Wang et al. (2025) and the subsequent analysis of Cieśliński et al. (2026) provide a useful illustration of why the physically operative quantum object is the complete global wavefunctional rather than a selected subsystem or postselected output sector. In the interwoven down-conversion architecture, interference depends on the coherent organization of alternative generation histories across the full optical network, while the physical interpretation of the resulting correlations depends additionally on how apparatus settings and the complete outcome space are represented. The continuing discussion over the appropriate Bell description is therefore itself informative: observable correlation structure cannot be inferred reliably from one reduced component of an experiment while discarding the global amplitudes, alternative paths and measurement map that give that component its physical meaning. This is precisely the role assigned to an ontic Ψ in sPNP. Ψ is not merely a catalogue of possible observations; its amplitude supplies objective relational distinction, its PhaSe structure supplies coherent transport and holonomy, and the complete system–apparatus wavefunctional together with the measurement interaction defines the branch geometry from which event-complete outcome probabilities are obtained. In this sense the global geometry is upstream of both subsystem decomposition and operational statistics. Wang’s path-identity architecture makes this global coherence unusually explicit, while the analysis of Cieśliński et al. emphasizes the importance of specifying the complete operational measurement structure before assigning Bell significance to observed interference. Neither point requires the present interpretive debate to be settled. Together they motivate the sharper geometric problem developed next: once event-complete correlations are derived from the ontic global Ψ and its apparatus coupling, why do their measurement directions form the common positive Gram geometry bounded by 2√2, and does finite relational resolution or reflexive state-dependent geometry preserve that structure?
Tsirelson Bound from Distinction–PhaSe Geometry
The Tsirelson bound is naturally interpreted as a geometric ceiling on physically realizable quantum correlations. For the CHSH combination S = E₀₀ + E₀₁ + E₁₀ − E₁₁, Bell-local correlations satisfy |S| ≤ 2, quantum correlations satisfy |S| ≤ 2√2, and the algebraic maximum is |S| ≤ 4. No-signaling alone does not determine the quantum ceiling, since Popescu–Rohrlich-type correlations can reach the algebraic maximum while preserving separated marginals. The relevant sPNP question is therefore stronger: whether the geometry carried by the global relational wavefunctional places physically realized Bell correlations in one common positive inner-product structure, and whether that structure remains protected under finite relational resolution and reflexive geometric feedback.
Distinction–PhaSe and Projective Quantum Geometry
For a multicomponent wavefunctional write locally Ψₐ(X) = R(X) exp[iS(X)/ℏ]uₐ(X), with u†u = 1. The amplitude R supplies scalar distinction and the resolved R-Fisher sector; S supplies the common U(1) transport phase; and u carries internal projective orientation, relative phase, Berry connection, holonomy and the quantum-geometric tensor. Defining 𝒜_I = i u†∂_Iu, D_Iu = (1 − |u⟩⟨u|)∂_Iu and Π_I = ∂_IS − ℏ𝒜_I, the complete derivative decomposes exactly as ∂_IΨ = exp[iS/ℏ][(∂_IR)u + (iR/ℏ)Π_Iu + RD_Iu]. Under the real Hilbert metric, the radial amplitude, common-phase and horizontal projective directions are mutually orthogonal, with P_u⊥exp[−iS/ℏ]∂_IΨ = RD_Iu for P_u⊥ = 1 − |u⟩⟨u|. Scalar R-Fisher geometry and projective quantum geometry can therefore interact dynamically without being identified.
The complete relational metric may be written schematically as Ĝ = M + αFᴿ,(Q₀) + βg^Q + …, while PhaSe/Berry structure enters through the complementary gauge-covariant connection. The R-Fisher tensor controls resolved amplitude distinction and contributes to relational distance and the Relational Kernel, whereas g^Q and the associated projective two-form describe the intrinsic geometry of the normalized state. This preserves the distinction between amplitude-defined relational geometry and the projective geometry of the full Ψ.
