after the new §7.10, then renumber the current §§7.11–7.13 upward by one.
7.11 Nodal Geometry and Finite-Resolution Regularization
The large variation of the local score norm can be understood analytically from the nodal geometry of the two-spin coherent-state wavefunction.
Introduce stereographic coherent-state coordinates
zᵢ = e⁻ⁱφⁱ tan(θᵢ/2).
For the real symmetric N=2 ground state
|Ψ₀⟩ = a|↑↑⟩ + b|↑↓⟩ + b|↓↑⟩ + d|↓↓⟩,
the coherent-state wavefunction takes the form
Ψ₀(z₁,z₂)
[a + bz₁ + bz₂ + dz₁z₂] / √[(1+|z₁|²)(1+|z₂|²)].
For the state used above,
a≈0.73054231,
b≈0.36048293,
d≈0.45432592.
Since the denominator is nonsingular within the stereographic patch, the nodal set is determined by the holomorphic polynomial
P(z₁,z₂)
a + bz₁ + bz₂ + dz₁z₂
through
P(z₁,z₂)=0.
Solving for one coordinate gives
z₂
−(a+bz₁)/(b+dz₁).
The determinant-like combination
ad−b²≈0.20195636
is nonzero. Consequently the two derivatives of P cannot vanish simultaneously on P=0, and the zero set is smooth. In CP¹×CP¹ it therefore defines a complex one-dimensional nodal submanifold, equivalently a real two-dimensional surface of real codimension two.
This structure explains the very large score observed at the configuration X_A of Section 7.10.
For
X_A=(1.1,0.4,2.0,3.2),
one has
ρ(X_A)≈0.00413056
and
q(X_A)
‖d lnρ‖²_g ≈930.3866.
Minimizing the native product Fubini–Study distance from X_A to the exact nodal curve gives a nearest nodal configuration approximately
X_N ≈ (1.088388,0.371600,1.965717,3.065717),
with normal distance
r_⊥
d_FS(X_A,N) ≈0.065346.
The combination
q(X_A)r_⊥² ≈3.973
is already extremely close to the universal simple-node limit
q r_⊥² → 4.
The origin of this behavior can be derived directly.
Near any smooth point of the nodal surface, choose locally orthonormal real coordinates (x,y) in the two-dimensional normal plane. Because P has a simple complex zero, the wavefunction can locally be written to leading order as
Ψ₀ ≈ C₀(x+iy),
after an admissible linear rotation and rescaling of the normal coordinates.
Hence
ρ
|Ψ₀|² ≈ C r_⊥²,
where
r_⊥²=x²+y²
and C>0 is smooth and nonzero at the node.
Therefore
lnρ ≈ lnC + 2lnr_⊥,
so that
d lnρ ≈ 2dr_⊥/r_⊥.
The invariant pointwise score norm consequently obeys
q
‖d lnρ‖²_g ≈ 4/r_⊥².
Thus the large value q≈930 at X_A is quantitatively explained by its proximity to the smooth nodal surface. The pointwise Fisher geometry becomes increasingly stiff as the node is approached.
The singularity of q does not, however, imply that the finite-resolution tensor is itself singular.
The quantity entering the numerator of the resolved tensor is not s⊗s alone but
ρ s⊗s,
with
s=d lnρ.
Near the node,
s ≈ 2n/r_⊥,
where n is the unit radial covector in the normal plane. Therefore
ρ s⊗s ≈ Cr_⊥² (4/r_⊥²) n⊗n
= 4C n⊗n.
The divergence cancels.
Although the pointwise logarithmic score is undefined at the node itself, the density-weighted tensor entering the finite-Q₀ construction remains bounded.
This cancellation allows the resolved tensor at the node to be calculated analytically in the leading short-distance approximation.
Let the local Relational Kernel be isotropic with variance
ℓ₀²=2Q₀²
in each orthonormal direction.
Because the nodal surface has real codimension two,
⟨r_⊥²⟩
2ℓ₀².
The resolved density at the node is therefore
ρ_Q₀(X_N) ≈ 2Cℓ₀².
Meanwhile angular averaging in the two-dimensional normal plane gives
⟨n⊗n⟩
½P_⊥,
where P_⊥ denotes the metric projector onto the normal plane.
Hence the numerator of the resolved tensor becomes
⟨ρ s⊗s⟩ ≈ 2C P_⊥.
Dividing by the resolved density yields
Fᴿ,(Q₀)(X_N) ≈ (1/ℓ₀²)P_⊥
= [1/(2Q₀²)]P_⊥
O(1).
This is a qualitatively different limit from the ordinary pointwise rank-one result.
Away from a node,
Q₀→0
gives
Fᴿ,(0)=s⊗s,
which is rank one whenever s≠0.
At a simple node, however, the score direction rotates through the entire two-dimensional normal plane as the node is approached from different angular directions. Finite resolution therefore produces an effective rank-two tensor,
Fᴿ,(Q₀) ∝ P_⊥,
that resolves the normal geometry of the nodal surface.
The effect can also be seen in the compatibility spectrum.
Because the nodal set is a complex submanifold of CP¹×CP¹, both its tangent plane and its orthogonal normal plane are J-invariant. To leading order at the node,
Fᴿ,(Q₀) ≈ f_Q P_⊥,
with
f_Q
1/(2Q₀²).
For the minimal benchmark metric
G
g + αFᴿ,(Q₀),
the normal Kähler plane is therefore scaled isotropically by
1+αf_Q,
while the tangent Kähler plane remains unchanged at leading order.
