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From Quantum Measurement to Gauge Structure: Minimal Response and Profile-Relative Canonicity Michael Tolbert Paper 3 Version 4.0 Abstract Generalized quantum measurements admit many purification and dilation realizations of the same operational state–measurement data. On each finite-dimensional regular fixed-rank stratum, this representation freedom forms a natural gauge structure: framed minimal purification and Naimark data constitute a smooth principal bundle whose compact fibers are exactly the auxiliary gauge orbits. Complete-POVM robustness controls perturbations of states, labeled measurement architectures, fixed coarse events, canonical normalization, and smooth paths. A right-invariant framed Hilbert–Schmidt response geometry assigns every operational tan- gent a unique minimum-response lift. These lifts define a smooth gauge-equivariant principal connectionH fr and a positive gauge-independent response metric on the operational base. The geometric construction establishes transport and relative canonicity independently of any informational identity premise. To address physical uniqueness, the paper introduces Quantum Informational Identity (QII) and Dynamical Quantum Informational Identity as explicit foundational assumptions. The generated carrier, labeled event algebra, and admitted covariance group are defined explicitly from the fixed framed architecture. Exact gauge intertwiners preserve this profile pointwise, and smooth gauge paths preserve its dynamical history. Under Dynamical QII and an explicit profile-closure restriction, all profile-closed admissible principal connections determine the same represented transport content. Relative to the framed Hilbert–Schmidt response geometry,H fr is the unique minimum-response representative within the profile-closed generated realization framework. This scoped conclusion is called profile-relative absolute canonicity; it does not claim that informational identity derives the Hilbert–Schmidt response geometry or excludes extensions with additional observables. Version status. Version 4.0 is the post-review correction and clarification revision. It corrects and completes the fixed-rank tangent-curve realization argument, aligns the quantifier scope of profile closure, adds an explicit failure-of-closure example and a compact dependency guide, removes the residual purification-space/event-algebra notation ambiguity, and sharpens the boundary between profile-relative content uniqueness and the open problem of characterizing the response metric itself. The established principal-bundle, robustness, response-geometry, and minimum-response results are retained. 1 Introduction Quantum measurements are operationally specified by states, labeled effects, and their event assignments, yet the purification and dilation data used to represent those measurements are not unique. This raises two questions: how is the freedom among realization descriptions organized geometrically, and is there a distinguished way to transport realizations along changing operational data? 1

The central claim is that realization freedom forms a natural gauge structure and that its framed geometry selects a unique minimum-response transport. On each regular fixed-rank state–POVM stratum, minimal purification amplitudes and framed Naimark realizations assemble into a smooth principal bundle. Its fibers are exactly the orbits of a compact auxiliary gauge group. Gauge structure therefore arises from the multiplicity of realization descriptions of the same operational measurement data. A right-invariant Hilbert–Schmidt metric separates vertical gauge motion from changes required by an operational tangent. Orthogonal projection selects the unique lift of least framed norm. The resulting horizontal distributionH fr is a smooth principal connection, and its squared horizontal norm descends to a positive response metric on the operational base. This construction is related in spirit to purification-bundle approaches to mixed-state parallel transport, while extending the realization space to include labeled Naimark data and complete measurement architectures [1,2,3]. The geometric result establishes relative canonicity. To address represented physical uniqueness, the paper defines a fixed generated carrier and a label-preserving covariance group, then introduces pointwise QII and Dynamical QII. Dynamical QII yields transport-content identity only within the explicitly profile-closed class of admissible connections. The framed Hilbert–Schmidt geometry selectsH fr as the unique minimum-response representative. The manuscript calls this scoped two-part result profile-relative absolute canonicity. 1.1 Roadmap of the argument The proof architecture is theorem-driven and separates construction from interpretation: 1. Operational domain. Regular fixed-rank state and labeled-POVM strata provide a smooth base with fixed auxiliary dimensions and stable event correspondence. 2.Gauge bundle. Framed purification–dilation data form a smooth principal bundle; each fiber is exactly one compact gauge orbit, and the vertical tangent space is the infinitesimal gauge action. 3.Robustness. Complete-POVM estimates control states, all labeled effects, fixed coarse grainings, canonical normalization, and smooth operational paths. 4.Minimal response. The framed Hilbert–Schmidt metric selects a unique minimum-norm lift of each operational tangent, producing the principal connectionH fr and the operational response metric. 5. Informational identity. Exact gauge transformations preserve the generated informational profile; smooth gauge paths preserve the corresponding dynamical profile. 6. Final theorem. Dynamical QII and explicit profile closure yield unique represented transport content, while the framed response geometry supplies its unique minimum-response represen- tative. These conclusions constitute profile-relative absolute canonicity in the scoped sense adopted here. The paper remains finite dimensional and restricted to regular fixed-rank strata, fixed outcome labels, minimal generated realizations, and the represented informational structures defined below. Rank-changing boundaries, nonregular POVMs, continuous outcomes, and a possible uniqueness derivation of the response geometry itself remain outside the present theorem. 2

2 Operational Parameter Space 2.1 Ambient operator spaces LetH S be a complex Hilbert space of finite dimensiond. LetHerm(H S ) be the real vector space of Hermitian operators on H S , equipped initially with the Hilbert–Schmidt inner product ⟨X,Y⟩ HS = Tr(XY ). Fix a finite labeled outcome set I ={1,...,n}. The labels are part of the operational specification and remain fixed along every admissible path considered in the main theorems. Definition 2.1 (Fixed-rank state stratum). For 1≤ r ≤ d, define D r (H S ) :={ρ∈ Herm(H S ) : ρ≥ 0, Trρ = 1, rankρ = r}. The support projection ofρis denotedp ρ . The full-rank case isr=d, but lower fixed ranks are retained because the minimal purification dimension is then constant on the stratum. Proposition 2.2 (Smooth state stratum). The fixed-rank density-operator setD r (H S ) is a smooth real manifold of dimension dimD r (H S ) = 2dr− r 2 − 1. Its tangent space at ρ is naturally identified with T ρ D r

n X = X † : (I− p ρ )X(I− p ρ ) = 0, TrX = 0 o . Proof sketch. Writeρ=AA † , whereA:C r →H S has full column rank andTr(AA † ) = 1. The unit sphere of full-column-rank amplitudes has real dimension 2dr−1. The free right actionA7→ AUof U(r) has dimensionr 2 and leavesAA † unchanged. The quotient therefore has dimension 2dr−r 2 −1; see, for example, standard quotient models of fixed-rank positive-semidefinite manifolds [4,5,6]. Differentiating a smooth rank-preserving curve gives the displayed support-block and trace conditions. Conversely, factorρ=AA † withA:C r →H S full column rank andTr(AA † ) = 1. For an admissible tangent X, set p := p ρ and define B := 1 2 pXp + (I− p)Xp, ̇ A := BA(A † A) −1 . Since p = A(A † A) −1 A † , one has ̇ AA † = Bp = 1 2 pXp + (I− p)Xp, and, using X = X † , A ̇ A † = pB †

1 2 pXp + pX(I− p). The tangent condition (I− p)X(I− p) = 0 gives the support-block decomposition X = pXp + pX(I− p) + (I− p)Xp. 3

Consequently, ̇ AA †

• A ̇ A † = X. Define A(t) := A + t ̇ A, b ρ(t) := A(t)A(t) † , and ρ(t) := b ρ(t) Tr b ρ(t) . Full column rank is an open condition, soA(t) has rankrfor sufficiently smallt. Henceρ(t) is positive semidefinite of rank r and has trace one. Moreover, d dt     t=0 Tr b ρ(t) = TrX = 0, so normalization does not change the first derivative and ̇ρ(0) =X. Thus every displayed tangent direction is realized by a smooth rank-preserving density-operator curve. 2.2 Fixed-rank POVM strata Choose a rank pattern r = (r 1 ,...,r n ),1≤ r i ≤ d. LetPos r i (H S ) denote the smooth manifold of positive semidefinite Hermitian operators of rankr i . Its real dimension is 2dr i − r 2 i . Consider the smooth summation map Σ : n Y i=1 Pos r i (H S )−→ Herm(H S ),Σ(E 1 ,...,E n ) = n X i=1 E i . The rank pattern alone does not guarantee that the level set Σ −1 (I) is smooth at every point. The manuscript therefore works on the regular locus of the summation constraint. Definition 2.3 (Fixed-rank POVM locus). Define POVM r (H S ) := ( (E 1 ,...,E n ) : E i ≥ 0, rank(E i ) = r i , X i E i = I ) . This locus need not be smooth at every point. Assumption 2.4 (Admissible nonempty rank pattern). Throughout the remainder of the paper, fix a rank patternr for which POVM reg r (H S )̸= ∅. In particular, P i r i ≥ dis necessary because the ranges of the effects must collectively spanH S when P i E i = I. Definition 2.5 (Regular fixed-rank POVM stratum). Define POVM reg r (H S ) := ( (E 1 ,...,E n ) : E i ≥ 0, rank(E i ) = r i , X i E i = I, dΣ E is surjective ) . Here dΣ E (X 1 ,...,X n ) = X i X i . 4

