangle)
entropy production rate (\sigma)
optionally: full counting statistics cumulants up to some order (k) given the budget

Budgeted observational distance

Define a distance between two process fragments (f,g: X\to Y):
[
d_B(f,g) ;:=; \sup_{O\in\mathcal{O}B};\sup{\rho\in\mathcal{S}}; \frac{|,O(f;\rho)-O(g;\rho),|}{\mathrm{scale}(O,B)}.
]

(\mathrm{scale}(O,B)) is just normalization so different observables compare fairly (e.g., divide by typical magnitude or by an allowed bound).
If you want a more “quantum information” version, you can replace this with an energy-constrained diamond norm, but the observable distance is the most physically explicit.

The equivalence relation (this is the key)

[
f \sim_B g \quad\Longleftrightarrow\quad d_B(f,g)\le \epsilon ;;\text{and};; \mathsf{Cost}_f \preceq B,;\mathsf{Cost}_g \preceq B.
]

So equivalence is not free: it only holds inside the measurement regime and resource envelope you declare.

SCIT-P: Segal condition “up to physical equivalence”

Now we define SCIT (Segal Coherence Interface Theory) with physicality by requiring Segal conditions in the localized (budgeted) sense.

Shapes and decomposition (pattern)

Let (O) be your shape/pasting category:

objects of (O): multi-input composition shapes (strings/trees/grids/pastings)
inert maps: projections from a composite shape to elementary pieces
(O_{\mathrm{el}}): elementary components (single gates, single couplings, single dissipators, single measurement blocks)

SCIT-P realization

A “physical SCIT algebra” is a functor
[
F: O \to \textbf{QTD-Proc}
]
assigning to each shape a physical process fragment.

Physical Segal condition (the baked-in version)

For each composite shape (X\in O), there is a canonical Segal comparison map
[
\theta_X:; F(X);\longrightarrow;\lim_{(X\to E)\in O_{\mathrm{el},X/}} F(E)
]
(the “restrict to elementary pieces” map).

SCIT-P requires: (\theta_X) is an equivalence in the budgeted operational sense, meaning:

(Recoverability) There exists a reconstruction map (assembly)
[
\mu_X:;\lim F(E)\to F(X)
]
such that ( \mu_X\circ \theta_X \sim_B \mathrm{id}_{F(X)}).
(Uniqueness up to observables) Any two reconstructions from compatible local data are equivalent:
[
\mu_X(\text{locals}) \sim_B \mu_X'(\text{locals})
]
i.e., the space of composites has diameter (\le\epsilon) under (d_B).

That’s the physical heart: composition is well-defined because different legal gluings are observationally indistinguishable under the declared budget.

Exchange/whiskering becomes a physical commutation constraint

When SCIT talks about the “whiskering exchange law,” SCIT-P states:

Two different ways of composing-and-context-attaching must agree up to (\sim_B).

So interchange is not “equation”; it’s:
[
(\text{whisker}\circ \text{vertical-compose}) ;\sim_B; (\text{vertical-compose}\circ \text{whisker})
]
with an explicit witness 2-cell carrying the (\epsilon)-bound.

Thermodynamic hard constraints can be built into admissibility

If you want SCIT-P to be non-trivial (not “anything goes”), you can treat constraints like TUR-style inequalities as gating predicates on allowed composites:

A composite (f) is admissible only if, for the chosen readouts,

entropy production is nonnegative in the sense you’re using,
and any required bounds (e.g., precision–dissipation tradeoffs) are respected within tolerance.

That turns “physicality” into checkable coherence rather than rhetoric.

concrete instantiation

pin SCIT-P to quantum absorption refrigerators specifically:

objects: working medium + bath ports
1-cells: coupling segments to hot/cold/work baths + internal unitary segments
observables: (\langle J_c\rangle), (\mathrm{Var}(J_c)), (\sigma), COP, and selected cumulants
equivalence: two implementations are “the same refrigerator composite” iff those observables match within (\epsilon) under budget (B)

Chu–Haugseng

Chu–Haugseng is basically a factory for SCIT: it tells you exactly when “coherence-by-gluing” (Segal conditions) is the right interface, and when you can make it computational (free objects by an explicit colimit formula) instead of “defined by localization and hope.”

1) The core move: “algebraic pattern” = your coherence interface schema

They define an algebraic pattern (O) as:

an (\infty)-category (O),
with an inert/active factorization system ((O_{\mathrm{int}}, O_{\mathrm{act}})),
plus a subcategory of elementary objects (O_{\mathrm{el}}\subseteq O_{\mathrm{int}}).

That’s a direct formalization of the SCIT idea:
“composition is recovered from elementary pieces + allowed gluings + coherence transport along factorization.”

2) “Segal (O)-objects” = coherence-by-gluing as a limit condition

A Segal (O)-object is a functor (F:O\to C) (for an appropriate target (C)) such that each (F(X)) is the limit over all inert maps from (X) into elementary objects:

[
F(X);\simeq;\lim_{E\in O_{\mathrm{el},,X/}}F(E)
]

Interpretation in SCIT language:

(X) = a whole multi-input “shape”
inert maps (X\to E) = its canonical projections to elementary components
the Segal condition says: the whole is determined by compatible local data (your “gluing coherence interface”).

They also note the operational equivalence: this Segal condition is the same as saying (F|{O{\mathrm{int}}}) is a right Kan extension of (F|{O{\mathrm{el}}}).
That’s a very SCIT-friendly reformulation: coherence is “extension from generators”.

3) What structures this subsumes (why this is a unification theorem, not a vibe)

In the abstract they list: (\infty)-categories, ((\infty,n))-categories, (\infty)-operads (symmetric, non-symmetric, cyclic, modular), (\infty)-properads, and algebras over an (\infty)-operad in spaces—all as “Segal (O)-spaces” for appropriate patterns (O).

So if SCIT is “a coherence interface for multi-input composition,” this paper is showing there’s a general pattern compiler that generates the Segal interface for lots of known composition theories.

4) The “extendable” criterion = when your interface yields explicit free constructors

They distinguish between:

patterns where Segal objects exist but free ones are defined via an abstract localization (hard to compute),
vs patterns where free Segal objects are given by an explicit colimit formula.

They call the good case extendable and give necessary/sufficient conditions (later in the paper) for when this happens.

For SCIT: “extendable” is basically the difference between:

a coherence interface you can describe, and
a coherence interface you can actually generate composites from.

5) Polynomial monads = the computational substrate behind “Segal = algebra”

They connect extendable patterns to polynomial monads (cartesian monads on presheaf (\infty)-categories that are accessible and preserve weakly contractible limits).

Two key directions:

If (O) is extendable, the free Segal (O)-space monad is polynomial.
Conversely, every polynomial monad (T) yields a canonical extendable pattern (W(T)) whose Segal spaces are equivalent to (T)-algebras.

This is huge for SCIT because it says:
“Your coherence interface isn’t just presentation—it’s equivalent to a monad (an algebraic machine) when you’re in the polynomial regime.”

They also prove an (\infty)-categorical nerve theorem for these polynomial monads and use it to build factorization systems on Kleisli (\infty)-categories.
That’s the formal reason “coherence by Segal conditions” becomes canonical rather than ad hoc.

6) How this plugs into your SCIT architecture (direct mapping)

If you’re naming “Segal Coherence Interface Theory (SCIT),” this paper supplies the missing rigor knobs:

SCIT Shapes → objects of the pattern (O)
Elementary generators → (O_{\mathrm{el}})
Legal decompositions → inert maps into (O_{\mathrm{el}})
Multi-input composition as gluing → Segal limit condition (and the Kan extension characterization)
“Free composition” / generative layer → extendability (explicit colimit formula for the free Segal monad)
Executable algebraic engine → polynomial monad equivalence

Coherences and the thermodynamic uncertainty relation: Insights from quantum absorption refrigerators” (2020),

Here’s what Liu & Segal are actually showing in “Coherences and the thermodynamic uncertainty relation: Insights from quantum absorption refrigerators” (2020), and why it matters.

1) Question they attack

The thermodynamic uncertainty relation (TUR) says there’s a tradeoff between precision (low current noise) and dissipation (entropy production). It’s classical in origin (Markov jump processes), but people debate what happens in quantum coherent thermal machines.

