The Geometry of Sameness: An ε-Equivalence of Translation and Distance

The Geometry of Sameness: An ε-Equivalence of Translation and Distance Categories of Realizations, Error-Budget Transfer, and a Hidden Variable Bee Rosa Davis NASA Mission Systems Engineer [email protected] Abstract At an intuitive level, the problems we study are like trying to understand the Earth from maps. The underlying semantic sameness structure S is the globe: a hidden world of entities and their relationships. One engineering tradition starts from the globe (a manifold): learn a space where geodesic distance reflects semantic change and read decisions off geodesic gaps—this is the geometry-first view used by systems like HERALD and VIDAR. A second tradition starts from overlapping flat maps: heterogeneous observation spaces with translators between them, stitched together just enough to reconcile telemetry or transactions—this is the translation-first view used by systems like AMC and PRISM. Our central claim is that, whenever the “maps” are smooth and consistent enough to be sewn into a globe, these are not competing descriptions but dual realizations of the same S: tools and guarantees can be moved across the seam. Formally, for a fixed semantic sameness structure S we define two categories of realizations: SamTrans(S), whose objects are translation-based systems on heterogeneous observation spaces with bounded translator drift, and SamGeom ( S ), whose objects are manifold-based systems with Riemannian metrics, path families, and configuration margins. We introduce ε -equivalence of categories tailored to detection, together with explicit functors FS : SamTrans0 ( S ) →SamGeom0 ( S ) and GS : SamGeom0 ( S ) →SamTrans0 ( S )that act like “stitching the maps into a globe” and “unfolding the globe back into maps” on well-behaved subcategories SamTrans0 ( S ) ,SamGeom0 ( S ) that satisfy a smooth chartability condition. Smooth chartability is the critical, falsifiable hypothesis of the framework: it is exactly the requirement that the local translators around S can be approximated by a compatible atlas whose transition maps behave like local diffeomorphisms, so that a globe Mactually exists. Our main result is an Error Budget Transfer Theorem. Each realization (translator or manifold) comes with a four-term detection error budget ( Egeom, Elink, ξ, ζ )for geometry, linkage, calibration, and abstention, together with an independence slack δindep and a finite-variance Cantelli term BCantelli linking statistic separation to posterior correctness. We show that for any T∈SamTrans0(S)and its geometric realization M=FS(T), ETSP tot (T) = 1 −(1 −ET geom)(1 −ET link)(1 −ξT)(1 −ζT)+δT indep and EMSP tot (M) = 1 −(1 −EM geom)(1 −EM link)(1 −ξM)(1 −ζM)+δM indep differ by at most an explicit, first-order slack |ETSP tot ( T ) −EMSP tot ( M ) | ≤ CF ( εtrans, εdist, δchart ), and the associated Cantelli bounds on high-risk alerts satisfy an analogous relation |BCantelli ( T ) − 1 BCantelli ( M ) |≤C′ F ( · · · ). A symmetric statement holds for M∈SamGeom0 ( S )and T = GS ( M ). In other words, once S passes the smooth chartability test, translator-based and manifold-based realizations of S share the same detection guarantees up to controlled slack: the choice of “flat maps vs. globe” becomes an engineering decision, not a conceptual one. Practically, this ε -equivalence has two consequences. First, it lifts theorems across the bridge: guarantees proved in a geometry-first system (e.g., Davis-style path-based bounds for HERALD or VIDAR) can be pulled back to translation-first architectures like AMC via GS , and vice versa. Second, it lifts tools: audits developed for translator graphs (e.g., AMC’s spectral stability checks) become curvature or chart-overlap diagnostics on manifolds via FS . We instantiate this framework on four systems—AMC and PRISM as translation-first, HERALD and VIDAR as geometry-first—to illustrate how chartability can be audited empirically and how error budgets and techniques transfer in practice. 1 Introduction Many modern systems need to decide whether two heterogeneous observations refer to the same underlying entity or event: is this telemetry packet semantically equivalent to that one after compression and format switching? Does this payment record match that ledger entry across rails? Do these viral variants represent the same antigenic state? Does this video clip belong to the same physical person as that enrollment image? Two families of architectures answer these questions in different languages: • Translator-based systems (e.g., AMC-style adaptive format pipelines, multi-rail reconciliation) build a graph of learned translators between embedding formats or modalities and assert sameness when translated representations agree within a drift budget. • Manifold-based systems (e.g., Davis-style HERALD and VIDAR) learn or inherit a Riemannian manifold on which geodesic distance along identity-preserving paths encodes semantic change, and assert sameness when points remain within a configuration margin in that geometry. Informally, one view speaks in translations and drift; the other in distances and curvature. Both solve the same kind of problem, but their guarantees, diagnostics, and engineering tools are typically developed in isolation. This work starts from a simple question: For a fixed notion of semantic sameness, are translator-based and manifold-based detection systems two views of the same underlying problem? If so, can guarantees and tools proven in one view be transported to the other without re-deriving them from scratch? We answer this in the affirmative under explicit regularity assumptions. The key move is to make the problem itself explicit. 1.1 Semantic sameness structures and realizations We model the underlying task as a semantic sameness structure S : latent identities, observation spaces, rendering maps, and a sameness relation on observations. For a fixed S , engineers can build many concrete systems. 2 •SamTrans ( S ): a category whose objects are translator-based realizations of S (observation encoders, format/rail translators, drift budgets, and detection logic in translator space), and whose morphisms are reparameterizations preserving translators up to controlled error. •SamGeom ( S ): a category whose objects are manifold-based realizations of S (Riemannian manifolds, chart atlases, path families, and Davis-style detection logic), and whose morphisms are quasi-isometries respecting configuration structure and charts. Crucially, these are two design spaces for the same S : translator systems and manifold systems are not being unified with each other directly, but as two families of realizations of the same hidden variable. On well-behaved subcategories SamTrans0 ( S ) ⊂SamTrans ( S )and SamGeom0 ( S ) ⊂SamGeom ( S ), defined by smooth chartability, bounded distortion, and non-vacuous configuration margins, we construct functors FS:SamTrans0(S)→SamGeom0(S), GS:SamGeom0(S)→SamTrans0(S), that turn translators into manifolds and manifolds into translator graphs. These functors are not magic; they succeed precisely when the chartability and distortion assumptions are satisfied. 1.2 Main contributions Our contributions are organized around three load-bearing claims and their practical implications. (0) Language: ε -equivalence of realization categories. We formalize translator and manifold realizations of a fixed semantic sameness structure S as categories SamTrans ( S )and SamGeom ( S )and introduce an ε -equivalence notion tailored to detection. Rather than claiming a strict equivalence of categories, we construct functors FS and GS and show that GS◦FS≃IdSamTrans0(S) and FS◦GS≃IdSamGeom0(S) up to quasi-isometry and bounded morphism error, with all slack terms explicit. Categorical language is not decorative here: it is the minimal formalism needed to express how error budgets transform under representation change. (1) Error Budget Transfer Theorem (main result). Our central technical result shows that detection guarantees transfer across the two realizations. Conceptually, we prove that the correctness bound for a translator system ( T ) is ε -equivalent to the correctness bound for the manifold system (M) that realizes the same problem S: LB(T)∼ =εLB(FS(T)) This main result, the Error Budget Transfer Theorem, is stated formally below. It shows that the full, multi-part error budgets for both systems are equivalent up to explicit, first-order slack terms. Error Budget Transfer Theorem (informal preview). Fix a semantic sameness structure Sand consider well-behaved realizations T∈SamTrans0(S)and M∈SamGeom0(S). Translator-side bound. The translator realization Tadmits a Davis-style correctness lower 3 bound LBT(T) = BCantelli(T)h(1 −ET geom)(1 −ET link)(1 −ξT)(1 −ζT)−δT indepi+−εT est. Manifold-side bound. The