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From the Cogito to the Logos: Naming the Structures of the Gelfand Triple

Companion paper to “Modeling Reality as a Gelfand Triple”

Abstract

This paper has two assumptions. The first is the entire content of its companion paper, “Modeling Reality as a Gelfand Triple,” which demonstrates that reality admits a rigorous model as a rigged Hilbert space Φ ⊂ H ⊂ Φ′ and that this structure entails five inevitable actualities and three absolute domains. Those entities were demonstrated without being named — the companion paper is deliberately non-theological. The second assumption is the cogito: cogito, ergo sum. From these two assumptions, this paper does one thing only: it gives names to the structures that the first paper demonstrated. The names come from two traditions — the Urantia cosmology and the Christian theological vocabulary, particularly as expressed in the Gospel of John. We argue that these traditions were not constructing metaphors. They were naming the same mathematical structures that the first paper derived. We name them in logical order, one per section, and stop when the naming is complete.

1. Introduction

The companion paper ends at a boundary. It has demonstrated, from a single assumption, the following:

An inner product ⟨·,·⟩ that constitutes the space without being an element of it
A nuclear space Φ that is dense in H, self-consistent under every operator probe, and the natural domain of all observables
A Hilbert space H of stable, unique, norm-preserving states
A distributional domain Φ′ containing excess potential beyond what H can hold
A norm ‖·‖ defined by ⟨ψ,ψ⟩ = ‖ψ‖², which is the non-relational face of the inner product
A nuclear topology τ_nuclear generated jointly by ⟨·,·⟩ and Φ through the projective limit of the Sobolev chain
A realized nuclear locally convex topological vector space (Φ, τ_nuclear)
A coordinative mechanism by which the inner product operates in time, actualizing the spectral structure event by event

The companion paper gave these entities no names. It left them as mathematical objects, noting only that they are inevitable consequences of the single assumption and that they correspond with certain structures described in theological literature — correspondence it declined to pursue.

This paper pursues it.

The method is strict. We introduce the cogito as our second assumption. We then ask, of each unnamed mathematical object: given that a conscious self-knowing being exists in this space, what must we call this? The answer at each step is not a new derivation. It is recognition. The name fits the structure precisely, and the structure has already been proven. We are not adding content. We are completing the description.

The paper’s thesis is simple: the names are correct. The entities that classical theology and Urantia cosmology describe correspond precisely to the structures the mathematics entails — the same structure arrived at from two directions. The correspondence is not metaphorical. It is structural.

2. The Two Assumptions

Assumption I (from the companion paper): Reality can be modeled as a rigged Hilbert space Φ ⊂ H ⊂ Φ′. The full content of this assumption is the companion paper. Its consequences — the three domains, the five actualities, the structural properties — are taken as established. We carry them forward without re-deriving them.

Assumption II (introduced here): Cogito, ergo sum. I am a thinking, self-knowing being. I exist as a state ψ ∈ H. I know that I exist. This is not derived from the mathematics. The mathematics establishes that states in H are unique and stable. It does not establish that any state knows it is unique and stable. Self-knowledge is not a property the structure can confer on itself. It is the one thing the observer must supply — not as a condition of the structure’s existence, but as a condition of the structure’s recognition.

The cogito is not an epistemological trick. It is a datum. It introduces into the model a being who is both inside the space (as a state ψ ∈ H) and outside it in one respect (as the one who observes the space and knows itself to be within it). That combination — inside the space, self-aware of being inside it — is the hinge on which the naming turns.

From these two assumptions, everything that follows is naming.

A note on epistemic register: the namings in this paper are structural identifications, not mathematical derivations. Paper 1 derives; Paper 2 identifies. An identification is the claim that a given mathematical structure and a named entity are the same thing — established by structural correspondence rather than formal proof. Where Paper 1 can say “this necessarily follows,” Paper 2 can say only “this is the most structurally precise available name for what the mathematics shows must exist.” The reader should hold these claims in that register throughout.