The compatibility spectrum becomes
κ_⊥ ≈ 2/[1+α/(2Q₀²)],
κ_∥ ≈ 2.
This differs from the point-limit rank-one formula
κ_-
2/√(1+η),
because the finite-resolution nodal tensor stretches both orthogonal axes of the normal Kähler plane equally rather than only one score axis.
The analytic prediction is reproduced by the local finite-resolution calculation. For example, at
Q₀=0.02,
one has
f_Q
1/(2Q₀²)
The numerically evaluated leading metric-normalized eigenvalues are approximately
1253
and
1243,
with the remaining two directions much smaller.
For
α=0.001,
the analytic compatibility prediction is
κ_⊥ ≈ 2/(1+1.25)
0.888889,
while the numerical calculation gives approximately
κ_⊥≈0.8885,
with the tangent pair remaining close to
κ_∥≈2.
The agreement confirms that the local finite-resolution geometry is resolving the normal complex plane of the nodal surface.
This result provides a second and independent motivation for Q₀-resolution.
Finite resolution does not merely collect several ordinary score directions and thereby increase the effective rank of the amplitude tensor. It also provides a well-defined geometry precisely where the pointwise logarithmic score becomes singular.
The sequence is therefore
smooth off-node region → rank-one local score geometry,
near-node region → q≈4/r_⊥² and strong pointwise stiffness,
exact node at finite Q₀ → finite rank-two resolved normal geometry.
In this sense the Relational Kernel converts a singular pointwise Fisher description into a regular geometric description of the nodal structure itself.
The nodal surface is therefore not an exceptional set that must simply be removed from the configuration manifold. Its surrounding amplitude structure leaves a finite and highly organized signature in the resolved Distinction tensor.
This observation will be important when extending the construction to larger interacting systems, where the nodal sets of many-body coherent-state wavefunctions can become substantially more complicated and may form an important part of the finite-resolution geometry of Ψ.
near the end of Section 8, just before the limitations/status material. It works best as the covariant validation of the local Gaussian benchmark.
8.10 Covariant Native-Geometry Cross-Check
The finite-Q₀ calculation above used a local Gaussian approximation in the tangent space at X₀. This was sufficient to demonstrate the basic mechanism of multidirectional score averaging, but it did not yet include the exact curvature, measure, or parallel transport of the native coherent-state manifold.
For
Q_rel^spin = CP¹×CP¹,
the native geometry is simple enough that this approximation can be removed without yet solving the full self-consistent sPNP fixed-point problem.
Each CP¹ factor carries
ds² = ¼(dθ² + sin²θ dφ²),
so it is a round two-sphere of radius
R=1/2.
The scalar Laplace–Beltrami spectrum is therefore
λ_ℓ = ℓ(ℓ+1)/R²
= 4ℓ(ℓ+1),
with degeneracy 2ℓ+1.
The exact scalar heat kernel on one CP¹ can consequently be written spectrally as
K_Q₀^(CP¹)(γ)
= (1/π) Σ_{ℓ=0}^∞ (2ℓ+1) e^[−4Q₀²ℓ(ℓ+1)] P_ℓ(cosγ),
where γ is the ordinary angular separation of the corresponding points on the unit sphere.
Because the native metric on CP¹×CP¹ is a direct product, the heat kernel factorizes,
K_Q₀(X,Y)
= K_Q₀^(CP¹)(γ₁) K_Q₀^(CP¹)(γ₂).
The resolved tensor can therefore be evaluated using the exact curved measure, the exact native heat kernel, and Levi-Civita parallel transport of each score covector along the shortest native geodesic from Y to X.
This calculation remains kinematic and uses the native metric gQ as the resolving geometry. It therefore does not yet implement the stronger self-consistency condition in which the heat kernel is constructed from the final state-shaped metric Ĝ. Its purpose is narrower: to test whether the finite-resolution effects found above survive replacement of the local tangent-space Gaussian by the actual covariant geometry of CP¹×CP¹.
At the representative point
X₀=(π/2,0.7,π/3,1.9),
the exact point-limit value remains
q(X₀)≈8.59178.
The covariant finite-Q₀ calculation gives the following metric-normalized eigenvalues of Fᴿ,(Q₀):
Q₀| covariant native-geometry eigenvalues 0.01| 8.58539, 0.001211, 0.000723, 0.000500 0.05| 8.43368, 0.03000, 0.01962, 0.01240 0.10| 7.97514, 0.11655, 0.09890, 0.04874 0.20| 6.38385, 0.58681, 0.40863, 0.19320
The same qualitative effect found in the local Gaussian calculation survives.
As Q₀ increases, the dominant pointwise score direction loses part of its relative spectral weight, while additional independent directions acquire nonzero stiffness.
The amount of redistribution is, however, smaller than in the flat local approximation.
For example, the participation rank
r_eff
(Tr𝔉)²/Tr(𝔉²)
becomes approximately
r_eff≈1.015 at Q₀=0.05,
r_eff≈1.067 at Q₀=0.10,
and
r_eff≈1.388 at Q₀=0.20.
The corresponding local-Gaussian values were approximately
1.038,
1.159,
and
1.75.
Native curvature and geodesic transport therefore do not remove finite-resolution multidirectionality, but they materially affect its quantitative strength.
This distinction is important.
The finite-Q₀ tensor is not determined solely by the radius of a Gaussian neighborhood. It depends on the actual geometry through the heat kernel, invariant measure, and transport of score directions.
The compatibility calculation shows the same robustness.