Proposition 2.6 (Smooth POVM stratum). The regular fixed-rank POVM setPOVM reg r (H S ) is a smooth embedded submanifold of Q i Pos r i (H S ). Its dimension is dim POVM reg r (H S ) = n X i=1 (2dr i − r 2 i )− d 2 . At E = (E i ), its tangent space is T E POVM reg r

( (X i ) n i=1 : X i = X † i , (I− p i )X i (I− p i ) = 0, X i X i = 0 ) , where p i is the support projection of E i . Proof. By definition,Iis a regular value of the restricted summation map at every point of POVM reg r (H S ). The regular-value theorem gives a smooth embedded level set of codimension dim Herm(H S ) =d 2 [7]. The tangent space is the kernel ofdΣ E inside the product of the fixed-rank positive tangent spaces. 2.3 The regular operational stratum Definition 2.7 (Regular operational stratum). For fixed d,n,r, andr, define O reg (r,r) :=D r (H S )× POVM reg r (H S ). An operational point is written o = (ρ,E), where E = (E 1 ,...,E n ). Its dimension is dimO reg = (2dr− r 2 − 1) + X i (2dr i − r 2 i )− d 2 . A convenient initial Riemannian metric is the product Hilbert–Schmidt metric ∥(δρ,δE)∥ 2 op =∥δρ∥ 2 HS + X i ∥δE i ∥ 2 HS . The later robustness theory may supplement this smooth metric with trace norm for states and operator norm for effects. The smooth geometry and the perturbation metric need not coincide, but both must be stated explicitly. 2.4 Label preservation and event correspondence Assumption 2.8 (Label-Preservation Hypothesis). Every operational path is a map t7−→ o(t) =

ρ(t),E 1 (t),...,E n (t)
∈O reg with one fixed labeled index setI. No discontinuous permutation or relabeling of outcomes is allowed inside an admissible chart. If a relabeling is physically intended, it must be represented by an explicit permutation action and carried through every event identification. This assumption separates smooth transport from an independent labeling convention. It prevents a path from silently identifying one outcome sector with an unrelated sector merely because both occupy the same numerical position in a list. 5

Definition 2.9 (Fine-event correspondence). For every subset A⊆ I and every t, define E A (t) := X i∈A E i (t). The same label subset A is used at every point of the path. Definition 2.10 (Fixed coarse-graining correspondence). Let Γ ={B 1 ,...,B m } be a fixed partition of I. The associated coarse measurement along the path is E Γ α (t) := X i∈B α E i (t), α = 1,...,m. A coarse event C ⊆{1,...,m} corresponds to the fixed fine-label union S α∈C B α . The main robustness and transport theorems treat Γ as fixed. Time-dependent merging or splitting of outcome blocks is a separate problem and is not included in the minimal paper. Proposition 2.11 (Smooth event correspondence). Ift 7→ E i (t) isC k for every labeli, then t7→ E A (t) and t7→ E Γ α (t) are C k for every fixed fine event A and every fixed coarse block B α . Proof. Finite summation is linear and therefore smooth. 2.5 Boundaries and scope The regular stratum excludes points where the rank ofρor an effect changes, where the regular-value condition fails, or where the chosen outcome correspondence ceases to be meaningful. Such points may require a stratified-space treatment. No global smoothness claim is made across those loci. 2.6 A two-outcome qubit example A simple example illustrates what is operational and what is gauge. LetH S =C 2 , choose a full-rank state ρ =

a0 0 1− a ! ,0 < a < 1, and a labeled two-outcome POVM E 1

p 0 0 q ! , E 2 = I− E 1 ,0 < p,q < 1. Both effects are full rank, and the pair lies in the regular full-rank stratum. One framed realization is obtained from X = ρ 1/2 , W 1 = E 1/2 1 , W 2 = E 1/2 2 , with W = (W 1 ,W 2 ) T . For arbitrary unitaries U,V 1 ,V 2 ∈ U (2), (X,W 1 ,W 2 )7−→ (XU,V † 1 W 1 ,V † 2 W 2 ) changes the auxiliary frames but leavesρ,E 1 , andE 2 unchanged. These descriptions therefore occupy one gauge fiber. 6

Now leta,p, andqvary smoothly with time while remaining inside (0,1). The operational path changes the state and measurement effects, while a realization lift must also choose how the auxiliary frames vary. Adding any smooth time-dependentU(t),V 1 (t),V 2 (t) changes that lift without changing the operational path. The vertical part of this change is gauge motion. The framed minimum-response connection removes the gratuitous vertical component relative tog fr and selects one horizontal lift. QII does not derive that metric; it identifies gauge-related lifted histories as carrying the same represented transport content. This example displays the two logical tasks of the final theorem: informational identity fixes the content class, and minimum response selects its framed representative. 3 Purification–Dilation Realization Space 3.1 Fixed model auxiliary spaces On O reg (r,r), the supported auxiliary dimensions are constant. Fix the model purification space A := C r . Fix the model generated Naimark space K := n M i=1 C r i , m := dimK = X i r i , with canonical coordinate projections P i onto the ith block. 3.2 Framed purification and dilation data The local construction is simplest when auxiliary realizations are represented by amplitudes and frames rather than by varying Hilbert spaces. Definition 3.1 (Purification amplitude). A minimal purification amplitude is a full-column-rank map X :A−→H S satisfying XX † = ρ,Tr(XX † ) = 1. The choice is not unique: X and XU represent the same state for every U ∈ U (r). Definition 3.2 (Amplitude–vector identification). Fix an orthonormal reference basis{|a⟩} r a=1 of A = C r . For an amplitude X :A→H S , define |φ X ⟩ := r X a=1 X|a⟩⊗|a⟩∈H S ⊗A. Then Tr A |φ X ⟩⟨φ X | = XX † . For U ∈ U (r), |φ XU ⟩ = (I H S ⊗ U T )|φ X ⟩, where the transpose is taken in the fixed reference basis. This convention fixes the purifying-factor gauge transformation used below. 7

Definition 3.3 (Framed minimal Naimark realization). ForE= (E i ) withrank(E i ) =r i , a framed minimal Naimark realization is a block isometry W =    W 1 . . . W n    :H S −→K such that W † i W i = E i , X i W † i W i = I. EachW i :H S → C r i has full row rank. The construction is the finite-dimensional labeled form of a minimal Naimark dilation [8, 9]. The fixed coordinate projectors satisfy W † P i W = E i . 3.3 The framed realization space Definition 3.4 (Framed Purification–Dilation Realization Space). LetA r be the normalized full-column-rank amplitude space A r := n X ∈ Hom(A,H S ) : rankX = r, Tr(XX † ) = 1 o , and let W r be the block-rank Stiefel stratum W r := ( (W 1 ,...,W n ) : rankW i = r i , X i W † i W i = I ) . Define the ambient compression map π amb :A r ×W r −→D r (H S )× POVM r (H S ) by π amb (X,W ) :=  XX † ,{W † i W i } n i=1  . The framed Purification–Dilation Realization Space over the regular operational stratum is R reg := π −1 amb (O reg ). The realization projection is the restriction π := π amb   R reg :R reg −→O reg . For o = (ρ,E)∈O reg , the fiber is F o := π −1 (o). Remark 3.5 (Why the regular-image condition is explicit). Fixed block ranks and the normalization P i W † i W i =Ido not by themselves guarantee that the compressed POVM belongs to the regular- value locus of Definition 2.5. The preimage condition in Definition 3.4 makes the codomain ofπ well defined and restricts the realization geometry to the same regular stratum used throughout the paper. This framed realization space provides a fixed finite-dimensional ambient model for the supported realization data of Paper 2. 8

3.4 Unified state and generated projectors GivenR= (X,W) overo= (ρ,E), let|φ X ⟩∈H S ⊗Abe the purification vector associated withX. Define |Ψ R ⟩ := (W ⊗ I A )|φ X ⟩, Q i := P i ⊗ I A . For every label event A⊆ I, set Q A := X i∈A Q i . The inherited refinement-derived typicality is represented by T R (A) :=∥Q A Ψ R ∥ 2 . Paper 2 proves that this value descends to the operational pair and is represented byTr(ρE A ) only after the projective abundance construction has been applied. Paper 3 retains that logical order. 3.5 Exact gauge action Let G := U (r)× n Y i=1 U (r i ). For g = (U,V 1 ,...,V n ), define the right action (X,W 1 ,...,W n )· g := (XU,V † 1 W 1 ,...,V † n W n ). This is a genuine right action: for g,h∈ G, (R· g)· h = R· (gh). It changes the auxiliary purification frame and the block outcome frames while leaving the compressed operational pair unchanged. The adjoints on the Naimark-frame factors reconcile their natural left unitary change of coordinates with the right-action convention used for principal realization geometry. Proposition 3.6 (Gauge invariance of the projection). For every R∈R reg and g ∈ G, π(R· g) = π(R). Proof. Right multiplication gives (XU )(XU ) † = XUU † X † = XX † . For every outcome block, (V † i W i ) † (V † i W i ) = W † i V i V † i W i = W † i W i . Thus the state and every labeled effect are unchanged. Because regularity is a property of the compressed operational point, the action also preserves R reg . Proposition 3.7 (Fibers are exact gauge orbits). LetR= (X,W) andR ′ = (X ′ ,W ′ ) lie inR reg . Then π(R) = π(R ′ ) if and only if there exists g ∈ G such that R ′ = R· g. 9