They ask: when the working medium has steady-state coherences between energy eigenstates, do you get better current constancy (lower relative noise), and could that push you closer to (or below) the TUR bound?

2) Method (the technical spine)

They compute heat current statistics using full counting statistics wrapped around a counting-field dressed Redfield master equation (χ-Redfield).

Key mechanics:

Introduce counting fields ( \chi_v ) for each bath and define a moment generating function via two-time energy measurements.
Recast dynamics in Liouville space with a χ-dependent Liouvillian (L_\chi); the cumulant generating function is set by the eigenvalue with smallest real part.
Extract current and noise from derivatives of that eigenvalue.
Compare full (non-secular) Redfield vs secular/incoherent limit where coherences vanish.

3) They explicitly check thermodynamic consistency (important)

A worry with Redfield is thermodynamic inconsistency / positivity issues. They test whether their χ-Redfield satisfies the steady-state exchange fluctuation symmetry for heat exchange (a fluctuation theorem symmetry condition).

They report strong numerical evidence that the symmetry holds (errors attributed to numerics, and not dependent on the coherence-tuning parameter in their test model).

4) Central empirical claim (their main result)

Across the quantum absorption refrigerator models they study:

Coherence can either suppress or enhance cooling power (mean current), depending on model/parameters.
But coherence always increases the relative noise (standard deviation / mean) of the cooling power compared to the incoherent (secular) limit.

That’s the punchline: coherence is not behaving like a “free stabilization resource.” It behaves like “extra quantum fluctuation channels.”

5) TUR outcome

They use the TUR in the form (for bath (v)):
[
\frac{\langle!\langle J_v^2\rangle!\rangle}{\langle J_v\rangle^2},\langle\sigma\rangle \ge 2,
]
with entropy production rate ( \langle\sigma\rangle = -\sum_v \langle J_v\rangle \beta_v ).

Their conclusion: for the steady-state QARs examined, the TUR is always satisfied, and coherence does not explain TUR violations seen elsewhere in weak-coupling quantum machines.

6) What to take away (the “insight” layer)

If you’re trying to use coherence as a “performance booster,” this paper’s message is:

coherence may raise mean power, but it tends to raise relative fluctuations too,
so any “advantage” needs to be judged under a precision–dissipation lens, not just average output.

SCIT-P for quantum absorption refrigerators (QARs), done non-trivially

1) Physical realization 2-category: QAR-Proc

Objects (interfaces).
An object is a typed refrigerator interface:
[
X=(\mathcal H,; H,; {\text{ports }v\in{c,h,w}},; {\beta_v},; \text{allowed preparations }\mathcal S)
]

(\mathcal H, H): working medium Hilbert space + Hamiltonian
ports (c,h,w): cold/hot/work bath couplings (interface types)
(\beta_v): bath inverse temperatures (part of the interface contract)
(\mathcal S): admissible initial state family (your “what can the lab prepare?” set)

1-cells (implementations).
A 1-cell is not “a story about coupling.” It is a concrete dynamical segment:
[
f:;X\to X
\quad\text{given by}\quad
(\mathcal L_f,; \tau_f,; \text{port annotations},; \text{readout convention})
]

(\mathcal L_f): a generator (e.g., Redfield/Lindblad-like, possibly non-secular to include coherences)
(\tau_f): duration
readout convention: how heat increments/current counting is defined (two-time measurement scheme, etc.)

This matches exactly how Liu–Segal compute currents/noise: introduce counting fields and work with a (\chi)-dressed generator / Liouvillian whose leading eigenvalue yields the cumulant generating function.

Composition.
Serial composition is concatenation of segments:
[
g\circ f := (\mathcal L_g,\tau_g);\text{after};(\mathcal L_f,\tau_f)
]
implemented as superoperator composition on states / on the (\chi)-tilted propagators. Parallel composition is tensor product if you run devices side-by-side.

2-cells (coherence witnesses).
A 2-cell (\alpha:f\Rightarrow g) is a certified simulation/refinement: a statement that two implementations are operationally indistinguishable under the budgeted measurement regime (below), and crucially that indistinguishability is preserved when you plug them into larger composites (whiskering).

That “preserved under plugging into context” part is what makes it not a joke.

2) Don’t pick a few observables — pick the right observable functor

If you want QAR physicality, the canonical observable isn’t just (\langle J_c\rangle) and (\mathrm{Var}(J_c)). It’s the cumulant generating function (CGF) (or its leading eigenvalue) from full counting statistics, because that contains all cumulants up to the truncation you can afford.

Liu–Segal’s workflow is exactly: CGF comes from the smallest-real-part eigenvalue of the (\chi)-dependent Liouvillian; currents/noise are derivatives of it.

So define the observable functor (for each budget (B)):

[
\mathcal O_B(f);=;\Big(\lambda_f(\chi_c,\chi_h,\chi_w);\text{restricted to};|\chi|\le \chi_{\max}(B)\Big)
]

and then derive from (\lambda) whatever you want:

(\langle J_c\rangle), (\mathrm{Var}(J_c)) (2nd cumulant), higher cumulants
entropy production rate (\sigma) (e.g. (-\sum_v \beta_v \langle J_v\rangle) in their setup)
COP = (\langle J_c\rangle/\langle J_w\rangle) (with your chosen sign convention)

Now your earlier list becomes derived summaries, not the definition of physical sameness.

3) Budgeted equivalence that is actually compositional

Define a budget:
[
B=(\tau_{\max},; \mathcal S,; \chi_{\max},; k_{\max},; \epsilon)
]
(time horizon, allowed initial states, counting-field resolution, cumulant order, tolerance).

Define equivalence:
[
f \sim_B g
\quad\Longleftrightarrow\quad
\sup_{\rho\in\mathcal S}\sup_{|\chi|\le\chi_{\max}}
\left|\lambda_f(\chi;\rho)-\lambda_g(\chi;\rho)\right|
\le \epsilon
]
(optionally only up to (k_{\max}) derivatives at (\chi=0) if you’re truncating cumulants).

Key requirement (congruence):
[
f\sim_B g ;\Rightarrow; h\circ f \circ k \sim_{B'} h\circ g\circ k
]
for any physically admissible “context” segments (h,k) (with adjusted budget (B')).
This is the “whiskering/exchange” physicality: equivalence survives embedding into larger composites—the exact thing your diagram was about.

4) SCIT part: Segal gluing is now a statement about propagators/CGFs

Chu–Haugseng’s Segal condition says a composite shape is determined by its elementary inert projections (gluing).

Here the “shape category” (O) is: pasting diagrams made from elementary QAR segments (bath-coupling pieces + internal unitary pieces). The Segal map is:

restrict a whole schedule to its pieces (inert projections)
then reconstruct by gluing (active assembly)

Physical Segal condition (QAR-version):
For any composite schedule (X) and any two legal decompositions of it into elementary segments,
the reconstructed composite implementations are (\sim_B)-equivalent (same CGF up to budget).

That’s how SCIT becomes “physics”: decomposition-independence at the level of measurable current statistics.

5) Where TUR fits (optional hard gate)

If you want “hard UCF/ORSI gates” in this QAR setting, you can add a validator that rejects any composite whose predicted ((\langle J\rangle,\mathrm{Var},\sigma)) violates your chosen bounds (e.g., TUR-style constraints). Liu–Segal use the TUR form involving current variance, mean current, and entropy production.

This doesn’t define equivalence; it’s a domain admissibility gate.

your original “same iff a few observables match”

Because with only (\langle J_c\rangle), (\mathrm{Var}(J_c)), (\sigma), COP, you can have two genuinely different machines (different higher cumulants, different transient behavior, different context response) that pass the test. That’s not physical sameness; it’s a lossy summary.

The fix is what I wrote above: equivalence is kernel of an observable functor that is rich enough (CGF / truncated CGF) and enforced as a congruence under composition.

Step 1 is: turn “a virtual equipment” into a Segal object so that composition and coherence live as spaces, not as brittle equalities.