manifold realization Madmits an analogous bound LBM(M)=BCantelli(M)h(1 −EM geom)(1 −EM link)(1 −ξM)(1 −ζM)−δM indepi+−εM est. Transfer. The functors FSand GStransport these guarantees: LBT(T)−LBM(FS(T))≤CF(εtrans, εdist, δchart), LBM(M)−LBT(GS(M))≤CG(εtrans, εdist, δchart), where CFand CGare explicit first-order slack functions. (2) Characterization via smooth chartability. The functor FS : SamTrans0 ( S ) →SamGeom0 ( S ) is not automatic; it succeeds only on translator systems with smooth chartability. We isolate this as the central, falsifiable hypothesis of the framework: locally linear approximants to each translator must exist with bounded Jacobian and small residual drift on in-regime data. This assumption is testable via concrete audit protocols (e.g., Jacobian-based linearization error histograms on AMCstyle translators) and is empirically supported in systems whose architectures already regularize for smoothness and stability. (3) Constructive functors and ε -equivalence. Under smooth chartability and bounded distortion, we give explicit constructions of FS (path-metric completion of a translator graph followed by a Riemannian smoothing) and GS (chart-transition translators induced by overlapping coordinate maps). We show that on SamTrans0 ( S )and SamGeom0 ( S )these functors form an ε -equivalence of categories: they are quasi-inverse up to explicitly bounded drift, curvature, and chart-overlap slack. This formalizes the intuition that translator and manifold views are dual coordinatizations of the same semantic sameness problem S. Implications: tool and theorem transfer. The practical consequence of this ε -equivalence is that tools and theorems proven in one view can now be reused in the other. • Theorems transfer. A guarantee proven in a manifold system (e.g., HERALD’s Davis-style dominance lower bound, VIDAR’s deepfake correctness bounds) can be pulled back via GS to a translator-based realization of the same S (e.g., AMC-style telemetry translators), with only first-order changes in the error budget. • Tools transfer. Engineering techniques developed for translator graphs (spectral stability audits, drift-triggered retraining, lineage-aware rollback) can be pushed forward via FS to become curvature and chart-stability diagnostics on manifolds, providing new audit tools for geometryfirst systems. 4 Engineers starting from a single semantic sameness structure S therefore have a principled choice of working in translator or manifold coordinates, knowing that both design spaces are linked by functors that preserve detection guarantees up to explicitly quantified slack. 1.3 Running examples We ground the theory in four concrete systems, which we treat as realizations of different semantic sameness structures S. AMC (telemetry format switching). AMC [ 1 ] is a telemetry-native embedding architecture for adaptive format switching and semantic translation in high-volume telemetry pipelines. It maintains a directed graph of learned format translators constrained by an ε -bounded semantic drift loss and uses spectral properties of a format-compatibility graph to select stable routes and fallbacks. In our language, AMC is a canonical object of SamTrans(Stele)for a telemetry sameness structure Stele. PRISM (payment reconciliation). PRISM [ 4 ] performs multi-rail payment reconciliation with learned translators between heterogeneous transaction formats, semantic alignment constraints, and a central latent reconciliation space. It naturally lives in SamTrans ( Spay )but exposes an implicit manifold structure over reconciled transaction identities. HERALD (viral antigenic drift). HERALD [ 2 ], introduced in prior work on Davis manifolds [ 3 ], builds a pullback Riemannian manifold on viral sequences where geodesic distance along epitoperestricted mutation paths approximates antigenic distance; a PIT-fused drift statistic and Cantelli bounds map geometric drift to dominance risk with an explicit error budget. HERALD is a prototypical object of SamGeom(Santigen). VIDAR (deepfake detection via identity trajectories). VIDAR treats face-recognition embeddings as points on the hypersphere, models short clips as identity trajectories on an identity manifold, and builds RIM features plus auxiliary detectors as inputs to a Davis-style scalar statistic and bound. 1VIDAR is a canonical object of SamGeom(Sid). Across these examples, the same pattern recurs: translator systems like AMC and PRISM live in SamTrans ( S ); geometry-first systems like HERALD and VIDAR live in SamGeom ( S ); and our functors FSand GSexplain how to move guarantees and tools between these views. 1.4 Scope and non-goals This manuscript is deliberately theoretical. It: •defines semantic sameness structures Sand the categories SamTrans(S)and SamGeom(S); • introduces smooth chartability and bounded-distortion path geometry as explicit, falsifiable assumptions defining subcategories SamTrans0(S)and SamGeom0(S); •constructs FSand GSand proves an ε-equivalence of realization categories; 1 See Davis, “The Davis Manifold: Geometry-First Detection with Compositional Error Budgets,” for the VIDAR instantiation. 5 • proves the Error Budget Transfer Theorem (Theorem 6.3) showing that Davis-style correctness bounds transfer across these views with first-order slack; and • sketches audit protocols for checking smooth chartability and error-budget components in systems like AMC, PRISM, HERALD, and VIDAR. We do not report new empirical performance, deployable calibration curves, or production latencies; those belong in system-specific papers. Our goal here is to expose the common hidden variable S , make the representation change between translator and manifold views explicit and functorial, and characterize when and how detection guarantees can be transported between them. 1.5 Roadmap The paper develops in three acts. Act I (Framework). Section 2 formalizes semantic sameness structures S and defines translator and manifold realizations, together with Davis-style error budgets and the notion of smooth chartability. Section 3 introduces the categories SamTrans ( S )and SamGeom ( S ), constructs the functors FSand GSat a high level, and defines ε-equivalence of realization categories. Act II (Constructions and transfer). Section 4 builds FS concretely by endowing translator graphs with a path metric and a smoothed Riemannian structure under smooth chartability, and analyzes the resulting distortion and chart-overlap slack. Section 5 constructs GS from manifold charts and transition maps and analyzes its stability. Section 6 proves the Error Budget Transfer Theorem (Theorem 6.3) by tracking how geometric, linkage, calibration, and abstention errors transform under FSand GS. Act III (Case studies and outlook). Section 7 instantiates the framework on AMC, PRISM, HERALD, and VIDAR, including concrete chartability audits and error-budget estimates. Section 8 discusses limitations, open problems, and the broader “Davis universality” conjecture for geometry-first detection in temporal semantic sameness problems. 2 Semantic sameness structures and realizations Throughout this section we fix a semantic sameness problem and make precise what it means to “realize” that problem either as a translator graph or as a Riemannian manifold. The hidden object is the sameness structure itself; translation-based and manifold-based systems are two different realizations of the same underlying task. We use boldface symbols such as Tand Mfor concrete systems (a particular TSR or MSR) and reserve plain letters for abstract spaces and maps. 2.1 Semantic sameness structures Intuitively, a semantic sameness structure records: (i) a latent space of entities u (viruses, identities, vehicles, . . . ), (ii) how each modality i renders those entities as observations xi , (iii) when two observations in possibly different modalities “are the same entity”, and (iv) which latent trajectories count as benign identity-preserving evolution. 6 Definition 1 (Semantic sameness structure).Asemantic sameness structure is a tuple S:= I, I,{Xi}i∈I,{πi}i∈I ,≈,{PS(L)}L>0, where: (i) I is a (typically high-dimensional) set of latent entities or states. A point u∈I represents the underlying “object of interest” (e.g., true antigenic state, true human identity, true vehicle configuration). (ii) Iis a finite index set of modalities (e.g., file formats, sensors, assays, views). (iii) For each i∈ I,Xiis an observation space for modality i. We write X:= G i∈I Xi for the disjoint union of all observations. (iv) For each i∈ I , πi : I→Xi is a (possibly partial) rendering map that produces an ideal observation of u in modality i when defined. 