Proof.One direction follows from Proposition 3.6. Conversely,XX † =X ′ X ′† withX,X ′ of full column rank implies X ′ = XU for some U ∈ U (r). For each i, the identities W † i W i = W ′† i W ′ i and the full row ranks imply, by polar decomposition on the common support, thatW ′ i =L i W i for some L i ∈ U (r i ). Set V i := L † i . Then W ′ i = V † i W i , so R ′ = R· (U,V 1 ,...,V n ). 3.6 Freeness, stabilizers, and quotient Proposition 3.8 (Free framed gauge action). The action of G on R reg is free. Proof.IfXU=XandXhas full column rank, thenU=I r . IfV † i W i =W i andW i has full row rank, thenV † i =I r i and henceV i =I r i . Therefore only the identity element fixes a framed realization. BecauseGis compact, its smooth action is proper [7]. Proposition 3.7 identifies the orbit space R reg /Gset-theoretically withO reg . A smooth principal-bundle conclusion is postponed until the next stage verifies the smooth structure ofR reg , the submersion property ofπ, and smooth local sections. 3.7 Exact gauge invariance of typicality Proposition 3.9 (Vertical invariance of refinement-derived typicality). IfR ′ =R· glies in the same exact gauge orbit and A⊆ I is a fixed label event, then T R ′ (A) = T R (A). Proof. By Definition 3.2, the purification-frame change X 7→ XU is implemented on the purifying factor by U T . The block-frame changes W i 7→ V † i W i combine into the block-diagonal unitary V K := n M i=1 V † i onK. BecauseV K commutes with every coordinate projectorP i , the unitaryV K ⊗ U T onK⊗A maps|Ψ R ⟩to|Ψ R ′ ⟩and intertwines every labeledQ i =P i ⊗ I A , hence everyQ A . Unitary norm preservation gives ∥Q A Ψ R ∥ =∥Q A Ψ R ′ ∥, which proves the claim. This is the realization-space form of the Representation Invariance Theorem of Paper 2. Remark 3.10 (Infinitesimal gauge directions). For the corrected right action, a Lie-algebra element (ξ,η 1 ,...,η n )∈ u(r)⊕ n M i=1 u(r i ) 10

generates the infinitesimal variation δX = Xξ, δW i =−η i W i . These directions are tangent to the exact gauge orbit and will be identified with the vertical tangent space after the local realization geometry establishes that the realization projection is a smooth submersion. 3.8 Lifted event correspondence Definition 3.11 (Lifted event correspondence). For a label-preserving operational patho(t) and a realization lift R(t), the event A⊆ I is represented in the fixed framed model by Q A := X i∈A Q i , Q i = P i ⊗ I A . For a fixed coarse partition Γ ={B α }, define Q Γ α := X i∈B α Q i . These coordinate projectors are independent oftbecauseK,A, and the labeled coordinate projections P i are fixed in the framed realization space. Thus event correspondence is carried by the fixed labels, not by motion of the coordinate projectors. The nontrivial transport problem concerns the amplitudesX(t), the Naimark isometries W(t), the unified states    Ψ R(t) E , and the gauge frames used to compare neighboring realizations. Under changes of framed chart, transition functions preserve the labeled projector decomposition. Assumption 3.12 (Event-Compatible Lift). An admissible realization liftt7→ R(t) must project too(t) and preserve the fixed outcome labels and coarse partition. In the fixed framed model, the coordinate event projectorsQ A remain constant. Gauge transformations along the lift must commute with the labeled coordinate projectors unless an explicit permutation action is included. Horizontal transport will therefore compareX(t),W(t), and    Ψ R(t) E relative to this fixed event correspondence. 3.9 Interim structural summary The preceding construction establishes: • a conservative smooth operational domainO reg defined by fixed ranks and a regular-value condition; • explicit label-preserving correspondence for fine and coarse events; • a fixed-dimensional framed Purification–Dilation Realization Space R reg ; • a projection π whose fibers are exactly the orbits of the compact gauge group G; • exact vertical invariance of the refinement-derived typicality; • a clearly stated transition from exact fiber-orbit structure to the local principal-bundle theorem proved in Section 4. 11

4 Local Realization Geometry This section proves the local geometric structure required for the realization bundle. All statements are restricted to one regular fixed-rank operational stratumO reg (r,r) and to the framed realization space R reg of Definition 3.4. No assertion is made across rank-changing or nonregular loci. 4.1 Smooth structure of the framed realization space Define the normalized amplitude manifold A r

n X ∈ Hom(A,H S ) : rankX = r, Tr(XX † ) = 1 o and the block-rank Stiefel stratum W r

n W = (W 1 ,...,W n ) : W † W = I, rankW i = r i o . Recall that R reg is the preimage of O reg under the ambient compression map π amb . Proposition 4.1 (Smooth framed realization manifold). The spaceR reg is a finite-dimensional smooth manifold. Its dimension is dimR reg = (2dr− 1) + (2md− d 2 ), m = n X i=1 r i . Equivalently, dimR reg = dimO reg

• dimG, where dimG = r 2

n X i=1 r 2 i . Proof. The full-column-rank condition onXis open inHom(A,H S ). On that open set, the smooth function F (X) = Tr(XX † ) has differential dF X ( ̇ X) = 2 Re Tr(X † ̇ X). BecausedF X (X) = 2Tr(X † X)>0, the value 1 is regular. HenceA r =F −1 (1) is a smooth hypersurface of real dimension 2dr− 1. Writing the blocks as one operator W :H S −→K, the conditionW † W=Idefines the complex Stiefel manifold of isometric embeddings ofH S into K, of real dimension 2md− d 2 [10,5]. The conditionsrankW i =r i are open, soW r is an open submanifold of that Stiefel manifold. Surjectivity of a linear map between finite-dimensional vector spaces is an open condition. Hence the regular-value condition definingPOVM reg r (H S ) is open in the fixed-rank POVM locus. Therefore O reg is open in D r (H S )× POVM r (H S ), and R reg = π −1 amb (O reg ) 12

is open in A r ×W r . This proves smoothness and the first dimension formula. For the second, dimO reg

• dimG =

2dr− r 2 − 1 + X i (2dr i − r 2 i )− d 2 ! +

r 2 + X i r 2 i ! = 2dr− 1 + 2d X i r i − d 2 = (2dr− 1) + (2md− d 2 ), where m = P i r i . 4.2 The realization projection Proposition 4.2 (Smooth surjective realization projection). The realization projection π :R reg −→O reg is smooth and surjective. Its differential at R = (X,W ) is dπ R ( ̇ X, ̇ W ) =  ̇ XX †

• X ̇ X † , n ̇ W † i W i
• W † i ̇ W i o n i=1  . Proof. The mapπis the restriction of a polynomial map in the real and imaginary matrix entries, so it is smooth, and direct differentiation gives the displayed formula. Leto= (ρ,E)∈O reg . Choose a full-column-rank amplitudeX:C r →H S withXX † =ρ. For each i, choose an isometry J i : C r i −→ suppE i onto the support of E i , and define W i := J † i E 1/2 i . Then W † i W i = E 1/2 i J i J † i E 1/2 i = E i , because J i J † i is the support projection of E i . Consequently, X i W † i W i = X i E i = I, so the stacked operatorWis an isometry. SinceE 1/2 i has rankr i andJ † i restricts to an isomorphism fromsuppE i ontoC r i , one hasrankW i =r i . The compressed operational point iso∈O reg . Thus (X,W )∈R reg and π(X,W ) = o. 4.3 Smooth local realization sections Lemma 4.3 (Local smooth support frames). LetA 0 ≥0 have rankk. There is a neighborhoodU of A 0 in the rank-k positive-semidefinite stratum and a smooth map J : U −→ Hom(C k ,H S ) such that J (A) † J (A) = I k and J (A)J (A) † is the support projection of A for every A∈ U. 13