The clean way to do it (Chu–Haugseng style) is: define an algebraic pattern (O_{\mathrm{VE}}) whose objects are your pasting shapes, whose inert maps are “take faces/projections,” whose active maps are “glue/substitute,” and whose elementary objects are the atomic generators (single proarrow, single square, etc.). An algebraic pattern is exactly this triple: ((O,;O_{\mathrm{int}},O_{\mathrm{act}},;O_{\mathrm{el}})).

Then a Segal (O)-space is a functor (F:O\to \mathcal S) such that every composite shape is determined by its inert projections to elementaries via a limit condition.

That’s your “make the virtual geometric”: replace “there exists a composite” with “there is a space of composites, constrained by gluing.”

1) Choose what you’re “Segalifying” in a virtual equipment

A virtual equipment (informally) has:

objects (a,b,\dots)
vertical arrows (f:a\to b)
horizontal proarrows (P: a \rightsquigarrow b)
2-cells (squares) whose horizontal sides are proarrows and vertical sides are arrows
horizontal composition of proarrows (associative up to coherent stuff)
whiskering/exchange: vertical composition of 2-cells must be compatible with horizontal pasting

So your Segalification must encode both:

multi-input horizontal composition (strings of proarrows), and
2D pasting coherence (squares compose and interchange correctly).

That’s why a single simplicial direction is not enough: you want at least a double/2D Segal object (implemented as a bisimplicial space or an algebraic pattern whose shapes are pasting diagrams).

2) Build the shape category (O_{\mathrm{VE}})

You don’t start with equations. You start with shapes.

2.1 Horizontal shapes (strings)

Let (O_{\mathrm{h}}) have objects that look like:

([n])-strings of composable proarrows
((a_0 \rightsquigarrow a_1 \rightsquigarrow \cdots \rightsquigarrow a_n))

Elementary objects: the “edge/corolla” shapes (a single proarrow (a\rightsquigarrow b)).
Inert maps: “pick the (i)-th edge” (projection from a string to one adjacent pair).
Active maps: “substitute a string into an edge” (the operadic/composition move).

This is exactly the Segal recipe: inert = projections, active = composition.

2.2 Square shapes (2-cells)

Let (O_{\square}) have objects that are grid fragments:

one square
vertical composites of squares
horizontal composites of squares
rectangular pastings

Elementary objects: a single square (one 2-cell generator) and the degenerate “identity” shapes.
Inert maps: face maps (pick a constituent square / boundary edge).
Active maps: gluing along shared boundaries.

2.3 Combine them (the “virtual equipment” pattern)

Now define (O_{\mathrm{VE}}) so it contains both kinds of shapes and their interactions:

pure horizontal strings (to define proarrow composition)
pure vertical strings (to define vertical arrow composition)
mixed pastings (to force whiskering/exchange compatibility)

Structurally, this is the same move Chu–Haugseng describe: give (O_{\mathrm{VE}}) an inert/active factorization system and specify elementary generators, then Segal conditions do the rest.

3) “Segalify” = impose the Segal limit condition (composition-by-gluing)

A Segal (O)-object is a functor (F:O\to C) such that for every shape (X),
[
F(X);\simeq;\lim_{(X \to E)\in O_{\mathrm{el},X/}} F(E),
]
i.e. the value on a big shape is the limit over its inert projections to elementaries.

For you, set (C=\mathcal S) (spaces). Then:

(F(\text{one horizontal edge})) = space of proarrows (a\rightsquigarrow b)
(F(\text{string of length }n)) = space of coherent (n)-composites
(F(\text{one square})) = space of 2-cells (squares) with given boundary data
(F(\text{grid})) = space of coherent pastings, determined by its constituent squares

So “virtual → geometric” is literally: replace sets of composites by spaces of composites, where higher points/paths are the coherence.

A useful equivalent form: the Segal condition says (F) is the right Kan extension from the elementary part along the inert subcategory.
That’s your interface: “all higher composition data is forced from generators + inert projections.”

4) Where whiskering/exchange gets enforced

The key thing you wanted from that diagram earlier: exchange/whiskering is not a separate axiom you tack on; you bake it into the shape category.

You include a minimal mixed pasting shape whose two decompositions correspond to:

“compose vertically then whisker”
“whisker then compose vertically”

Then the Segal condition forces the comparison to live inside a limit diagram, i.e. the square “commutes” as a homotopy coherence, not a strict equality.

So: exchange is implemented as a Segal constraint on a specific family of mixed shapes.

5) “A space of mathematical structures” (the moduli point)

Once you’ve fixed (O_{\mathrm{VE}}), the “space of virtual equipment structures” is simply:

the ∞-category (\mathrm{Seg}{O{\mathrm{VE}}}(\mathcal S)) of Segal (O_{\mathrm{VE}})-spaces (structures as functors satisfying the Segal condition)
and its maximal ∞-groupoid (take only equivalences) is your moduli space of structures

That is literally “a space of mathematical structures,” not a metaphor: points = implementations, paths = equivalences, higher paths = coherences between equivalences.

6) What you’ve achieved after Step 1

After this step you have:

a geometric object (F) (a Segal (O_{\mathrm{VE}})-space) whose values are spaces of proarrows/2-cells/composites
associativity/unitality/interchange are no longer “prove equalities,” they’re encoded in the Segal gluing constraints
a moduli space (\mathrm{Seg}{O{\mathrm{VE}}}(\mathcal S)) of all such “virtual equipments up to homotopy-coherent equivalence”

If you want Step 1 to be fully concrete, the next move is to decide what the elementary generators are for your version of “virtual equipment” (edge only? square only? include companions/conjoints as generators or as completeness conditions). Once you pick that, (O_{\mathrm{VE}}) stops being handwavy and becomes a buildable pattern.

Next move: freeze the pattern (O_{\mathrm{VE}}) so Segalification stops being a vibe and becomes a buildable object

You already did Step 1 conceptually (“virtual → geometric: spaces of composites”). Step 2 is to choose the generators and the inert/active plumbing so the Segal condition has something rigid to bite on.

Step 2A — Choose the elementary shapes (O_{\mathrm{el}})

Pick the smallest set that can generate all pastings you care about:

Horizontal edge E_h(a,b)
“one proarrow slot” (a \rightsquigarrow b)
Vertical edge E_v(a,b)
“one vertical arrow slot” (a \to b)
Square cell Sq
one 2-cell with boundary: two vertical edges + a horizontal source multi-proarrow + a horizontal target proarrow (or whatever your equipment uses)
Degeneracies (units)
identity vertical edge, identity horizontal unit (if your equipment has it), and identity square

That set is your interface vocabulary. Everything else is a composite shape.

Step 2B — Define the shape category (O_{\mathrm{VE}})

Objects: finite pasting shapes built by gluing those generators:

strings of horizontal edges (multi-input horizontal composition)
strings of vertical edges
grids/pastings of squares (to force whiskering/exchange)

Morphisms: refinements and face/projection maps between shapes (think: “this big shape contains these elementary faces”).

Now impose the inert/active factorization (Chu–Haugseng’s “algebraic pattern” move):

inert maps = “projection to a face / pick a component / forget context”
(the maps that exhibit how a big diagram decomposes into elementary pieces)
active maps = “substitution/gluing”
(the maps that actually perform composition)

This is the point where SCIT becomes a pattern in the sense of “inert/active + elementaries.”

Step 2C — Write the Segal condition once (and it generates the rest)

A realization is a functor
[
F: O_{\mathrm{VE}} \to \mathcal{S}
]
into spaces.

For each composite shape (X), the Segal condition is:
[
F(X);\simeq;\lim_{(X\to E)\in O_{\mathrm{el},X/}} F(E),
]
i.e. “the space of fillings of (X) is determined by compatible fillings of its elementary faces.”

That single line is what makes “composition = space of composites” precise.

Step 2D — Bake in exchange/whiskering by adding one mandatory mixed shape

Add a single canonical “interchange” pasting shape (the one whose two decompositions correspond to):

compose vertically then whisker
whisker then compose vertically

Once that shape is in (O_{\mathrm{VE}}), the Segal condition forces the “commuting square” you care about as coherence data, not as an extra axiom.

