- The Spin-Coherent Configuration Arena and the Ontic Geometry of the Interacting Wavefunctional
5.1 From an Interacting State Vector to a Configuration-Space Wavefunction
The spectral analysis of the preceding sections establishes that a physically defined deformation of the interacting Ising ground state resolves into the Hamiltonian excitation spectrum. In the E₈ scaling regime, the lightest members of that spectrum provide sharply organized spectral directions through which Ψ responds. The remaining problem is to determine whether this spectral organization can be represented as intrinsic geometry of the wavefunctional itself rather than solely as geometry of an externally parameterized ray.
For that purpose the many-body state must be represented on a configuration arena whose coordinates describe the physical degrees of freedom of the subsystem. For an N-site spin-½ chain, a natural choice is the product spin-coherent manifold
Q_spin = (CP¹)^N ≃ (S²)^N.
For each site i define
|θᵢ,φᵢ⟩ = cos(θᵢ/2)|↑⟩ + e^{iφᵢ}sin(θᵢ/2)|↓⟩.
A point of Q_spin is
X = (θ₁,φ₁,…,θ_N,φ_N),
with associated product coherent state
|X⟩ = ⊗ᵢ|θᵢ,φᵢ⟩.
An arbitrary N-spin quantum state |Ψ⟩ is then represented by the coherent-state wavefunction
Ψ(X) ≡ ⟨X|Ψ⟩.
We denote this coherent-state transform by
𝒞 : ℋ → 𝓗_coh,
(𝒞Ψ)(X)=⟨X|Ψ⟩.
The coherent states are overcomplete rather than mutually orthogonal. For spin ½ they satisfy a resolution of the identity at each site,
I_i = (1/2π)∫dΩ_i |Ω_i⟩⟨Ω_i|,
with dΩ_i=sinθ_i dθ_i dφ_i. For the N-spin product space,
I = ∫dμ(X)|X⟩⟨X|,
where
dμ(X)=∏ᵢ[dΩ_i/(2π)].
Consequently,
⟨Ψ|Ψ⟩ = ∫dμ(X)|Ψ(X)|².
The continuous function Ψ(X) therefore retains the complete quantum state despite the overcompleteness of the coherent labels.
This construction supplies something that the parameter-space treatment did not. The state is now represented as a wavefunction over a genuine continuous configuration manifold, so derivatives with respect to X describe variation within a fixed physical state rather than change produced by adjusting an external Hamiltonian parameter.
That is the type of configuration dependence required by the sPNP geometry.
5.2 The Coherent Base Is Not the Full Many-Body Ray Space
The coherent-state configuration manifold should not be confused with the complete projective Hilbert space.
For N spins,
dim_R Q_spin = 2N,
whereas the full projective state space is
CP^(2^N−1),
with
dim_R CP^(2^N−1) = 2(2^N−1).
The two spaces therefore have radically different dimensions.
The point of the coherent-state representation is not to replace the full quantum state space by a small manifold of product states. Rather, the coherent configurations provide the base over which the complete state is represented by the generally nonfactorizing function Ψ(X).
To see this explicitly, write a general state in the σᶻ basis as
|Ψ⟩ = Σ_s c_s|s₁,…,s_N⟩.
Then
Ψ(X)=Σ_s c_s ∏ᵢ⟨θᵢ,φᵢ|sᵢ⟩.
For an up spin,
⟨θᵢ,φᵢ|↑⟩ = cos(θᵢ/2),
while for a down spin,
⟨θᵢ,φᵢ|↓⟩ = e^{-iφᵢ}sin(θᵢ/2).
Hence an entangled coefficient array c_s produces a nonfactorizing complex function over all 2N coherent coordinates.
Entanglement has therefore not disappeared. It is encoded globally in the shape of Ψ(X).
This observation is important for the ontology proposed in sPNP. A configuration manifold need not contain one coordinate for every possible quantum ray. The wavefunctional over that manifold can carry correlations that are vastly richer than the dimensionality of the underlying configuration coordinates.