Proof.Letp(A) denote the support projection ofA. On a sufficiently small neighborhood ofA 0 in the rank-kpositive-semidefinite stratum, the nonzero spectrum remains separated from zero. Choose a positively oriented contourγenclosing the nonzero spectrum ofA 0 and excluding zero. After shrinking the neighborhood if necessary, the support projection is the Riesz projection p(A) = 1 2πi Z γ (zI− A) −1 dz, and therefore depends smoothly on A by smooth spectral functional calculus [11]. Fix an isometry J 0 : C k −→ suppA 0 . Define B(A) := p(A)J 0 . At A = A 0 , one has B(A 0 ) = J 0 , so B(A 0 ) † B(A 0 ) = I k . By continuity, B(A) † B(A) remains positive definite on a smaller neighborhood. Set J (A) := B(A)  B(A) † B(A)  −1/2 . The inverse square-root map is smooth on the positive-definite cone, henceJ(A) is smooth. Direct calculation gives J (A) † J (A) = I k . Moreover,ranB(A)⊆ ranp(A) and both spaces have dimensionk, soranJ(A) =ranp(A). There- fore J (A)J (A) † = p(A). Lemma 4.4 (Smooth fixed-rank square-root factors). On the neighborhoodUof Lemma 4.3, the maps A7−→ A 1/2 J (A)and A7−→ J (A) † A 1/2 are smooth. Proof. Write H(A) := J (A) † AJ (A). This is a smooth family of positive-definitek× kmatrices. Its spectrum stays in a compact subset of (0,∞) after shrinkingU. The positive square-root map is smooth on positive-definite matrices, so H(A) 1/2 varies smoothly. Because ranJ (A) = suppA, the identity A = J (A)H(A)J (A) † holds on the full Hilbert space. Consequently, A 1/2 J (A) = J (A)H(A) 1/2 and J (A) † A 1/2 = H(A) 1/2 J (A) † . Both maps are therefore smooth. 14

Theorem 4.5 (Smooth local realization sections). For everyo 0 = (ρ 0 ,E 0 )∈O reg , there is an open neighborhood U ⊆O reg of o 0 and a smooth map s : U −→R reg such that π◦ s = id U . Proof.Apply Lemma 4.3 toρ 0 and to every effectE 0 i . After shrinking to a product neighborhood U, obtain smooth support isometries J ρ (ρ) : C r → suppρ and J i (E i ) : C r i → suppE i . Define X(ρ) := ρ 1/2 J ρ (ρ) and W i (E) := J i (E i ) † E 1/2 i . Lemma 4.4 gives smoothness. Moreover, X(ρ)X(ρ) † = ρ 1/2 J ρ (ρ)J ρ (ρ) † ρ 1/2 = ρ 1/2 p ρ ρ 1/2 = ρ, and W i (E) † W i (E) = E i . SinceJ ρ (ρ) is an isomorphism ontosuppρ,X(ρ) has rankr. Similarly,J i (E i ) † is an isomorphism fromsuppE i ontoC r i , soW i (E) has rankr i . Because P i E i =I, the stackedW(E) is an isometry. Hence s(ρ,E) :=

X(ρ),W 1 (E),...,W n (E)
lies in R reg and satisfies π(s(ρ,E)) = (ρ,E). Corollary 4.6 (Submersion property). The realization projectionπ:R reg → O reg is a smooth surjective submersion. Proof. At every o∈ U, differentiation of π◦ s = id U gives dπ s(o) ◦ ds o = id T o O reg , so dπ s(o) is surjective. Every R∈ π −1 (o) has the form R = s(o)· g by Proposition 3.7. Let R g :R reg −→R reg , R g (R ′ ) = R ′ · g. Because π◦ R g = π, differentiation gives dπ R ′ ·g ◦ d(R g ) R ′ = dπ R ′ . The mapR g is a diffeomorphism with inverseR g −1 , sod(R g ) R ′ is invertible. Hencedπ R ′ ·g anddπ R ′ have the same rank. Taking R ′ = s(o) shows that dπ R is surjective at every R. 15

4.4 Local principal realization geometry Theorem 4.7 (Local principal realization geometry). Leto 0 ∈O reg . There is an open neighborhood U of o 0 and a G-equivariant diffeomorphism Φ s : U × G−→ π −1 (U ),Φ s (o,g) = s(o)· g, where the right action on U × G is (o,g)· h := (o,gh). The trivialization satisfies π◦ Φ s = pr U . Consequently, π :R reg −→O reg is a smooth principalG-bundle on each regular fixed-rank stratum in the standard local-triviality sense [12, 7]. Proof.Letsbe the local section from Theorem 4.5. Smoothness of Φ s follows from smoothness ofs and of the right action. Proposition 3.7 gives surjectivity, and freeness from Proposition 3.8 gives injectivity. It remains to prove that the inverse is smooth. Let s(o) =

X s (o),W s,1 (o),...,W s,n (o)
and let R = (X,W 1 ,...,W n )∈ π −1 (U ), o = π(R). The matricesX s (o) have constant full column rankr, and the matricesW s,i (o) have constant full row rank r i . Moore–Penrose inversion is smooth on a fixed-rank matrix manifold [13]. Define U R := X s (o) + X and V i,R :=  W i W s,i (o) +  † . By Proposition 3.7, there is a unique unitary gauge elementg= (U,V 1 ,...,V n ) such thatR=s(o)·g. Multiplying the identities X = X s (o)U and W i = V † i W s,i (o) by the corresponding one-sided pseudoinverses shows that the recovered matrices above equal those unitary gauge coordinates. Thus U R ∈ U (r) and V i,R ∈ U (r i ), and Φ −1 s (R) = (π(R), (U R ,V 1,R ,...,V n,R )) is smooth. The identity π◦ Φ s = pr U is immediate. Finally, Φ s ((o,g)· h) = s(o)· (gh) = (s(o)· g)· h = Φ s (o,g)· h, so Φ s is G-equivariant. 16

Corollary 4.8 (Vertical tangent space). For R = (X,W )∈R reg , kerdπ R ={(Xξ,−η 1 W 1 ,...,−η n W n ) : ξ ∈ u(r), η i ∈ u(r i )}. In particular, relative to the framed gauge action, the vertical tangent space is canonically identified with the Lie algebra g = u(r)⊕ n M i=1 u(r i ). Proof.The local trivialization of Theorem 4.7 identifies the fiber throughRwith the rightG-orbit. Differentiating the orbit map at the identity gives δX = Xξ, δW i =−η i W i . Becauseπis a submersion, its kernel is the tangent space to the fiber. Freeness makes the infinitesimal-action map from g injective, so these directions exhaust kerdπ R . Remark 4.9 (Scope of the bundle statement). Theorem 4.7 applies only to the framed minimal realization space over one regular fixed-rank stratum. It does not assert a single bundle across rank-changing boundaries, nonregular POVMs, varying outcome number, continuous outcome spaces, or unused ancillary sectors. Those extensions may require stratified or groupoid-based geometry. 5 Complete-POVM Robustness Theory 5.1 Setting and norms Let H S be a complex Hilbert space of dimension d, and let E = (E 1 ,...,E n ) be a finite POVM: E i ≥ 0, n X i=1 E i = I. For a density operator ρ and an event A⊆ I :={1,...,n}, define E A := X i∈A E i and the operational typicality T(ρ,E;A) := Tr(ρE A ). In the unitary-refinement program this trace formula represents the already-derived projective typicality after purification and Naimark realization; it is not taken here as an independent probabilistic axiom. Definition 5.1 (Complete-measurement distances). For statesρ,σand two labeled finite POVMs E = (E i ) and F = (F i ), define d st (ρ,σ) :=∥ρ− σ∥ 1 , d max (E,F ) := max i ∥E i − F i ∥ ∞ , 17

and d Σ (E,F ) := n X i=1 ∥E i − F i ∥ ∞ . Define the event norm d event (E,F ) := sup A⊆I ∥E A − F A ∥ ∞ . For a fixed partition Γ ={B 1 ,...,B m }, define d Γ (E,F ) := max α ∥E B α − F B α ∥ ∞ . The event norm gives the sharp measurement term for uniform control over the fixed labeled event algebra. The sum and maximum distances provide convenient computable upper bounds: d event (E,F )≤ d Σ (E,F )≤ nd max (E,F ). The partition distance can be much smaller when fine-outcome perturbations cancel inside fixed coarse blocks. Remark 5.2 (Trace-norm convention). Throughout this section,∥·∥ 1 is the unnormalized trace norm. We do not include the conventional factor 1/2 used in the quantum-information trace distance. 5.2 Joint state-and-measurement robustness Theorem 5.3 (Single-event joint perturbation bound). Letρ,σbe density operators and letE,F be labeled finite POVMs. For every event A⊆ I, |T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥ 1 +∥E A − F A ∥ ∞ . Consequently, |T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥ 1 + X i∈A ∥E i − F i ∥ ∞ . Proof. Insert and subtract Tr(σE A ): Tr(ρE A )− Tr(σF A ) = Tr((ρ− σ)E A ) + Tr(σ(E A − F A )). Because 0≤ E A ≤ I, one has ∥E A ∥ ∞ ≤ 1. Trace duality and ∥σ∥ 1 = 1 give |Tr((ρ− σ)E A )|≤∥ρ− σ∥ 1 and |Tr(σ(E A − F A ))|≤∥E A − F A ∥ ∞ . The second bound follows from the operator-norm triangle inequality. Corollary 5.4 (Uniform event robustness). Under the hypotheses of Theorem 5.3, sup A⊆I |T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥ 1