2 When πi ( u )is undefined, modality i simply does not observe u. (v) ≈is a semantic sameness relation on X, defined by x≈x′⇐⇒ ∃u∈I, ∃i, j ∈ I such that xis an observation “of” uin Xi, x′is an observation “of” uin Xj. In the noiseless idealization this reduces to: x≈x′ if there exist u∈I and ( i, j )with x = πi ( u ) and x′ = πj ( u ). In practice we blur small observation noise and treat x as an observation “of” uwhenever it lies in a pre-specified neighborhood of πi(u). (vi) For each L > 0, PS ( L )is a family of benign latent paths γ : [0 , 1] →I indexed by a semantic horizon L . Each γ∈ PS ( L )represents an identity-preserving evolution of latent state (e.g., realistic antigenic drift over one season, smooth identity motion over a short clip). We require that L7→ PS(L)is monotone: if L1≤L2then PS(L1)⊆ PS(L2). Examples (semantic structures). • Format-heterogeneous telemetry (AMC [ 1 ]). I is the space of underlying system states; each Xi is a particular log or telemetry format; πi renders a state into that format; γ∈ PS ( L )is a short-time state trajectory. • Antigenic drift (HERALD). I are viral lineages; Xseq are amino-acid sequences; Xneut are neutralization measurements; πseq encodes the canonical genome for a lineage; πneut renders idealized assay readouts; PS(L)are mutation paths with at most Lepitope substitutions. • Identity trajectories (VIDAR). I are person identities; Xvideo are short talking-head clips; Xaudio are speech segments; each πi renders a clip of the same person; PS ( L )are natural motion trajectories over windows of bounded duration. 2 Formally, one may model πi as a Markov kernel from I to Xi , capturing stochastic observation noise; for notational simplicity we write πias a map and treat noise separately in the detector error budget. 7 We will later view realizations of a fixed S as objects of two categories: a translation-based category SamTrans(S)and a geometry-based category SamGeom(S). 2.2 Translation-based realizations (TSRs) A translation-based realization starts from modality-specific feature spaces and implements sameness comparisons via learned translators between those spaces. Definition 2 (Translation-based semantic realization (TSR)).Let S be a semantic sameness structure as in Definition 1. A translation-based semantic realization (TSR) of Sis a tuple T:= {Vi}i∈I,{φi}i∈I , GT,{Tij}(i,j)∈ET, εtrans(·), where: (i) For each i∈ I,Viis a finite-dimensional inner-product space (feature space for modality i). (ii) φi:Xi→Viis a feature extractor (e.g., a neural encoder). We denote ideal features by vi(t):=φiπi(γ(t))for γ∈ PS(L), t ∈[0,1], whenever πi(γ(t)) is defined. (iii) GT = ( I, ET )is a directed graph whose vertices are modalities and whose edges i→j indicate the availability of a learned translator Tij :Vi→Vj. (iv) For any path p= (i0, . . . , ik)in GTwe write Tp:= Tik−1ik◦ · · · ◦ Ti0i1and |p|:= k. (v) εtrans : N→ [0 ,∞ )is a translator drift profile such that, for any benign latent path γ∈ PS ( L ), any time t∈ [0 , 1], any modalities i, j with both πi ( γ ( t )) and πj ( γ ( t )) defined, and any GT -path p:i→jof length k, we have  Tpvi(t)−vj(t) ≤εtrans(k).(1) This inequality measures representation error of the translators at a single latent time, decoupled from any semantic drift along γ. We write TSR(S)for the collection of all TSRs that realize S. Examples (TSRs). • AMC-style telemetry [ 1 ]. Vi are format-specific embedding spaces; φi encode logs into these spaces; Tij are learned format translators; εtrans bounds accumulated translator drift over paths in the format graph. • PRISM-style multi-signal systems [ 4 ]. Vi are detector-specific representations; Tij fuse or reconcile outputs between detectors; εtrans captures how far translations can drift while still preserving semantic identity. 8 2.3 Manifold-based realizations (MSRs) A manifold-based realization represents semantic sameness by embedding latent entities into a Riemannian manifold and requiring that feature-space coordinates act as approximate charts. Definition 3 (Manifold-based semantic realization (MSR)).Let S be as in Definition 1. A manifold-based semantic realization (MSR) of Sis a tuple M:= {Vi}i∈I,{φi}i∈I ,(M, g),{ψi}i∈I , εdist(·), where: (i) {Vi}i∈I and {φi:Xi→Vi}i∈I are feature spaces and encoders as in a TSR. (ii) ( M, g )is a d -dimensional smooth Riemannian manifold of semantic states with geodesic distance dg. (iii) For each i∈ I , ψi : Vi→ M is a smooth parameterization whose image Ui := ψi ( Vi )is an open subset of M. We require that the inverse maps χi:= ψ−1 i:Ui→Vi exist and are smooth; the family { ( Ui, χi ) }i∈I forms a (possibly partial) chart atlas on the portion of Mwe actually use. (iv) There exists a latent state map F:I→ M such that, whenever πi(u)is defined, F(u) = ψiφi(πi(u)). This ensures that, on ideal data, all modalities agree on the same manifold point for a given latent entity. (v) εdist : (0 ,∞ ) → [0 ,∞ )is a metric distortion profile such that, for any chart ( Ui, χi ), any two points za, zb∈Uiwith dg(za, zb)>0, and chart distance δi(za, zb) :=  χi(za)−χi(zb) Vi, we have  δi(za, zb) dg(za, zb)−1≤εdistdg(za, zb).(2) This εdist measures representation error of the charts χirelative to the underlying metric g. We write MSR(S)for the collection of all MSRs that realize S. Examples (MSRs). • HERALD-style geometry. ( M, g )is an antigenic manifold; Vi is the latent space of a sequence encoder; ψi is the learned map into M ; εdist captures geodesic–Euclidean distortion along antigenic trajectories. • VIDAR-style identity manifold. ( M, g )is the hypersphere Sd−1 with its standard metric; Vvideo is the pre-normalized ArcFace embedding space; ψvideo is ℓ2 -normalization; εdist is small in 9 objects (they are pseudo-metrics rather than metrics). They deliberately ignore implementation details that do not affect geometric behavior on in-regime entities. 3.4 ε-equivalence of realization categories We next formalize the sense in which SamTrans ( S )and SamGeom ( S )will be treated as approximately equivalent. The key idea is that we do not expect strict categorical equivalence, but rather an equivalence up to small slack in the error gauges. Definition 11 (ε-equivalence of categories of realizations).Let (C,∆C)and (D,∆D)be categories equipped with object-level pseudo-metrics (error gauges) on Ob ( C )and Ob ( D ), respectively. Let C0⊆ C and D0⊆ D be full subcategories. We say that C0and D0are ε-equivalent if there exist functors F:C0→ D0, G :D0→ C0, and non-negative slack functions CF, CGsuch that: 1. For every object C∈Ob(C0), ∆CG(F(C)), C≤CG(εC), where εCcollects the relevant distortion parameters of C(e.g., εT trans,εF(T) dist , chart slack, etc.). 2. For every object D∈Ob(D0), ∆DF(G(D)), D≤CF(εD), with εDdefined analogously. 3. (First-order property.) The slack functions are first order in the distortion parameters: when all distortion and slack terms (e.g., εtrans , εdist , chart-overlap slack δchart , translator asymmetry δasymm) tend to zero, CFand CGalso tend to zero. In our setting, C0 and D0 will be well-behaved subcategories SamTrans0 ( S ) ⊆SamTrans ( S )and SamGeom0 ( S ) ⊆SamGeom ( S )on which smooth chartability and bounded-distortion conditions hold. The functors F and G will be the forward and reverse constructions FS and GS that move between translators and manifolds. 3.5 Structural equivalence theorem and implications We are now ready to state the main structural theorem of this section in informal form. It says that, under a smooth chartability assumption, the TSR and MSR design spaces for a fixed sameness structure Sare ε-equivalent in the sense of Definition 11. Theorem 1 (Realization categories are ε -equivalent (informal)).Fix a semantic sameness structure Sand an in-regime latent subset Ωideal ⊆I. There exist: •well-behaved subcategories SamTrans0(S)⊆SamTrans(S),SamGeom0(S)⊆SamGeom(S), 16 consisting of TSRs and MSRs that satisfy smooth chartability and bounded-distortion conditions; •functors FS:SamTrans0(S)→SamGeom0(S), GS:SamGeom0(S)→SamTrans0(S), that: 1. preserve the underlying sameness structure Sand feature spaces {Vi, φi}i∈I; 2. send translators to manifolds by quotienting and gluing charts (Section 4); 3. send manifolds to translators by extracting chart transitions (Section 5); • slack functions CF, CG which are first order in the distortion and chartability parameters (translator drift εtrans , metric distortion εdist , chart-overlap slack δchart , and translator asymmetry δasymm), such that: ∆SamTransT, GS(FS(T))≤CGεT trans, εFS(T) dist , δchart(T, FS(T)), δasymm(T)for all T∈SamTrans0(S), ∆SamGeomM, FS(GS(M))≤CFεM dist, εGS(M) trans , δchart(GS(M),M), δasymm(GS(M))for all M∈SamGeom0(S). In particular, as all distortion and slack terms tend to zero, the round-trip errors in the gauges ∆ SamTrans and ∆ SamGeom tend to zero: SamTrans0 ( S )and SamGeom0 ( S )are ε -equivalent categories of realizations of S. The detailed constructions of FS and GS and the proof of Theorem 1 are given in Sections 4–5. At a conceptual level: • For a fixed sameness structure S , TSRs and MSRs are two different realizations of the same problem: SamTrans0 ( S )and SamGeom0 ( S )are two design spaces for the same semantic sameness structure. • The functors FS and GS witness an ε -equivalence between these design spaces, showing that— under smooth chartability—they are dual up to controlled slack. An engineer who starts from only S can choose either a translator-based or manifold-based architecture, with the assurance that the theoretical guarantees of the resulting detector are comparable. • Because the equivalence is functorial and comes with explicit slack bounds, theorems and tools transfer: risk bounds or stability results proven in the manifold view can be pulled back to translator systems via GS , and translator-side diagnostics (e.g., spectral audits on format graphs) can be pushed forward to furnish new manifold-side checks via FS . This transfer principle underlies the Error Budget Transfer Theorem in Section 6. 