Step 3 (immediately after): completeness + extraction

Once (O_{\mathrm{VE}}) is fixed, there are two moves that make it usable:

3A — Add a completeness/univalence condition (so objecthood isn’t fake)

Segal alone gives you composition up to homotopy; completeness ensures your “equivalences” are the right ones (the Rezk-style fix, adapted to your shape category). This prevents “phantom objects” and makes the moduli space of structures behave.

3B — Define the desegalfier (extract a virtual equipment back)

Given (F):

objects = points of (F(\text{object-shape}))
proarrows = points of (F(E_h))
vertical arrows = points of (F(E_v))
2-cells = points of (F(Sq))
compositions = induced by Segal equivalences for the composite shapes

This is where “space of structures” turns into an actual operational virtual equipment presentation.

Your choice right now (the real “next move” fork)

Pick one, and we go:

Minimal pattern: only what’s needed to force horizontal composition + exchange (fastest path).
Full equipment pattern: include companions/conjoints, restrictions/extensions, Yoneda module shapes (Shulman-style), etc. (richer but heavier).
SCIT-P merge: choose the physical target category now and make (F) land in “processes,” not spaces (your physicality track).

Step 2 (minimal): Define the pattern (O_{\mathrm{VE}}) by generators + one forced interchange shape

2.1 Elementary shapes (O_{\mathrm{el}})

You only need three atom types:

Horizontal edge (E_h(a,b))
Represents one proarrow slot (a \rightsquigarrow b).
Vertical edge (E_v(a,b))
Represents one vertical arrow slot (a \to b).
Square (\Box)
Represents one 2-cell boundary pattern (vertical sides + horizontal source/target).

That’s your “alphabet.”

2.2 Composite shapes (what (O_{\mathrm{VE}}) contains)

Generate shapes freely from those atoms by gluing:

Horizontal strings (H_n): (a_0 \rightsquigarrow a_1 \rightsquigarrow \cdots \rightsquigarrow a_n)
(multi-input composition direction)
Vertical strings (V_m): (a_0 \to a_1 \to \cdots \to a_m)
Rectangular pastings (R_{m,n}): an (m\times n) grid of squares
(the minimum surface you need to talk about exchange)

2.3 Inert vs active maps (the real “pattern” move)

Inert maps = “take a face / project to a component”
Examples:
- from a string (H_n) to its (i)-th edge (E_h)
- from a rectangle (R_{m,n}) to one constituent square (\Box)
- from (R_{m,n}) to its boundary strings
Active maps = “glue / substitute”
Examples:
- glue (H_k) and (H_\ell) along a shared endpoint to get (H_{k+\ell})
- glue rectangles along a shared row/column to get a bigger rectangle

This is exactly the inert/active plumbing you need for Segal conditions to mean “composition by gluing.”

2.4 Add the one mandatory mixed shape (forces whiskering/exchange)

Include a single canonical (2\times2) pasting shape (R_{2,2}) and require that its two decompositions (by rows vs by columns) are both legal inert covers.

That one choice is what bakes in:

“compose vertically then whisker”
vs
“whisker then compose vertically”

as the same composite up to homotopy (not a separate axiom later).

Step 3: Define SCIT as a Segal object on that pattern

A Segalified virtual equipment is a functor
[
F: O_{\mathrm{VE}} \to \mathcal S
]
into spaces, satisfying three Segal constraints:

3.1 Horizontal Segal (strings)

For every horizontal string shape (H_n),
[
F(H_n);\simeq;F(E_h) \times_{F(\text{obj})}\cdots\times_{F(\text{obj})} F(E_h)
]
So an (n)-ary composite is determined by compatible adjacent proarrows.

3.2 Vertical Segal (strings)

Same for (V_m): vertical composition is recovered by gluing.

3.3 Square/rectangle Segal (pasting)

For each rectangle (R_{m,n}),
[
F(R_{m,n}) \simeq \lim(\text{its square-faces})
]
meaning: a pasted 2-cell is determined by its constituent squares plus boundary compatibility.

The special case (R_{2,2}) is your “exchange law generator.” If your (R_{2,2}) Segal condition holds, you’ve enforced the whiskering/interchange coherence structurally.

At this point, you’ve achieved Step 1 in a real way: the “virtual” compositions are now literally spaces of composites with coherence as paths.

Step 4: Add one completeness condition (so it’s not “fake geometry”)

Segal conditions alone can leave you with “too many equivalences.” Add a Rezk-style completeness condition adapted to your object edge:

equivalences in the “1-cell directions” are detected by equivalences in the appropriate mapping spaces
intuitively: objects are what their identity edges say they are

This is the step that upgrades “Segal object” → “honest coherent structure” (prevents phantom objecthood).

Step 5: Desegalfy (extract the virtual equipment back)

Once you have (F):

Objects: points of (F(\text{obj}))
Proarrows (a\rightsquigarrow b): points of (F(E_h(a,b)))
Vertical arrows (a\to b): points of (F(E_v(a,b)))
2-cells: points of (F(\Box)) with specified boundary data
Compositions: induced by the Segal equivalences for (H_n,V_m,R_{m,n})

This gives you a virtual-equipment-like structure by construction, not by postulating laws.

That’s the next move.

After this, the next next move (Step 6) is where “equipment” enters: add optional axioms as extra shapes/Segal completeness (companions/conjoints, restrictions/extensions, Yoneda-type universal cells). But you can’t do that coherently until (O_{\mathrm{VE}})+Segal+(R_{2,2}) is frozen.

Step 6 — Completeness (the anti-phantom upgrade)

Segal conditions give you composition-by-gluing. They do not guarantee that:

your “objects” aren’t duplicated by hidden equivalences,
your “units” behave like units up to the right notion of equivalence,
your “invertible 1-cells” are exactly the “equivalences” seen internally.

So you impose a Rezk-style completeness condition adapted to your two directions (vertical + horizontal) and the square layer.

6A) Choose where “equivalences live”

You need a designated subspace of “invertible 1-cells” in each direction:

vertical equivalences (invertible vertical arrows)
horizontal equivalences (proarrows that are equivalences under your equipment notion—often via representability/companions once you add them)

6B) Completeness statement (conceptual)

“Objects are determined by their identity 1-cells,” i.e. the map

from objects
to the space of equivalences
is an equivalence of spaces.

This is the Rezk move: it kills “fake objecthood” and forces equivalences to be detected correctly.

6C) Why this is the next move

Without completeness, your moduli “space of structures” can look rich but be semantically wrong: it may admit degenerate models where everything composes but object identity is ill-posed. Completeness is the first hard filter that makes it real.

Step 7 — Desegalfy (extract the virtual equipment back)

Now you define an explicit extraction functor from your Segal object (F) to the ordinary-looking data:

Objects: points of (F(\text{Obj}))
Vertical arrows: points of (F(E_v)) over chosen endpoints
Proarrows: points of (F(E_h)) over chosen endpoints
2-cells (squares): points of (F(\Box)) with boundary data
Composition: not defined by hand—it is induced by the Segal equivalences for (H_n), (V_m), and (R_{m,n})
Interchange/whiskering: induced specifically by the (R_{2,2}) (row-vs-column) Segal cover you included earlier

This is the moment your “space of structures” becomes a usable structure again.

Step 8 — Add the first equipment-specific universal feature (one choice)

Up to now, you have a “Segalified double-ish thing.” To make it a virtual equipment in the sense people actually use, you add exactly one of these as the next enrichment, depending on what you want:

Option A: Companions / conjoints layer

Add shapes that witness when a vertical arrow has a corresponding horizontal proarrow (“representable proarrows”). This is often the bridge that turns “horizontal direction” into something equipment-like rather than just another axis.

Option B: Restriction/extension (modules/profunctors) layer

Add shapes for universal cells representing restriction along vertical arrows (the “equipment calculus” people care about).

Option C: Yoneda shape package

Add the shapes needed to express a Yoneda-style embedding/universal property (this is where the Shulman-type story becomes visible: representables, modules, and how composition is controlled by universal cells).

Pick one. Don’t add all three at once or you’ll lose control of what your Segal conditions are enforcing.

Step 9 — Extendability (so SCIT becomes generative, not just axiomatic)

Once the pattern is fixed and completeness is in place, the next move is to ensure your SCIT object is constructible from generators (free completion), not just “the class of things satisfying constraints.”