For
α=0.01
and
Q₀=0.20,
the local Gaussian benchmark gave approximately
κ_-≈1.94232,
κ_+≈1.99297.
The covariant native calculation instead gives approximately
κ_-≈1.93348,
κ_+≈1.99392.
Both Kähler pairs remain displaced from the native value κ=2.
Thus the principal conclusion of the finite-resolution construction survives: once the resolved amplitude tensor develops appreciable support outside the pointwise score-selected Kähler plane, the second compatibility pair is no longer exactly protected.
The curved calculation also clarifies the redistribution effect.
The native heat kernel retains more of the original dominant score stiffness than the flat approximation, so the initially active compatibility pair remains somewhat more strongly deformed. At the same time, sufficient transverse score structure is still resolved to move the second pair away from κ=2.
Finite resolution therefore remains both smoothing and multidirectional, but the balance between those effects is geometrically controlled.
Covariant Check at the Nodal Surface
The nodal calculation of Section 7 provides an even sharper test.
At a smooth node of the two-spin coherent wavefunction, the local analysis predicted
Fᴿ,(Q₀) ≈ [1/(2Q₀²)]P_⊥,
where P_⊥ projects onto the real two-dimensional normal Kähler plane of the nodal CP¹.
This result can be tested using the same exact spherical heat kernel and geodesic parallel transport.
The leading metric-normalized eigenvalues are:
Q₀| 1/(2Q₀²)| covariant leading eigenvalues 0.01| 5000| 5000.61, 4999.45 0.02| 1250| 1250.16, 1249.86 0.05| 200| 200.034, 199.986 0.10| 50| 50.0383, 50.0294
The agreement is essentially exact at small Q₀.
The remaining two eigenvalues are strongly suppressed, confirming that the resolved amplitude tensor selects the normal complex plane of the nodal surface.
The compatibility prediction is equally accurate.
At
Q₀=0.02
and
α=0.001,
the leading analytic nodal result gives
κ_⊥ ≈ 2/[1+α/(2Q₀²)]
= 0.888889,
while the covariant native-geometry calculation gives
κ_⊥≈0.888884.
The tangent Kähler pair remains close to
κ_∥≈1.99988.
This agreement establishes that the rank-two nodal geometry derived in Section 7 is not an artifact of the local flat approximation.
Interpretation
The covariant calculation therefore supports two distinct finite-resolution effects.
Away from nodes, finite Q₀ converts a nearly rank-one pointwise score geometry into an increasingly multidirectional tensor because transported score directions rotate across the resolution cell.
At a smooth node, finite Q₀ performs a stronger role: the pointwise logarithmic score is singular, but its density-weighted second moment resolves into a finite rank-two geometry associated with the normal complex plane of the nodal surface.
Both phenomena survive exact native curvature and parallel transport.
The hierarchy of calculations is therefore
point limit,
→ local finite-Q₀ Gaussian resolution,
→ exact covariant finite-Q₀ native geometry,
→ eventual self-consistent Ĝ-based resolution.
The present cross-check completes the third step.
It does not yet solve the final self-measuring fixed-point problem because the heat kernel continues to be generated by gQ rather than by the converged state-shaped metric Ĝ★.
Nevertheless, it removes the possibility that the principal finite-resolution effects reported in this section arise solely from flattening CP¹×CP¹ into a local Euclidean chart.
The remaining quantitative differences between the local and covariant calculations should instead be interpreted as evidence that the geometry of the resolution kernel itself is physically significant.
- Exact N=2 Point-Limit Compatibility Theorem
7.1 Why Two Spins Are the Minimal Nontrivial Case
The coherent-state construction developed above gives the intrinsic spin configuration manifold
Q_rel^spin = (CP¹)^N.
For N=1 the real tangent space has dimension two. A positive metric paired with a nondegenerate two-form on a real two-dimensional space possesses only one compatibility scale. A single spin can therefore exhibit a shifted value of κ, but it cannot display direction-dependent internal κ-splitting.
The first nontrivial case is
Q₂ = CP¹×CP¹,
with real dimension four.
At the undeformed level, the native Fubini–Study metric g and Berry curvature ω satisfy
−(g⁻¹ω)² = 4I₄,
so the compatibility spectrum is
κ = (2,2,2,2).
This is the rank-four null benchmark established in Section 6.
The simplest state-dependent deformation occurs in the point limit Q₀→0. In this limit the resolved amplitude tensor reduces to the local score dyad,
Fᴿ,(Q₀) → d,
with
d = s⊗s,
s = d lnρ.
Whenever s≠0, this tensor has rank one. Although this is the simplest possible amplitude correction, four real dimensions are already sufficient for it to change one PhaSe-active plane differently from another.
The resulting problem can be solved exactly.
7.2 Pointwise State-Shaped Metric
Let g denote the native Fubini–Study metric on CP¹×CP¹ and ω its native projective two-form. Define the native complex structure J by
g⁻¹ω = 2J,
so that
J² = −I.
Let
ρ(X)=|Ψ(X)|²
be the coherent-state probability density of the interacting ground state and define the score one-form
s = d lnρ.
Raise its index with the native metric,
v ≡ s^♯ = g⁻¹s.
Its invariant squared norm is
q ≡ g(v,v)
= g⁻¹(s,s)
= ‖d lnρ‖²_g.
In the Q₀→0 benchmark, take the state-shaped symmetric ruler to be
G = g + αs⊗s,
with α≥0.
The compatibility operator is
A = G⁻¹ω.