4 Smooth chartability and the functor FS Fix a semantic sameness structure S and the associated categories SamTrans ( S )and SamGeom ( S ) from Section . Throughout this section, we work with ideal latent entities u∈ Ω ideal ⊆I and their encoder outputs vi ( u ) := φi ( πi ( u )) ∈Vi as in Section 2. We write ∥·∥ for the natural inner-product norm in the relevant feature or ambient space. 17 Our goal is to construct a functor FS:SamTrans0(S)−→ SamGeom0(S) on a well-behaved subcategory SamTrans0 ( S ) ⊆SamTrans ( S ), sending a translation-based realization (TSR) Tto a manifold-based realization (MSR) FS (T), and a morphism R :T → T ′ to a morphism FS ( R ) : FS (T) →FS (T ′ ). Crucially, FS must be constructed from the TSR data, not defined by fiat. We proceed in three steps: 1. We build a canonical translator metric quotient (f MT,˜ dT)from any TSR T(Section 4.1). 2. We define an intrinsic notion of smooth chartability for TSRs that depends only on Titself and its latent coverage (Section 4.2), and use it to carve out the subcategory SamTrans0(S). 3. We impose a single, explicit assumption of Riemannian realizability for these metric spaces and use it to define FSon objects and morphisms (Section 4.3). This isolates the genuinely geometric hypothesis—that the translator-induced metric space is quasi-isometric to a smooth manifold— from the intrinsic regularity properties of the TSR itself. 4.1 The translator metric quotient Let T∈SamTrans(S)be a TSR with components T=S, {VT i}i∈I,{φT i}i∈I,{TT ij }(i,j)∈ET, GT, εT trans(·), as in Definition 2. For each modality i∈ I, recall the in-regime feature subset Ωreg i(T) := vi(u) = φT i(πi(u)) : u∈Ωideal, πi(u)defined⊆VT i, and define the disjoint union XT:= G i∈I Ωreg i(T), x = (i, vi)∈XT. Local edge costs. We endow XTwith two families of elementary edges: • Within-chart edges. For any i∈ I and any vi, v′ i∈ Ω reg i (T)we introduce an edge e : ( i, vi ) → ( i, v′ i ) with cost cintra(e):=∥vi−v′ i∥. • Translator edges. For any ( i, j ) ∈ET and any vi∈ Ω reg i (T)such that TT ij ( vi )lies in Ω reg j (T), we introduce an edge e: (i, vi)−→ (j, TT ij (vi)) with cost ctrans(e) :=  TT ij (vi) . (Any consistent bounded positive edge-weighting based on the translator action would suffice; the specific choice is not essential for the subsequent theory.) 18 Path pseudo-metric on XT .Afinite edge-path in XT is a sequence p = ( e1, . . . , em )of composable edges. We define its cost len(p) := m X r=1 c(er), where c ( e )is cintra ( e )or ctrans ( e )depending on the edge type. For any two points x, y ∈XT , we define the path pseudo-metric ˜ draw T(x, y) := inflen(p):pis a finite edge-path from xto y. Standard arguments show that ˜ draw T is a pseudo-metric: it is symmetric, nonnegative, and satisfies the triangle inequality, but may assign zero distance to distinct points. Definition 12 (Translator metric quotient).Define an equivalence relation ∼0on XTby x∼0y⇐⇒ ˜ draw T(x, y)=0. The translator metric quotient of Tis the metric space f MT,˜ dT obtained as the quotient XT/∼0with metric ˜ dT([x],[y]) := ˜ draw T(x, y), which is well-defined and satisfies the metric axioms. For each modality i∈ I there is a canonical quotient parameterization ˜ ψT i: Ωreg i(T)→f MT,˜ ψT i(vi) := [(i, vi)]. Intuitively, f MT is the “patchwork quilt” you obtain by gluing together the in-regime feature charts along the translator edges, with ˜ dT measuring the shortest-path cost induced by chart distances and translator hops. The construction is canonical (up to isometry) and depends only on the TSR T. 4.2 Intrinsic smooth chartability The previous construction associates a metric space ( f MT,˜ dT )to any TSR T. However, for an arbitrary Tthis space may be pathological: it need not resemble a manifold at any scale. We now isolate a set of purely intrinsic conditions on Tunder which the translator metric quotient behaves like a well-controlled, finite-dimensional geometric object. Definition 13 (Intrinsic smooth chartability).A TSR Tis intrinsically smoothly chartable if the following conditions hold on an in-regime subset Ωideal and the associated feature sets Ωreg i(T): 1. Translator regularity. For each ( i, j ) ∈ET , the translator TT ij : VT i→VT j is C2 on an open 19 neighbourhood of Ωreg i(T), with uniformly bounded Jacobian and inverse Jacobian: sup vi∈Ωreg i(T) DTT ij (vi) ≤CJac,sup vi∈Ωreg i(T) DTT ij (vi)−1 ≤CJac. 2. Round-trip asymmetry bound. There exists δT asymm <∞ such that for every ( i, j ) ∈ET and every vi∈Ωreg i(T)with TT ij (vi)∈Ωreg j(T)and TT ji (TT ij (vi)) ∈Ωreg i(T)we have  TT ji (TT ij (vi)) −vi ≤δT asymm. 3. Cocycle bound. There exists δT cocycle <∞ such that for any triple ( i, j, k )with ( i, j ) , ( j, k ) , ( i, k ) ∈ ETand any vi∈Ωreg i(T)lying in the domain of all three compositions,  TT jk(TT ij (vi)) −TT ik (vi) ≤δT cocycle. This ensures that multi-hop translator paths are consistent, up to a controlled slack, with direct translations. 4. Latent coverage and drift control. There is an operational horizon L⋆>0such that: • every benign latent path γ∈ PS ( L⋆ )projects to in-regime observations πi ( γ ( t )) and hence to in-regime features vi(γ(t)) = φT i(πi(γ(t))); • the drift profile εT trans ( k )is finite for all k and satisfies the per-time-step bound of Definition 2 on these paths. Definition 14 (Well-behaved subcategory SamTrans0 ( S )).We define SamTrans0 ( S )as the full subcategory of SamTrans ( S )whose objects are intrinsically smoothly chartable TSRs in the sense of Definition 13, and whose morphisms are exactly the morphisms of SamTrans ( S )between such objects. By construction, membership in SamTrans0 ( S )is a property of the TSR Tand its translators alone. No manifold is mentioned in Definition 13: it is a falsifiable hypothesis about the regularity, consistency, and coverage of the translators. 4.3 Riemannian realizability of the translator metric The translator metric quotient ( f MT,˜ dT )from Definition 12 is a canonical metric space built from Talone. Intrinsic chartability ensures that this space behaves like a finite-dimensional, locally well-behaved metric space: translators are smooth with bounded Jacobians, roundtrip and cocycle inconsistencies are small, and latent data populate a connected, bounded region. To bridge from metric geometry to Riemannian geometry we isolate the following explicit assumption, which is the only place where we posit the existence of a smooth manifold “shadow.” Assumption 1 (Riemannian realizability of translator metric).For every T ∈SamTrans0 ( S ), the translator metric quotient ( f MT,˜ dT )admits a Riemannian realization: there exists a smooth, finite-dimensional Riemannian manifold (MT, gT)and a quasi-isometry qT:f MT,˜ dT−→ MT, dgT 20 with distortion bounded by a function εT realize ( · )that depends continuously and monotonically on the intrinsic slack parameters εT trans(·),δT asymm and δT cocycle. Moreover, we assume that qT is coarse-biLipschitz on the in-regime region populated by Ω ideal , so that for all x, y in this region dgTqT(x), qT(y) ˜ dT(x, y)−1≤εT realize˜ dT(x, y). We do not attempt to prove Assumption 1 in this work; it is our single, explicit geometric hypothesis. In practice, it is an empirical question whether a given TSR Tsatisfies this assumption, and Section 8 discusses audit procedures for probing Riemannian realizability. 