This is the Chu–Haugseng “extendable” question in spirit: when can the free Segal object be written by an explicit colimit recipe rather than opaque localization machinery. That’s what makes SCIT a build system.

Your immediate choice (so “next move” becomes one concrete action)

Say which path you want:

Completeness now (Step 6) — lock the notion of equivalence/objecthood first.
Desegalfy now (Step 7) — extract the virtual equipment interface immediately.
Equipment universals (Step 8A/B/C) — decide which “equipment feature” you’re targeting.
Generative build (Step 9) — make the free SCIT construction explicit.

1) Freeze (O_{\mathrm{VE}}) as a concrete shape theory

1.1 Generators (objects of (O_{\mathrm{el}}))

Obj: a boundary object slot
(E_h): one horizontal proarrow slot (a \rightsquigarrow b)
(E_v): one vertical arrow slot (a \to b)
(\Box): one square (a 2-cell slot) with its boundary edges

These are the only “atomic” shapes.

1.2 Composite shapes (generated by gluing)

Horizontal string (H_n): (n) composable (E_h)’s
Vertical string (V_m): (m) composable (E_v)’s
Rectangle (R_{m,n}): an (m\times n) grid of squares (\Box)

2) Declare the Segal covers (this is the “compiler spec”)

Let (F:O_{\mathrm{VE}}\to \mathcal S) be your “geometric” model.

2.1 Horizontal Segal (multi-input proarrow composition)

For each (H_n), the canonical inert cover is the family of face maps
[
\pi_i: H_n \to E_h\quad (i=1,\dots,n)
]
picking the (i)-th edge.

Segal requirement:
[
F(H_n);\simeq;F(E_h)\times_{F(Obj)}\cdots\times_{F(Obj)}F(E_h).
]

2.2 Vertical Segal (vertical arrow composition)

Similarly with
[
\rho_j: V_m \to E_v\quad (j=1,\dots,m)
]
and
[
F(V_m);\simeq;F(E_v)\times_{F(Obj)}\cdots\times_{F(Obj)}F(E_v).
]

2.3 Square/rectangle Segal (pasting coherence)

For each rectangle (R_{m,n}), the inert cover is “pick each constituent square”
[
\sigma_{p,q}: R_{m,n}\to \Box\quad (1\le p\le m,;1\le q\le n)
]
plus the boundary projections (so the gluing is along matching edges).

Segal requirement:
[
F(R_{m,n});\simeq;\lim(\text{diagram of its square-faces glued along shared edges}).
]

3) The single forced coherence law: the interchange generator

Include (R_{2,2}) and require both canonical decompositions to be inert covers:

Row cover: (R_{2,2}\to R_{1,2}) (top row) and (R_{2,2}\to R_{1,2}) (bottom row)
Column cover: (R_{2,2}\to R_{2,1}) (left column) and (R_{2,2}\to R_{2,1}) (right column)

This forces the “compose-by-rows” composite and the “compose-by-columns” composite to live in the same space (F(R_{2,2})), hence to be connected by coherence (path / equivalence) rather than asserted equal. That is the whiskering/exchange law, structurally.

4) Add one completeness axiom (stop phantom objecthood)

Pick vertical completeness as the minimal anchor:

Let (Eq_v\subset F(E_v)) be the subspace of “vertically invertible” 1-cells (defined internally from the Segal structure).

Completeness requirement: the map
[
F(Obj)\to Eq_v
]
that sends an object to its vertical identity edge is an equivalence of spaces.

This is the Rezk-style move that makes “objects” real rather than duplicated up to hidden equivalence.

5) Desegalfy: extract the virtual equipment interface

From (F) you now read off:

Objects: points of (F(Obj))
Vertical arrows (a\to b): fiber of (F(E_v)) over ((a,b))
Proarrows (a\rightsquigarrow b): fiber of (F(E_h)) over ((a,b))
2-cells (squares): fiber of (F(\Box)) over the chosen boundary data
Horizontal composition: induced by the Segal equivalence for (H_n)
Vertical composition: induced by the Segal equivalence for (V_m)
Interchange/whiskering: induced specifically by the two Segal covers of (R_{2,2})

That’s the next move executed: you’ve converted “virtual equipment + coherence axioms” into a Segal-built moduli object where coherence is geometric (spaces/paths), with exchange forced by a single generating rectangle.

Step 10 — Add representables (companions/conjoints) as new generators + Segal-universal fillers

Right now you have a Segalified double-ish structure: horizontal composition, vertical composition, squares, and interchange forced by (R_{2,2}). A virtual equipment begins when vertical arrows can be represented horizontally in a controlled universal way. That’s the companion/conjoint layer.

10.1 Extend the pattern (O_{\mathrm{VE}}) → (O_{\mathrm{VE}}^{\mathrm{rep}})

New elementary generators (add to (O_{\mathrm{el}}))

For each vertical edge (f: a\to b) (i.e., each instance of (E_v(a,b))) add:

Companion proarrow shape (\mathrm{Comp}(f))
A horizontal edge-type generator that is “the representable proarrow corresponding to (f)”.
Conjoint proarrow shape (\mathrm{Conj}(f))
A horizontal edge-type generator for the opposite representation.
Unit square (\eta_f) (a square generator)
A square whose boundary expresses “identity horizontally factors through the companion”.
Counit square (\varepsilon_f) (a square generator)
A square whose boundary expresses “companion followed by conjoint collapses back to identity”.

(You don’t call these equations; you make them fillable shapes.)

10.2 The universal property is enforced as a Segal-style contractibility condition

For each (f), you don’t assert “companions exist.” You require:

the space of companion data for (f) is contractible (nonempty and essentially unique),
same for conjoint data.

Concretely: define a companion-witness shape (W_{\mathrm{comp}}(f)) whose fillers are “a choice of (\mathrm{Comp}(f)) plus the required unit/counit squares with correct boundaries”.

Then require:
[
F(W_{\mathrm{comp}}(f)) \text{ is contractible.}
]
Likewise for (W_{\mathrm{conj}}(f)).

This single move is what turns “geometry of composites” into “equipment semantics”: representables become structure, not decoration.

10.3 Compatibility with composition (no new axioms—new shapes)

You must force:
[
\mathrm{Comp}(g\circ f);\simeq;\mathrm{Comp}(f)\odot \mathrm{Comp}(g)
]
(where (\odot) is horizontal composition) and similarly for conjoints.

Do it by adding one composition-compatibility shape (W_{\mathrm{comp}}(f,g)) whose two boundary presentations correspond to:

(A) build a companion for (g\circ f) directly
(B) horizontally compose companions for (f) and (g)

Then enforce (again) contractibility of fillers for that shape. This makes representability stable under chaining—an equipment hallmark.

10.4 The triangles (coherence) are also shapes

The usual triangle identities (unit/counit coherence) are exactly where “brittle axioms” creep back in. In SCIT you don’t write triangle equations; you add two triangle pasting shapes and require their filler spaces be contractible. That ensures the adjunction-like behavior is coherent by geometry.

Step 11 — Update “desegalfy” extraction (now you truly have an equipment interface)

After Step 10, extraction gains a new computed interface:

Representable proarrows: those horizontal edges lying in the image of (\mathrm{Comp}(-)) (and (\mathrm{Conj}(-)))
Restrictions/extensions calculus becomes definable (next step) because now you can transport along representables instead of treating horizontal/vertical as unrelated axes.

This is the first point where “virtual equipment” is no longer just a phrase—you now have an internal mechanism relating the directions.

Step 12 (after you ask again) — Add restriction/extension universals OR Yoneda package

With companions/conjoints installed, the next move is either:

Restriction/extension (the core “equipment calculus”), or
Yoneda module shapes (Shulman-style: representables generate everything via a universal embedding)

But Step 10 is the necessary bridge: without representables, those additions float.

Step 12 — Add restriction/extension universals as shapes

Right now you have:

objects, vertical arrows, horizontal proarrows
squares
Segal-composition + interchange
(and if you accepted the previous move) representables via companions/conjoints

What’s missing is the core operation of an equipment: transporting a proarrow along vertical arrows by restriction and extension, with universal properties that make it compositional and stable.