It is convenient to define the dimensionless local deformation strength
η ≡ αq
= α‖d lnρ‖²_g.
This is the invariant quantity controlling the local departure from the native metric. The bare coefficient α has no independent geometric meaning at a given point without the accompanying score norm q.
If s=0, then q=η=0 and G=g, so the native spectrum κ=(2,2,2,2) is recovered immediately.
For s≠0, the rank-one structure permits an exact solution.
7.3 Exact Inverse Metric
The Sherman–Morrison identity gives
G^{IJ}
g^{IJ} − [α/(1+αq)]v^Iv^J.
Equivalently, as an endomorphism of the tangent space,
G⁻¹g
I − [α/(1+η)]v⊗s,
where
(v⊗s)(Y)=v s(Y).
Since
ω = 2gJ,
the compatibility operator may be written
A = G⁻¹ω
= 2(G⁻¹g)J.
The problem is therefore reduced to understanding how the rank-one operator G⁻¹g acts relative to the native complex structure J.
7.4 The Score-Selected Kähler Plane
Compatibility of g and J implies
g(JX,JY)=g(X,Y)
and
g(X,JX)=0.
In particular,
g(v,Jv)=0.
The score therefore selects the J-invariant real two-plane
E_s ≡ span{v,Jv}.
Its g-orthogonal complement is
E_s^⊥.
Because E_s is J-invariant and g is Hermitian with respect to J, E_s^⊥ is also J-invariant. Thus
T_XQ₂ = E_s ⊕ E_s^⊥
is a decomposition into two orthogonal native Kähler planes.
The rank-one correction acts only through
s(Y)=g(v,Y).
For every Y∈E_s^⊥,
s(Y)=0.
Hence
G⁻¹gY=Y
on the entire orthogonal plane.
It follows that
AY=2JY,
and therefore
−A²Y=4Y.
The compatibility scale on E_s^⊥ remains exactly
κ_+=2.
The pair of eigenvalues pinned at the native value is therefore required by the geometry; it is not a numerical accident.
7.5 Exact Compatibility Scale on the Active Plane
Now restrict A to E_s.
Choose the g-orthonormal basis
e₁ = v/√q,
e₂ = Je₁.
Then
s(e₁)=√q
and
s(e₂)=g(v,Je₁)=0.
Consequently,
(G⁻¹g)e₁ = e₁/(1+η),
while
(G⁻¹g)e₂ = e₂.
The complex structure acts as
Je₁=e₂,
Je₂=−e₁.
Therefore
Ae₁=2e₂,
and
Ae₂=−2e₁/(1+η).
In the ordered basis (e₁,e₂),
A|_{E_s}
2 [[0, −1/(1+η)], [1, 0]].
Squaring gives
−A²|_{E_s}
[4/(1+η)]I₂.
The compatibility scale on the score-selected plane is therefore
κ_- = 2/√(1+η),
or equivalently
κ_- = 2/√[1+α‖d lnρ‖²_g].
The complete compatibility spectrum is
κ = (κ_-,κ_-,2,2).
This result is exact.
7.6 Rank-One Compatibility Theorem
The calculation may be summarized independently of the particular coordinates of CP¹×CP¹.
Rank-One Compatibility Theorem. Let (V,g,J,ω) be a 2n-dimensional Kähler vector space normalized by
ω=2gJ.
Let s be a covector, define v=s^♯, and let
G=g+αs⊗s,
with α≥0. Set
q=‖s‖²_g
and
η=αq.
For s≠0, V decomposes as
V = span{v,Jv} ⊕ span{v,Jv}^⊥.
For
A=G⁻¹ω,
one has
−A²
[4/(1+η)]I
on span{v,Jv},
while
−A²=4I
on its J-invariant orthogonal complement.
Hence the compatibility scale is
κ_-=2/√(1+η)
on the Kähler plane generated by the score, while all 2n−2 orthogonal real directions retain the native value
κ=2.
For n=2,
κ=(κ_-,κ_-,2,2).
Thus a rank-one symmetric amplitude deformation modifies exactly one native Kähler plane.
This higher-dimensional statement is useful because the result is not peculiar to the two-spin coordinate system. It follows entirely from the interaction between a rank-one positive metric update and an initially compatible Kähler structure.
7.7 Geometric Interpretation
At a fixed configuration X, the score
s=d lnρ
selects the local direction in which the logarithmic amplitude varies.
Because the native projective geometry pairs this direction with Js^♯, the amplitude correction naturally identifies a full Kähler plane,
span{s^♯,Js^♯}.
Adding
αs⊗s
to the symmetric ruler stretches only the score axis of this plane. The native antisymmetric form ω is left unchanged.
The result is therefore not a destruction of the underlying PhaSe geometry. It is a change in the relative normalization between the symmetric ruler and the native two-form.
The compatibility scale of the affected plane decreases from
κ=2
to
κ=2/√(1+η),
while the orthogonal Kähler plane remains exactly at κ=2.
The state-shaped amplitude geometry has therefore produced genuine direction-dependent compatibility in the smallest projective spin manifold on which such splitting is possible.
7.8 Weak- and Strong-Deformation Limits
The exact formula also makes the limiting behavior transparent.
For
η≪1,
one has
κ_-
2/√(1+η)
= 2−η+3η²/4+O(η³).
Thus weak state-shaped deformation produces a linear reduction of the affected compatibility scale.
For
η≫1,
κ_- ≈ 2/√η.
The affected plane becomes increasingly mismatched relative to the fixed native two-form, while the orthogonal plane remains exactly compatible at κ=2.
