Author: Jack Kowalski <jack@entropment.com> https://entropment.com

Patent Pending


Hierarchical Projection Functional (HPF)
The presented construction is a hierarchical projection functional defined on discrete data with a graph structure. It was created as a tool for analyzing the local significance of data components, evaluating representational dominance under projection from a chosen vertex, and ordering the hierarchy of couplings in coherent structures.

The construction is based on representing data as a multi-level vector, assigning exponentially increasing damping weights to successive levels, defining a radial functional that aggregates the contributions of the levels, and analyzing the relationship between the dominant part and the projected part.
The hierarchy of exponents (QCO) ensures separation of influences between levels and stability with respect to the dominance of any single component.

The functional transforms a local view of the data into a scalar radial measure, identifies dominant components,
exponentially suppresses higher-order contributions, and generates a secondary geometric structure without assuming a continuous space.
The resulting “geometry” is not presupposed, but induced by the functional.

The construction follows from three operational assumptions:
Data are discrete and relational (graph-based).
Higher-order couplings should have lower representational strength.
The measure of significance must be robust against the dominance of any single level.
No a priori continuous topology, specific spatial ontology, or privileged physical interpretation is assumed.

The goal was to create a tool for reconstructing local structure in coupled data, assessing the significance of projections relative to an observation point, and analyzing the distribution of “representational energy” in systems without natural geometry.
Any cosmological, physical or algebraic interpretations are secondary to this purpose.

=====0=====
0. Conventions
All linear spaces over ℂ.
Bounded linear operators.
N < ∞ until passing to the general version.
Standard Euclidean norms.

1. Definition of the space H
Let α > 0.
We define:
H := { Ψ = (xₙ)ₙ≥₀ : xₙ ∈ ℂ^{dₙ},   ∑_{n=0}^∞ α^{-n} ‖xₙ‖² < ∞ }.

With the inner product:
⟨Ψ, Φ⟩_H = ∑_{n=0}^∞ α^{-n} ⟨xₙ, yₙ⟩.

This is a Hilbert space:
H = ⨁_{n≥0}^{ℓ²(α^{-n})} ℂ^{dₙ}.

2. Embedding of the operator
Let:
W ∈ End(ℂᴺ),  u ∈ ℂᴺ.

We define:
Ψ(W, u) = (Wⁿ u)_{n≥0}.

3. Embeddability condition
Ψ(W, u) ∈ H  ⟺  ∑_{n=0}^∞ α^{-n} ‖Wⁿ u‖² < ∞.

Theorem 1 (spectral criterion)
Let ρ(W) denote the spectral radius.
Then:
Ψ(W, u) ∈ H  ⟺  ρ(W)² < α   on the cyclic subspace K(W, u).

Here it is necessary to formally note that I mean
sup{ |λ|² : λ ∈ σ(W|K(W,u)) }
and not the global ρ(W).


Proof
Jordan decomposition:
W = V J V⁻¹.

Each block:
Jᵢ = λᵢ I + Nᵢ.

We have:
Wⁿ u = ∑ᵢ λᵢⁿ Pᵢ(n),

where Pᵢ(n) are polynomials of degree ≤ mᵢ − 1.

Thus:
‖Wⁿ u‖² ∼ n^{2 m_max − 2} |λ_max|^{2n}.

Convergence of the series:
∑ α^{-n} n^k |λ|^{2n}

holds if and only if:
|λ|² < α.

QED.

4. Operator algebra
We define algebraically:
A := alg̅ {S},

where the shift operator:
(S Ψ)_n = x_{n+1}.

In the representation generated by Ψ(W, u):
S Ψ(W, u) = Ψ(W, W u).

I use operator algebras everywhere in the purely algebraic sense, not C*.
The C*-version would require taking the closure and would be closer to the Toeplitz algebra.
Since I am doing this only for the purpose of an implementational construction, I do not develop the obviousness of this closure further. Jokingly, I can add — I leave this as an exercise for the inquisitive reader.


Theorem 2
The algebra generated by S on the cyclic subspace is isomorphic to the algebra:
ℂ[z] / (m_W(z)),

where m_W is the minimal polynomial of the operator restricted to K(W, u).

Proof
Sᵏ Ψ(W, u) = Ψ(W, Wᵏ u).

A linear relation:
∑_{k=0}^m a_k Wᵏ u = 0

is equivalent to:
∑_{k=0}^m a_k Sᵏ Ψ(W, u) = 0.

The minimal polynomial is precisely the relation in the operator algebra.
QED.

5. Spectral invariant of the representation
We define the spectral measure with respect to u:
μ_u := ∑_{λ ∈ spec(W)} ‖P_λ u‖² δ_λ.