On an even-dimensional coherent projective sector E, let g_E and ω_E denote the symmetric metric and projective two-form and define Aᴵⱼ = (g_E)ᴵᴷ(ω_E)ₖⱼ. Compatibility is characterized by −A²eₐ = κₐ²eₐ. When −A² = c²I_E, the normalized operator J = A/c satisfies J² = −I_E, so symmetric distinction and antisymmetric PhaSe geometry form a compatible Hermitian structure.
The qutrit CP² reconstruction gives an explicit nontrivial example. For independent component probabilities p = (p₁,p₂) and relative phases ϑ = (β,δ), define C = [[p₁(1−p₁), −p₁p₂], [−p₁p₂, p₂(1−p₂)]]. The Fubini–Study metric is G_FS = diag(¼C⁻¹,C), while the projective two-form has matrix Ω = [[0,I₂],[−I₂,0]]. Hence A = G_FS⁻¹Ω = [[0,4C],[−C⁻¹,0]], so −A² = 4I₄. Therefore κ₁ = κ₂ = 2 and J_Q = A/2 satisfies J_Q² = −I₄. Component-probability Fisher geometry and conjugate relative-PhaSe geometry thus combine into the native Fubini–Study Kähler structure beyond the exceptional two-real-dimensional qubit case.
Amplitude-Only Boundary
The scalar R-Fisher sector by itself defines distinguishability but does not generate Bell-nonclassical correlations. If the four binary responses are represented only by real configuration-space functions a_i(X), b_j(X) ∈ {−1,+1}, then E_ij = ∫dX |Ψ(X)|²a_i(X)b_j(X). Since the pointwise CHSH combination a₀b₀ + a₀b₁ + a₁b₀ − a₁b₁ has absolute value at most 2 for every X, integration gives |S| ≤ 2. This provides a structural boundary inside sPNP: scalar R-distinction supplies the geometry by which configurations become distinguishable, while access to the genuinely quantum region 2 < |S| ≤ 2√2 requires the complete apparatus-coupled PhaSe/projective geometry of Ψ. The original section already identified this division between common scalar responses and the full phase-sensitive measurement geometry.
Measurement-Generated Gram Geometry
Let A be a Hermitian measurement observable acting on the normalized global system–apparatus state and define ΔA = A − ⟨A⟩ and X_A = −(i/ℏ)ΔA|Ψ⟩. The projective complex structure gives J_QX_A = (1/ℏ)ΔA|Ψ⟩. For Hermitian A and B one has ℏ²g_Q(X_A,X_B) = ½⟨{ΔA,ΔB}⟩, and for separated Alice and Bob observables [A,B] = 0, so ⟨AB⟩ = ⟨A⟩⟨B⟩ + ℏ²g_Q(X_A,X_B).
Introduce the extended real measurement space ℰ_Ψ = ℝ ⊕ T_[Ψ]P(ℋ) with metric G_ext((a,V),(b,W)) = ab + g_Q(V,W), and associate to A the vector 𝐮_A = (⟨A⟩,ℏX_A). The physical correlation is then E_AB = G_ext(𝐮_A,𝐮_B). For a binary observable A² = I, its norm is ‖𝐮_A‖² = ⟨A⟩² + ℏ²g_Q(X_A,X_A) = ⟨A²⟩ = 1. The four CHSH settings therefore determine four unit vectors satisfying Eᵢⱼ = G_ext(𝐮_Aᵢ,𝐮_Bⱼ).
Writing S = 𝐮_A₀·(𝐮_B₀ + 𝐮_B₁) + 𝐮_A₁·(𝐮_B₀ − 𝐮_B₁), Cauchy–Schwarz gives |S| ≤ ‖𝐮_B₀ + 𝐮_B₁‖ + ‖𝐮_B₀ − 𝐮_B₁‖. If p = ‖𝐮_B₀ + 𝐮_B₁‖ and q = ‖𝐮_B₀ − 𝐮_B₁‖, the parallelogram identity gives p² + q² = 4, and therefore (p + q)² ≤ 2(p² + q²) = 8. Hence |S| ≤ 2√2. The numerical Tsirelson value follows directly from the unit-Gram geometry generated by binary measurement directions of the complete global wavefunctional.