The local effect is therefore controlled entirely by η.
A small value of α does not necessarily imply a weak deformation if the score norm is large, while a comparatively large α can remain mild in a region where the amplitude varies slowly.
This point is important when numerical values are quoted.
7.9 Corrected Interacting Two-Spin Example
Consider the N=2 transverse-plus-longitudinal Ising Hamiltonian
H
−σᶻ₁σᶻ₂ −hₓ(σˣ₁+σˣ₂) −h_z(σᶻ₁+σᶻ₂),
with
hₓ=1,
h_z=0.15.
In the ordered σᶻ basis
(|↑↑⟩,|↑↓⟩,|↓↑⟩,|↓↓⟩),
the normalized ground state is
|Ψ₀⟩ ≈ 0.730542|↑↑⟩ + 0.360483|↑↓⟩ + 0.360483|↓↑⟩ + 0.454326|↓↓⟩.
The coefficients are real and positive, as expected for the static stoquastic Hamiltonian.
Now represent this state on CP¹×CP¹ through
Ψ₀(X)=⟨X|Ψ₀⟩.
At the representative configuration
θ₁=π/2,
φ₁=0.7,
θ₂=π/3,
φ₂=1.9,
one finds, using the coherent-state convention introduced in Section 5,
Ψ₀(X₀) ≈ 0.437359−0.345621i.
Thus
ρ(X₀)
|Ψ₀(X₀)|² ≈ 0.310737,
and
arg Ψ₀(X₀) ≈ −0.668765.
The sign of this local phase depends on the coherent-state convention, whereas the amplitude geometry below does not.
Analytic differentiation of Ψ₀(X) gives
s_I(X₀)
∂_I lnρ(X₀)
≈ (−0.411626, −0.564012, −0.636183, −0.970439).
With the native product Fubini–Study metric,
g
¼ diag(1,sin²θ₁,1,sin²θ₂),
the invariant score norm is
q
g⁻¹(s,s)
≈ 8.59178.
The dimensionless deformation strength is therefore
η≈8.59178α.
For example, at
α=0.01,
η≈0.085918,
and the exact theorem gives
κ_-
2/√(1+η)
≈ 1.91925.
Direct diagonalization of
−(G⁻¹ω)²
gives
κ ≈ (1.91925,1.91925,2.00000,2.00000),
in exact agreement with the theorem to numerical precision.
A few representative values are
α| η| κ_- 0.0001| 0.000859| 1.99914 0.001| 0.008592| 1.99146 0.01| 0.085918| 1.91925 0.1| 0.859178| 1.46680 0.5| 4.29589| 0.86908 2.0| 17.1836| 0.46902
The second compatibility pair remains exactly at κ=2 for every α in the point-limit model.
The numerical calculation therefore verifies the analytic result rather than supplying the result itself.
7.10 Why the Earlier Large Splitting Was Spurious
An earlier exploratory finite-difference evaluation of the same point reported a score norm near
q≈930,
which would have implied a much stronger deformation, including κ_-≈0.62 already at α=0.01.
That value does not survive independent checking.
Analytic differentiation of the coherent-state wavefunction gives
q≈8.59178,
and direct evaluation of the compatibility matrix agrees with the corrected closed-form prediction.
The discrepancy arose in the numerical score evaluation rather than in the Sherman–Morrison theorem.
The corrected result is physically more reasonable and methodologically useful: the exact formula supplies a stable benchmark against which later finite-resolution numerical procedures can be tested.
7.11 What the Point Limit Establishes
The Q₀→0 calculation is the first fully intrinsic compatibility calculation in the present spin construction.
Unlike the parameter-space quantity κ_param of Section 4, every index here belongs directly to
T_X(CP¹×CP¹).
The native metric g, score s, state-shaped ruler G, Berry curvature ω, complex structure J, and compatibility operator A all inhabit the same configuration tangent space.
No external Hamiltonian parameter has been relabeled as a relational direction.
The calculation therefore establishes several points.
First, the native coherent-state geometry reproduces the compatible null spectrum
κ=(2,2,2,2).
Second, the amplitude of an actual interacting ground state supplies a nontrivial intrinsic score one-form.
Third, a state-shaped rank-one metric update produces genuine rank-four compatibility splitting.
Fourth, the form of that splitting is fixed exactly:
κ
(2/√(1+η), 2/√(1+η), 2, 2).
Finally, the result depends only on the invariant scalar
η=α‖d lnρ‖²_g
and not on the coordinate representation used to calculate it.
This is an analytic existence proof for the minimal compatibility mechanism proposed in the parent construction.
7.12 Why the Point Limit Is Not the Full Finite-Resolution Geometry
The exact solvability of the point limit follows from
Fᴿ,(0)=s⊗s.
The parent construction introduces finite Q₀ because one pointwise score direction cannot contain the full directional structure visible across a physically resolved neighborhood.
At finite Q₀,
Fᴿ,(Q₀)_IJ(X)
= [1/ρ_Q₀(X)] ∫dμ_G(Y) K_Q₀^(G)(X,Y) ρ(Y) s̃_I(Y;X)s̃_J(Y;X),
where s̃(Y;X) denotes the score at Y after covariant transport into T*_XQ.
If the transported score directions remain nearly parallel throughout the Q₀-cell, the tensor remains effectively rank one.
If the score rotates across the neighborhood, its second moment acquires additional independent directions.
The point-limit theorem consequently gives an exact null prediction for the finite-resolution calculation:
as Q₀→0,
one compatibility pair must approach
κ_-=2/√(1+η),
while the other must approach
κ_+=2.