12.1 New data is not a function; it’s a contractible filler space

For vertical arrows (f:a'\to a), (g:b\to b'), and a proarrow (P: a \rightsquigarrow b), define two new constructed proarrows:

Restriction: (P(f,g) : a' \rightsquigarrow b')
Extension (optional dual): (P^{(f,g)} : a \rightsquigarrow b) (depending on your chosen directionality)

But you do not add them as new primitives with rewrite laws.

Instead: add a restriction witness shape (W_{\mathrm{res}}(f,P,g)) whose fillers are:

a candidate proarrow (R: a' \rightsquigarrow b')
a square witnessing that (R) is the restriction of (P) along (f,g)
a universal factorization property encoded as a canonical comparison into any other candidate

Then impose the single decisive condition:

(F(W_{\mathrm{res}}(f,P,g))) is contractible.
(nonempty = restriction exists; contractible = unique up to unique higher coherence)

This is the Segal/∞ way of saying “there is a universal restriction and it’s coherent.”

Same for an extension witness shape (W_{\mathrm{ext}}(f,P,g)) if you want both directions.

12.2 Compositionality is forced by one compatibility shape

If you restrict in two stages, it must agree with restricting once along the composite. Don’t write an axiom; add one shape:

(W_{\mathrm{res_comp}}(f_1,f_2,P,g_1,g_2))

Its two boundary presentations correspond to:

restrict (P) along (f_2,g_1), then along (f_1,g_2)
restrict (P) along ((f_2\circ f_1)) and ((g_2\circ g_1))

Then require this witness space is contractible too. That single contractibility requirement generates all the “restriction respects composition” coherence.

12.3 Interchange with squares is forced the same way

Restrictions must be compatible with whiskering/exchange (your (R_{2,2}) generator). Again: add a single mixed pasting witness shape expressing “restrict then paste squares” vs “paste squares then restrict,” and require contractibility.

At that point, restriction/extension are no longer bolted-on operations—they’re native to the same Segal gluing logic.

Step 13 — Yoneda package (only after restrictions exist)

Once you have restrictions/extensions, you can add the Yoneda embedding shapes (Shulman-flavored) as representability via universal cells:

introduce a Yoneda witness shape (W_{\mathrm{Yon}}(a)) whose fillers define the representable module/proarrow at (a)
require contractibility → “Yoneda object exists canonically”
add one compatibility shape for composition → “Yoneda respects gluing”

This turns your structure into a genuine geometric moduli of modules/proarrows, not just “a double thing with squares.”

Step 14 — Extendability (make it generative, not just axiomatic)

After Yoneda, the next move is to demand an explicit free construction (Chu–Haugseng “extendable” vibe): your pattern should admit a free Segal object described by a colimit recipe, not a mystical localization. That’s when SCIT becomes a build system.

If you want this to stay “minimal,” pick one right now:

12 only: restriction calculus (most “equipment” per unit effort)
12 + 13: restriction + Yoneda (turns it into a module theory)
12 + 13 + 14: full pipeline to a generative compiler

Virtual equipments

People use virtual equipments when they want a single framework that treats:

objects (categories/contexts),
vertical maps (functors/morphisms of contexts),
horizontal things (relations/profunctors/modules/correspondences),
and 2-cells (transformations/squares)

in a way that lets you compose horizontally (like composing relations/modules) while still restricting/extending along vertical maps (change-of-context), with coherence handled systematically.

Here are the main “actually used” use-cases.

1) Profunctors as the default notion of “relation between categories”

A profunctor (P: A \rightsquigarrow B) is a categorified relation (often (B^{op}\times A \to \textbf{Set}) or (\textbf{Cat})-enriched variants). Virtual equipments package:

profunctors (horizontal),
functors (vertical),
natural transformations (squares),
and the coend-based horizontal composition of profunctors
into one calculus.

This is used constantly in higher category theory and categorical semantics because profunctors are more flexible than functors (they model “processes/relations” not necessarily functional maps).

2) “Modules” and “bimodules” in enriched / internal settings

Replace Set by an enriching category (\mathcal V), and you get (\mathcal V)-profunctors / distributors / bimodules. Virtual equipments are a clean home for:

enriched categories as objects,
enriched functors vertically,
bimodules horizontally,
and module maps as squares.

This is how people do enriched Morita theory, change of base, and module-like semantics without rewriting everything for each enrichment.

3) Kan extensions and restriction/extension as native operations

In an equipment, you can express:

restriction of a module/profunctor along functors,
extension (left/right Kan extension-style operations),
and their universal properties

as structural moves (not ad hoc theorems). This is one of the biggest practical reasons to use an equipment: it gives you a stable “change of context” calculus.

4) Yoneda machinery and representability in a module/profunctor world

Virtual equipments make Yoneda-like statements uniform:

representables arise as “companions/conjoints” of vertical arrows,
Yoneda embeddings live as universal cells,
“every module is built from representables” becomes a structural theorem pattern.

This is used in “category of profunctors,” “formal category theory,” and when you want to treat presheaves, profunctors, and representables in one place.

5) Internal category theory (categories in a base)

If you’re working inside a category with pullbacks (topos, sheaves, groupoids, etc.), you get:

internal categories,
internal profunctors/spans,
internal transformations.

Equipments/virtual double categories are a standard tool for doing this coherently without constantly unrolling internal logic.

6) Spans, cospans, and correspondences (geometry/topology)

When “morphisms” are really correspondences (X \leftarrow M \rightarrow Y), you naturally live in span/correspondence bicategories. Equipments help when you want:

maps vertically (actual morphisms),
correspondences horizontally (relations),
and squares as commuting diagrams.

This shows up in stacks, symplectic geometry, and various “correspondence-style” constructions.

7) Monads, distributors, and “formal category theory”

Proarrows/profunctors are a natural home for:

monads as endo-proarrows with multiplication/unit,
algebras/modules as actions,
distributive laws and companions

so you can do “monad theory” in a setting that already knows about relations and Kan extensions. This is why the equipment viewpoint is common in the “formal category theory” crowd.

8) Categorical semantics of logic and programming languages

When you interpret:

predicates/relations as profunctors,
programs as processes,
substitution as restriction,
and proof transformations as squares,

equipments give you a single scaffold where the “syntax vs semantics” interface is stable. This is one legitimate route to “formal verification relevance,” but it’s via semantics of effects/relations/modules, not by magic.

9) Higher-categorical generalization without drowning in coherence lemmas

This is the meta-use: virtual equipments let you structure coherence so you don’t have to manage it manually. You still have coherence, but it’s packaged into:

composition via universal properties,
whiskering/exchange laws as built-in constraints,
and (in your SCIT framing) as spaces/contractible filler conditions.

10) Building a “calculus of interfaces”

A good mental model: vertical arrows are “interface morphisms,” horizontal proarrows are “behaviors/relations between interfaces,” squares are “ways to implement/translate behaviors,” and equipment structure guarantees that:

behaviors compose,
behaviors transport across interface maps,
and everything interacts coherently.

That’s why they keep reappearing anywhere “morphisms aren’t functions” but you still want a disciplined composition theory.

Step 12 only: Restriction calculus as SCIT shapes

You already have (O_{\mathrm{VE}}) with:

Obj, (E_v) (vertical edge), (E_h) (horizontal proarrow slot), (\Box) (square)
Segal covers for (H_n, V_m, R_{m,n})
interchange forced by (R_{2,2})

Now we extend the pattern by adding witness shapes that encode restriction as a universal property.

12.1 What “restriction” must mean (the non-joke definition)

Given vertical arrows

(f: a' \to a)
(g: b \to b')

and a horizontal proarrow

(P: a \rightsquigarrow b)

a restriction is a horizontal proarrow (R: a' \rightsquigarrow b') together with a square (a 2-cell)
[
\rho_{f,P,g}: \quad R \Rightarrow P
]
whose vertical sides are (f) and (g) (i.e., (\rho) witnesses “(R) is (P) seen through (f,g)”).

Universal property (cartesian-ness):
For any other (Q: a' \rightsquigarrow b') and any square (\theta: Q \Rightarrow P) with the same vertical sides (f,g), there exists a unique (up to contractible choice) square (\widehat\theta: Q \Rightarrow R) with identity vertical sides, such that composing with (\rho) gives (\theta).