Theorem 3 (classification of representations)
Two pairs (W₁, u₁), (W₂, u₂) give unitarily equivalent representations in H if and only if:
1. The spectral measures are identical,
2. The Jordan block structures of the active eigenvalues are identical.

6. Fullness of the representation
Definition:
A representation is full when:
span {Wᵏ uⱼ} = ℂᴺ.

Theorem 4
The minimal number of starting vectors needed for a full classification of the operator equals the number of Jordan blocks.

Proof
Each Jordan block requires a vector whose component in the highest vector of the chain is nonzero.
The minimal set of such vectors must intersect every block.
QED.

7. Boundary cases
(A) ρ(W)² = α
The series behaves like:
∑ nᵏ
— divergent.
No embeddability.

(B) ρ(W)² > α
Exponential explosion.
No embeddability.

(C) Nilpotent operator
Wᵐ = 0.
Finite embedding.
Finite-dimensional algebra.

8. K-theory
Consider the C*-algebra:
A = C*(S).

If the spectrum of the shift is contained in the disk:
spec(S) ⊂ D_{√α},

then:
A ≅ C(Dr̅).

In that case:
K₀(A) ≅ ℤ,   K₁(A) ≅ 0.

In the crossed product case:
(∏_n End(Aₙ)) ⋊ ℤ,

K-theory is given by the Pimsner–Voiculescu exact sequence.

9. Infinite-dimensional limit
If W is a bounded operator on a Hilbert space:
Embeddability condition:
r_spec(W)² < α.

The spectral measure becomes a Borel measure.
Classification of representations is given by equivalence of measures.

10. Summary of the algebraic core
1. The embedding is a moment functional of the spectral measure.
2. The representation depends only on the restriction of the operator to the cyclic subspace.
3. Full classification requires covering all Jordan blocks.
4. The projection horizon is a function of the spectral radius.
5. The K-theory of the shift algebra reduces to the classical Toeplitz / crossed product case.



=====tower construction from other notes; just to mark what construction is operational from QCO=====
Definition Tower space
Let (E_k, ‖·‖_k) be a sequence of normed spaces over ℝ, for k ≥ 0, where:
  E_0 = ℝ,
  for k ≥ 1   E_k is a linear space (finite- or infinite-dimensional).
1. Carrier space
The tower space is defined as the Cartesian product
  T = ∏_{k=0}^∞ E_k
whose elements we write as
  x = (a, B_1, B_2, B_3, …)    where a ∈ ℝ, B_k ∈ E_k
In the case of finite height N:
  T_N = ∏_{k=0}^N E_k
2. Radial functional
  β: T → [0,∞]
  β(x) = ( ∑_{k=1}^∞ ‖B_k‖_k ^{2^k} )^{1/2}
In the case of a finite tower:
  β_N(x) = ( ∑_{k=1}^N ‖B_k‖_k ^{2^k} )^{1/2}
3. Admissible space
  T_adm = { x ∈ T : β(x) < ∞ }
In particular: if ‖B_k‖_k ≤ 1 for all k, then
  β(x) ≤ ( ∑_{k=1}^∞ ‖B_k‖_k ^2 )^{1/2}
(the exponents 2^k ensure hierarchical damping of higher levels)
4. Conical functional
For x = (a, B_1, B_2, …) ∈ T_adm we define:
  Δ(x) =
    min( |a| / β(x) , β(x) / |a| )    if a ≠ 0 and β(x) ≠ 0
    0                                 otherwise
This function is symmetric with respect to the interchange a ↔ β(x)
and measures the deviation from the cone diagonal |a| = β(x).
5. Boundary class
Points satisfying a = 0  or  β(x) = 0
form the boundary class of the radial singularity.
The functional Δ is well-defined on
  T_adm \ {a=0 and β=0}
and extendable by continuity to the boundary.
6. Property of hierarchical damping
For every k ≥ 1 and ‖B_k‖_k ≤ 1 we have:
  ‖B_k‖_k ^{2^k} ≤ ‖B_k‖_k ^2
which implies the contractive character of higher tower levels
and the separation of contributions between levels.





=====1=====
1. Action of ℤ on a C*-algebra
Let A be a C*-algebra (separable).
Let α ∈ Aut(A) be a *-automorphism.
We define the action:
αⁿ := α ∘ ⋯ ∘ α (n times),    α^{-n} := (α^{-1})ⁿ.

This gives a group homomorphism ℤ → Aut(A),   n ↦ αⁿ.

2. Algebraic crossed product
We define the *-algebra
A ⋊_α^{alg} ℤ = { ∑_{n∈ℤ} aₙ Uⁿ : aₙ ∈ A, finite sum },

with multiplication and *-operation:
(a Uᵐ)(b Uⁿ) = a αᵐ(b) U^{m+n},
(a Uⁿ)^* = α^{-n}(a*) U^{-n}.