Saturation is itself geometric. Equality requires p = q = √2, which for unit 𝐮_B₀ and 𝐮_B₁ implies 𝐮_B₀·𝐮_B₁ = 0 in the relevant real correlation geometry; Alice's optimal directions then align with the normalized sum and difference of Bob's directions. The value 2√2 is therefore the extremum of a definite geometric configuration rather than an independently imposed numerical rule.
For each actual setting pair the outcome statistics remain setting-conditioned: P(a,b|i,j) ≥ 0, Σₐ,ᵦP(a,b|i,j) = 1 and Eᵢⱼ = Σₐ,ᵦabP(a,b|i,j). No classical joint probability over counterfactual settings is assumed. The four correlations arise instead from one complete global system–apparatus measurement construction. Common Gram geometry is consequently stronger than no-signaling alone: no-signaling fixes separated marginals, while positive Gram structure constrains the joint four-correlation geometry.
Exact Preservation Under Finite Relational Resolution
Let 𝐮_r(Y), with r ∈ {A₀,A₁,B₀,B₁}, be unit measurement vectors transported into a common real Hilbert fiber, and let w_Q₀(Y) ≥ 0 be one normalized Relational-Kernel weight satisfying ∫dμ w_Q₀ = 1. Define Γ_rs^(Q₀) = ∫dμ(Y) w_Q₀(Y)G_ext(𝐮̃_r(Y),𝐮̃_s(Y)). Since every local vector remains unit, Γ_rr^(Q₀) = 1. For arbitrary real coefficients c_r one has c_rc_sΓ_rs^(Q₀) = ∫dμ w_Q₀‖Σ_r c_r𝐮̃_r‖² ≥ 0. Hence Γ^(Q₀) ⪰ 0 with diag Γ^(Q₀) = 1.
The resolved matrix is therefore itself a unit Gram matrix, and its CHSH correlations satisfy |S_Q₀| ≤ 2√2 exactly. Ordinary finite relational resolution preserves the quantum correlation body whenever the four measurement directions remain represented through one common positive normalized second-moment resolver. Q₀-resolution may rotate, attenuate or redistribute accessible structure, but it cannot raise the CHSH ceiling.
Ceiling and Operational Accessibility
Preservation of the outer Gram body does not imply that every resolved experiment must saturate its boundary. Intrinsic finite-Q₀ resolution preserves the common positive correlation geometry, while Gaussoherence has the separate dynamical role of contracting unresolved transverse structure. The causal Relational Kernel then maps surviving relational records into localized operational regions of emergent spacetime. The conceptual sequence is therefore full projective quantum geometry → 2√2 outer Gram ceiling → positive finite-Q₀ resolution preserves that ceiling → Gaussoherence determines which coherent structures survive → causal projection localizes their records in spacetime. This preserves the useful distinction already present in the original section between the geometry establishing the quantum ceiling and the downstream processes controlling operational accessibility.
These downstream operations may change the experimentally realized point within the protected quantum body and can reduce the ability of a particular setup to approach saturation, but no monotonic decrease of every individual CHSH value need be assumed. The robust statement is simply |S_accessible| ≤ 2√2 as long as the common positive Gram representation survives. Q₀ is therefore not a parameter inserted into the numerical Tsirelson value; it is a relational resolution scale.
