This map records, for each framework given universal foundational scope, where it relies on primitive structural input not derived from within the system — assessed against one criterion, closed-world admissibility: a quantity should be invariant under internal symmetries, independent of imposed subsystem cuts, and independent of non-dynamical representational choices.

Read it with three cautions, which are part of the claim, not disclaimers on it. First, every entry is conditional: a structure counts as an import only when a framework is given closed-world scope and treats that structure as primitive or ontological. The same structure — used as a derived result, as eliminable bookkeeping, or borrowed inside an avowedly local model — imports nothing. The entries are not defects in successful equations. Second, “import” and “error” are technical terms: a mismatch between primitive input and the closure criterion, not a claim that a theory is empirically wrong or that anyone blundered. Third, and most important, these are proposed structural diagnoses, not settled classifications. Applying a taxonomy is a judgment call: whether a given theory is best read under one axis rather than another, whether a structure is genuinely primitive there, whether a given repair is really bookkeeping — these are exactly the questions the four-axis framework is meant to open, not foreclose. A reader who rejects the closure premise can coherently decline any row; the disagreement then lies at the premise. A single framework may instantiate several axes at once, since the four are logically independent roles, not exclusive bins.

The four columns are the axes of the four-axis taxonomy (Foundational Closure and Primitive Structural Input, Definitions 4.5–4.8): Externalization — primitive evaluation, state, arena, time, or measure placed outside the system; Artificial factorization — a subsystem decomposition imposed rather than derived; Premature globalization — global carrier structure imposed before relational compatibility is verified; Reification of repair — a compensating structure introduced for an obstruction, then treated as ontology rather than bookkeeping. A dash means “not characteristically the locus of this axis,” not a clean bill.

Framework	Externalization	Artificial factorization	Premature globalization	Reification of repair
Classical (Newtonian) mechanics	When absolute space and external time are treated as primitive arena, the inertial frame given rather than relational (Mach’s critique).	When body or particle individuation is taken as a primitive partition.	When a single global Euclidean space and universal simultaneity are imposed as holding everywhere.	—
Lagrangian / Hamiltonian mechanics	When an external time parameter, along which the action is extremized, is treated as primitive.	When the choice of generalized coordinates or configuration partition is taken as given.	When global phase space or a single symplectic manifold is assumed available before relational compatibility is shown.	—
Special relativity	When Minkowski spacetime and the inertial-frame class are treated as fixed external arena.	When the observer or frame split is imposed for description.	When global Lorentz structure is assumed to extend over all of spacetime.	—
General relativity	When the differentiable manifold, its dimension, and the metric signature are treated as background primitive. The metric is dynamical; the manifold is posited.	When a 3+1 foliation or time-slicing is imposed — non-invariant, and the locus of the problem of time (DeWitt).	When global hyperbolicity, global topology, or a single atlas is assumed before internal compatibility is established.	—
Quantum mechanics	Primary. When an external observer, apparatus, or classical clock supplies a primitive evaluation map — the measurement problem. Collapse, Everettian branching, decoherence, and relational quantum mechanics are best read as competing strategies for removing, relocating, or internalizing this evaluator — attempts to discharge externalization — rather than as a separate axis.	When the system/apparatus cut and the tensor-product decomposition are imposed rather than derived.	When a single global Hilbert space or universal wavefunction structure is presupposed.	Applies only where a specific interpretation reifies a collapse mechanism as ontology; otherwise the question sits under externalization above.
Quantum field theory	When background spacetime, fixed causal structure, or borrowed asymptotic in/out regions are treated as primitive.	When local-region or subalgebra subsystem cuts are imposed — the Haag obstruction shows naive factorization fails (Haag).	When continuum or global field-configuration structure is imposed before relational compatibility — the reading under which renormalization is associated with this axis, the global carrier assumed too early.	When compensating structures — renormalization counterterms, ghost fields — are treated as ontology rather than bookkeeping, specifically when scale-dependent renormalization-group flow is treated as independently fundamental. Renormalization is not flatly “repair”; it instantiates this axis only under that ontological reading, and the previous axis under the continuum reading.
Gauge theory	—	—	When a global gauge section is assumed to exist — against the Gribov–Singer obstruction, which shows no global section does (Singer).	Primary. When gauge-fixing terms and Faddeev–Popov ghosts — repair structures for redundancy — are treated as ontology rather than controlled bookkeeping; on the taxonomy’s reading this also absorbs the assumed-global-gauge-fixing case.
Statistical mechanics	When an external coarse-graining or macrostate partition, or a primitive ensemble measure such as equiprobability, is imposed.	When the system/bath boundary is taken as primitive.	—	When coarse-graining is promoted from bookkeeping to physical irreversibility — the arrow of time treated as ontological.
The closure program	Aims to discharge it by construction, deriving evaluation as internal — a comparison is a relation among states, invariant under the symmetries it constitutes (intrinsicality), with no external evaluator, arena, time, or measure.	Aims to discharge it by deriving subsystem decomposition from the relational structure (rectangular completeness and profile structure) or showing it non-invariant; the two-locus theorem characterizes what attribution survives.	Aims to discharge it by reaching global structure only through the finite-to-global compactness boundary, itself derived from intrinsicality rather than assumed.	Aims to discharge it by introducing no compensating structure: the transport obstruction is recorded as curvature, not repaired and reified.