4.4 Constructing FSon objects With the metric quotient and realizability assumption in hand, we can now define FS on objects in a non-circular way. Definition 15 (Functor FS on objects).Let T ∈SamTrans0 ( S )be an intrinsically smoothly chartable TSR. Consider its translator metric quotient ( f MT,˜ dT )and a Riemannian realization ( MT, gT ) with quasi-isometry qT given by Assumption 1. For each modality i∈ I , define the parameterization ψT i:VT i⊇Ωreg i(T)−→ MT, ψT i(vi) := qT˜ ψT i(vi), where ˜ ψT iis the quotient map from Definition 12. We define FS(T) := MT:= S, {VT i}i∈I,{φT i}i∈I,(MT, gT),{ψT i}i∈I, εT dist(·), where the distortion profile εT dist is defined by pulling back the quasi-isometry distortion: δT i(za, zb) dgT(za, zb)−1≤εT distdgT(za, zb), za, zb∈ψT i(Ωreg i(T)), with chart distances δT i ( za, zb ) := ∥χT i ( za ) −χT i ( zb ) ∥ and chart maps χT i := ( ψT i ) −1 on their images. By construction, M T is an MSR of S in the sense of Definition 3, and we have FS (T) ∈ SamGeom0(S). Importantly, FS (T)is now determined (up to quasi-isometry) by Titself: the only freedom lies in the choice of Riemannian realization ( MT, gT )of the canonical metric quotient. Different choices of qT give rise to MSRs that are close in the error gauge ∆ SamGeom of Definition 10, and Theorem 1 will show that this slack is first-order in the intrinsic errors of T. 4.5 Constructing FSon morphisms We next define FS on morphisms. Let R :T → T ′ be a morphism in SamTrans0 ( S )as in Definition 8, with reparameterizations Ri : VT i→VT′ i satisfying the encoderand translator-compatibility bounds on ideal data. 21 The TSR morphism Rinduces a map between the translator metric quotients, ˜ hR:f MT−→ f MT′, defined on representatives by ˜ hR˜ ψT i(vi):= ˜ ψT′ iRi(vi), and extended by continuity. The compatibility conditions in Definition 8 ensure that ˜ hR is welldefined up to the metric identification ∼0 and is Lipschitz (with constants depending on the morphism error tolerances (ηenc, ηtrans)). Using the quasi-isometries qT and qT′ from Assumption 1, we then define the manifold-level map hR:MT−→ MT′, hR:= qT′◦˜ hR◦(qT)−1, understanding (qT)−1as a coarse inverse on the in-regime region. Definition 16 (Functor FS on morphisms).For a morphism R :T → T ′ in SamTrans0 ( S ), we define FS(R):=HR:= hR, R:MT−→ MT′, where MT=FS(T)and MT′=FS(T′), and hRis the manifold map defined above. The encoder-compatibility condition for HR,  RiφMT i(πi(u))−φMT′ i(πi(u)) ≤η′ enc, is inherited from R, while the chart-compatibility condition,  RiφMT i(πi(u))−χMT′ ihR(zT(u)) ≤η′ chart, with zT ( u )the latent state in MT , follows from the construction of hR and the quasi-isometry bounds. Thus HRis a morphism in SamGeom(S)in the sense of Definition 9. Proposition 1 (Functoriality and stability of FS ).The assignment FS of Definitions 15 and 16 defines a functor FS:SamTrans0(S)−→ SamGeom0(S) satisfying: 1. For every object T∈SamTrans0(S),FS(idT) = idFS(T). 2. For any composable morphisms R:T→T′and R′:T′→T′′ in SamTrans0(S), FS(R′◦R) = FS(R′)◦FS(R). 3. There exists a universal, increasing function CF such that for any T , T ′∈SamTrans0 ( S )the error gauge satisfies ∆SamGeomFS(T), FS(T′)≤CF∆SamTrans(T,T′), δchart(T), δchart(T′), where δchart (T)is the chart-overlap slack of Tfrom Definition 5. Moreover, CF is first-order in 22 its arguments: it vanishes as ∆SamTrans →0and δchart →0. Proof sketch. Functoriality of FS on objects and morphisms follows from the definitions of the translator metric quotient and the functoriality of the quasi-isometry construction in Assumption 1. The stability bound in (iii) leverages the fact that the distortion profile εT dist is controlled by the intrinsic errors of Tvia the quasi-isometry, and that the error gauge ∆ SamGeom is built from these distortion profiles and chart disagreements. Full details are deferred to Section 6, where we prove the stronger Error Budget Transfer Theorem. 4.6 Discussion: what FSreally does The functor FS no longer hides a manifold inside the definition of smooth chartability. Instead, it proceeds in two conceptually clean stages: • From flat maps to a quilt. For any intrinsically well-behaved TSR T, we canonically construct the translator metric quotient ( f MT,˜ dT )that encodes how features in different modalities can be transported and compared via translators. This is the “patchwork quilt” of all flat maps stitched together along their edges, with a well-defined metric. • From the quilt to the globe. The single geometric hypothesis of Riemannian realizability asserts that this quilt is quasi-isometric to some smooth manifold ( MT, gT ). The functor FS then declares this manifold, together with the induced parameterizations ψT i , to be the MSR FS(T). In this way, FS is a genuine construction rather than a renaming: it builds a metric geometry from the translators, and only then—under an explicit, isolated assumption—passes to a smooth Riemannian geometry. The error gauge stability in Proposition 1 will feed directly into the Error Budget Transfer Theorem in Section 6, where we show that detection guarantees can be transported across this functor with only first-order slack. 5 From manifolds back to translators: the functor GS Section 4 constructed a functor FS : SamTrans0 ( S ) →SamGeom0 ( S )by starting from a smoothly chartable TSR T, forming its translator metric quotient ( f MT,˜ dT ), and—under a Riemannian realizability assumption—obtaining a manifold-based realization FS (T)whose charts approximate the original translators. In this section we build the reverse functor GS:SamGeom0(S)→SamTrans0(S), which takes a well-parameterized manifold-based realization Mand extracts a canonical translatorbased realization GS (M). At the level of the Globe–Flat Maps analogy, FS turns a consistent atlas of flat maps into a globe; GS takes a globe with well-behaved charts and recovers a translator graph encoding the chart transitions. Throughout this section we fix a semantic sameness structure S and work within the realization categories SamGeom ( S )and SamTrans ( S )from Section . Norms ∥·∥ denote the natural inner-product norms on the relevant feature spaces, as in Section . 23 5.1 Geometrically well-parameterized MSRs We first specify which manifold-based realizations admit a stable translation view. Intuitively, an MSR is geometrically well-parameterized when its chart maps behave like well-conditioned coordinate systems on the region traced out by benign latent paths, with controlled distortion and overlap. Definition 17 (Geometrically well-parameterized MSR).Let Mbe an MSR of S in the sense of Definition 3: M=S, {VM i}i∈I,{φM i}i∈I,(MM, gM),{ψM i}i∈I, εM dist(·). Write UM i:= ψM i(VM i)and χM i:= (ψM i)−1:UM i→VM i, and define zM(u)as in Definition 4. We say that Mis geometrically well-parameterized if the following conditions hold. (i) Shared feature model. For each modality i∈ I the feature spaces and encoders agree (up to isometry) with those used in the TSR view: there exist inner-product-preserving isomorphisms Qi : VM i→Vi such that, after identifying VM i with Vi via Qi , we may write φM i = φi and use the common notation Vi,φi. (ii) Chart regularity and bounded distortion. Each parameterization ψi : Vi→ MM is C2 on an in-regime region Vreg i⊆Vi that contains the image of the benign observation set under φi◦πi . The corresponding chart maps χi : Ui→Vi with Ui := ψi ( Vreg i )satisfy the distortion profile of Definition 3: for all za, zb∈Uiwith dgM(za, zb)>0,  δi(za, zb) dgM(za, zb)−1≤εM distdgM(za, zb), δi(za, zb) :=  χi(za)−χi(zb) Vi.(9) (iii) Chart overlap control. For any i, j with Ui∩Uj = ∅ and any z∈Ui∩Uj , the local chart transition χj◦ψi : Vreg i→Vj is well-defined on a neighborhood of χi ( z )and has uniformly bounded Jacobian and inverse Jacobian there. Equivalently, the family {χi} forms a uniformly well-conditioned atlas on the in-regime region of MM. (iv) Benign path coverage. There exists an operational horizon L⋆ such that for every benign latent path γ∈ PS ( L⋆ ), the induced manifold path zM ( · )remains in SiUi and crosses only finitely many chart overlaps. The semantic drift profile σM S(L⋆)from Definition 4 is finite. Condition (i) ensures that the manifold view and the translator view share the same feature model; (ii)–(iii) guarantee that chart coordinates are well-behaved and that chart transitions are controlled; and (iv) ensures that the benign path family never leaves the region where charts are valid. Taken together, these conditions are the geometric mirror of the smooth chartability conditions imposed on TSRs in Section 4. Definition 18 (Well-behaved geometric subcategory).Let SamGeom0 ( S )denote the full subcategory of SamGeom ( S )whose objects are geometrically well-parameterized MSRs in the sense of Definition 17, and whose morphisms are those of SamGeom ( S )(Definition 9) between such objects. 5.2 Constructing GSon objects We now define GS on objects by “unfolding” a well-parameterized manifold back into a TSR whose translators are given by chart transitions. 