Resolved Rank and State-Bundle Rank
Finite Q₀ can make both amplitude and projective geometry highly multidirectional. For example, F_IJᴿ,(Q₀) = ⟨s̃_Is̃_J⟩Q₀ can acquire rank greater than one when neighboring transported score directions span several independent relational covectors. Likewise, the resolved projective frame operator obtained from transported D_Iu directions can possess a multidimensional real range E_Q₀ ⊂ T[u(X)]CPⁿ⁻¹. These enlarged ranks are ranks of resolved distinction or tangent geometry; they are not ranks of the quantum-state bundle.
At each X the normalized multicomponent state u(X) selects the single complex line 𝓛_X = span_C{u(X)}, so dim_C𝓛_X = 1. A finite-Q₀ support projector acts on projective tangent directions, whereas a genuine rank-r quantum multiplet would require a Hilbert-space projector P_r(X) = Σₘ₌₁ʳ|u_m(X)⟩⟨u_m(X)| with rank_C P_r = r. Thus rank Fᴿ,(Q₀) > 1 does not imply rank 𝓛 > 1, and higher rank of the resolved projective tangent operator likewise does not imply rank 𝓛 > 1.
Finite relational resolution can therefore increase covariance rank inside an existing fiber without increasing the dimension of the state fiber itself. A nondegenerate multicomponent pure state retains the U(1) ray bundle selected by u(X). A genuine U(r) Wilczek–Zee connection requires an independently preserved degenerate r-dimensional multiplet rather than merely multidirectional QGT or Fisher structure. The manuscript explicitly distinguishes these two cases.
This distinction between resolved tangent rank and state-bundle rank is central to the reflexive Tsirelson result.
Canonical Bell Measurement Through the Fisher Ruler
Consider the CHSH-saturating state |Ψ₀⟩ = |Φ⁺⟩φ_A(x)φ_B(y), with |Φ⁺⟩ = (|00⟩ + |11⟩)/√2, and choose A₀ = σ_z, A₁ = σ_x, B₀ = (σ_z + σ_x)/√2 and B₁ = (σ_z − σ_x)/√2. Then E₀₀ = E₀₁ = E₁₀ = 1/√2 and E₁₁ = −1/√2, while ⟨A_i⟩ = ⟨B_j⟩ = 0.
Take the ideal von Neumann interaction H_int^(ij) = g_AA_iP_x + g_BB_jP_y, with short pointer translations ε_A = g_AΔt and ε_B = g_BΔt. Writing f(x) = |φ_A(x)|² and g(y) = |φ_B(y)|², the joint pointer density expands as ρ_ij = fg − ε_A⟨A_i⟩f′g − ε_B⟨B_j⟩fg′ + ε_Aε_BE_ijf′g′ + ⋯. Because the local means vanish, ∂{ε_A}ρ_ij|₀ = ∂{ε_B}ρ_ij|₀ = 0. The local interaction therefore changes the PhaSe/projective state at first order while leaving the scalar pointer density unchanged at that order.
The first scalar-amplitude response is genuinely joint. Defining s_x = ∂xlnf and s_y = ∂ylng gives χ_ij ≡ ∂{ε_A}∂{ε_B}lnρ_ij|₀ = E_ijs_xs_y. The ordinary Bell correlation therefore appears directly as the coefficient of the first joint amplitude distinction. In sPNP terms, the projective quantum correlation becomes part of the amplitude geometry from which the relational Fisher ruler is subsequently constructed.
For Gaussian pointer densities f(x) ∝ exp[−x²/(2σ_A²)] and g(y) ∝ exp[−y²/(2σ_B²)], define a = σ_A⁻² and b = σ_B⁻². Then s_x = −ax, s_y = −by and χ_ij = E_ijabxy. If τ_A² and τ_B² are the Gaussian variances after multiplication by the centered Relational Kernel, the complete normalized mixed Fisher variation at the packet center is δ_ABF_ijᴿ,(Q₀) = −E_ijC_Q₀ dx⊙dy, where C_Q₀ = ab[aτ_A² + bτ_B² − abτ_A²τ_B²]. For the kernel-narrowed Gaussian cell 0 < aτ_A² < 1 and 0 < bτ_B² < 1, so C_Q₀ > 0.