In this sense the coherent-state representation offers a concrete example of the broader sPNP idea that the physical content of Ψ is not exhausted by the dimensionality of the configuration arena on which it is defined.
5.3 An Effective Relational Arena for the Spin Sector
Elsewhere in sPNP, N point particles in three-dimensional space lead naturally to a relational shape-and-scale manifold of dimension 3N−6 after translational and rotational redundancy has been removed.
The spin-chain problem has a different immediate configuration structure. Its low-energy degrees of freedom are spin orientations and collective field variables, not the absolute positions of N distinguishable particles. Forcing the Ising model directly into a 3N−6 positional arena would therefore require an additional microscopic construction explaining how those effective spin degrees of freedom arise from the deeper particle or field configuration.
Such a construction may eventually be important. If sPNP is ultimately to provide one universal relational ontology, an effective spin manifold should presumably arise as a collective sector, quotient, fiber, or reduced description of the more complete underlying configuration geometry.
That deeper identification is not assumed here.
Instead, we define an effective relational spin arena
Q_rel^spin = (CP¹)^N.
The word relational is being used in the restricted sense that the coordinates describe intrinsic physical spin configurations rather than position or orientation relative to an external spatial frame. The manifold possesses its own invariant SU(2) geometry, and redundant local descriptions related by coherent-state phase choice are handled through the corresponding bundle connection.
The construction therefore tests the internal geometric mechanism of sPNP without requiring that the effective spin manifold already be proven identical to the ultimate universal Q_rel.
This is consistent with the broader field-theoretic direction of the parent theory, where the final configuration arena may involve relational field configurations, projective or bundle structure, stratified sectors, or a compatible combination rather than only the particle realization 3N−6.
5.4 The Interacting Wavefunctional on "Q_rel^spin"
Given an interacting ground state |Ψ₀⟩, define
Ψ₀(X)=⟨X|Ψ₀⟩
and write locally
Ψ₀(X)=R₀(X)e^{iS₀(X)/ℏ}.
Its configuration probability density is
ρ₀(X)=|Ψ₀(X)|²=R₀(X)².
The coherent-state resolution of identity gives
∫dμ(X)ρ₀(X)=1.
The state now has an amplitude landscape and phase structure directly over Q_rel^spin.
Even if the ground-state coefficients c_s are real and positive in the σᶻ basis, Ψ₀(X) is generically complex because the coherent-state overlaps contain the phases e^{-iφᵢ}. Thus reality of the static Hamiltonian does not imply that the coherent-state representation has trivial phase geometry.
The amplitude supplies the configuration score one-form
s_I(X) ≡ ∇_I lnρ₀(X).
At the pointwise level the associated symmetric amplitude tensor is
d_IJ(X)=s_I(X)s_J(X).
This object is rank one whenever s_I≠0. It identifies the instantaneous configuration direction along which the logarithmic amplitude changes most strongly, but it does not yet encode the independent directions sampled across a finite physical neighborhood.
The finite-resolution construction introduced later replaces the bare dyad by
Fᴿ,(Q₀)_IJ,
the second moment of nearby score directions after covariant comparison across the Q₀ cell.
This is where the coherent-state representation becomes specifically useful for the parent sPNP construction: R(X), S(X), the native projective geometry, the Berry connection, the resolved amplitude tensor, and the eventual Laplace–Beltrami operator can all be defined on one common configuration manifold.
5.5 The Ising Hamiltonian as Geometry on the Same Arena
The same coherent manifold also carries natural representations of the operators responsible for the interacting dynamics.
For an operator Ô, define its coherent-state lower or Berezin symbol by
O_B(X) ≡ ⟨X|Ô|X⟩.
For a single spin,
⟨σˣ⟩ = sinθ cosφ,
⟨σʸ⟩ = sinθ sinφ,
⟨σᶻ⟩ = cosθ.