At finite Q₀, the second pair may move only if the resolved amplitude tensor develops appreciable components outside the original score-selected Kähler plane.
This provides a direct diagnostic for whether finite resolution is doing more than merely rescaling the pointwise dyad.
7.13 Significance for the E₈ Program
The N=2 theorem does not itself contain the E₈ scaling spectrum. A two-spin system is far from the continuum magnetic Ising regime.
Its importance is methodological.
Before comparing E₈ spectral states with compatibility eigendirections, one must first establish that the intrinsic configuration-space operator can be constructed consistently and that an interacting wavefunction can genuinely deform its spectrum.
The N=2 point-limit calculation establishes this under analytic control.
The mechanism is explicit:
interacting ground state
→ coherent-state wavefunction Ψ(X)
→ amplitude density ρ(X)
→ intrinsic score s=d lnρ
→ state-shaped symmetric ruler G
→ native Berry two-form ω
→ compatibility spectrum of −(G⁻¹ω)².
In the simplest nontrivial rank-four setting, every step is mathematically well typed and the final spectrum is known exactly.
The later E₈ comparison therefore does not begin from a purely formal analogy. It begins from a concrete configuration-space mechanism that already works in an interacting spin system.
The next question is what changes when finite resolution samples more than one local score direction.
- Finite-Q₀ Resolution and Multidirectional Amplitude Geometry
8.1 From the Point Limit to Finite Resolution
Section 7 established an exact result in the point-resolution limit. When
Q₀→0,
the amplitude-derived tensor reduces to
Fᴿ,(0) = s⊗s,
with
s=d lnρ.
At any point where s≠0 this tensor is rank one. On the four-dimensional coherent-spin manifold CP¹×CP¹, the corresponding state-shaped metric therefore modifies only the native Kähler plane generated by s^♯ and Js^♯, leaving the orthogonal Kähler plane exactly at its undeformed compatibility scale κ=2.
The finite-resolution construction is designed to contain more information.
The score direction varies across configuration space. If neighboring scores are compared covariantly and averaged over a physical Q₀-neighborhood, the resulting second moment need not remain aligned with the score at the central point. Independent score directions can then contribute to the same resolved tensor.
The parent construction defines
Fᴿ,(Q₀)_IJ(X)
= [1/ρ_Q₀(X)] ∫dμ_G(Y) K_Q₀^(G)(X,Y)ρ(Y) s̃_I(Y;X)s̃_J(Y;X),
where
ρ_Q₀(X) = ∫dμ_G(Y)K_Q₀^(G)(X,Y)ρ(Y),
and s̃_I(Y;X) is the score covector at Y after parallel transport into T*_XQ.
The tensor is positive semidefinite and its resolved directional content is controlled by the span of the transported score covectors sampled inside the Q₀-cell.
This provides the mechanism by which a scalar amplitude can generate a multidirectional geometry at finite resolution.
The purpose of the present section is not yet to solve the complete covariant fixed-point problem. Instead, we test this mechanism in the same local short-time approximation used as a pedagogical benchmark in the parent construction.
8.2 Local Gaussian Approximation
Let X₀ denote the representative configuration used in Section 7,
θ₁=π/2,
φ₁=0.7,
θ₂=π/3,
φ₂=1.9.
At X₀ the native product Fubini–Study metric is
g₀ = ¼ diag(1, sin²θ₁, 1, sin²θ₂).
The short-time heat kernel of the native metric is locally Gaussian. Neglecting curvature corrections and replacing geodesic normal coordinates by the local coordinate chart gives the approximation
δX = Y−X₀ ∼ N(0, 2Q₀²g₀⁻¹).
Within the same approximation, parallel transport is replaced by the identity,
s̃_I(Y;X₀) ≈ s_I(Y).
The finite-resolution tensor becomes
Fᴿ,(Q₀)_IJ(X₀)
≈ [⟨ρ(Y)s_I(Y)s_J(Y)⟩_K] / [⟨ρ(Y)⟩_K],
where the expectation is taken over the local Gaussian kernel.
This approximation retains the two ingredients needed to test the proposed rank-raising mechanism:
the score direction is evaluated at different physical configurations Y,
and those differently oriented score covectors contribute to one common second moment at X₀.
It omits the specifically covariant corrections arising from curvature, nontrivial parallel transport, and the state dependence of the full metric Ĝ.
The results below should therefore be read as a local finite-resolution benchmark rather than the final sPNP geometry.
8.3 Independent Check of the Point Limit
Before evaluating finite Q₀, it is useful to verify the local data analytically.
For the N=2 ground state used in Section 7,
|Ψ₀⟩ ≈ 0.730542|↑↑⟩ +0.360483|↑↓⟩ +0.360483|↓↑⟩ +0.454326|↓↓⟩,
the coherent-state wavefunction at X₀ is, with the convention
Ψ(X)=⟨X|Ψ⟩,
approximately
Ψ(X₀) ≈ 0.437359−0.345621i.
Hence
ρ(X₀) ≈ 0.310737.
Analytic differentiation of the coherent-state wavefunction gives the logarithmic-amplitude score
s_I(X₀) ≈ (−0.411626, −0.564012, −0.636183, −0.970439).
Using the native Fubini–Study metric,
q ≡ g₀⁻¹(s,s)
≈ 8.59178.
Thus the point-limit tensor has one nonzero generalized eigenvalue,
λ_F = q ≈ 8.59178,
when F is measured relative to the native metric.