Thus the Bell correlations genuinely enter the state-dependent Fisher ruler through δF_ijᴿ,(Q₀) ∝ −E_ij dx⊙dy. Finite resolution is important here because a second-moment Fisher construction can retain nonzero directional distinction at a smooth amplitude extremum even when the pointwise score vanishes.
Reflexive Laplace–Beltrami Protection
The reflexive sPNP regime allows the state-dependent relational geometry to feed back into the effective generator, schematically Ψ → Ĝ[Ψ] → Ĥ[Ĝ,Ψ]Ψ; the manuscript already separates this stronger regime from the kinematic linear theory. The minimal scalar realization is Ĥ_ref[ρ] = −(ℏ²/2M_R)(Δ_Ĝ[ρ] − Δ_G₀).
In the two-pointer sector the mixed Bell-pattern Fisher deformation produces δĜ_ij = −κE_ij dx⊙dy, with κ = αC_Q₀ε_Aε_B > 0. For G₀ = I this gives Ĝ_ij⁻¹ = [[1, κE_ij],[κE_ij,1]] + O(κ²), and at the symmetric packet center Δ_Ĝij = ∂_x² + ∂_y² + 2κE_ij∂_x∂_y + O(κ²). Since P_xP_y = −ℏ²∂_x∂_y, the leading reflexive Hamiltonian correction is δĤ_ref^(ij) = (κ/M_R)E_ijP_xP_y + O(κ²). The Bell correlation has therefore modified the physical apparatus/configuration geometry through a definite cross-kinetic term.
For fixed settings define the ideal Bell branch projectors Π_ab^(ij) = Π_a^(A_i)Π_b^(B_j)⊗I_ptr. The von Neumann interaction obeys [H_int^(ij),Π_ab^(ij)] = 0, while the scalar reflexive Hamiltonian obeys [H_ref[ρ],Π_ab^(ij)] = 0 because it acts on relational/configuration variables and as the identity on the internal Bell outcome space. Consequently d⟨Π_ab^(ij)⟩/dt = (i/ℏ)⟨[H_ij,Π_ab^(ij)]⟩ = 0.
The ideal Bell branch weights therefore remain P_reflex(a,b|i,j) = P_QM(a,b|i,j), which gives E_ij^reflex = E_ij^QM and hence D_ij = D_CHSH = 0. The state-dependent Fisher geometry genuinely responds to the Bell correlation while remaining unable to reweight the ideal outcome sectors.
Native Berry-Covariant Protection
The same result survives inclusion of the native PhaSe/Berry connection. Because the single nondegenerate multicomponent state u(X) selects the complex line 𝓛_X, its Berry connection 𝒜_I = i u†∂Iu is U(1). A Berry-covariant reflexive operator may therefore be written L(Ĝ,𝒜) = −|Ĝ|⁻¹ᐟ²D_I[|Ĝ|¹ᐟ²ĜᴵᴶD_J] + 𝓔, with D_I = ∂_I + i𝒜_I.
On the state line the Berry connection acts as 𝒜_I = a_I(X)I_internal, so [𝒜_I,Π_ab^(ij)] = 0 and therefore [D_I,Π_ab^(ij)] = 0. If the zeroth-order endomorphism 𝓔 is likewise scalar or block-preserving on the ideal Bell sectors, then [L_(Ĝ,𝒜),Π_ab^(ij)] = 0. Native PhaSe/Berry covariance therefore does not open a Bell branch-reweighting channel. Finite Q₀ cannot convert this U(1) state connection into a non-Abelian one because it increases resolved tangent rank without enlarging the underlying state line.