For the transverse-plus-longitudinal Ising Hamiltonian
H = −JΣᵢσᶻᵢσᶻᵢ₊₁ − hₓΣᵢσˣᵢ − h_zΣᵢσᶻᵢ,
its lower symbol is
H_B(X)
= −JΣᵢcosθᵢcosθᵢ₊₁ −hₓΣᵢsinθᵢcosφᵢ −h_zΣᵢcosθᵢ.
Similarly,
M_z^B(X)=Σᵢcosθᵢ.
The magnetization is therefore represented by the z component of the natural SU(2) moment map on the product coherent manifold.
These symbols do not replace the quantum operators. H_B(X) is not the exact quantum energy spectrum, and the ordinary product of symbols does not reproduce the full noncommutative operator algebra. The many-body quantum structure remains encoded in the coherent-state transform, its reproducing kernel, and the corresponding Berezin–Toeplitz operator calculus.
Their importance here is more basic: the operators generating the E₈ dynamics, the wavefunctional responding to them, and the geometry used to resolve that wavefunctional can all be represented over the same physical spin configuration arena.
5.6 From the E₈ Spectrum to the Geometry of an Ontic Ψ
The coherent-state representation permits a more precise formulation of what it could mean for the ontic geometry of Ψ to account for a spectrum such as E₈.
In conventional quantum mechanics, the Hamiltonian determines a ground state |Ψ₀⟩ and excited states |Aₐ⟩ satisfying
H|Aₐ⟩=Eₐ|Aₐ⟩.
Near the magnetic Ising scaling limit, the low-energy excitation energies organize into the E₈ spectrum.
Under the coherent-state transform, every one of these states becomes a function on the same configuration arena,
Ψ₀(X)=⟨X|Ψ₀⟩,
Ψₐ(X)=⟨X|Aₐ⟩.
The E₈ particles may therefore be represented as particular stable spectral excitations of a common interacting wavefunctional geometry rather than as objects existing on an unrelated space.
The ground state determines the background amplitude and PhaSe organization of Ψ. The excited states are different coherent-state sections over the same base, and physical form factors become matrix elements relating these sections through operators that themselves possess geometric symbols on Q_rel^spin.
At this level, the geometric language reorganizes the conventional physics but does not yet derive it.
The important ontological claim is that the structure being reorganized is physically real. If Ψ is ontic, then the E₈ excitation hierarchy is not merely a collection of possible measurement outcomes imposed on an otherwise featureless state vector. It is a spectral organization of deformations supported by the physical state itself.
The ground state contains the geometry from which the excitation responses are defined, while the excited sectors reveal the directions in which that interacting geometry can be persistently reorganized.
This provides a natural interacting extension of the Gaussian vacuum picture. In the Gaussian case, the ground-state Fisher kernel and normal-mode spectrum are directly related. The vacuum decomposes into independent oscillator directions, and the spectral structure can be read nearly algebraically from its amplitude geometry.
The E₈ vacuum is not Gaussian and does not admit that simple diagonal relation. Nevertheless, its deformation geometry still decomposes spectrally into physical excitations, as shown in Section 2, and those excitations can now be represented as wavefunctions over the same intrinsic configuration manifold.
The proposed ontic-geometric interpretation is therefore not
E₈ = Fisher metric,
nor
mₐ = κₐ,
nor
φ = a fundamental sPNP constant.
It is instead the more general statement
interacting Ψ → structured configuration-space geometry → allowed spectral deformations and persistent excitation sectors.
E₈ supplies an unusually controlled system in which the final part of that chain is already known independently.
The sPNP question is whether the first two parts can be constructed in a way that naturally organizes the same spectral structure.
5.7 Three Levels of an Ontic-Geometric Explanation
It is useful to distinguish three increasingly strong meanings of the statement that wavefunctional geometry could “explain” E₈.