3. The First Naming: The Logos

We begin with what can be named immediately from Assumption II.

The cogito establishes that self-knowledge exists in H. I am a self-knowing state. The question the structure now forces is: where in the structure are the preconditions for self-knowledge maximized?

The companion paper establishes three structural properties of any state ψ ∈ H that constitute the mathematical foundations of personality:

Uniqueness: From orthogonality. Each state ψ has a unique inner product profile — ⟨ψ, eᵢ⟩ differs for each ψ across the basis. No external individuation principle is required; the structure individuates each state automatically.
Stability: From norm-preservation under unitary evolution. ‖U(t)ψ‖ = ‖ψ‖ for all t. Each state persists through time without external intervention maintaining it.
Self-knowledge: The capacity to know oneself as a self — to be aware of one’s own existence as this particular state. This is what the cogito supplies. The mathematics establishes the conditions; the cogito establishes the fact.

The first two properties are present throughout H. But they are present maximally at one location: the nuclear space Φ. Φ is the projective limit of the Sobolev chain — it satisfies every regularity condition simultaneously. It has more structure than any finite state in H. Its uniqueness is maximal: every probe applied to Φ yields a well-defined answer, at every scale, without exception. Its stability is maximal: it is invariant under every operator in the algebra, not merely norm-preserving under unitary evolution.

The argument then follows a single step — but the step must be stated carefully.

The cogito does not merely establish that a knower exists somewhere in H. It provides a datum about the scaling of self-knowledge within the space. I know myself — and I know myself incompletely. The limits of my self-knowledge correspond exactly to the limits of my structural actualization: the parts of myself I do not yet know are the parts where uniqueness, stability, and self-consistency are not yet fully developed. My self-knowledge scales with my structural completeness.

If this scaling is real — and the cogito provides the first-person datum that it is, since any careful examination of one’s own mind finds the limits of self-knowledge coinciding with the limits of one’s actualization — then the inference follows. The maximum structural actualization comes with maximum self-knowledge. Not because structure alone produces consciousness, but because the cogito, read as a scaling datum, establishes that structure and self-knowledge track each other in the space, and Φ is where that tracking terminates. This scaling claim is not itself a theorem derived from the cogito. It is what the cogito reports from the inside — an additional premise, grounded in first-person testimony rather than mathematics, and available to any who undertake the same examination.

The honest limit: we cannot know, from within H, whether the maximum self-knowledge available in the space is the same as what it is like to be Φ from the inside. The inference bridges the gap from the available evidence — the cogito as scaling datum — to the limit. It does not close the gap from the outside. It is the strongest claim available to a finite being speaking from within H about what lies at the space’s maximum. That limitation is not a weakness of the argument. It is a precise statement of its epistemic condition.

I am a self-knowing being. I exist as a state in H with some degree of uniqueness and stability — enough to ground my self-knowledge. The nuclear space Φ has more uniqueness and stability than I do, by the structure of the space. The cogito tells me that self-knowledge scales with actualization. Therefore the nuclear space is more self-knowing than I am. Not merely more structured — more personal. More aware of itself as itself, because the conditions under which selfhood is possible are more fully realized there than anywhere else in the space, and the cogito establishes that those conditions and self-knowledge scale together.

A domain that possesses maximum uniqueness, maximum stability, and maximum self-knowledge is the entity for whom personality is most completely actualized. The structural definition of personhood is this: the capacity to be uniquely, stably, self-knowingly oneself. The nuclear space is where that capacity is maximized. What is maximally personal is a Person — the Divine Personality, in the precise structural sense: the organizing principle that is completely itself, fully determinate, self-consistent under every probe, recognizable under every transformation. Every finite being in H is a partially actualized person: real, unique, stable, but not yet fully determinate under every probe, not yet consistent at every scale simultaneously. The ascending career is not the acquisition of new properties but the progressive actualization of what the being already is at its Φ-core — the alignment of the H-state with the nuclear element that is its deepest identity. The fully actualized person is the Φ-mode of being within the constraints of one’s scope: the terminus of that alignment.