24 Definition 19 (Functor GS on objects).Let M ∈SamGeom0 ( S )with data as in Definition 17. We define the TSR GS(M) =: TMby TM=S, {VM i}i∈I,{φM i}i∈I,{TM ij }(i,j)∈EM, GTM, εM trans(·), where: (i) Feature spaces and encoders. We reuse the feature spaces and encoders of M: VM i and φM ibecome the feature spaces and encoders of TM. (ii) Translator graph. The vertex set of GTM is I , and we include a directed edge ( i, j )whenever Ui∩Uj is non-empty on the in-regime region traced by benign paths. Thus translators exist exactly where chart overlaps do. (iii) Chart-transition translators. For each edge ( i, j ) ∈EM we define the translator TM ij : VM i→VM jon the in-regime domain Vreg iby TM ij (vi):=χM jψM i(vi)whenever ψM i(vi)∈Ui∩Uj.(10) On points not in Vreg i or not mapping into Ui∩Uj we leave TM ij undefined; the TSR semantics restrict attention to in-regime usage as in Definition 2. (iv) Drift profile. We define the drift profile εM trans ( k )for k -hop translator paths by composing the one-hop distortions induced by the chart maps. Informally, for k = 1 and an in-regime latent entity uwith zM(u)∈Ui∩Ujand vi:= χM i(zM(u)),  TM ij (vi)−χM j(zM(u)) = 0, and departures from this ideal arise only through observational and encoder noise (captured in Elink ) and chart distortion (Equation (9) ). For general k , we define εM trans ( k )as a nondecreasing function of k whose leading-order term is a linear accumulation of the one-hop distortion contributions; Section 6 makes this dependence explicit and shows that under the bounded-distortion regime of Definition 3, εM trans ( k )is first-order in the distortion profile εM dist . By construction, GS (M)is a TSR of S in the sense of Definition 2. Intuitively, GS forgets the manifold and keeps only the chart-transition structure, turning the Globe back into a translation graph whose edges represent coordinate changes between overlapping charts. 5.3 Constructing GSon morphisms We now define GS on morphisms by discarding the manifold map and retaining only the feature-space reparameterizations. Definition 20 (Functor GS on morphisms).Let H :M → M ′ be a morphism in SamGeom0 ( S )in the sense of Definition 9, with H=h, R, h :MM→ MM′, R ={Ri:VM i→VM′ i}i∈I. We define GS(H):=R:GS(M)→GS(M′), 25 By realizability, Γ F real (T) <∞ and, for chartable TSRs, it is controlled by the intrinsic translator quantities: ΓF real(T)≲a1sup k≥1 εT trans(k)+a2δasymm(T), for constants a1, a2depending only on the ambient architecture and the chosen path horizon. Dually, for M ∈SamGeom0 ( S )with chart family {χM i} and distortion profile εM dist ( · ), we define ΓG real(M) := sup z,z′∈MM  δi(z, z′) dgM(z, z′)−1,(24) where δi ( z, z′ ) := ∥χM i ( z ) −χM i ( z′ ) ∥Vi on any chart Ui containing z, z′ . By Definition ?? , this is bounded by the distortion profile: ΓG real(M)≲sup ℓ>0 εM dist(ℓ). Both slacks are first-order: whenever the intrinsic translator errors (for T) or chart distortions (for M) tend to zero, the corresponding ΓF real or ΓG real tends to zero. We can now define the gauge-level slack functions that appear in the functorial stability theorem. Definition 23 (Gauge-stability slack functions).Let ∆ SamTrans and ∆ SamGeom be the error gauges on SamTrans0(S)and SamGeom0(S)from Definition 10. We define: CF(T,T′):=αF∆SamTrans(T,T′)+βFΓF real(T)+ΓF real(T′),(25) CG(M,M′):=αG∆SamGeom(M,M′)+βGΓG real(M)+ΓG real(M′),(26) for fixed constants αF, βF, αG, βG> 0that depend only on the ambient architectures and on the choice of path horizon and operational region, not on particular realizations S,T, or M. By construction these are first-order in the gauges and slacks: if ∆SamTrans(T,T′)→0,ΓF real(T),ΓF real(T′)→0, then CF(T,T′)→0, and likewise for CGwith ∆SamGeom and ΓG real. 7.2 Main structural theorem: gauge stability of FSand GS We now prove the main structural result: the functors FS and GS are Lipschitz with respect to the error gauges, with slack functions CF and CG from Definition 23. This theorem is the “machine” that all error-budget transfer statements in Section 6.1 rest on. Theorem 4 (Gauge stability of FS and GS ).Fix a sameness structure S and realizability constants as above. 1. Functorial stability of FS .For any T , T ′∈SamTrans0 ( S ), let M:= FS (T)and M ′ := FS (T ′ ) be the manifold-based realizations constructed via the translator metric quotient and Riemannian realizability. Then ∆SamGeomM,M′≤CFT,T′,(27) 32 with CF as in (25) . In particular, if ∆ SamTrans (T , T ′ ) → 0and the realizability slacks of T , T ′ tend to zero, then ∆SamGeom(M,M′)→0. 2. Functorial stability of GS .For any M , M ′∈SamGeom0 ( S ), let T:= GS (M)and T ′ := GS(M′)be the translator-based realizations constructed in Section 5. Then ∆SamTransT,T′≤CGM,M′,(28) with CG as in (26) . In particular, if ∆ SamGeom (M , M ′ ) → 0and the realizability slacks of M , M ′ tend to zero, then ∆SamTrans(T,T′)→0. Proof sketch. We sketch the argument for FS ; the case of GS is dual with translator/metric roles switched. Let T , T ′∈SamTrans0 ( S ), and let M= FS (T),M ′ = FS (T ′ )be obtained by: (i) forming translator metric quotients ˜ MT,˜ dT and ˜ MT′,˜ dT′ as in Definition 12; (ii) invoking realizability to obtain quasi-isometries qT : ˜ MT→ MT , qT′ : ˜ MT′→ MT′ ; and (iii) defining charts ψT i, ψT′ i by composing the canonical quotient maps with qT, qT′. Profile part. The distortion profiles εM dist and εM′ dist compare chart distances δi to geodesic distances dgT and dgT′ . For any pair of latent entities u, u′ in the in-regime set, we compare the corresponding manifold distances via the chain dgT(z(u), z(u′)) −dgT′(z′(u), z′(u′))≤A+B+C, where A=dgT(z, z′)−˜ dT([v],[v′]), B=˜ dT([v],[v′]) −˜ dT′([v′],[v′′]), C=˜ dT′([v′],[v′′]) −dgT′(z′, z′′). Here [ v ] , [ v′ ] , [ v′′ ]are the equivalence classes of feature vectors encoding u, u′ in the translator quotients; the exact indexing is not important. Terms Aand Care controlled by the realizability slacks: by definition of ΓF real, A≲ΓF real(T), C ≲ΓF real(T′). Term B is controlled by the translator gauge ∆ SamTrans (T , T ′ ): the quotient metric ˜ dT is defined as an infimum over path costs built from the translators TT ij , and likewise for ˜ dT′ and TT′ ij . By construction of ∆ SamTrans , the per-hop deviations ∥TT ij ( v ) −TT′ ij ( v′ ) ∥ are bounded by ∆ SamTrans (T , T ′ ) on in-regime latent points, and a standard path-comparison argument in metric geometry then yields B≲∆SamTrans(T,T′). Combining the three terms and taking suprema over in-regime latent pairs shows that the profile component of ∆SamGeom(M,M′)is bounded by a constant multiple of CF(T,T′). Operational part. The operational component of ∆ SamGeom compares the induced manifold distances and chart coordinates along benign paths, again over in-regime latent entities u . The 33 same triangle-inequality decomposition |dgT−dgT′| ≤ |dgT−˜ dT|+|˜ dT−˜ dT′|+|˜ dT′−dgT′| controls these discrepancies uniformly over the operational region, with the first and last terms again bounded by Γ F real and the middle term by ∆ SamTrans . This yields the desired bound for the operational piece and thus for ∆SamGeom(M,M′)as a whole. Collecting constants into αF, βF produces (27) . The argument for GS follows the same template with chart distances and manifold distances swapped and ΓG real in place of ΓF real, yielding (28). 7.3 Error-budget transfer as corollaries We now explain how the main error-budget transfer results in Section 6 (Theorems 2 and 3) follow from the gauge-stability theorem 4 together with the Lipschitz dependence of detector error budgets on the gauges. Recall that Section 6 introduces aligned detectors DR for R ∈SamTrans0 ( S )or SamGeom0 ( S ), each with an error budget ER geom, ER link, ξR, ζR, δR indep, and an induced lower bound on correctness for high-risk alerts LB(R)of the Davis form. The key analytic input from Section 6 is that these budgets are Lipschitz in the gauges: there exist functions ΛSamTrans and ΛSamGeom such that, for any pair of realizations T,T′and M,M′, ET geom −ET′ geom≤ΛSamTrans∆SamTrans(T,T′),and likewise for Elink, ξ, ζ, δindep,(29) EM geom −EM′ geom≤ΛSamGeom∆SamGeom(M,M′),(30) with ΛSamTrans,ΛSamGeom first-order at the origin. We now combine these Lipschitz properties with Theorem 4. Corollary 2 (Error-budget equivalence along FS ).Let T , T ′∈SamTrans0 ( S )and M= FS (T), M ′ = FS (T ′ ). Let DT and DT′ be aligned detectors on Tand T ′ with Davis-style error budgets and lower bounds LB (T), LB (T ′ ), and let DM , DM′ be their aligned MSR detectors with bounds LB(M),LB(M′). Then there exists a first-order function Γ det F (depending only on the Lipschitz moduli in (29) – (30) and on the constants in CF) such that LB(M)−LB(M′)≤Γdet F∆SamTrans(T,T′),ΓF real(T),ΓF real(T′), and similarly with (M,M′)and (T,T′)reversed: LB(T)−LB(T′)≤Γdet F∆SamTrans(T,T′),ΓF real(T),ΓF real(T′). In particular, if ∆ SamTrans (T , T ′ )and the realizability slacks of T , T ′ tend to zero, then all four lower bounds LB(T),LB(T′),LB(M),LB(M′)converge together. Proof. Apply the Lipschitz property (30) to the pair (M , M ′ )and then use the gauge-stability 34 bound (27): LB(M)−LB(M′)≲ΛSamGeom∆SamGeom(M,M′)≤ΛSamGeomCF(T,T′). Since