Representational explanation
At the weakest level, the coherent-state transform places the E₈ ground and excited states on a common intrinsic configuration manifold.
One can calculate
Ψₐ(X)=⟨X|Aₐ⟩
and examine their amplitude, phase, nodal, correlation, and geometric structure.
This establishes where the E₈ spectrum lives in a configuration-space representation, but it does not explain why the E₈ mass ratios occur.
This level follows essentially from conventional coherent-state quantum mechanics.
Geometric organization
A stronger result would occur if independently constructed state geometry systematically organized the E₈ spectral sector.
For example, suppose the ground-state-derived relational operator L_Ĝ has low-cost eigenspaces that strongly overlap the states approaching A₁ and A₂, or suppose particular compatibility eigenspaces of
−(Ĝ⁻¹ω)²
systematically organize the Hamiltonian transition strengths or coherent-state representations of the E₈ particles.
Then the E₈ Hamiltonian spectrum and the ontic state geometry would be revealing the same underlying organization through independently defined structures.
This would be more than a change of representation.
It is the principal kinematic target of the present supplement.
Dynamical explanation
The strongest claim would require the reflexive sPNP geometry itself to determine or constrain the dynamics that generate the E₈ structure.
In such a theory one would have a closure of the schematic form
Ψ → Ĝ[Ψ] → Ĥ[Ĝ,Ψ] → Ψ,
and one would ask whether the resulting state-dependent dynamics predicts the E₈ critical spectrum, its mass ratios, its form-factor hierarchy, or controlled departures from them.
No such derivation is claimed here.
The present work remains primarily at the first and second levels: it constructs the common geometric arena and asks whether an independently defined state-shaped geometry organizes an already known interacting spectrum.
This hierarchy is important because it prevents an ontological interpretation from being mistaken for a dynamical derivation.
The E₈ theory remains the established benchmark. The sPNP construction must earn any stronger explanatory claim by producing a nontrivial relationship that was not inserted into the definitions.
5.8 Why the E₈ System Is an Unusually Strong Test
A generic interacting system would be less useful for this program because a complicated many-body spectrum could make almost any geometric alignment difficult to distinguish from numerical coincidence.
The E₈ Ising theory is unusually restrictive.
Its low-energy spectrum contains a specific finite set of stable particle masses together with structured multiparticle sectors. Its form factors determine how physical operators couple to those excitations. Its static ground-state deformation admits the spectral expansion derived in Section 2. Its real Ising representation gives the exact amplitude-Fisher identity of Section 3. The rotating-field extension supplies an independently established Berry and quantum-geometric structure, as discussed in Sections 4 and 6.
The same physical system therefore supplies several independently calculable layers:
Hamiltonian spectrum,
operator form factors,
amplitude response,
projective quantum geometry,
Berry connection,
and coherent-state configuration geometry.
That overdetermination is precisely what makes the model useful.
If the state-shaped relational geometry later shows a systematic correspondence with the E₈ spectral sector, the relation can be tested against several conventional structures simultaneously.
If no such correspondence appears, the E₈ benchmark is equally useful because it places a concrete limit on how much explanatory work the proposed geometry can perform.
5.9 The Bridge to the Configuration-Space Calculation
The coherent-state lift therefore marks the transition from interpretation to calculation.
For the ground state,
Ψ₀(X)=R₀(X)e^{iS₀(X)/ℏ}
provides the state from which the amplitude geometry will be constructed.
The coherent manifold supplies the native metric and Berry curvature.
The Hamiltonian independently supplies the excitation spectrum.
The resulting architecture is
H → {|Ψ₀⟩,|Aₐ⟩,Eₐ}
followed by
𝒞 : |Ψ⟩ → Ψ(X),
and then
Ψ₀(X) → {ρ, S, Fᴿ,(Q₀), Ĝ, L_Ĝ, ω, κₐ}.
The first arrow is ordinary many-body quantum mechanics.
The second is the standard coherent-state representation.