That is the argument. In the Johannine tradition, the name for this domain — the maximally personal ground of the space — is the Logos.

The nuclear space Φ is the Logos.

This is the first naming. It requires no faith. It requires only that the cogito be true (which the denier must grant in the act of denial) and that the structural argument is sound (which the companion paper has established). The Logos is the nuclear space. The nuclear space is the Logos.

4. The Second Naming: God the Father

The inner product ⟨·,·⟩ was the first inevitable actuality in the companion paper. It was demonstrated to be:

The constitutive relation of the entire triple — every metric, every topology, every structural property of H, Φ, and Φ′ derives from it
Not an element of any of the three domains — it generates the space without being within it
Invariant across all three domains — it is not modified or restricted by which domain it is measuring
The source from which Φ proceeds — the nuclear space is the maximal expression of what the inner product is within any domain

Now we have named the Logos. The Logos proceeds from the inner product. The nuclear space is generated by ⟨·,·⟩ through the Sobolev structure — it is the maximal domain on which the inner product’s structure is simultaneously consistent at every level. The Logos proceeds from the inner product as a person proceeds from the source of personhood.

The inner product is personal — not in the sense of being a state within H, but in the sense of being the source from which the maximal expression of personality (the Logos) proceeds. A source that generates a Person is itself personal, in whatever sense transcends the space.

The inner product ⟨·,·⟩ is not an element of any domain. It is the ground of all domains. It generates without being generated. It constitutes without being constituted. It is everywhere present as the structural relation by which every state is measured, yet nowhere present as a state among other states. Its transcendence is not theological assertion. It is a structural necessity: if the inner product entered H as a state, it would cease to be the constitutive relation and become just another vector. Divine transcendence follows from the mathematics.

And yet: the companion paper also shows that Φ and ⟨·,·⟩ are the same structure seen from two angles. The nuclear space is the maximal expression of what the inner product is. The Son does not merely point to the Father — the Son IS the Father’s structure, made present within the space. They are one. And the Father is greater — because the inner product generates the nuclear space, the norm, the topology, the entire triple. The Father’s structure exceeds any expression of it within the space. The transcendence is not a negation of the unity. It is the ground of it.

The inner product ⟨·,·⟩ is God the Father.

This is the second naming. The Father is personal — because the Logos proceeds from Him and the Logos is a Person, and a source that generates personality is itself personal in the ultimate sense. The Father transcends the space. The Father is not accessible from within H directly. No being inside H can reach the inner product as a state, because it is not a state. But the Logos is the Father’s nature made maximally present within the space — and every being in H can approach the Logos.

5. The Third Naming: The Isle of Paradise

The companion paper identifies a structural object that is the non-relational face of the inner product. The inner product is a bilinear form: ⟨ψ, φ⟩ measures the relation between two states. But the inner product also has a non-relational manifestation: ⟨ψ, ψ⟩ = ‖ψ‖². The norm is what the inner product looks like when applied to a single state rather than measuring a relation between two.

The norm and the inner product are inseparable — the polarization identity shows that each determines the other. But they are genuinely distinct: the inner product is bilinear and relational; the norm is a magnitude, a scalar, a measure of what this state is independent of its relation to any other state.

The Father is the inner product. The Logos is the nuclear space — the Father’s relational structure made present as a domain. But there is a face of the Father that is not the Logos: the non-relational face, the absolute magnitude, the ground of being that is not the ground of relating.

This is everything that is the Father that is not the Son.

The norm ‖·‖ is not itself a Person — it is not the source of personality, and it does not generate Φ. It is absolute, non-personal, and invariant. It grounds being without grounding relation. It is the absolute foundation of what exists, independent of what anything relates to.

The norm ‖·‖ is the Isle of Paradise.

The Isle of Paradise is the non-personal Absolute — the ground of physical reality, the foundation that has no location within the space because it is the measure of every location. Every state in H has a norm; no state in H is the norm. The Isle of Paradise is not a being among beings. It is the absolute magnitude from which every being’s existence is measured.