CF is first-order in ∆ SamTrans and Γ F real , and Λ SamGeom is first-order at the origin, the composite Γdet F:= ΛSamGeom ◦CFhas the claimed first-order dependence. The same reasoning applied directly to (T , T ′ )via (29) bounds |LB (T) −LB (T ′ ) | by a (possibly different but comparable) first-order function of ∆ SamTrans and Γ F real ; we absorb both into the single Γdet F. Corollary 3 (Error-budget equivalence along GS ).Let M , M ′∈SamGeom0 ( S )and T= GS (M), T ′ = GS (M ′ ). With aligned detectors and Davis-style error budgets as in Corollary 2, there exists a first-order function Γdet Gsuch that LB(M)−LB(M′),LB(T)−LB(T′)≤Γdet G∆SamGeom(M,M′),ΓG real(M),ΓG real(M′). In particular, if ∆ SamGeom (M , M ′ )and the realizability slacks of M , M ′ tend to zero, all four lower bounds converge together. Proof. Identical in structure to Corollary 2, using (28) and the Lipschitz property (29) for TSR error budgets. Together, Theorem 4 and Corollaries 2–3 formalize the idea that the TSR and MSR descriptions of a given sameness structure S are equivalent as detection substrates: small gauge distance between two TSRs implies small gauge distance between their manifold realizations, and vice versa, and aligned detectors on either side inherit this equivalence at the level of error budgets and lower correctness bounds. 8 Operational audits and evaluation protocols This section specifies empirical audits and evaluation protocols for Davis systems. It is deliberately procedural and does not report numerical results, case studies, or deployment performance. The goal is to make the theoretical assumptions of Sections 3–5 falsifiable in practice: each audit targets a specific part of the geometry, separation, or error budget, and is designed to be run on held-out data or prospective streams. Throughout, we fix a Davis system D=(M, g, δ), P(L), c, h, S, A, error budget in some domain, with distortion profile ε ( L ), configuration margins ( κhard, κsoft ), and error-budget components ( Egeom, Elink, ξ, ζ, δindep )as defined earlier. All protocols assume strict train/validation splits and time-asymmetric estimation where applicable (no look-ahead). 8.1 Evaluation principles and pre-registration Before running any audit or reporting any number, we recommend pre-registering: 35 (i) Primary endpoints. Which quantities will be assessed? Examples: (a) distortion tails ε ( L⋆ ) at the proposed operational horizon, (b) effective separation gap κsoft − 2 Rε ( L⋆ ), (c) empirical Cantelli bound tightness, (d) calibration error ξ in a high-risk band, (e) abstention coverage and residual failure ζ. (ii) Path and slice selection. How will benign paths in P ( L )and evaluation slices be sampled (e.g., by time, region, demographic group, capture condition)? Which slices count as indistribution Rvalid? (iii) Thresholds and vacuity criteria. Pre-specify acceptable ranges for distortion (e.g., ε(L⋆)≤ε⋆), error-budget sums Egeom +Elink +ξ+ζ≤τvac, and independence slack δindep. (iv) Baselines and ablations. Define baselines (e.g., flat Euclidean models, models without smoothness regularization, non-geometric detectors), and which components will be ablated (e.g., remove path features, remove auxiliary detectors, remove abstention). (v) Data governance and leakage. Fix time windows for training vs evaluation, forbid re-using evaluation data for tuning, and record configuration hashes for all runs to support auditability. All audits below should be run on held-out validation data (retrospective or prospective) that is not used to train fθ, fit S, or tune the abstention policy A. 8.2 Distortion audits: verifying the bounded-distortion regime These audits target the geometric assumptions behind the distortion profile ε ( L )and the path-horizon choice L⋆(Theorem 1 and the non-vacuity condition κsoft −2Rε(L⋆)>0). Protocol 1: geodesic–Euclidean distortion vs. path length. (1) Sample benign paths. On validation data, construct a set of identity-preserving paths {γm}M m=1 ⊂P ( Lmax )via natural temporal sequences (e.g., time windows, short trajectories) or controlled perturbations. Record their lengths L(γm). (2) Subdivide into segments. For each γm , sample pairs of times 0 ≤s<t≤ 1with arc-length distance dgz(γm(s)), z(γm(t))spanning a grid of target lengths ℓ∈ {ℓ1, . . . , ℓK}up to Lmax. (3) Approximate geodesic distance. For each pair (s, t), approximate dgusing either: • the path length of the interpolated trajectory in the embedding (if g is induced by a pullback metric), or • a discrete geodesic approximation (e.g., graph-based shortest paths) constrained to lie within a local neighborhood. (4) Compute distortion ratios. Let δ be the ambient chord metric (Euclidean distance in the embedding). For each pair, compute the distortion ratio ρ(s, t) = δz(γm(s)), z(γm(t)) dgz(γm(s)), z(γm(t)). Aggregate ρinto bins by geodesic length ℓ. 36 (5) Summarize distortion. For each length bin ℓ , estimate E [ |ρ− 1 | | dg≈ℓ ]and upper quantiles (e.g., 95th/99th percentile). Plot or tabulate these as a “distortion curve” b εgeom(ℓ). (6) Choose L⋆ and validate. Select an operational horizon L⋆ such that b εgeom ( ℓ ) ≤ε⋆ for all ℓ≤L⋆ , with ε⋆ chosen so that the effective soft margin κsoft − 2 Rε⋆ remains positive. Document any slices (e.g., specific conditions or subpopulations) where distortion tails are heavy; these may be excluded from Rvalid or treated as out-of-distribution. Protocol 2: curvature and smoothness along paths. To empirically test the smoothness assumptions used in Section 3: (1) Approximate discrete curvature along each path by the angles between successive segments in the embedding, or by second differences of z(t)in arc-length parameterization. (2) Confirm that curvature statistics (mean, 95th percentile) remain below a pre-registered threshold for paths with length ≤L⋆. (3) Flag regimes where curvature spikes (e.g., hard scene cuts, recombination events, abrupt control changes) as candidates for abstention or separate modeling. These two audits jointly support (T1) in Section 6.2: they test whether the bounded-distortion regime is empirically valid on the intended deployment distribution. 8.3 Configuration and margin audits These audits target the configuration map c , coarse labels h , and the separation margins ( κhard, κsoft ) used in the Cantelli bound and non-vacuity conditions. Protocol 3: separation in geodesic and ambient distance. (1) Collect labeled pairs. On validation data, assemble a set of endpoint pairs ( xa, xb )with coarse labels hc ( z ( xa ))  and hc ( z ( xb ))  , distinguishing “similar” vs. “changed” vs. “ambiguous” configurations. (2) Measure distances. For each pair, compute both the approximate geodesic distance dgz ( xa ) , z ( xb )  and ambient distance δz(xa), z(xb). (3) Estimate margins. For each label pair (e.g., similar–similar, similar–changed, changed–changed), estimate the empirical distribution of distances. From these, estimate: b κhard ≈inf (a,b)cross-config dg(za, zb),b κsoft ≈quantileq(dg(za, zb)) for a pre-registered quantile q(e.g., 5th percentile of cross-config pairs). (4) Check non-vacuity under distortion. Using the distortion estimates from the previous subsection, verify that b κsoft − 2 Rb εgeom ( L⋆ ) > 0on the slices of interest. If this inequality fails or is marginal, treat the corresponding slice as ambiguous and rely on abstention or auxiliary detectors. 37 Protocol 4: stability across slices. Repeat Protocol 3 across pre-registered slices (e.g., demographics, capture conditions, time periods) and compare the estimated margins; large variation may indicate bias in fθ or in the configuration definition. Where gaps appear, consider slice-specific abstention policies or separate retraining. 8.4 Feature–geometry linkage audits These audits target the monotone link ∆ S≥g (∆ g )between configuration movement in geometry and changes in the scalar statistic S. Protocol 5: monotone linkage in bins of geodesic displacement. (1) Define path segments and labels. For each benign path γ∈P ( L⋆ )and time pair ( s, t ), compute the geodesic displacement ∆ g = dgz ( γ ( s )) , z ( γ ( t ))  and the corresponding featurebased change in the statistic, e.g., ∆S=St−Ssor Stalone if Sis cumulative. (2) Bin by geodesic distance. Partition the range of ∆ g into bins B1, . . . , BK on an operational interval [0, gmax]. For each bin, compute: ∆Sk=E[∆S|∆g∈Bk],d Var(∆S|∆g∈Bk). (3) Test monotonicity. Fit a monotone regression b g of ∆Sk on the midpoints of Bk , and measure deviations from monotonicity (e.g., number and magnitude of violations). Pre-register an acceptable tolerance for such violations. (4) Report linkage slack. Summarize a linkage slack term b ηlink capturing how far the empirical relation deviates from an ideal monotone link; this feeds into the linkage component of the error budget Elink. Protocol 6: variance and finite-moment checks. To justify finite-variance Cantelli bounds, estimate second moments of S (or ∆ S ) conditional on configuration changes. If heavy tails are observed, tighten the operational range or incorporate robust statistics (e.g., winsorized S ) and re-audit. 