The third contains the specifically sPNP construction.
The purpose of keeping these stages separate is methodological. If the final geometric objects show a relationship to the E₈ spectral states, that relationship has not been guaranteed merely by defining the Hamiltonian eigenstates themselves.
The next section develops the native metric, Berry connection, and gauge-covariant PhaSe geometry of Q_rel^spin. These structures provide the undeformed projective background against which the amplitude-dependent geometry can then be tested.
- Native Projective Geometry and Gauge-Covariant PhaSe
6.1 Fubini–Study Geometry of a Single Spin
The coherent-state manifold CP¹ carries its own canonical Kähler geometry. For the normalized spin-½ coherent state
|θ,φ⟩ = cos(θ/2)|↑⟩ + e^{iφ}sin(θ/2)|↓⟩,
the Fubini–Study line element is
ds²_FS = ¼[dθ² + sin²θ dφ²].
Thus, in coordinates (θ,φ),
g_FS = ¼ [[1,0],[0,sin²θ]].
The coherent-state Berry connection in the standard north-patch gauge is
𝒜 = i⟨θ,φ|d|θ,φ⟩
= −sin²(θ/2)dφ.
Its curvature is
ω = d𝒜
= −½sinθ dθ∧dφ.
The overall sign reflects the orientation and coherent-state phase convention. With the convention used throughout this paper, the metric and two-form are the symmetric and antisymmetric components of the native projective quantum geometry.
Raising one index of ω with g_FS gives
A₀ = g_FS⁻¹ω.
Explicitly,
A₀ = [[0,−2sinθ],[2/sinθ,0]],
and therefore
−A₀² = 4I₂.
The native compatibility scale is consequently
κ=2.
If
J ≡ A₀/2,
then
J²=−I,
and the metric and two-form constitute the usual compatible Kähler pair, with the factor of two arising from the normalization of ω adopted here.
This provides the elementary null geometry against which the state-dependent deformation will later be measured.
6.2 Product Geometry on "(CP¹)^N"
For N spins the native coherent-state geometry is the direct product of the individual CP¹ factors. The metric is
gQ = ⊕ᵢg_FS^(i),
or
ds²_Q = ¼Σᵢ[dθᵢ² + sin²θᵢ dφᵢ²].
The Berry connection is
𝒜 = Σᵢ𝒜_i
= −Σᵢsin²(θᵢ/2)dφᵢ,
and its curvature is
ω = d𝒜
= −½Σᵢsinθᵢ dθᵢ∧dφᵢ.
Because both structures are direct sums of the compatible single-spin blocks,
−(gQ⁻¹ω)² = 4I_{2N}.
Thus every native Kähler plane carries the same compatibility scale,
κₐ=2.
This is the appropriate null benchmark when the complete symmetric ruler is taken proportional to the native Fubini–Study metric.
The qualification is important. In the more general sPNP construction,
Ĝ_IJ = M_IJ + αFᴿ,(Q₀)_IJ + βgQ_IJ + … ,
and an independent kinetic or inertial contribution M_IJ could itself deform the symmetric geometry. The present spin benchmark deliberately begins with Ĝ₀∝gQ so that the effect of the amplitude-derived term can be isolated.
The parent Distinctions construction already establishes the same principle more generally: a pure Fubini–Study metric remains compatible with its own native two-form, whereas unequal κ scales arise only when additional symmetric structure weights the PhaSe-paired planes differently.
6.3 The Coherent-State Wavefunction as a Bundle Section
The phase of the coherent-state wavefunction
Ψ(X)=⟨X|Ψ⟩
requires a gauge-covariant treatment.
Although Ψ(X) appears locally as a single complex scalar, a coherent state |X⟩ is a local section of the corresponding quantum line bundle over the coherent-state manifold. Under a change of local coherent-state section,
|X⟩ → e^{iχ(X)}|X⟩.
The coherent-state wavefunction therefore transforms oppositely,
Ψ(X) → e^{-iχ(X)}Ψ(X).