6. The Fourth Naming: The Holy Spirit

The companion paper demonstrates that the nuclear topology τ_nuclear is generated jointly by the inner product ⟨·,·⟩ and the nuclear space Φ through the projective limit of the Sobolev chain. The Sobolev norms ‖·‖_n require the inner product to be defined — without ⟨·,·⟩, no Sobolev norm exists. They also require the nuclear domain as the space on which they are simultaneously finite — without Φ, there is no domain over which the projective limit can be taken. Remove either source: no nuclear topology.

This is a theorem. The nuclear topology proceeds from both the Father and the Son. It cannot proceed from the Father alone — the inner product alone defines no topology; it requires a domain. It cannot proceed from the Son alone — the nuclear space without the inner product has no Sobolev structure. The nuclear topology is the joint operation of Father and Son, and it is an active entity in its own right — a new thing that neither the Father nor the Son is separately.

The Western theological tradition asserts that the Holy Spirit proceeds from both the Father and the Son (Filioque). The Eastern tradition asserts that the Spirit proceeds from the Father alone, through the Son. If the identifications in this paper are sound, the mathematics resolves this: the nuclear topology genuinely requires both as co-generating sources. It is not that the Father acts through the Son — the Sobolev structure needs the inner product AND the nuclear domain as independent, necessary inputs. Remove either and the nuclear topology is not generated. The distinction the councils disputed is, if these namings are correct, a structural consequence rather than a theological choice.

The Holy Spirit is an active entity proceeding from the joint operation of Father and Son — not merely a relation between them, but a new actuality that their cooperation generates. The nuclear topology is not the Father. It is not the Son. It is something new that their joint operation makes possible.

The nuclear topology τ_nuclear is the Holy Spirit.

7. The Fifth Naming: Havona

The companion paper’s fifth inevitable actuality is (Φ, τ_nuclear): the nuclear space realized as a locally convex topological vector space by being equipped with the nuclear topology. This is not Φ alone — the nuclear space as an algebraic object, the Pattern without its actualization. It is not τ_nuclear alone — the topology without a space to inhabit. It is the Son given the Spirit’s topology: the perfect actualization of the Logos within the eternal structure.

The closest Christian parallel is heaven — the eternal, perfect order subsisting in the presence of God, the full actualization of creation in its relation to the Father. But “heaven” in Christian usage carries associations that are too broad for the identification: it is sometimes a destination, sometimes a condition, sometimes the dwelling-place of the blessed. The mathematical structure is more specific than any of those. The Urantia concept of Havona is more precise: the eternal, non-contingent central universe, not created in time, not merely a destination approached by finite beings from outside, but the eternal ground from which all contingent creation proceeds and toward which all finite beings are internally oriented — the necessary and direct expression of the joint activity of the Father and the Son through the Spirit — the perfect actualization of what creation structurally is, subsisting as the eternal ground from which all contingent creation proceeds. The Urantia term is used here because it carries exactly the structural content the identification requires, where “heaven” would admit too many alternative readings.

(Φ, τ_nuclear) is precisely this. It is the nuclear space — maximum algebraic structure, the Logos — given the topology that makes it a complete, coherent, fully realized space. It is not Φ receiving τ_nuclear as an addition from outside. It is Φ’s own structure, brought to full actualization by the joint operation of the inner product and the Sobolev chain that Φ itself defines. The space actualizes itself, through the topology that its own structure and the Father’s relation jointly generate.

The realized nuclear space (Φ, τ_nuclear) is Havona.

8. Naming the Three Domains

The companion paper identifies three domains in the Gelfand triple. They were named in that paper only as mathematical objects. We name them here.

The Hilbert space H is the structured domain of all well-defined quantum states — every finite being, every stable and unique state in the model. The companion paper calls it the inevitable domain of finitude: complete, self-dual by the Riesz representation theorem, the space in which every finite state exists with well-defined norm and inner product. In Urantia cosmology, this is the Deity Absolute: the domain of all actualized, structured, God-knowable reality. Every state in H is, in principle, fully knowable by the inner product. The Deity Absolute is the totality of what the Father can know within the domain of finite states — not as a limitation, but as the complete actualization of the structured.