• d event (E,F ). Consequently, sup A⊆I |T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥ 1
• d Σ (E,F ) and sup A⊆I |T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥ 1
• nd max (E,F ). 18

Proof. Apply Theorem 5.3 and take the supremum over A. The remaining bounds follow from d event (E,F )≤ d Σ (E,F )≤ nd max (E,F ). Remark 5.5. The factornis a worst-case consequence of controlling each effect separately. It should not be interpreted as optimal. The event-level norm∥E A − F A ∥ ∞ or a partition-adapted norm is preferable when available. 5.3 Fixed coarse graining Let Γ ={B 1 ,...,B m } be a fixed partition of the outcome labels. Define the coarse POVM E Γ α := E B α . Proposition 5.6 (Coarse-event robustness). For every coarse event C ⊆{1,...,m},    T(ρ,E Γ ;C)− T(σ,F Γ ;C)    ≤∥ρ− σ∥ 1 + X α∈C

E Γ α − F Γ α

∞ . Hence sup C    T(ρ,E Γ ;C)− T(σ,F Γ ;C)    ≤∥ρ− σ∥ 1

• md Γ (E,F ). Proof. Apply Theorem 5.3 to the coarse POVMs E Γ and F Γ . Remark 5.7. Coarse graining is not merely a change of notation. A fine-measurement perturbation can be large ind Σ while the induced coarse perturbation remains small because signed operator changes cancel within a block. 5.4 Canonical normalization of approximately complete families Let F = (F 1 ,...,F n ), F i ≥ 0, be a labeled positive family with frame operator S := n X i=1 F i . Assume S is positive definite. Define the canonical normalized effects N S (F i ) := E i = S −1/2 F i S −1/2 . Proposition 5.8 (Exact normalization and coarse-graining compatibility). The normalized family E = (E i ) is a POVM. For every fixed block B ⊆ I, E B = S −1/2 F B S −1/2 , F B := X i∈B F i . Thus canonical normalization commutes exactly with every fixed coarse-graining sum. 19

Proof. Positivity is preserved by congruence. Moreover, X i E i = S −1/2

X i F i ! S −1/2 = S −1/2 SS −1/2 = I. Linearity of finite summation gives the block identity. Lemma 5.9 (Inverse-square-root deviation). If ∥S− I∥ ∞ ≤ δ < 1, then (1− δ)I≤ S ≤ (1 + δ)I,

S −1/2

∞ ≤ (1− δ) −1/2 , and

S −1/2 − I

∞ ≤ (1− δ) −1/2 − 1. Proof. The first inequality follows from the spectral characterization of the operator norm. Apply the decreasing scalar functionx7→ x −1/2 to the spectrum ofS. The maximal deviations occur at 1− δ. Theorem 5.10 (Effect-wise normalization bound). Assume ∥S− I∥ ∞ ≤ δ < 1. Let c δ := (1− δ) −1/2 − 1. Then for every i, ∥E i − F i ∥ ∞ ≤∥F i ∥ ∞ c δ  (1− δ) −1/2

• 1  . Equivalently, ∥E i − F i ∥ ∞ ≤∥F i ∥ ∞ δ 1− δ . The same estimate holds with F i ,E i replaced by any coarse sums F B ,E B . Proof. Set A = S −1/2 . Then AF i A− F i = (A− I)F i A + F i (A− I). Therefore ∥E i − F i ∥ ∞ ≤∥A− I∥ ∞ ∥F i ∥ ∞ (∥A∥ ∞
• 1). Apply Lemma 5.9. Finally,  (1− δ) −1/2 − 1  (1− δ) −1/2
• 1  = δ 1− δ . The coarse statement follows from Proposition 5.8. Corollary 5.11 (Typicality cost of normalization). For every state ρ and event A⊆ I, |Tr(ρE A )− Tr(ρF A )|≤∥F A ∥ ∞ δ 1− δ . Proof. Use trace duality, ∥ρ∥ 1 = 1, and Theorem 5.10 for the coarse sum A. 20

5.5 Perturbing an entire normalized architecture Consider two positive families F = (F i ) and G = (G i ) with frame operators S = X i F i , T = X i G i , and normalized POVMs E i = S −1/2 F i S −1/2 , H i = T −1/2 G i T −1/2 . Assumption 5.12 (Standing uniform frame-gap hypothesis for this section). Fix a constanta >0, independent of the perturbation pair, such that S ≥ aI, T ≥ aI. Lemma 5.13 (Inverse-square-root Lipschitz bound). Under Assumption 5.12,

S −1/2 − T −1/2

∞ ≤ 1 2a 3/2 ∥S− T∥ ∞ . Proof. Use the standard functional-calculus integral representation [14, 11] X −1/2

1 π Z ∞ 0 λ −1/2 (X + λI) −1 dλ and the resolvent identity. Since S,T ≥ aI,

(S + λI) −1

≤ (a + λ) −1 and similarly for T. Hence

S −1/2 − T −1/2

∞ ≤ ∥S− T∥ ∞ π Z ∞ 0 λ −1/2 (a + λ) 2 dλ. With the substitution λ = at, Z ∞ 0 λ −1/2 (a + λ) 2 dλ = a −3/2 Z ∞ 0 t −1/2 (1 + t) 2 dt. The remaining integral is B  1 2 , 3 2 

Γ(1/2)Γ(3/2) Γ(2)

π 2 , so the original integral equals π/(2a 3/2 ). Theorem 5.14 (Complete normalized-POVM perturbation bound). Under Assumption 5.12, define M i := max{∥F i ∥ ∞ ,∥G i ∥ ∞ }. Then for every i, ∥E i − H i ∥ ∞ ≤ 1 a ∥F i − G i ∥ ∞ + M i a 2 ∥S− T∥ ∞ . Because ∥S− T∥ ∞ ≤ X j ∥F j − G j ∥ ∞ , on every subset satisfyingS,T ≥ aIandM i ≤ M, theith normalized effect is Lipschitz in the labeled positive family with an explicit constant depending on a and M. 21

Proof. Add and subtract S −1/2 G i S −1/2 : E i − H i = S −1/2 (F i − G i )S −1/2

• (S −1/2 − T −1/2 )G i S −1/2
• T −1/2 G i (S −1/2 − T −1/2 ). Now

S −1/2

∞ ,

T −1/2

∞ ≤ a −1/2 . Apply Lemma 5.13 to the final two terms. Their combined coefficient is at most 2  1 2a 3/2  a −1/2 = a −2 . Remark 5.15. The theorem controls simultaneous variation of every effect through the common frame operators. This is stronger than applying an effect-wise perturbation estimate while ignoring normalization coupling. 5.6 Smooth-path bounds Let t7−→ o(t) = (ρ(t),E(t)) be a C 1 operational path with a fixed outcome labeling and fixed coarse partition. Proposition 5.16 (Pathwise typicality derivative). For every fixed event A, d dt T(ρ(t),E(t);A) = Tr( ̇ρ(t)E A (t)) + Tr(ρ(t) ̇ E A (t)). Consequently,     d dt T(ρ(t),E(t);A)     ≤∥ ̇ρ(t)∥ 1 +

̇ E A (t)

∞ . Proof. Differentiate the trace product and use 0≤ E A (t)≤ I, trace duality, and ∥ρ(t)∥ 1 = 1. Corollary 5.17 (Integrated pathwise stability). For 0≤ s≤ t≤ 1, |T(ρ(t),E(t);A)− T(ρ(s),E(s);A)| ≤ Z t s  ∥ ̇ρ(τ )∥ 1 +

̇ E A (τ )

∞  dτ. Proof. Integrate Proposition 5.16 and apply the triangle inequality. 5.7 Scope and remaining quantitative problems The trace, normalization, coarse-graining, and pathwise bounds proved above apply to arbitrary finite-dimensional positive effects and therefore do not require rank-one measurements. Additional hypotheses are needed only when one seeks quantitative control of support projectors, local support frames, or realization-space lifts. On a compact subset of a fixed-rank stratum, uniform lower bounds on the nonzero state and effect spectra support local projector and frame estimates. Standard invariant-subspace perturbation 22

theory, including Davis–Kahan type control, provides the natural background for such spectral-gap- dependent estimates [15]. Establishing explicit constants adapted to the present realization charts remains separate. The next section nevertheless proves a compact-chart response bound of the form ∥ds o (u)∥ fr ≤ L sec

