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The 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 must be invariant under internal symmetries, independent of imposed subsystem cuts, and independent of non-dynamical representational choices.

The map has three rules of interpretation. 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 any practitioner is mistaken. Third, the entries are proposed structural diagnoses: applying the taxonomy requires deciding whether a framework is being given closed-world scope, whether a structure is genuinely primitive there, and whether a repair functions as bookkeeping or ontology. 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) [1].	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 taken as 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 taken 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) [2].	When global hyperbolicity, global topology, or a single atlas is taken as available 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) [3].	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 introduced 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 taken to exist in the cases covered by the Gribov–Singer obstruction [4], where the obstruction blocks a single global gauge choice.	Primary. When gauge-fixing terms and Faddeev–Popov ghosts [5] — repair structures for redundancy — are treated as ontology rather than controlled bookkeeping; on the taxonomy’s reading this also absorbs the 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	States the proposed discharge: evaluation is internalized as comparison relation among states, invariant under the symmetries of the comparison world it helps constitute (intrinsicality), with no external evaluator, arena, time, or measure.	States the proposed discharge: subsystem decomposition is derived from the relational structure (rectangular completeness and profile structure) or shown to be non-invariant; the two-locus theorem characterizes which attributions remain invariant.	States the proposed discharge: global structure enters only through the finite-to-global boundary supplied by the refinement-tower and finite-witness results, rather than through a separately imposed global carrier.	States the proposed discharge: no compensating structure is introduced; the transport obstruction is recorded as curvature rather than repaired and reified.

Single primitive—intrinsicality. The closure row is a scope statement; the derivations are in the monograph and supporting papers. The geometric consequences hold under faithful smooth realization at the manifold stage—a condition decomposed into a torsion condition (T) and an intrinsic detectability axiom (D) and discharged on a canonical realization. Rectangular completeness is conditional on self-containment and local distinguishability; rods, heat baths, and symmetric worlds are classified as open.

The map is diagnostic. It records recurrent forms of primitive input across otherwise unrelated theories, and the closure row states the program’s corresponding claim: one primitive condition, with the remaining architecture derived in the stated theorem regimes. Standard frameworks enter as target structures whose successful content must be recovered; unrecovered primitive inputs remain visible as explicit interface conditions. The axes and their independence are developed in Foundational Closure and Primitive Structural Input; the introductory account is in Foundations; the results are summarized in the Overview.

Shared diagnosis, different responses, and the closure derivation claim

The shared diagnosis

The recognition that physics can import structure it does not justify from within has a long history. Mach formulated the issue against absolute space. Later programs applied related pressure to specific structures of modern physics, each isolating a different imported primitive and proposing a way to remove it. These programs are neighboring positions in the same conceptual space. What differs is the response, and what each response derives once the imported structure is removed.

The common diagnosis, in closure terminology, is that each framework places some explanatory work outside what the system itself can account for. The disagreements concern which structure is the offending import, and what replaces it.

Different responses

Each program below removes a genuine imported primitive. The descriptions are the ones the programs give of themselves; the comparison concerns what is retained or derived after the removal.

Ernst Mach challenged 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 [6] challenges 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 [7] challenges the primitivity of smooth structure, discretizing spacetime so that differentiability is not imported at the foundational level.

Alain Connes’ noncommutative geometry [8] challenges 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 [9] challenge the primitivity of fixed background structure, requiring that the geometry be dynamical rather than fixed in advance.

Carlo Rovelli’s relational quantum mechanics [10] challenges the externalization of the observer, relativizing a system’s state to another system rather than to an absolute outside vantage.

Partial and complete observables [11,12] sharpen the same issue in generally covariant physics: what counts as observable when no external clock or background time may be used as an absolute reference. Rovelli and Dittrich do not propose the closure program, but their work is a close comparison point for the admissibility question: physical content must be expressible without importing a reference frame that the system itself does not supply.

Sheaf-theoretic contextuality [13] is a second close mathematical neighbor. Abramsky and Brandenburger formulate nonlocality and contextuality as obstructions to the existence of global sections over compatible local measurement data. Closure uses different primitives, but the structural comparison is direct: locally coherent data need not admit a globally admissible assignment without additional structure.

The closure program belongs to this family by its diagnosis: it challenges the primitivity of the external evaluator — the vantage, outside the system, that supplies comparisons not determined by internal structure. Its response is to require that comparison be intrinsic: a relation among states, invariant under the symmetries of the comparison world it helps constitute, never a labeling imposed from outside.