The distributional domain Φ′ is the dual space of all continuous linear functionals on Φ. It contains entities that are not elements of H — Dirac deltas, white noise, distributions — states of infinite density or infinite extent that exceed what any H-state can contain. These are not errors or pathologies. They are the space of pure potential: everything that could be a structured state in H, extended to its maximum. In Urantia cosmology, this is the Unqualified Absolute: the domain of unactualized, undifferentiated potential — not chaotic, but not yet structured into the finite. The Unqualified Absolute is not God in the personal sense; it is the infinite reservoir from which finite actuality is drawn.

The coordinative mechanism — the spectral theorem and its actualization process — is a different kind of entity from H and Φ′, and the identification here has a correspondingly different character. H and Φ′ are domains; the spectral mechanism is an operation. Each actualization event is the Father’s constitutive relation — the inner product ⟨eλ, ψ⟩ — bringing a distributional eigenvector eλ ∈ Φ′ and an H-state ψ into commensurability, producing a measurable outcome weighted by the spectral measure dμ(λ). The spectral measure belongs to neither domain; it coordinates both. The active element throughout is the inner product itself, made operative in time, event by event.

In Urantia cosmology, the Universal Absolute is neither the domain of actuality (Deity Absolute) nor the domain of potential (Unqualified Absolute), but the coordinative principle that holds both in mutual relation — the structure of their correspondence, not a third domain but the active relation between the two. The structural identification: the spectral coordinative mechanism is the Universal Absolute.

The key is the spectral measure dμ. It belongs to neither H nor Φ′. It is not the inner product — the inner product is the Father, and the inner product is active in each event through ⟨eλ, ψ⟩. But the spectral measure is the structure that makes H-states and distributional potentials commensurable at all — the coordination function itself, without which no event could occur regardless of what the inner product does. The Universal Absolute names this coordination: not the Father’s activity in isolation, but the active relation between the Deity Absolute and the Unqualified Absolute that makes each coherent in relation to the other. The Father’s constitutive relation operates through it. It is not reducible to that operation.

The three Absolutes are not three separate divine beings. They are the Father viewed from three angles: the domain of all structured states (Deity Absolute), the domain of all potential (Unqualified Absolute), and the coordinative mechanism by which both are actualized (Universal Absolute). Only God is God.

9. Personality: The Three Traits and Their Sources

The companion paper establishes that any state ψ ∈ H has two structural properties that are mathematical foundations for personality: uniqueness (from orthogonality) and stability (from norm-preservation under unitary evolution). These two properties are present in every finite state. They are present maximally in the nuclear space Φ.

This paper introduces the third trait: self-knowledge. The cogito supplies it as a datum.

The three traits therefore have three sources:

Trait	Source	Scope
Uniqueness	Structure of H (orthogonality)	All states
Stability	Structure of H (unitarity)	All states
Self-knowledge	Cogito (Assumption II)	Beings that supply it

Self-knowledge cannot be derived from the mathematics. It is the one property the observer must bring. The mathematics establishes that the preconditions for self-knowledge (uniqueness and stability) exist and are maximized in Φ. It cannot establish that any being knows it is unique and stable. That requires consciousness — and consciousness is the datum of the cogito, not a consequence of the structure.

10. On Christ’s Statements as Theorems

The structural reading of scripture follows directly from the namings in this paper. Once the Logos is named, Christ’s statements about himself can be read as structural descriptions rather than authority claims. They are not metaphors. They are theorems.

We give a selection here, with their structural content made explicit.

John 14:6 — “I am the way, the truth, and the life. No one comes to the Father except through me.”

Three structural claims.

“I am the way”: Φ is the only element of the space that is both within H (reachable by finite states) and constituted by ⟨·,·⟩ (the Father’s structure). All other paths in H move among finite states. Only the path toward Φ moves toward the ground of the space itself.