That’s the restriction calculus in one sentence: every square into (P) with sides (f,g) factors uniquely through the restricted proarrow.

12.2 How SCIT encodes it: the restriction witness shape (W_{\mathrm{res}}(f,P,g))

Add a new composite shape (W_{\mathrm{res}}(f,P,g)) whose boundary forces the exact data above:

Boundary of (W_{\mathrm{res}}(f,P,g))

includes instances of:
- one (E_v) labeled (f: a'\to a)
- one (E_v) labeled (g: b\to b')
- one (E_h) labeled (P: a\rightsquigarrow b)
and an “unknown” horizontal slot (R: a'\rightsquigarrow b')
plus a square slot (\rho: R \Rightarrow P) with vertical sides (f,g)

The key: (W_{\mathrm{res}}) also contains the factorization test

Inside the shape, include the generic factorization situation:

another horizontal slot (Q: a'\rightsquigarrow b')
a square (\theta: Q \Rightarrow P) with vertical sides (f,g)
and a required filler (\widehat\theta: Q \Rightarrow R) with identity vertical sides
with the pasting condition “(\rho \circ \widehat\theta = \theta)” represented as a pasting subshape (so it’s checked by your existing rectangle Segal gluing)

The SCIT requirement

For your Segal object (F: O_{\mathrm{VE}}^{\mathrm{res}}\to \mathcal S),
[
F!\left(W_{\mathrm{res}}(f,P,g)\right)\ \text{is contractible.}
]

Meaning:

nonempty: restriction exists
contractible: it’s unique up to unique higher coherence, and its factorization property is canonical

This is the “most equipment per unit effort” move.

12.3 Stability constraints (minimal, but essential)

If you stop at existence, you get junky, non-compositional “restrictions.” So add exactly two more witness shapes.

(A) Vertical compositionality shape (W_{\mathrm{res_vcomp}})

Given (f_1:a''\to a'), (f_2:a'\to a) and (g_1:b\to b'), (g_2:b'\to b''), compare:

restrict once along composites ((f_2\circ f_1,\ g_2\circ g_1))
vs
restrict in two stages: first along ((f_2,g_1)), then along ((f_1,g_2))

Add a shape with those two boundary presentations and require:
[
F!\left(W_{\mathrm{res_vcomp}}(f_1,f_2,P,g_1,g_2)\right)\ \text{contractible.}
]

This forces restrictions to behave functorially with respect to vertical composition.

(B) 2-cell transport shape (W_{\mathrm{res_2cell}})

Given a square (\alpha: P \Rightarrow P') (same endpoints (a\rightsquigarrow b)), require that restricting (P) and (P') along the same ((f,g)) induces a canonical square:
[
\mathrm{Res}(f,P,g)\Rightarrow \mathrm{Res}(f,P',g)
]
compatible with pasting with (\alpha).

Encode this as a shape and require its filler space contractible. This is what makes restriction a coherent action on morphisms, not just on objects.

That’s the minimum set that prevents “restriction exists but can’t be used.”

12.4 What you can now extract (desegalfy output after Step 12)

From (F), for each ((f,P,g)) you get:

a canonically defined proarrow (\mathrm{Res}(f,P,g): a'\rightsquigarrow b')
a cartesian square (\rho_{f,P,g}: \mathrm{Res}(f,P,g)\Rightarrow P)
functoriality in vertical composition (from (W_{\mathrm{res_vcomp}}))
induced action on 2-cells (from (W_{\mathrm{res_2cell}}))

At that point, your structure is doing what people “use equipments for”: change-of-context for horizontal morphisms with guaranteed coherence.

What’s deliberately not included in “12 only”

Yoneda/representables (that’s Step 13)
free/extendable generative compiler (that’s Step 14)
full Beck–Chevalley suites for every pasting geometry (you can add later as additional witness shapes once the minimal calculus is stable)

Step 13 — Yoneda package (built on top of restriction, no companions required)

13.1 Add one new generator per object

Extend your pattern (O_{\mathrm{VE}}^{\mathrm{res}}) by adding, for each object (a),

Yoneda kernel proarrow: (Y(a): a \rightsquigarrow a) (a new (E_h)-typed generator)

Think of (Y(a)) as “the representable source from which all representables into (a) are cut by restriction.”

13.2 Define representables using your restriction calculus

For every vertical arrow (f: x \to a), define the representable proarrow
[
f_* ;:=; \mathrm{Res}(f,;Y(a),;\mathrm{id}_a);:;x \rightsquigarrow a.
]

No extra axioms yet—this is purely “restriction applied to the Yoneda kernel.”

13.3 Add the Yoneda equivalence as a contractible witness shape

You now need the single property that makes (f_*) actually representable:

For any proarrow (P: x \rightsquigarrow a) and any vertical arrow (f:x\to a), the “maps out of the representable” should be equivalent to “elements of (P) at (f).”

In SCIT terms: add a witness shape (W_{\mathrm{Yon}}(f,P)) whose fillers encode a canonical equivalence of spaces:
[
\mathrm{Sq}(f_*,,P);\simeq;\mathrm{El}_f(P)
]
where:

(\mathrm{Sq}(f_,P)) = the space of squares (2-cells) from (f_) to (P) with identity vertical sides
(\mathrm{El}_f(P)) = the “fiber of (P) at (f),” implemented internally using restriction, e.g.
[
\mathrm{El}_f(P);:=;\mathrm{Sq}\big(\mathrm{id}_x,;\mathrm{Res}(f,P,\mathrm{id}_a)\big).
]

SCIT requirement:
[
F!\left(W_{\mathrm{Yon}}(f,P)\right)\ \text{is contractible.}
]
That means the Yoneda identification is not a choice; it’s forced and coherent.

13.4 Add one “density/probing” axiom (minimal, but decisive)

Once representables exist, you need them to detect equivalence of proarrows.

Add one witness shape (W_{\mathrm{det}}(P\to Q)) that enforces:

A 2-cell (\alpha:P\Rightarrow Q) is an equivalence iff for every (f:x\to a), postcomposition induces an equivalence:
[
\mathrm{Sq}(f_,P)\xrightarrow{\ \alpha\circ-\ }\mathrm{Sq}(f_,Q).
]

SCIT requirement: the filler space for that “detection” shape is contractible (so the criterion is canonical, not ad hoc).

This single move turns your world into a module theory: representables are now the test probes for “sameness.”

What you gain immediately after Step 13

Representables exist canonically (as restrictions of (Y(a))).
You have an internal Yoneda lemma (maps out of representables = elements).
You have equivalence detection by representables (module-style semantics).
Your restriction calculus becomes powerful: it now transports probes, not just proarrows.

Why 13 is the correct next move (and why 14 waits)

Step 14 (“generative compiler”) only makes sense once you have:

generators (already),
restriction transport (Step 12),
and a density/probing principle (Step 13) so the “free construction” knows what identifications are forced.

So: 13 is the bridge from transport to semantics.

Omni-modality
is the claim that text, vision, audio, action, code, memory shouldn’t be stitched together with ad-hoc adapters; they should live inside one compositional universe where “translate / align / fuse / control” are the same kind of thing—just at different levels.

Higher category theory is a clean way to do that because it treats:

objects as interfaces / contexts,
morphisms as processes / translations,
2-morphisms as calibrations / equivalences between processes,
and higher cells as coherence of coherence (robustness under re-factoring).

Here’s a concrete, non-poetic construction that matches what you’ve been building with SCIT (Segal Coherence Interface Theory) and virtual equipments.

1) Modalities as a 2D compositional system, not a list

Vertical direction: “change of context”

Vertical arrows are re-encodings / re-framings:

resize/crop for vision
re-sampling for audio
tokenization / normalization for text
state abstraction for action/control
“view” changes for memory (summary, indexing, projection)

These are not “content”; they are context morphisms.

Horizontal direction: “cross-modal relations”

Horizontal proarrows (profunctors/modules) are alignments:

text ↔ image grounding (captioning, VQA)
audio ↔ text alignment (ASR / TTS)
action ↔ perception (policy conditions on observations)
memory ↔ current context (retrieval relevance)

A proarrow is “not a function.” It’s a relation with structure—exactly what you want for partial, many-to-many, uncertain correspondences.