8.5 Calibration, abstention, and error-budget estimation These audits estimate the non-geometric terms ( ξ, ζ )and link theoretical error budgets to empirical behavior. Protocol 7: calibration in the high-risk region. (1) Fit and freeze calibration. Using historical training data, fit a monotone calibration map b π = gcal ( S )from the statistic to an interpretable risk quantity (e.g., probability of class change within horizon T). Freeze gcal before calibration auditing. (2) Evaluate on validation. On held-out or prospective data, compute calibration metrics (Brier score, expected calibration error, reliability curves) on: •the full support of S, and 38 •a high-risk band (e.g., Sabove a pre-registered threshold where actions are taken). (3) Estimate ξ .Define b ξ as the maximum deviation between empirical and nominal probabilities in the high-risk band, or as a pre-registered functional of the calibration curves. This quantity enters the error budget as the calibration term. Protocol 8: abstention behavior and ζ. (1) Record abstentions. Under a candidate abstention policy A , log for each evaluation sample whether the system abstained and the rationale (e.g., large distortion, high OOD score, large predictive interval). (2) Estimate coverage. For each slice, estimate coverage 1 −A (fraction of non-abstained cases) and the conditional error rates among non-abstained decisions (e.g., misclassification rate or violation of a domain-specific safety criterion). (3) Estimate ζ .Define b ζ as the frequency with which the system fails to abstain in regimes where distortion or OOD scores exceed pre-registered safety thresholds. This captures abstention failures in the error budget. Protocol 9: empirical error-budget decomposition. On validation data where ground-truth outcomes are available, label each failure with its dominant source (geometry drift, linkage failure, calibration error, abstention failure, or uncontrollable noise). Estimate empirical frequencies of each error type and compare to the theoretical components ( Egeom, Elink, ξ, ζ ); large discrepancies signal either model misspecification or unmodeled dependencies. 8.6 Independence slack and compositional bounds The compositional dominance bound in Theorem 2 assumes approximate conditional independence of the error sources; when this fails, the theory recommends a conservative union bound. This subsection sketches how to empirically assess the independence slack δindep. Protocol 10: independence diagnostics. (1) Error indicators. For each sample, define Bernoulli indicators Egeom, Elink, Ecal, Eabst for the occurrence of each error type, based on the audits above and domain-specific thresholds. (2) Estimate joint vs. product. For selected pairs or triples of error types (e.g., ( Egeom, Elink ), (Elink, Ecal)), estimate: p12 =P(E1∧E2), p1=P(E1), p2=P(E2), and the deviation from independence ∆12 =p12 −p1p2. (3) Define δindep .Aggregate the magnitudes | ∆ 12| across pairs into an independence slack b δindep (e.g., maximum or suitable norm), and compare the compositional bound using products to the conservative union bound using sums. If b δindep is large, prefer the union bound in practice. 39 8.7 Slicing, OOD monitoring, and lifecycle audits Finally, we outline cross-cutting audits that track geometry and performance over time and across slices. Protocol 11: slice-wise Davis audits. For each pre-registered slice (e.g., region, demographic group, capture condition, hardware profile): (1) Re-run Protocols 1–9 restricted to that slice. (2) Compare distortion curves, margins, linkage slack, calibration error, and abstention behavior across slices. (3) Flag slices where any component of the error budget becomes vacuous (e.g., Egeom + Elink + ξ + ζ > τvac), and treat them as out-of-regime for the current system. Protocol 12: OOD detection and retraining triggers. Monitor distributions of: •distortion ratios ρ(s, t), •path curvature statistics, •feature distributions used in S, •OOD or density scores from auxiliary detectors. If these drift beyond pre-registered control limits, increase abstention, freeze thresholds, and consider retraining fθor redefining P(L)and L⋆before resuming full operation. Section scope note. This section specifies audit protocols and reporting templates for future empirical work with Davis systems. By design, it contains no numerical results, plots, or case studies. In practical deployments, we recommend that these protocols be instantiated as part of a comprehensive system dossier, with results refreshed on a fixed cadence and after major system changes. 9 Case study: a geometry-first multi-modal threat detector The abstract framework of Sections 1.1–7 was developed independently of any particular domain. In parallel, the author built a deployed system—codenamed KRAKEN in a separate, application-focused manuscript—for real-time, multi-modal maritime threat detection. 3 In hindsight, KRAKEN can be understood as a concrete instance of the TSR–MSR equivalence and error-budget decomposition developed in this paper. This section sketches that correspondence. It is not required for the theoretical results, but serves as a sanity check: a complex, safety-critical system built before the formalism was articulated nonetheless fits naturally into the semantic-sameness, realization, and functorial picture. 3 The present section intentionally omits operational details (sensor layouts, campaign locations, adversary tactics) and focuses only on the structural aspects relevant to the TSR–MSR theory: semantic sameness, translator graphs, geometric realizations, and error budgets. 40 9.1 Operational setting (informal) At a high level, the system fuses heterogeneous sensor streams to detect and track subsurface threats and undersea infrastructure hazards in coastal waters. Representative modalities include: • Kinetic channels: distributed acoustic sensing (fiber strain), passive SONAR spectrograms, and related vibration measurements; • Electromagnetic channels: synthetic aperture radar and optical/IR imagery of the sea surface, plus RF emissions from platforms and shore; • Contextual channels: shipping lanes, bathymetry, environmental metadata, and operator annotations. Each modality has its own units, SNR regime, failure modes, and coverage gaps. Operators care about entities: specific submarines, vessels, and patterns of behavior. The core question is semantic: when is a faint acoustic trace, a marginal image artifact, and a weak RF transient evidence of the same latent threat, and when are they unrelated clutter? 9.2 Semantic sameness structure for maritime threats We briefly sketch a semantic sameness structure Smar = ( I, {Xi}i∈I ,{πi}i∈I ,≈,PS ( L )) in the sense of Section 1.1 that captures this setting. • Latent space I .Elements u∈I represent latent maritime threat states at an instant: a tuple ( C, E, Q )of threat class (platform or behavior type), environment (sea state, sound-speed profile, background traffic), and configuration (speed, depth, machinery state, loadout). • Modalities {Xi} .Each i∈ I indexes a sensor family (e.g., DAS fiber segments, SONAR beams, satellite tiles, RF channels). Xi is the space of idealized observations from that family over a short time window. • Rendering maps πi : I→Xi .Given a latent state u = ( C, E, Q ), the map πi ( u )encodes what an ideal, noise-free sensor of modality i would observe: an acoustic spectrum induced by propeller cavitation, a surface wake pattern in SAR, a strain pattern along a buried fiber, an RF emission mask, and so on. In practice, real observations include noise, occlusion, and quantization; as in Section 1.1, we blur notation and treat xi∈Xi as an observation “of” u when it lies within an acceptable noise ball around πi(u). • Sameness relation ≈ .Two observations x∈Xi , x′∈Xj are semantically equivalent, x≈x′ , if they are rendered from the same latent entity within a short temporal tolerance: there exists a path γ∈ PS ( L )and time t such that x≈πi ( γ ( t )), x′≈πj ( γ ( t )). Intuitively, they are evidence about the same contact rather than coincident clutter. • Benign path family PS ( L ).Paths γ : [0 , 1] →I describe physically plausible threat trajectories over an operational horizon L : slow drifts in position and depth, machinery state changes, and routine environmental variation. This family encodes both how fast threats can move in latent space and which directions are benign. KRAKEN was designed without this formal vocabulary, but its informal threat models and scenario design align closely with Smar. 41