∥ ̇ρ∥ 1 + X i

̇ E i

∞ ! , with a finite chart-dependent constant; no explicit spectral-gap formula is asserted here. Remark 5.18 (Robustness bounds do not select a connection). The bounds in this section constrain operational typicalities and the conditioning of canonical normalization. They do not, by themselves, select a horizontal complement tokerdπ R inT R R reg . Every lift of a fixed operational tangent has the same first-order scalar typicality derivatives because the typicality functions descend through π. The preceding bounds estimate those base-space derivatives and finite differences, but they contain no criterion that distinguishes two lifts differing by a vertical gauge vector. Selecting a connection therefore requires realization-level structure beyond scalar robustness. The framed response geometry of Section 6 supplies that structure. Remark 5.19. This section supplies operational and conditioning scales. Section 6 constructs the transport law independently, and Section 7 establishes its informational canonicity within the generated realization framework. 6 Robustness-to-Realization Response Geometry The complete-POVM estimates of Section 5 control operational variation on the base manifold. This section lifts that control to the framed realization bundle and establishes a baseline transport law. The construction uses the simplest right-invariant Hilbert–Schmidt metric on the purification and dilation frames. It is established independently of the informational canonicity argument developed later. 6.1 Operational and framed response norms For an operational tangent vector u = ( ̇ρ, ̇ E 1 ,..., ̇ E n )∈ T o O reg , define ∥u∥ rb :=∥ ̇ρ∥ 1 + n X i=1

̇ E i

∞ . The subscript “rb” identifies the robustness norm and distinguishes it from the smooth Hilbert– Schmidt metric introduced earlier on the operational stratum. For framed realization tangent vectors v = ( ̇ X, ̇ W 1 ,..., ̇ W n ), w = ( ̇ Y , ̇ Z 1 ,..., ̇ Z n ), define g fr R (v,w) := Re Tr( ̇ X † ̇ Y ) + n X i=1 Re Tr( ̇ W † i ̇ Z i ) and ∥v∥ fr := q g fr R (v,v). 23

Proposition 6.1 (Gauge invariance of the framed metric). For every g ∈ G and v,w ∈ T R R reg , g fr R·g

d(R g ) R v,d(R g ) R w
= g fr R (v,w). Proof. For g = (U,V 1 ,...,V n ), the differential of the right action sends ( ̇ X, ̇ W 1 ,..., ̇ W n )7−→ ( ̇ XU,V † 1 ̇ W 1 ,...,V † n ̇ W n ). Left and right multiplication by fixed unitaries preserve the Hilbert–Schmidt inner product. Summing the purification and labeled dilation-block contributions gives the result. 6.2 Compact-chart realization response Let s : U →R reg be any smooth local section supplied by Theorem 4.5. Theorem 6.2 (Compact-chart realization-response bound). LetK⋐ Ube compact. Then there is a finite constant L K,s

0 such that ∥ds o (u)∥ fr ≤ L K,s ∥u∥ rb for every o∈ K and u∈ T o O reg . Proof.The differentialdsis a continuous bundle map. In a finite-dimensional local trivialization, the operator norm o7−→∥ds o ∥ rb→fr is continuous. Compactness of K implies that its supremum is finite. Taking L K,s := sup o∈K ∥ds o ∥ rb→fr gives the displayed estimate. Corollary 6.3 (Pathwise section response). Leto: [a,b]→ KbeC 1 and setR s (t) :=s(o(t)). Then

̇ R s (t)

fr ≤ L K,s ∥ ̇o(t)∥ rb and Length fr (R s )≤ L K,s Length rb (o). Proof. The chain rule gives ̇ R s (t) = ds o(t) ̇o(t). Apply Theorem 6.2 pointwise and integrate. Remark 6.4 (Section dependence). The response of a chosen section is not gauge invariant. If s ′ (o) =s(o)· g(o), thends ′ o (u) contains an additional vertical term induced bydg o (u). Theorem 6.2 proves quantitative control for each smooth gauge choice; the next construction removes that dependence. 24

6.3 Minimum-response connection Recall the vertical tangent space V R := kerdπ R . Define its framed orthogonal complement by H fr R := V ⊥ g fr R . Theorem 6.5 (Gauge-invariant minimum-response connection). For everyR ∈ R reg andu ∈ T π(R) O reg , the affine lift space L R (u) :={v ∈ T R R reg : dπ R (v) = u} is nonempty and contains a unique vector of minimum framed norm. Denote it by h fr R (u) := arg min v∈L R (u) ∥v∥ 2 fr . Then:

1. u7→ h fr R (u) is linear;
2. h fr R (u)∈ H fr R ;

T R R reg = V R ⊕ H fr R ; 4. R7→ H fr R is smooth; 5. for every g ∈ G, d(R g ) R H fr R = H fr R·g and h fr R·g (u) = d(R g ) R h fr R (u). Consequently, H fr is a principal connection on π :R reg →O reg . Proof. The submersion property of Corollary 4.6 makesdπ R surjective, soL R (u) is nonempty. The positive-definite framed metric gives the orthogonal direct sum T R R reg = V R ⊕ V ⊥ g fr R . The restriction dπ R   H fr R : H fr R −→ T π(R) O reg is injective because its kernel isV R ∩ H fr R ={0}. It is surjective because the orthogonal projection of any lift ontoH fr R has the same image underdπ R . Hence the restriction is a linear isomorphism, and its inverse is the unique minimum-norm lift h fr R . The vertical tangent spaces form a smooth subbundle, and orthogonal complements with respect to a smooth metric form a smooth subbundle. Proposition 6.1 shows that the right action preserves the framed metric. Gauge invariance of π gives d(R g ) R V R = V R·g . 25

Therefore the right action carries framed orthogonal complements into framed orthogonal comple- ments. Right equivariance of the minimum lift follows from uniqueness. These properties are exactly the requirements for a principal connection. Corollary 6.6 (Connection form and curvature). The horizontal distributionH fr determines a principal connection one-form ω fr ∈ Ω 1 (R reg ; g) with kerω fr = H fr . Its curvature is Ω fr = dω fr + 1 2 [ω fr ,ω fr ]. Proof. This is the standard correspondence between smooth right-equivariant horizontal distributions and principal connection one-forms. 6.4 Operational response geometry Definition 6.7 (Minimum-response quadratic form). Foro ∈ O reg , choose anyR ∈ π −1 (o) and define q o (u,w) := g fr R

h fr R (u),h fr R (w)
. Write q o (u) := q o (u,u). Theorem 6.8 (Gauge-independent operational response geometry). The formq o is independent of the representative R∈ π −1 (o). It defines a smooth positive-definite quadratic form on TO reg . Proof. Let R ′ = R· g. By Theorem 6.5, h fr R ′ (u) = d(R g ) R h fr R (u). Proposition 6.1 therefore gives g fr R ′

h fr R ′ (u),h fr R ′ (w)
= g fr R·g

d(R g )h fr R (u),d(R g )h fr R (w)
= g fr R

h fr R (u),h fr R (w)
. Thus the form descends to the base. Smoothness follows from smoothness of the horizontal inverse of dπ. If u̸= 0, then h fr R (u)̸= 0 because dπ R h fr R (u) = u. Hence q o (u) > 0. Theorem 6.9 (Robustness-to-response bound). Lets:U → R reg be a smooth local section and K⋐ U compact. Then q q o (u)≤∥ds o (u)∥ fr ≤ L K,s ∥u∥ rb for every o∈ K and u∈ T o O reg . Consequently, there is a finite constant L K such that q q o (u)≤ L K ∥u∥ rb on K. 26

Proof. Differentiating π◦ s = id U gives dπ s(o) ds o (u) = u. Thusds o (u) is an admissible lift in the minimum definingh fr s(o) (u). Minimum-norm optimality gives q q o (u) =

h fr s(o) (u)

fr ≤∥ds o (u)∥ fr . Theorem 6.2 gives the second inequality. Set L K = L K,s . Corollary 6.10 (Minimum-response path bound). For every C 1 operational path o : [a,b]→ K, Z b a q q o(t) ( ̇o(t))dt≤ L K Z b a ∥ ̇o(t)∥ rb dt. Proof. Apply Theorem 6.9 pointwise and integrate. 6.5 Status of canonicity The connectionH fr is canonical relative to the framed Hilbert–Schmidt response geometry. It is independent of local section, gauge equivariant, smooth, and introduces no branch weights or regularization parameter. Theorem 6.5 therefore establishes relative canonicity before any informational identity principle is introduced. Relative canonicity alone does not identify the quantum-physical status of gauge-related al- ternative lifts. Section 7 addresses that distinct question. It defines the generated informational profile, proves pointwise and pathwise preservation under the established gauge action, states Quantum Informational Identity, and derives absolute canonicity within the generated realization framework. Scalar typicality derivatives are not used to select among lifts; they enter only as one representation-invariant component of the generated profile. 6.6 Scope The results in this section apply on one regular fixed-rank framed realization bundle. They do not assert a single response connection across rank-changing boundaries, nonregular POVMs, changes in outcome number, or other singular loci outside the regular framed model. Explicit spectral-gap dependence of L K,s is also not derived here; only compact-chart finiteness is proved. Dependency guide for the final argument The final interpretive theorem uses the following chain. Each arrow records a distinct logical dependence rather than a new physical postulate: principal-bundle geometry =⇒ gauge relation between lifts ⇓ generated-profile preservation =⇒ Dynamical QII ⇓ profile-closed represented observables =⇒ common represented transport content ⇓ framed Hilbert–Schmidt geometry =⇒ unique minimum-response representative 27