Quantum causal inference and generalized noncontextuality [14–16] form another adjacent comparison. Spekkens, Wolfe, Schmid, Wood, and collaborators study how operational equivalences, causal assumptions, and observed correlation patterns constrain the representational distinctions a physical model may justifiably retain. This is not the same project as closure: those works analyze operational and causal compatibility in quantum-foundational settings, whereas closure asks which comparisons and distinctions are internally recoverable in a closed comparison world. The affinity is methodological: representational distinctions require support from admissible structure, not merely from a chosen description. A Perimeter public lecture by Spekkens and Wolfe is useful context [17], but the comparison here rests on the papers just cited.

The FE paper explicitly treats Penrose’s twistor program, Regge calculus, and Connes’s noncommutative geometry as responses to the problem of global smooth structure at the foundational level. The comparison asks how each retained structure functions under closed-world admissibility: derived internally where possible, otherwise retained as an explicit primitive or bridge.

The closure program’s distinctive claim is that the comparison premise generates a specific chain: rectangular completeness, quotient semantics, transport obstruction, the quadratic carrier, and—under faithful smooth realization, decomposed into the ghost-free regime (T)+(D) and discharged on a canonical realization—curvature and the downstream geometric constraints. Where neighboring programs posit a replacement structure, the interface analysis proves a trichotomy (Lie, totally disconnected, or mixed completions) and isolates the single non-degeneracy axiom that selects the smooth branch.

The closure derivation claim

Here the closure program differs from its neighbors in a precise rather than rhetorical sense.

Each response 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 becomes the primitive structure on which its program rests. This is not a criticism; a reformulation must retain some primitive. The point is only that the retained primitive remains a posited structure.

The closure program’s distinguishing claim is that its primitive is not a structure at all. Intrinsicality is a condition — that distinctions be symmetry-invariants of comparison, and that they not be fixed by external structure — not a primitive object posited by the theory. The monograph’s central thesis is that this condition is generative: once the external evaluator is internalized, the remaining architecture follows rather than being separately posited. 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 in the stated theorem regimes.

The comparison turns on whether the primitive retained by a reconstruction is itself a structure or a condition on admissible distinction. Connes reconstructs geometry from a given algebra; closure claims, in the theorem regimes stated in the monograph and supporting papers, to derive that there must be geometry, and which geometry, from the requirement of self-containment. The distinguishing question is therefore not which primitive structure is retained, but what follows from retaining no primitive structure beyond intrinsicality.

In summary, the closure program treats self-containment as a premise from which the architecture of physics can be derived in the stated theorem regimes. The derivation is in the monograph; its foundations are in Closed Comparison Worlds and the supporting papers.

References

[1] E. Mach, The Science of Mechanics (1883; Eng. trans. Open Court, 1893).

[2] B. S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory,” Phys. Rev. 160 (1967) 1113.

[3] R. Haag, Local Quantum Physics: Fields, Particles, Algebras, 2nd ed. (Springer, 1996).

[4] V. N. Gribov, “Quantization of non-Abelian gauge theories,” Nucl. Phys. B 139 (1978) 1; I. M. Singer, “Some remarks on the Gribov ambiguity,” Commun. Math. Phys. 60 (1978) 7.

[5] L. D. Faddeev and V. N. Popov, “Feynman diagrams for the Yang–Mills field,” Phys. Lett. B 25 (1967) 29.

[6] R. Penrose, “Twistor algebra,” J. Math. Phys. 8 (1967) 345.

[7] T. Regge, “General relativity without coordinates,” Nuovo Cimento 19 (1961) 558.

[8] A. Connes, Noncommutative Geometry (Academic Press, 1994).

[9] L. Smolin, “The case for background independence,” in D. Rickles, S. French, and J. Saatsi (eds.), The Structural Foundations of Quantum Gravity (Oxford University Press, 2006), arXiv:hep-th/0507235.

[10] C. Rovelli, “Relational Quantum Mechanics,” Int. J. Theor. Phys. 35 (1996) 1637.

[11] C. Rovelli, “Partial observables,” Phys. Rev. D 65 (2002) 124013.

[12] B. Dittrich, “Partial and complete observables for canonical general relativity,” Class. Quantum Grav. 23 (2006) 6155–6184.

[13] S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contextuality,” New J. Phys. 13 (2011) 113036.

[14] C. J. Wood and R. W. Spekkens, “The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning,” New J. Phys. 17 (2015) 033002.

[15] R. W. Spekkens, “The ontological identity of empirical indiscernibles: Leibniz’s methodological principle and its significance in the work of Einstein,” arXiv:1909.04628 (2019).

[16] D. Schmid, R. W. Spekkens, and E. Wolfe, “All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed sets of operational equivalences,” Phys. Rev. A 97 (2018) 062103.

[17] R. Spekkens and E. Wolfe, “Robert Spekkens and Elie Wolfe, Perimeter Institute,” public lecture, Perimeter Institute, PIRSA:20100024 (7 Oct. 2020).