Pure-State Reflexive Tsirelson Protection Theorem
The preceding results give a structural theorem for the presently defined pure-state architecture. Suppose the ontic state is one normalized pure multicomponent wavefunctional Ψₐ = R exp[iS/ℏ]uₐ with u†u = 1; finite-Q₀ resolution performs positive normalized averaging and covariant transport of amplitude and projective tangent data without replacing that ray by a higher-rank state multiplet; reflexive feedback enters through the positive relational metric Ĝ[Ψ], its Laplace–Beltrami operator and the native U(1) Berry connection; any additional endomorphism terms remain scalar or block-preserving with respect to the ideal Bell outcome sectors; and measurement is represented by a complete ideal system–apparatus interaction with binary Hermitian outcomes.
Under these conditions [Ĥ_ref^(ij),Π_ab^(ij)] = 0 for every setting pair and outcome sector. The reflexive dynamics therefore cannot reweight the ideal Bell branches, so P_reflex(a,b|i,j) = P_QM(a,b|i,j), E_ij^reflex = E_ij^QM and |S_sPNP| ≤ 2√2. For the standard Bell state and optimal settings, S_sPNP = 2√2.
The protection mechanism is stronger than positivity of the Fisher tensor alone. Mixed Fisher responses need not themselves have definite sign. What fixes the physical Bell result is that the actual scalar and native-U(1) geometric generators remain reducible with respect to the measurement outcome decomposition.
Structural Interpretation
Tsirelson's bound is therefore not introduced as an independent numerical postulate of the present pure-state sPNP architecture. Instead, the correlation hierarchy follows from the same global wavefunctional in three stages. Scalar amplitude responses alone supply a common configuration-space representation and hence the Bell-local ceiling |S| ≤ 2. The complete apparatus-coupled Distinction–PhaSe/QGT geometry supplies the unit-Gram quantum correlation body with |S| ≤ 2√2. Positive finite relational resolution and the specified scalar/U(1) reflexive geometry then preserve that outer quantum body rather than replacing it with a different correlation structure.
Finite Q₀ can make amplitude and projective geometry highly multidirectional without multiplying the ontic state. The canonical Bell interaction first changes PhaSe/projective structure and then imprints its joint quantum correlation into the amplitude/Fisher ruler at mixed order. The resulting Fisher geometry changes the relational ruler and apparatus dynamics, but scalar Laplace–Beltrami and native U(1) Berry-covariant feedback remain block-preserving with respect to the ideal Bell sectors. Their branch weights therefore retain their ordinary quantum values.
The positive structural chain is therefore R-only scalar response → Bell-local configuration geometry → |S| ≤ 2; full Distinction–PhaSe/QGT measurement geometry → unit measurement vectors → common positive Gram geometry → |S| ≤ 2√2; positive finite-Q₀ resolution → multidirectional resolved geometry without state-bundle rank growth → Bell correlations enter the Fisher ruler → scalar/U(1) reflexive feedback → ideal branch-weight preservation → Tsirelson ceiling remains intact. Gaussoherence and causal projection then determine which part of that protected correlation geometry remains operationally accessible and how its surviving records appear in emergent spacetime.
This completes the objective already stated in the earlier Tsirelson section: deriving the Gram representation from the phase-sensitive apparatus-coupled wavefunctional and determining how it behaves under relational coarse-graining and projection. The result is stronger than the original target. The common Gram representation is explicit, positive finite resolution preserves it, the canonical Bell interaction feeds its correlations into the Fisher ruler, and the resulting scalar and native-U(1) reflexive geometry remains unable to reweight the ideal Bell branches.
A beyond-Tsirelson extension would consequently require additional internal ontology or dynamics not generated by the present finite-Q₀ pure-state construction—for example, a physically preserved higher-rank state multiplet carrying a genuinely non-Abelian connection with Bell-sector-mixing matrix elements. Such a structure may be studied as an extension of sPNP, but it is not produced merely by stronger Fisher curvature, higher resolved tangent rank or finite relational resolution. Under the stated pure-state assumptions, the structural result is therefore |S_sPNP| ≤ 2√2.