Pointwise QII concerns individual realizations, Dynamical QII concerns histories, profile closure specifies the represented observables under comparison, and the framed metric supplies the variational selection. None of these informational premises derives the framed metric. 7Quantum Informational Identity and Profile-Relative Absolute Canonicity The preceding sections construct the framed response connection and prove its relative canonicity independently of any informational identity premise. This section addresses the distinct interpretive question of when gauge-related realization histories represent the same quantum-physical transport content. To make that claim verifiable, the generated carrier, event algebra, and admitted covariance group are now defined directly from the fixed framed architecture. TermRole in the argument Generated informational profile Declared finite-dimensional represented information attached to a realization. QIIPointwise identity premise for profile-isomorphic realizations. Dynamical QIIHistory-level identity premise for isomorphic dynamical profiles. Profile closureRequirement that represented transport observables be invariant on dynamical-profile isomorphism classes. Dynamical LWMRestatement of the Dynamical QII consequence for admissible histories, not a metric-selection principle. Profile-relative absolute canonicity Common represented content within the profile-closed class, plus the unique minimum-response representative relative to the stated metric. 7.1 The fixed generated carrier and covariance structure Definition 7.1 (Generated framed carrier and labeled event algebra). On a fixed regular stratum, define the generated framed carrier H gen :=K⊗A and the fixed labeled projectors Q i := P i ⊗ I A , i = 1,...,n. For B ⊆ I, write Q B := P i∈B Q i . The labeled event algebra is A := C ∗ (Q 1 ,...,Q n )⊆B(H gen ). For every realization R∈R reg , the unified state |Ψ R ⟩ defined in Section 3 lies in H gen . Definition 7.2 (Admitted label-preserving covariance group). The admitted covariance group of the generated framed architecture is U gen :={Z ∈ U (H gen ) : ZQ i Z † = Q i for every i}. Equivalently,U gen is the unitary group of the commutant of the labeled event algebra. This group is fixed on the entire regular stratum; it is not selected separately for each realization. 28

Proposition 7.3 (Gauge implementation preserves the generated architecture). Forg= (U,V 1 ,...,V n )∈ G, define b V g :=

n M i=1 V † i ! ⊗ U T ∈ U (H gen ). Then, for every R∈R reg , b V g |Ψ R ⟩ =|Ψ R·g ⟩, b V g Q i b V † g = Q i , b V g A b V † g = A, and b V g U gen b V † g =U gen . Proof. The state identity is the implementation calculation from Proposition 3.9. The block-diagonal factor L i V † i preserves each coordinate block ofK, whileU T acts only onA; hence b V g commutes with everyQ i =P i ⊗ I A . It therefore fixesAunder conjugation and belongs toU gen . SinceU gen is a group, conjugation by b V g ∈U gen maps it onto itself. 7.2 Generated informational profiles Definition 7.4 (Generated realization system). For R∈R reg , define G R :=

H gen ,|Ψ R ⟩,{Q i } n i=1 , A,U gen , T R
. Thus the carrier, labeled projectors, event algebra, and admitted covariance group are fixed by the framed architecture; realization dependence enters through|Ψ R ⟩and the represented event assignment T R . Definition 7.5 (Generated informational profile). The generated informational profile Prof gen (R) consists exactly of:

1. the state ω R (A) =⟨Ψ R |A|Ψ R ⟩ on A;
2. the labeled fine- and coarse-event structure i7→ Q i and B 7→ Q B ;
3. the finite event correlations ω R (A 1 ···A k ) for A j ∈ A;
4. the admitted covariance action (Z,A)7→ ZAZ † , together with Z|Ψ R ⟩, for Z ∈U gen ;
5. the induced event values T R (B). No claim is made about informational variables outside this finite-dimensional generated profile. Definition 7.6 (Generated profile isomorphism). The profiles ofRandR ′ are generated-profile isomorphic if there exists V ∈ U (H gen ) satisfying V |Ψ R ⟩ =|Ψ R ′ ⟩, V Q i V † = Q i for every i, VU gen V † =U gen , and T R (B) = T R ′ (B) for every labeled event B. Theorem 7.7 (Preservation of the generated informational profile). IfVimplements a generated profile isomorphism from R to R ′ , then Prof gen (R) ∼ = Prof gen (R ′ ). 29

Proof. For A∈ A, ω R ′ (V AV † ) =⟨Ψ R |A|Ψ R ⟩ = ω R (A). The identitiesV Q i V † =Q i preserve all fine and coarse labeled events and makeAd V a unital ∗-automorphism ofA. Products satisfy (V A 1 V † )···(V A k V † ) =V(A 1 ···A k )V † , so all listed finite correlations are preserved. Conjugacy ofU gen preserves the admitted covariance action, and equality of T R and T R ′ preserves the induced event values. Corollary 7.8 (Pointwise gauge-profile invariance). For every R∈R reg and g ∈ G, Prof gen (R) ∼

Prof gen (R· g). Proof.UseV= b V g . Proposition 7.3 supplies the state, event, algebra, and covariance identities. Proposition 3.9 supplies T R (B) = T R·g (B). 7.3 Pointwise and dynamical informational identity Assumption 7.9 (Quantum Informational Identity). Within the generated realization framework, Prof gen (R) ∼

Prof gen (R ′ ) =⇒ R≡ phys R ′ . QII is a pointwise identity premise. It does not by itself identify entire transport histories. Definition 7.10 (Dynamical generated informational profile). For a smooth realization path e γ : J →R reg , define Prof dyn gen ( e γ) :=

t7→ Prof gen ( e γ(t))
. The fixed structuresH gen ,{Q i },A, andU gen are part of every pointwise profile and need not be listed a second time as a realization-dependent history. Definition 7.11 (Isomorphism of dynamical generated profiles). Two dynamical generated profiles are isomorphic if there exists a smooth mapV:J → U(H gen ) such thatV(t) implements a generated profile isomorphism at every t∈ J. Theorem 7.12 (Pathwise preservation of the dynamical generated profile). Let e γ 1 , e γ 2 :J →R reg be smooth lifts of the same operational path and suppose e γ 2 (t) = e γ 1 (t)· g(t) for a smooth gauge path g : J → G. Then Prof dyn gen ( e γ 1 ) ∼

Prof dyn gen ( e γ 2 ). Proof.SetV(t) = b V g(t) . Smoothness ofgimplies smoothness ofV. Proposition 7.3 and Proposi- tion 3.9 show that V (t) implements a generated profile isomorphism at every t. Assumption 7.13 (Dynamical Quantum Informational Identity). Isomorphic dynamical generated informational profiles represent the same quantum-physical transport content within the generated realization framework. Definition 7.14 (Dynamical Principle of Least Wasted Motion). The Dynamical Principle of Least Wasted Motion states that two admissible realization histories with isomorphic dynamical generated profiles cannot represent distinct quantum-physical transports within the generated framework. 30

Theorem 7.15 (Dynamical QII implies Dynamical LWM). Dynamical Quantum Informational Identity implies the Dynamical Principle of Least Wasted Motion. Proof. This is the direct specialization of Assumption 7.13 to admissible realization histories with isomorphic dynamical generated profiles. Lemma 7.16 (Smooth gauge relation between lifts). Let e γ 1 , e γ 2 :J →R reg be smooth lifts of the same smooth operational path. Then there is a unique smooth map g : J → G such that e γ 2 (t) = e γ 1 (t)· g(t) for every t∈ J. If the lifts share the same initial representative at t 0 , then g(t 0 ) = e. Proof.At eacht, the lifts lie in the same fiber, which is a free and transitiveG-orbit. This gives a unique pointwiseg(t). In any local principal-bundle trivialization along the base path,g(t) is obtained from the two smooth fiber-coordinate maps and is therefore smooth. Uniqueness makes these local expressions agree on overlaps. The initial statement follows from freeness. Corollary 7.17 (Pathwise quantum-physical identity). Under Dynamical QII, any two smooth lifts of the same operational path represent the same quantum-physical transport content within the generated framework. 7.4 Worked comparison of two nonidentical connections LetH fr be the minimum-response connection. On a principal-bundle trivialization overU ⊆O reg , choose a nonzero smoothg-valued one-formα ∈Ω 1 (U;g). The affine structure on the space of principal connections supplies a second connection whose local connection form is A ′ = A fr