Single primitive — intrinsicality. The closure row states the program’s aim, not a settled result: that these structures are derived rather than primitive is what the monograph and papers argue, and a critic may fairly ask whether a principle such as rectangular completeness is itself a repair principle — a question the work must answer rather than assume. Scope: the geometric consequences hold under faithful smooth realization at the manifold stage, and rectangular completeness is conditional on self-containment and local distinguishability, not universal — rods, heat baths, and symmetric worlds are genuinely open.

The value of the map is not that every cell is beyond dispute — several are interpretive, and the framework invites argument at exactly that level. Its value is that the same four kinds of primitive input recur across otherwise unrelated theories, and that the closure program’s distinctive claim is visible in one line: a single primitive, with the rest proposed as derived. Whether that proposal succeeds is what the papers are for. The axes and their independence are developed in Foundational Closure and Primitive Structural Input; the plain-language account is in The Idea; the results in the Overview.

Shared diagnosis, different cures, and what closure claims to derive beyond them

The shared diagnosis

Closure did not discover, and does not claim to have discovered, that physics imports structure it cannot justify from within. That recognition has a long line. Mach pressed it against absolute space. A family of later programs pressed it against the specific structures of modern physics, each isolating a different imported primitive and proposing to remove it. These programs are not rivals to be defeated; they are neighboring moves in the same conceptual space, and the closure program shares their starting diagnosis. What differs is the cure, and — the point of this section — what each cure is able to derive once the import is removed.

The common diagnosis, in the vocabulary of this site: somewhere, each framework places explanatory work outside what the system itself can account for. The disagreements are about which structure is the offending import, and what to put in its place.

Different cures

Each program below removes a genuine imported primitive. The descriptions are the ones the programs give of themselves; none is in dispute.

Ernst Mach attacked the primitivity of absolute space, arguing inertial structure should derive from the relations among masses rather than from a fixed external arena.

Roger Penrose’s twistor program attacks the primitivity of the spacetime point, recasting spacetime structure in terms of twistor space and complex-geometric data rather than ordinary points.

Tullio Regge’s calculus attacks the primitivity of smooth structure, discretizing spacetime so that differentiability is not imported at bedrock.

Alain Connes’ noncommutative geometry attacks the primitivity of the smooth manifold itself, replacing it with a spectral triple — geometry reconstructed from algebraic and operator data.

Lee Smolin and the background-independence programs attack the primitivity of fixed background structure, requiring that the geometry be dynamical rather than a stage set in advance.

Carlo Rovelli’s relational quantum mechanics attacks the externalization of the observer, relativizing a system’s state to another system rather than to an absolute outside vantage.

The closure program belongs to this family by its diagnosis: it attacks the primitivity of the external evaluator — the vantage, outside the system, that supplies comparisons the system cannot supply for itself. Its cure is to require that comparison be intrinsic: a relation among states, invariant under the symmetries it constitutes, never a labeling imposed from outside.

What closure claims to derive beyond them

Here the closure program parts from its neighbors, and the difference is precise rather than rhetorical.

Each cure above is, structurally, a substitution: it removes one imported primitive and installs a different one in its place, taken as given. The spectral triple is posited; twistor space is posited; the discretized complex, the dynamical-background framework, the relational state — each is the new floor on which its program stands. This is no criticism; a reformulation must stand on something, and these are deep and fruitful reformulations. But the floor remains a posited structure.

The closure program’s distinguishing claim is that its floor is not a structure at all. Intrinsicality is a condition — that distinctions be symmetry-invariants of comparison, that the system owe its distinctions to nothing outside itself — not an object handed to the theory. And the claim the monograph sets out to establish is that this condition, alone, is generative: once the external evaluator is internalized, nothing further need be posited, because the rest is forced. Rectangular completeness, quotient semantics, the obstruction to subsystem attribution, transport-obstruction curvature, the dimension, and the geometry of a genuinely closed three-dimensional system follow as consequences rather than further assumptions.

This is the line between a reformulation and a candidate resolution. A program that removes a primitive and substitutes another is a response to the shared problem; it sits alongside its neighbors. A program that removes the primitive and then derives the structure others install by hand is making a different kind of claim. Connes reconstructs geometry from an algebra he is given; the closure program aims to derive that there must be geometry, and which geometry, from the bare requirement of self-containment. Whether that derivation succeeds is precisely what the monograph and the supporting papers are for — it is not asserted here, it is the work. But it is the right question to ask of the program, and the one that distinguishes it from the family it otherwise belongs to: not which primitive it retains, but what it proves from retaining none beyond the condition itself.

The shared diagnosis is old and widely held. The cures are many. What the closure program offers, and asks to be judged on, is the claim that the diagnosis has a consequence — that self-containment is not only a discipline to observe but a premise from which the architecture of physics can be derived. The derivation is in the monograph; its foundations, in Closed Comparison Worlds and the supporting papers.