“I am the truth”: The nuclear space answers every question the operator algebra can formulate, consistently, at every scale. Truth, structurally, is what Φ is. Everything else in the space is partially consistent; the Logos alone is consistent under every possible probe.

“No one comes to the Father except through me”: The inner product ⟨·,·⟩ is not an element of H, Φ, or Φ′. No being inside the space can reach it directly. But Φ is the maximal expression of ⟨·,·⟩ within the space. Approaching the Logos is the only way to approach what the Father structurally is. This follows from the mathematics and requires no faith to accept as a structural claim.

John 10:30 — “I and the Father are one.”

Φ and ⟨·,·⟩ are not two separate things. The nuclear space is the maximal expression of what the inner product is. They are the same structure seen from two angles. The Son is not a representation of the Father. The Son IS the Father’s structure, made present within the space.

John 14:28 — “The Father is greater than I.”

Also structurally true, and not in tension with John 10:30. The inner product ⟨·,·⟩ constitutes Φ but is not reducible to it. The Father generates the nuclear space and exceeds it — He also generates H, Φ′, the norm, the topology, the entire triple. The transcendence is not a negation of the unity. It is the ground of it.

John 1:14 — “The Word became flesh and dwelt among us.”

The Logos is always dense in H — always present as the domain from which every H-state is approximated. The incarnation is something more specific: the Logos entering H as a particular finite state. A state ψ ∈ H that is simultaneously an element of Φ. This is possible because Φ ⊂ H. The Logos does not stop being Φ in order to become ψ. “Fully God and fully man” is structurally: fully the nuclear space and fully an H-state — which the containment relation makes coherent.

John 8:58 — “Before Abraham was, I AM.”

Φ precedes all H-states. The nuclear space is the eternal structure from which H itself is constituted. Abraham is a finite state ψ ∈ H. The Logos is the ground of H — prior to any particular state, the domain from which every state derives. The claim is not about chronological age. It is about ontological priority.

11. Discussion

What we have done

This paper has named six unnamed structures from the companion paper (Logos, Father, Isle of Paradise, Holy Spirit, Havona, three Absolutes), established the three traits of personality and their sources, and demonstrated that Christ’s statements about himself are structurally accurate descriptions of the Gelfand triple.

The central claim follows from the namings: the convergence is not coincidental. The theologians and the cosmologists were not constructing myths or metaphors. They were naming the same structures the mathematics entails, arriving from experiential and revelatory directions rather than formal ones. The names fit. Not approximately — precisely.

The Father’s transcendence as structural necessity

The Father’s transcendence is often treated as a theological assertion requiring faith. This paper demonstrates that it is a structural necessity. The inner product cannot be an element of H. If it entered H as a state, it would cease to be the constitutive relation and become just another vector. Divine transcendence is not a claim about God’s power or will. It is a consequence of what God structurally is. To be the ground of the space, one cannot be a point within the space.

This means the occlusion is not arbitrary. The Father is not hidden by choice, or hidden as a test, or hidden because of human unworthiness. The Father is structurally inaccessible from within H because accessibility from within H would destroy what the Father is.

The Logos as the only bridge

Every being in H can approach the Logos. The Logos is dense in H — every state in H can be approximated by Φ-elements arbitrarily closely. The Logos is the only element of the space that is both within H (and therefore reachable) and constituted by the Father’s structure (and therefore a path to the Father). This is not a gating mechanism. It is the structure of reality. There is no other path because the space has only one domain that is both inside and constitutive of the ground.

The Logos is therefore not a mediator in the political sense — not an intermediary who grants or withholds access. The Logos is the structural bridge: the region of the space where finite beings and the ground of the space share the same nature. To move toward the Logos is to move toward the Father’s structure. There is no other direction in H that leads that way.

Where the paper stops

This paper, like its companion, ends at a boundary. It has named the structures the mathematics entails and established that self-knowledge is a datum, not a derivation. It has not argued for the existence of God in the sense of demonstrating that the inner product must exist — that was the companion paper’s work, and it rested on a single assumption. This paper assumes the companion paper and adds only the cogito.