• α. Leto:J → Ube an operational path for whichα o(t) ( ̇o(t)) is nonzero somewhere, and fix a common initial realization. TheH fr -horizontal lift e γ fr and theH ′ -horizontal lift e γ ′ then have different lifted velocities and arise from nonidentical connections. Lemma 7.16 nevertheless gives a unique smooth gauge path g : J → G satisfying e γ ′ (t) = e γ fr (t)· g(t). Theorem 7.12 therefore gives Prof dyn gen ( e γ ′ ) ∼ = Prof dyn gen ( e γ fr ). Every observable satisfying Definition 7.18 assigns the same value to these histories even though the connections and horizontal lifts differ. By contrast, a quantity defined from the additional vertical gauge pathg(t) itself could distinguish them; such a quantity would fail the factorization criterion and enlarge the represented theory. This comparison makes the closure hypothesis independently testable and displays its exact role in the final theorem. 7.5 Profile-relative absolute canonicity Definition 7.18 (Profile-closed represented observables and admissible transport). Fix a declared familyO rep of represented transport observables, each assigning a value to every smooth realization history. The familyO rep is profile closed if, for everyO ∈O rep and every pair of smooth histories e γ 1 , e γ 2 with isomorphic dynamical generated profiles, Prof dyn gen ( e γ 1 ) ∼ = Prof dyn gen ( e γ 2 )=⇒ O( e γ 1 ) =O( e γ 2 ). 31

A smooth, label-preserving principal connectionHis profile-closed admissible relative toO rep when O rep is profile closed and the represented histories selected byHremain within the fixed generated realization framework. Equivalently, for everyO ∈O rep there exists a functionalF O on dynamical generated-profile isomorphism classes such that O( e γ) = F O  [Prof dyn gen ( e γ)]  for every smooth realization history e γin the comparison class. A proposed transport theory fails profile closure as soon as one represented observable distinguishes two profile-isomorphic histories. Adding such an observable enlarges O rep and defines an extension of the profile-closed theory. Remark 7.19 (What profile closure excludes). Profile closure excludes from the present comparison class any additional holonomy label, path-memory variable, ancillary frame coordinate, connection identifier, or other represented quantity that is not invariant under dynamical generated-profile isomorphism. It does not assert that such extensions are impossible; it states that they are not observables of the profile-closed theory under comparison. Example 7.20 (An observable that fails profile closure). Let e γ 1 and e γ 2 be lifts of the same operational path with common initial representative, and letg:J → Gbe the unique gauge path satisfying e γ 2 (t) = e γ 1 (t)· g(t). After fixing any auxiliary norm ongand writingg −1 ̇gfor the gauge velocity, the assignment O vert ( e γ 1 , e γ 2 ) := Z J

g(t) −1 ̇g(t)

2 g dt can distinguish histories whose dynamical generated profiles are isomorphic. It therefore does not factor through the dynamical generated-profile isomorphism class and is not profile closed. Treating O vert as represented information would define an extension of the theory rather than an alternative transport within the present comparison class. Definition 7.21 (Profile-relative absolute canonicity). A transport is profile-relatively absolutely canonical if all profile-closed admissible principal connections determine the same quantum-physical transport content, while a stated response geometry selects a unique representative of that common content by a specified variational criterion. Theorem 7.22 (Profile-relative physical uniqueness and minimum-response selection). Assume Dynamical Quantum Informational Identity. On each regular fixed-rank framed realization bundle, every profile-closed admissible principal connection determines the same quantum-physical transport content for a fixed operational path and initial realization. Relative to the framed Hilbert–Schmidt response geometry,H fr is the unique minimum-response representative of that content. HenceH fr is profile-relatively absolutely canonical in the sense of Definition 7.21. Proof. LetH ′ be a profile-closed admissible principal connection. Fix a smooth operational path and a common initial realization, and let e γ ′ and e γ fr be the lifts determined byH ′ andH fr . Lemma 7.16 supplies a unique smooth gauge path relating them. Theorem 7.12 identifies their dynamical generated profiles, and Dynamical QII identifies their represented transport content. SinceH ′ was arbitrary within the profile-closed class, the content conclusion holds for every such connection. By Theorem 6.5, every affine lift space contains a unique vector minimizing the framed Hilbert– Schmidt norm, and these minimizers defineH fr . ThusH fr is the unique minimum-response representative relative to the stated response geometry. 32

Remark 7.23 (Deductive role of the final theorem). The content conclusion combines four separately stated ingredients: the principal-bundle fact that lifts of one operational path are smoothly gauge related, preservation of the dynamical generated profile under that relation, Dynamical QII, and the profile-closure restriction on represented observables. The theorem is therefore conditional and classificatory within the declared comparison class. It is not a characterization theorem for all conceivable transport observables or response geometries. Remark 7.24 (Pointwise versus dynamical roles). QII identifies pointwise profile-isomorphic realiza- tions. Dynamical QII, not pointwise QII alone, supports the transport-content conclusion. Neither premise derives the framed Hilbert–Schmidt metric. Remark 7.25 (Claim boundary). The theorem is conditional on Dynamical QII and on closure of the represented content under the explicitly defined generated profile. It does not exclude extensions with additional observables, apply across singular rank-changing loci, or assert that informational identity uniquely forces the Hilbert–Schmidt response geometry. Remark 7.26 (Separate metric-characterization problem). Complete canonicity of the response geometry would require a separate theorem showing that every admissible realization metric satisfying specified covariance, additivity, smoothness, and structural compatibility axioms is a global multiple of the framed Hilbert–Schmidt metric, followed by an independent normalization or proof that the remaining scale is physically irrelevant. No such metric-classification theorem is claimed here. Corollary 7.27 (Canonicity chain). On each regular fixed-rank framed realization bundle, Dynamical QII plus profile closure =⇒ unique represented transport content, while the framed Hilbert–Schmidt response geometry gives the unique minimum-response representa- tive H fr . 8 Discussion The principal-bundle theorem, complete-POVM robustness, minimum-response connection, and informational identity theorem now form one continuous argument. Exact representation freedom is encoded by vertical gauge directions. Operational perturbations are controlled by the robustness norm. The framed Hilbert–Schmidt response converts operational motions into unique minimum- response lifts, whose squared norms descend to a positive response geometry on the operational base. The canonicity result has two distinct components. The geometric component proves that the framed response functional has a unique minimum-response lift. The informational component now defines the carrier and admitted covariance structure explicitly and proves that exact gauge alterna- tives preserve the represented profile pointwise and pathwise. Pointwise QII addresses realizations; Dynamical QII addresses histories. Under the stated profile-closure restriction, Dynamical QII identifies gauge-related histories as the same represented transport content. Relative to the framed Hilbert–Schmidt geometry, minimum response supplies the unique representative of that content. Dynamical LWM summarizes this identity consequence; it does not derive the response metric. The meaning of “absolute” is profile-relative and explicitly scoped. The theorem applies within the finite-dimensional generated realization framework on each regular fixed-rank bundle, under Dynamical QII, and among profile-closed admissible connections. It does not exclude extensions with additional represented variables or claim one smooth structure across singular rank changes, 33

nonregular architectures, or continuous outcomes. A stronger, separate characterization theorem would be needed to prove that intrinsic covariance, additivity, and admissibility principles force the framed Hilbert–Schmidt metric itself, up to an overall scale. That metric-canonicity problem is not solved or assumed here. 9 Conclusion Regular state–POVM strata admit framed purification–dilation realization bundles with compact gauge fibers. The realization projection is a smooth principal bundle, and its vertical tangent spaces are precisely the infinitesimal purification- and labeled outcome-frame transformations. Complete-POVM robustness controls simultaneous perturbations of states and measurement architectures, including fixed coarse events, canonical normalization, and smooth paths. The framed Hilbert–Schmidt response geometry defines unique minimum-response lifts. Their horizontal spaces form a smooth principal connection, and their squared norms descend to a gauge-independent positive response geometry. On compact charts, minimum response is bounded by the operational robustness norm. The generated informational profile is built on the fixed carrierH gen :=K ⊗A, the fixed labeled event algebra, and the explicitly defined label-preserving covariance group. Exact gauge transformations preserve this profile, and smooth gauge paths preserve its dynamical history. Under Dynamical QII and profile closure, profile-isomorphic histories determine the same represented transport content. Relative to the framed Hilbert–Schmidt geometry,H fr is the unique minimum- response representative. Together these conclusions constitute profile-relative absolute canonicity within the generated realization framework. Program Linkage Paper 1 develops the refinement-derived conditional representation theorem for the Born measure [16]. Paper 2 extends the unitary-refinement program to generalized quantum measurements through purification–dilation representation invariance and finite robustness [17]. The present Paper 3 supplies regular operational and realization geometries, complete finite-POVM robustness, the framed minimum-response connection, the generated informational profile, and the QII-based profile- relative canonicity theorem. Future work includes extension beyond regular finite-dimensional generated realization bundles and the separate metric-characterization problem of determining whether intrinsic structural principles uniquely force the framed Hilbert–Schmidt response geometry up to scale. References [1]Armin Uhlmann. Parallel transport and quantum holonomy along density operators. Reports on Mathematical Physics, 24(2):229–240, 1986. [2]Armin Uhlmann. Transition probability (fidelity) and its relatives. arXiv preprint arXiv:1106.0979, 2011. Revised 2016. [3]Xu-Yang Hou and Hao Guo. Uhlmann geometry of fixed-rank density matrices: A fiber-bundle approach. arXiv preprint arXiv:2609.21875, 2026. 34

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