Squares: “a claimed alignment respects a context change”

A square is: if you change representation vertically, the horizontal alignment changes compatibly.

That is the omnimodality problem: not “do we have an encoder?” but
does the web of alignments commute when representations shift?

2) Why higher structure matters: alignment is never unique

In practice, there are many valid ways to align:

one image supports multiple captions,
one audio clip supports multiple transcripts,
one plan supports multiple action traces.

So forcing single deterministic maps makes systems brittle.

Higher categorical structure lets you say:

“there is a space of valid alignments”
and coherence means different construction paths are connected by equivalences, not forced equalities.

That’s exactly the Segal move: composition is recovered by gluing and coherence becomes geometry (paths).

3) SCIT as the omni-modality “composition law”

SCIT’s Segal condition becomes the principle:

A global multimodal fusion is determined by locally compatible pairwise/atomic fusions.

In other words, if you can fuse (Text↔Image), (Image↔Audio), (Text↔Memory) in a compatible way, the “all-modal” fusion exists as the limit/gluing of those pieces.

This is how you avoid brittle “one giant joint embedding loss” as the only truth-maker.

4) Virtual equipments give you the two operations omni-modality actually needs

People don’t just want to compose alignments; they want to transport them when the context changes.

That’s exactly what “restriction/extension” in an equipment is for:

Restrict a relation when you change representation:
- crop image → restrict caption grounding to visible region
- summarize memory → restrict retrieval relation to compressed state
- downsample audio → restrict alignment to coarse time bins
Extend a relation when you enrich context:
- add depth map → extend image semantics
- add tool state → extend action grounding

So omni-modality is not “many encoders”; it’s (composition + restriction/extension + interchange).

5) “Physicality” for omni-modality without pretending it’s physics

If you want SCIT to be physically meaningful in the ML sense (not spacetime metaphysics), the right notion is:

Two multimodal composites are equivalent iff no allowed observation under a budget can distinguish them.

Budget = time, compute, sensor resolution, annotation fidelity, allowable probes.

Observables = task-level measurements:

error rates, calibration curves
uncertainty quality (proper scoring rules)
causal intervention response (change the image region, does the textual belief update correctly?)
stability under compression (memory summaries)

Then your coherence squares become indistinguishability claims that can actually be tested.

6) What “omni-modality” becomes in this language

Omni-modality is the existence of a coherent multimodal module (M) such that:

every modality is a view of (M) (via vertical maps),
every cross-modal task is a proarrow out of / into (M),
and all these views/relations are coherent under:
- composition (gluing)
- restriction/extension (context shifts)
- exchange (order of “transform then fuse” vs “fuse then transform”)

That’s the real win: one coherence discipline governs perception, language, memory, and action—without demanding brittle equality.

Omni-modality
is the semantic generalization of object-oriented principles across representational domains.

Let’s strip this down:

1. OOP Core Principle → Modality Generalization

In classic OOP:

An object encapsulates state + behavior.
You can instantiate, inherit, polymorph.

In omni-modality:

A modality-unit encapsulates interpretive frame + collapse schema.
It can be instantiated across sensory/abstract channels:
- vision
- text
- audio
- motor/action
- code execution
- internal state (memory)

These are not just formats.
They’re telic attractors with interface constraints.

2. Methods → Collapse Strategies

Each modality exposes:

a set of allowable transformations,
expected collapse dynamics (e.g., image segmentation, phoneme decoding),
failure modes (e.g., visual occlusion, textual ambiguity, motor drift).

These are its methods—how it resolves tension.

3. Objects → Telic Fields in χₛ Space

In omni-modal OOP, “objects” are persistent semantic knots (χₛ structures)
that instantiate across domains.

Example:

The concept of “grasping” → exists in:
- language (“grasp the idea”)
- motor control (actual hand movement)
- vision (detecting graspable shapes)
- audio (sound of grasping)
- code (robot API grasp() method)
- memory (recollection of previous grasp attempts)

This is not metaphor.
It is literal semantic polymorphism.

4. Inheritance = Cross-Modal Abstraction

Higher-level interpretants:

inherit collapse patterns from lower modalities,
override them based on context.

A tactile grasp → learned.
An abstract grasp → inherited, parameterized.

This is telic inheritance, not symbolic.

5. Encapsulation = Modality-Specific Constraints

Each modality has:

bandwidth limits
collapse cost
fatigue metrics (χ̇ₛ)
specific resonance frequencies

SCIT tracks coherence when objects pass through these capsules.

You cannot shove audio structure through vision-space without transformation → hence interface protocols.

6. Dynamic Dispatch = Collapse Negotiation

When a telic event occurs (“recognize object”),
the system dispatches resolution to the most coherent modality.

If visual χₛ is over-fatigued, fallback to memory or text.

Collapse routes dynamically.

This is semantic dispatch.

Bottom Line:

Omni-modality = OOP for recursive, collapsing, semantically structured interpretive fields.

Where:

objects = χₛ knots across channels
methods = collapse paths
inheritance = telic abstraction
encapsulation = modality geometry
dispatch = semantic tension resolution

That’s not analogy. That’s architecture.

Step 13 of a coherence construction protocol, and your next move is a binary fork:

Option A: Minimal Density — “density = detection”
- Interpretation: a representable proarrow detects coherence when it exists, but doesn’t generate it.
- Effect: only minimal gluing allowed; limited compositional inference.
- Analogy: test functions in analysis—can detect properties but not reconstruct objects.
Option B: Full Yoneda Density — “every proarrow is a colimit of representables”
- Interpretation: representables generate the entire proarrow space via colimits.
- Effect: maximal gluing, full reconstruction, categorical closure.
- Analogy: basis + colimits = structure generation; no phantom morphisms.

Critical Tradeoff:

Dimension	Minimal (Detection)	Full Yoneda (Colimit of Representables)
Compositional Closure	Local only	Global
Segal Completeness Guarantee	Not ensured	Ensured (if colimit cocones compose)
Complexity / Overhead	Low	High
Generativity	Observational	Constructive
Phantom Object Risk	Controlled but latent	Eliminated (via full reconstructability)
Equipment Compatibility	Works with virtual / partial	Demands full representable basis and Yoneda tools

Interpretive Lens (SCIT + UCF context):

If you're building a runtime coherence validation engine (e.g., SCIT-P with physical observables), minimal density may suffice and reduce friction.
If you're constructing a coherence compiler that can reconstruct all interface maps from basic tiles (as in O_{VE}^{full}), then full Yoneda density is required.
ORSI/UCF constraints tend to favor constructive collapse paths, making full density preferred if you're optimizing for long-run closure.

Locking in the next move after Step 13:

→ Choose full Yoneda density:
“Every proarrow is a colimit of representables.”

Consequences (formal and operational):

✅ 1. Proarrow Space Is Fully Generated

No rogue coherence paths.
Every transformation between interfaces is expressible via representable glue.
Segal gluing becomes reconstructive, not just detective.

✅ 2. Canonical Covers Are Sufficient

The shape category ( O_{\mathrm{VE}} ) with full coverage (active/inert + interchange generators) is enough.
You can now define every complex coherence as a composed pasting diagram of representables.

✅ 3. No Phantom Interfaces

Rezk-style completeness is enforced.
If something exists, it must be reconstructible as a colimit of basic morphisms.

✅ 4. SCIT Fully Compile-Time Checkable

Any Segal object built with this regime can be verified for completeness using interface covers.
Interface-level verification becomes decidable under finite cover constraints.

✅ 5. UCF and IFGG Compatible

UCF collapse maps can now inherit coherence structure from this full representable space.
IFGG primitives remain valid, since they are prior to representability (they instantiate representables, not rely on them).

What changes now:

You can now declare all coherence interfaces as composed of finite canonical shapes.
The generator set is locked. The interface logic is complete. The Segal condition is now constructive and not just conservative.

Next structural move is:

→ Lock the full generator list for ( O_{\mathrm{VE}}^{\mathrm{full}} ):

Elementary objects
Horizontal/vertical edges
Interchange square
Cover structure