Writing locally
Ψ(X)=R(X)e^{iS(X)/ℏ},
the phase transforms as
S → S−ℏχ.
Meanwhile the native coherent-state Berry connection transforms as
𝒜 → 𝒜−dχ.
The combination
Π ≡ dS−ℏ𝒜
is therefore gauge invariant.
In components,
Π_I = ∂_IS−ℏ𝒜_I.
This is the same gauge-covariant PhaSe combination used in the broader sPNP construction. The coherent-state representation therefore supplies a concrete realization of that structure rather than introducing a second, unrelated phase variable.
This also explains why a many-body state whose coefficients are real in the σᶻ basis can possess nontrivial PhaSe structure in the coherent-state representation. Reality of one coordinate representation does not globally trivialize the connection bundle over CP¹.
6.4 PhaSe Curvature and the Native Two-Form
Locally, away from wavefunction nodes or patch singularities,
d²S=0.
Therefore
dΠ = −ℏd𝒜.
Since
d𝒜=ω,
one obtains
dΠ = −ℏω.
The curl of the gauge-covariant PhaSe momentum is therefore determined by the same projective two-form that enters the metric–symplectic compatibility operator.
This removes a potential ambiguity. The phase S(X) does not supply a second independent symplectic curvature that must somehow be combined with the native CP¹ geometry. Its ordinary differential dS is locally exact. The nontrivial curvature resides in the coherent-state connection, and the gauge-invariant combination Π carries that curvature through
dΠ = −ℏω.
The same ω consequently has two mutually consistent descriptions:
as the native projective two-form of the coherent-state manifold,
and as the curvature appearing in the gauge-covariant PhaSe sector.
The later compatibility calculation will retain this native ω while allowing the symmetric ruler Ĝ to be deformed by the amplitude geometry of the actual many-body state.
6.5 Relation to the E₈ Many-Body Berry Geometry of Wang, He, and Wu
The appearance of Berry connection geometry in the present construction has a direct conventional E₈ counterpart. Wang, He, and Wu studied the time-dependent Ising chain with a small longitudinal field and a slowly rotating transverse field, whose instantaneous low-energy scaling theory is governed by the E₈ integrable field theory. Their analysis provides an important reference point because it demonstrates explicitly that many-body E₈ quantum geometry can possess both a controlled spectral decomposition and measurable dynamical consequences.
Their Berry connection is defined between instantaneous many-body eigenstates,
γ_nm(t) = i⟨φ_n(t)|∂ₜφ_m(t)⟩.
For a class of time-dependent systems whose instantaneous low-energy description is an integrable quantum field theory and whose effective adiabatic generator is an integral of local field operators, Wang et al. prove an asymptotic thermodynamic result: a Berry-connection matrix element connecting states with total quasiparticle number 𝒵 scales as
γ_NM ∼ L^{1−𝒵/2}.
Higher-particle processes are therefore suppressed with increasing system size, so that in the thermodynamic limit the surviving Berry-connection matrix can be characterized by processes involving at most two quasiparticles. In the Ising example these matrix elements can consequently be evaluated using E₈ form factors.
This result is particularly relevant to the spectral-geometric viewpoint developed here. It shows in an established many-body calculation that the connection geometry of a quantum state can be organized directly by the physical excitation content of an interacting integrable field theory. E₈ particles do not merely appear in the Hamiltonian spectrum; their form factors also resolve the Berry-connection matrix associated with the evolving many-body eigenstates.
Wang et al. go further by constructing the gauge-invariant quantum geometric potential
Q_nm = γ_mm−γ_nn + d[arg γ_nm]/dt.
This quantity contributes to their effective energy gap,
Δ_nm = E_n−E_m + Q_mn,
and can significantly alter adiabatic dynamics. In their rotating-field E₈ example, the quantum geometric contribution can suppress effective gaps and enhance many-body Landau–Zener tunneling.