This provides a stringent numerical check on the finite-Q₀ calculation:
as Q₀→0, its dominant metric-normalized eigenvalue must approach 8.59178 while all orthogonal eigenvalues vanish.
The local Monte Carlo calculation satisfies this limit.
8.4 Metric-Normalized Spectrum of the Resolved Tensor
Because Fᴿ_IJ is a covariant tensor, ordinary matrix eigenvalues of its coordinate components are not invariant quantities.
We therefore characterize its directional spectrum through the generalized eigenvalue problem
Fᴿ_IJ u^J = λ g₀_IJ u^J,
or equivalently through the symmetric operator
𝔉 = g₀^−1/2 Fᴿ g₀^−1/2.
The eigenvalues of 𝔉 measure the amplitude stiffness relative to the native metric.
Using analytic score derivatives and local Gaussian sampling gives the following representative values:
Q₀| metric-normalized eigenvalues of Fᴿ,(Q₀) 0.001| 8.5916, 4.32×10⁻⁵, 1.69×10⁻⁵, 3.62×10⁻⁶ 0.01| 8.5784, 4.31×10⁻³, 1.69×10⁻³, 3.63×10⁻⁴ 0.05| 8.2749, 0.1050, 0.0410, 0.00937 0.10| 7.4049, 0.3898, 0.1501, 0.0407 0.20| 4.8986, 1.1351, 0.4585, 0.1895
The Q₀=0.001 result is already extremely close to the exact point-limit value q≈8.5918.
The remaining eigenvalues are strongly suppressed but nonzero at finite Q₀.
As Q₀ increases, spectral weight is redistributed away from the single dominant score direction and into additional independent directions.
This is the finite-resolution effect sought in the construction.
8.5 Effective Rank Rather Than Literal Rank Jumps
The finite-Q₀ behavior requires a small conceptual refinement.
It is tempting to summarize the preceding table as a sequence
rank 1 → rank 2 → rank 3 → rank 4.
That description is useful intuitively but is not mathematically exact.
For a generic analytic score field, any nonzero Gaussian neighborhood samples an open set of configurations. If the score direction changes in sufficiently many independent ways within that neighborhood, the second-moment tensor can already be strictly full rank for arbitrarily small Q₀, although several eigenvalues may be parametrically tiny.
The more precise statement is therefore that finite resolution produces an increase in the effective directional rank of the amplitude geometry.
At Q₀=0.001, the spectrum is overwhelmingly dominated by one eigenvalue.
At Q₀=0.20, all four eigenvalues are substantial relative to their small-Q₀ values.
A convenient scalar measure is the participation rank
r_eff ≡ (Tr𝔉)²/Tr(𝔉²).
For the representative calculation,
r_eff ≈ 1.000 at Q₀=0.001,
r_eff ≈ 1.001 at Q₀=0.01,
r_eff ≈ 1.038 at Q₀=0.05,
r_eff ≈ 1.159 at Q₀=0.10,
and
r_eff ≈ 1.75 at Q₀=0.20.
Thus the finite-resolution tensor evolves continuously from an almost purely one-directional object toward an increasingly multidirectional one.
The precise values depend on the state, point, metric, and local approximation. The structural observation is that finite resolution changes not only the magnitude of the amplitude tensor but also the number of configuration directions carrying appreciable amplitude stiffness.
8.6 Why Finite Resolution Raises the Directional Content
The mechanism can be understood locally.
Near X₀, write schematically
s(Y) ≈ s₀ + H·δX + … ,
where
H_IJ = ∇_J s_I
is the Hessian of lnρ.
The point-limit tensor is
s₀⊗s₀.
At finite Q₀ the Gaussian average also samples the variation H·δX.
Its second moment therefore contains terms schematically of the form
⟨(H·δX)⊗(H·δX)⟩
in addition to the original dyad.
Since
⟨δX^IδX^J⟩ ∼ 2Q₀²g₀^{IJ},
the leading new directional structure is controlled by the Hessian of the logarithmic amplitude and appears at order Q₀² in the local expansion.
If H maps the resolved neighborhood into several linearly independent score directions, the finite-resolution tensor develops corresponding eigenvalues outside the original rank-one plane.
Thus the numerical broadening has a direct differential-geometric interpretation:
the pointwise gradient gives one local direction,
while finite resolution also detects how that direction turns across configuration space.
This is precisely the information absent from the bare dyad s⊗s.
8.7 Effect on the Compatibility Spectrum
The next question is whether this multidirectional amplitude geometry actually deforms more than one native Kähler plane.
At each Q₀ define the local benchmark metric
G(Q₀) = g₀ + αFᴿ,(Q₀),
while retaining the native Berry curvature ω₀.
The compatibility operator is
A(Q₀)=G(Q₀)⁻¹ω₀.
In the strict point limit, Section 7 proved that one compatibility pair moves while the other remains exactly at κ=2.
At finite Q₀, no such rank-one protection remains once Fᴿ has appreciable support in additional directions.
For the representative Q₀=0.20 calculation one obtains:
α| lower κ pair| upper κ pair 0.0001| 1.99940| 1.99993 0.001| 1.99405| 1.99929 0.01| 1.94232| 1.99297 0.1| 1.55691| 1.93226
Each entry occurs with multiplicity two, as expected from the two real compatibility planes.
Both pairs now move away from the native value.
This verifies the central qualitative prediction of the finite-resolution construction:
once the resolved amplitude tensor acquires appreciable components outside the pointwise Fisher-active plane, the second PhaSe-paired plane can also be deformed.