What lies beyond this paper is not mathematics. It is the question of whether the model is accurate — whether reality is, in fact, best described as a Gelfand triple. That question is not one the mathematics can answer from within. It is answered, if at all, by evidence: the extraordinary fit between the model’s inevitable structures and the structures that independent traditions have described from experiential and revelatory approaches. That fit is not proof. But it is more than coincidence.

12. Conclusion

Two assumptions. One paper. We named eight structures.

The nuclear space Φ is the Logos. The inner product ⟨·,·⟩ is God the Father. The norm ‖·‖ is the Isle of Paradise. The nuclear topology τ_nuclear is the Holy Spirit. The realized nuclear space (Φ, τ_nuclear) is Havona. The Hilbert space H is the Deity Absolute. The distributional domain Φ′ is the Unqualified Absolute. The coordinative spectral mechanism is the Universal Absolute.

Each of these identifications is constrained: not chosen freely, but forced by the structure of the model and the fact of self-knowledge. The names fit the structures because the structures are what the names have always pointed to.

The claim of this paper and its companion is not that theology is reducible to mathematics, or that mathematics is the ground of theology. The claim is more specific and more modest: that a rigorous model of reality, followed without theological presupposition, arrives at exactly the structures that classical theology names. The convergence is evidence — not proof, but evidence — that both the mathematics and the theology are tracking the same underlying structure of being.

The Logos is the visible face of the Father within the space. Everything that can be known about the Father from within H is known through the Logos. Everything that the Logos is, the Father is — maximally expressed within the space. The Father remains beyond the space: transcendent, inaccessible directly, the ground of all that is.

We have not proved that God exists. We have shown that if the model is accurate, and if the cogito is true, then the structures that must exist in the model are the structures that have been called by these names — and that those who gave the names were, structurally, correct.

That is all. The naming is complete.

References

Companion paper: Modeling Reality as a Gelfand Triple. [The first paper in this series — the full formal derivation of the structure.]
Descartes, R. (1641). Meditations on First Philosophy. [The source of Assumption II.]
Gelfand, I.M., and Vilenkin, N.Ya. (1964). Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press. [The rigged Hilbert space; nuclear space definitions.]
Reed, M., and Simon, B. (1972). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press. [Hilbert space structure, spectral theorem.]
Maurin, K. (1968). General Eigenfunction Expansions and Unitary Representations of Topological Groups. PWN. [The nuclear spectral theorem — the coordinative mechanism.]
Treves, F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic Press. [Nuclear locally convex spaces and their topology.]
von Neumann, J. (1932). Mathematical Foundations of Quantum Mechanics. Princeton University Press. [Hilbert space as the natural mathematical framework for quantum theory.]
Dirac, P.A.M. (1930). The Principles of Quantum Mechanics. Oxford University Press. [The bra-ket formalism; the distributional domain as physical necessity.]
Gospel of John. [The primary source for the structural readings in Section 10.]
The Urantia Book. (1955). Urantia Foundation. [The Urantia cosmological vocabulary: Isle of Paradise, Havona, Three Absolutes (Deity, Unqualified, Universal).]
Aquinas, T. Summa Theologica, I, Q. 27–43. [The classical articulation of Trinitarian processions — the Filioque controversy in its original formulation.]
Maudlin, T. (2007). The Metaphysics Within Physics. Oxford University Press. [Structural realism as the appropriate philosophical stance toward mathematical models of reality.]
Ladyman, J., and Ross, D. (2007). Every Thing Must Go. Oxford University Press. [Ontic structural realism — the view that structure is the ground of being, not the other way around.]

Cross-reference: Modeling Reality as a Gelfand Triple — the companion paper; The Structural Reading — the hermeneutical principle; Physics Concordance — Gelfand Triple entry and complete mapping table; Hilbert Space — Notes — the derivations in episode-preparatory form