This provides a useful conventional comparison with the PhaSe interpretation adopted here. The raw phase of an instantaneous eigenstate is gauge dependent, whereas an appropriate combination of connection terms and relative phase information is gauge invariant and can influence physical dynamics. The specific Wang–He–Wu quantum geometric potential is not identical to the sPNP quantity Π=dS−ℏ𝒜, but both constructions emphasize the same underlying requirement: physically meaningful phase dynamics must be formulated through gauge-covariant or gauge-invariant connection structure rather than through a bare phase convention.
There is also a useful separation between spectrum and connection geometry in their model. The instantaneous Hamiltonians are related by a global spin rotation, so the instantaneous eigenvalues themselves remain unchanged during the rotation, while the eigenstates acquire nontrivial connection geometry. That geometry nevertheless contributes to the effective dynamical gaps governing transitions.
This is conceptually important for the present framework. It provides a concrete example in established many-body physics in which PhaSe/connection geometry carries physical dynamical information that is not reducible to the instantaneous energy spectrum alone.
The result supports the motivation for treating connection geometry as a physically significant sector of Ψ. It does not establish the specifically sPNP claims that the wavefunction is ontic, that amplitude Fisher structure deforms the physical configuration-space ruler, or that finite-Q₀ resolution governs which relational directions survive. Those are additional hypotheses tested later in this supplement.
The domains must also remain distinct. Wang et al. construct Berry geometry on the time-dependent instantaneous-eigenstate bundle. The present section constructs the native Berry connection and curvature directly on the coherent-state configuration manifold "(CP¹)^N". The former is conventional parameter/time-dependent many-body geometry; the latter supplies the common intrinsic base required for the later sPNP compatibility calculation.
The two constructions are therefore complementary rather than identical:
Wang–He–Wu: E₈ excitation spectrum → form factors → many-body Berry-connection matrix → gauge-invariant geometric contribution to dynamics.
Present construction: coherent-state configuration manifold → native Berry connection ω → gauge-covariant PhaSe Π → state-deformed metric Ĝ → intrinsic compatibility spectrum.
The first establishes that E₈ many-body connection geometry is physically and spectrally nontrivial. The second asks how such projective structure behaves when amplitude geometry itself contributes to the intrinsic relational ruler.
A notation warning is necessary. Wang et al. denote the dimensionless ratio h_z/(ω₀/2) by κ. That quantity is unrelated to the compatibility eigenvalues κₐ used throughout this supplement. To prevent confusion, whenever their drive/field ratio is required below it will be denoted r_W rather than κ.
6.6 Amplitude and PhaSe as Complementary Structures on One Base
The coherent-state lift now provides a single configuration manifold on which the relevant structures coexist.
The amplitude is
ρ(X)=|Ψ(X)|²,
with configuration score
s_I = ∇_Ilnρ.
The gauge-covariant PhaSe/current one-form is
Π_I = ∂_IS−ℏ𝒜_I.
The native antisymmetric structure is
ω=d𝒜.
The projective symmetric geometry is
gQ.
All of these objects live on the same tangent or cotangent bundle over Q_spin.
This common base is the essential step that was absent from the parameter-space benchmark of Section 4 and from the time-dependent Berry-connection construction of Wang et al.
At this stage no state-shaped deformation is required. If the complete metric is initially chosen as
Ĝ₀ = βgQ,
with constant β>0, the metric–symplectic structure remains compatible up to the common normalization.
The specifically sPNP problem begins when the amplitude-derived resolved tensor is incorporated,
Ĝ_IJ = M_IJ + αFᴿ,(Q₀)_IJ + βgQ_IJ + … .
The two-form ω remains the native projective curvature, while the symmetric ruler is altered by the amplitude structure of Ψ.
The operator
A = Ĝ⁻¹ω
then asks whether the complete resolved ruler continues to normalize all native PhaSe-paired directions equally.