8.8 Finite Resolution Redistributes Rather Than Simply Increases the Deformation
The finite-Q₀ result contains an additional feature worth emphasizing.
Finite resolution does not simply increase every incompatibility.
In the point limit, the full amplitude stiffness is concentrated in one Kähler plane. For the present state,
q≈8.5918.
At α=0.01 the exact point-limit theorem gives
κ_-≈1.91925,
κ_+=2.
At Q₀=0.20 the same α gives approximately
κ_-≈1.94232,
κ_+≈1.99297.
Thus the previously active plane actually moves slightly back toward native compatibility, while the previously protected plane moves away from it.
Finite resolution has redistributed the amplitude geometry across multiple directions.
This is more informative than a simple monotonic “washing out” picture.
The Q₀-cell can smooth the strongest local score anisotropy while simultaneously revealing weaker independent score directions that were invisible at one point.
The resulting geometry is therefore both smoother and more multidirectional.
This behavior is exactly what one would expect from a second-moment construction rather than from a scalar damping factor.
8.9 Relation to the Parent Resolved-Fisher Construction
The numerical benchmark directly realizes one of the main motivations for introducing Fᴿ,(Q₀) in the parent theory.
A scalar wavefunction amplitude has only one gradient covector at a single point, so the bare score dyad is rank one.
But finite physical resolution samples a neighborhood rather than an infinitesimal point. If the gradient direction changes throughout that neighborhood, the resolved second moment can contain several independent directions.
The present N=2 calculation supplies a minimal explicit example of that mechanism.
At small Q₀ the geometry is almost indistinguishable from the pointwise rank-one tensor.
As Q₀ increases, the additional metric-normalized eigenvalues become progressively larger.
Correspondingly, the compatibility spectrum changes from
one deformed pair + one exactly native pair
toward
two independently deformed pairs.
This is the first finite-resolution numerical realization in the present paper of the sequence
pointwise amplitude gradient → multidirectional resolved amplitude geometry → multichannel metric–PhaSe compatibility.
8.10 What the Local Calculation Does Not Yet Include
The result remains deliberately limited.
First, the sampling uses the short-time Gaussian kernel of the native metric evaluated at X₀.
The complete construction requires the heat kernel of the state-dependent metric Ĝ.
Second, score covectors at Y are compared using the local coordinate identification rather than true parallel transport along Ĝ-geodesics.
The exact tensor requires
s̃_I(Y;X₀) = Π_I{}^A(X₀←Y)s_A(Y).
Third, curvature corrections to the heat kernel and volume measure have been neglected.
Fourth, the metric has not been solved self-consistently.
In the full construction,
Fᴿ,(Q₀)
changes Ĝ,
Ĝ changes the geodesic distance, connection, heat kernel, and parallel transport,
and these changes feed back into Fᴿ,(Q₀).
The final geometry therefore requires an iteration of the form
Ĝ^(n) → K_Q₀^[Ĝ^(n)] → Fᴿ,(Q₀)[ρ;Ĝ^(n)] → Ĝ^(n+1).
Fifth, the calculation has been performed at one representative configuration point rather than over the full coherent-state manifold.
Finally, Q₀=0.20 is already large enough that the local-coordinate Gaussian should be regarded as illustrative rather than quantitatively precise on the curved CP¹×CP¹ manifold.
These limitations prevent the present numbers from being interpreted as invariant predictions of the full sPNP geometry.
They do not undermine the narrower conclusion being tested here: finite resolution of a varying score field can produce multidirectional amplitude geometry and consequently deform more than one compatibility plane.
8.11 Status of the N=2 Result
The N=2 analysis now contains two levels of evidence.
The point-resolution result is analytic:
Fᴿ,(0)=s⊗s
implies exactly
κ = (2/√(1+αq), 2/√(1+αq), 2, 2).
The finite-resolution result is numerical and approximate:
local Gaussian averaging of the actual interacting coherent-state amplitude produces additional resolved eigenvalues, and the compatibility calculation correspondingly moves both κ-pairs away from their native values.
The first establishes the exact minimal mechanism.
The second demonstrates that the additional finite-Q₀ mechanism proposed in the parent theory occurs in an explicit interacting spin state.
The remaining step is to replace the local approximation by the actual covariant and self-consistent geometry.
That problem is addressed in the following section.
8.12 Implication for the Later E₈ Comparison
The result also clarifies what Q₀ can contribute once the chain is enlarged toward the E₈ scaling regime.
At Q₀=0, the amplitude geometry is tied to an infinitesimal score direction at each configuration.
At finite Q₀, the geometry incorporates how the wavefunctional changes throughout a resolved neighborhood.
Consequently, the relevant question for an E₈ excitation will not simply be whether its coherent-state representation aligns with the local gradient of the ground-state amplitude.
It can instead be asked whether the spectral state lies within, overlaps, or is preserved by the multidirectional structures selected by the resolved geometry.
This makes finite Q₀ potentially important for comparing an ontic wavefunctional geometry with an interacting excitation spectrum.
The E₈ spectrum is global and collective.
A strictly pointwise rank-one description would be too impoverished to encode a rich set of independent spectral directions.
Finite-resolution amplitude geometry provides a concrete mechanism by which a scalar ontic Ψ can nevertheless generate a higher-rank relational ruler capable of supporting such comparisons.
The present two-spin calculation does not establish that this mechanism explains E₈.
It establishes the smaller but necessary result that the mechanism is mathematically operative before the many-site E₈ limit is attempted.