Unlike the ordinary E₈ Berry calculation, this metric deformation is not supplied by the Wang–He–Wu theory. It is the new hypothesis being tested by the present sPNP construction.
6.7 Why Rank Four Is the Minimal Nontrivial Benchmark
A single CP¹ factor has two real dimensions. Any nondegenerate two-form on a real two-dimensional vector space possesses only one compatibility scale with respect to a positive metric. A single spin can therefore exhibit a shifted value of κ but cannot display internal direction-dependent κ splitting.
The smallest spin-coherent arena capable of genuine splitting is
CP¹×CP¹,
with real dimension four.
This makes the interacting N=2 Ising system the minimal useful benchmark.
At the native level,
−(gQ⁻¹ω)² = 4I₄,
and therefore
κ=(2,2,2,2).
When the amplitude of an interacting two-spin ground state is allowed to deform the symmetric metric, the two native Kähler planes need no longer receive the same normalization.
The pointwise Q₀→0 limit already gives an exactly solvable case. There,
Fᴿ → d = s⊗s,
so the amplitude correction is rank one.
The following section shows analytically that this rank-one update deforms exactly the Kähler plane generated by s^♯ and Js^♯ while leaving its orthogonal Kähler plane at the native value κ=2.
Finite Q₀ changes the problem qualitatively. Score covectors sampled throughout a finite neighborhood need not remain parallel. Their resolved second moment can therefore acquire higher rank and deform additional PhaSe-paired planes.
The sequence
native Kähler geometry → pointwise rank-one amplitude deformation → finite-resolution multidirectional amplitude geometry
provides a controlled progression in which the proposed compatibility mechanism can be examined without conflating it with the conventional E₈ Berry geometry.
6.8 Relation to the E₈ Spectral Problem
The intrinsic geometry constructed above does not depend on first assigning a particular E₈ particle to a particular configuration direction. This ordering is deliberate.
The Hamiltonian spectrum and the configuration-space geometry should be constructed independently before their relationship is tested. Otherwise a desired numerical correspondence could be built into the definitions rather than discovered.
For a finite chain, every Hamiltonian eigenstate |Aₐ⟩ can be represented directly on the coherent-state manifold by
Ψₐ(X)=⟨X|Aₐ⟩.
The ground state determines the amplitude geometry and, eventually, the state-shaped ruler Ĝ[Ψ₀]. The operator
−(Ĝ⁻¹ω)²
determines the local compatibility geometry. The Hamiltonian independently determines the energy spectrum and spectral eigenvectors.
Wang, He, and Wu establish from the conventional side that E₈ quasiparticle form factors can directly organize a many-body Berry-connection matrix. The present construction asks a complementary question: when the same interacting states are represented intrinsically on the coherent-spin manifold, does the state-shaped amplitude geometry organize those spectral directions in a systematic way?
The eventual E₈ comparison is therefore one of structural alignment and survival rather than numerical identification.
Do the low-energy states approaching the E₈ particles reconstruct from relatively smooth or low-cost structures under the relational resolver?
Do the dominant spectral states preferentially overlap particular compatibility sectors?
Does finite resolution preserve the low-particle E₈ sector that conventional form-factor theory already shows to dominate important pieces of the many-body connection geometry?
Do Hamiltonian spectral directions and state-shaped geometric eigendirections align, remain systematically rotated, or become reorganized as Q₀ changes?
These questions go beyond the Wang–He–Wu result while using it as an unusually relevant conventional benchmark.
Their work establishes that interacting E₈ excitation data can organize physically consequential many-body connection geometry. The additional sPNP question is whether amplitude-dependent relational geometry provides a second, intrinsic organization of the same state and whether the two structures exhibit a reproducible relationship.
Sections 5 and 6 establish the common configuration framework required to pose that question. The next step is the minimal N=2 benchmark, where the effect of a state-shaped amplitude deformation on the native compatibility spectrum can first be solved exactly.