Covariance relation:
U a U^{-1} = α(a).

3. Norms and completion
3.1. Covariant representations
A pair (π, V), where:
π : A → B(H) is a *-representation,
V unitary on H,

satisfies the covariance condition:
V π(a) V^{-1} = π(α(a)).

Each such pair induces a representation:
(π ⋊ V) ( ∑ aₙ Uⁿ ) = ∑ π(aₙ) Vⁿ.

3.2. Norms
Maximal crossed product: supremum norm over all covariant representations.
Reduced crossed product: via the regular representation on
ℓ²(ℤ, H_A),

where H_A is the GNS space for A.

For amenable actions (and ℤ is amenable):
A ⋊_α^{max} ℤ = A ⋊_α^r ℤ.

Thus we simply denote:
A ⋊_α ℤ.

4. Universal property
The C*-algebra A ⋊_α ℤ is universal for covariant pairs:
every pair (π, V) factors uniquely through a *-homomorphism
π ⋊ V : A ⋊_α ℤ → B(H).

5. Identification in our formalism
5.1. Shift operator
In the space
H = ⨁_{n≥0}^{ℓ²(α^{-n})} ℂ^{dₙ}

we define the shift:
(S Ψ)_n = x_{n+1}.

If we consider the two-sided space ⨁_{n∈ℤ}, then S is unitary.

5.2. Fiber algebra
Let:
A = ∏_{n∈ℤ} End(Aₙ)

(with supremum norm; C*-product).

We define the shift automorphism:
(α(a))_n := a_{n-1}.

This is a *-automorphism of A.

5.3. Crossed product
We consider:
A ⋊_α ℤ = ( ∏_{n∈ℤ} End(Aₙ) ) ⋊_α ℤ.

The generator U implements the fiber shift:
U a U^{-1} = α(a).

This is precisely the algebra generated by:
local operators at each level,
global level shift.

6. Representation on H
We define the representation:
π(a) Ψ = (aₙ xₙ),
V Ψ = S Ψ.

Covariance is satisfied:
V π(a) V^{-1} = π(α(a)).

Thus there exists a representation:
π ⋊ V : A ⋊_α ℤ → B(H).

7. Simplicity (simplicity criterion)
Classical theorem:
If:
1. the action α is minimal (no nonzero α-invariant ideals in A),
2. the action is aperiodic (no nonzero elements satisfying αⁿ(a) = a for n ≠ 0),

then the crossed product is simple.

In the case of a fiber product:
minimality corresponds to the absence of invariant subsets of indices,
periodicity appears when the sequence Aₙ is periodic.

If Aₙ = A constant and α = id, then:
A ⋊_{id} ℤ ≅ A ⊗ C(𝕋),

which is not simple.

8. K-theory (Pimsner–Voiculescu)
For any automorphism α:

     K₀(A)  →^{id - α_*}  K₀(A)  →  K₀(A ⋊_α ℤ)
       ↑∂                                 ↓∂
K₁(A ⋊_α ℤ)  ←             K₁(A)  ←^{id - α_*}  K₁(A)

This is the exact six-term sequence.

8.1. Case of matrix product
If:
A = ∏_{n∈ℤ} M_{kₙ}(ℂ),

then:
K₀(A) = ∏_{n∈ℤ} ℤ,    K₁(A) = 0.

The shift automorphism acts as index shift on ∏ ℤ.

Then:
K₀(A ⋊_α ℤ) ≅ coker(id - α_*),
K₁(A ⋊_α ℤ) ≅ ker(id - α_*).

This gives a topological invariant of the shift structure.

9. Limits and pathological cases
1. If α has period p, the crossed product reduces to an algebra with factor C(𝕋).
2. If A has nonzero invariant ideals, the crossed product is not simple.
3. In the infinite-dimensional version, separability is required for standard classification theorems.

10. Structural conclusion in the formalism:
algebra of local operators = fiber product,
dynamics of levels = shift automorphism,
full representation algebra = crossed product,
its K-theory classifies the global structure of transitions between levels.


=====2=====
1. Setup
Let:
H — separable Hilbert space,
W ∈ B(H) — bounded operator,
α > 0,
u ∈ H.

We define the embedding:
Ψ(W, u) := (Wⁿ u)_{n≥0}.

Norm:
‖Ψ‖_H² = ∑_{n=0}^∞ α^{-n} ‖Wⁿ u‖².

2. Embeddability condition
Let r(W) denote the spectral radius.
Theorem 1
Ψ(W, u) ∈ H  ⟺  sup_{λ ∈ supp μ_u} |λ|² < α,

where μ_u is the spectral measure associated with the pair (W, u).

3. Spectral measure (normal case)
We now assume:
W is a normal operator.

From the spectral theorem:
W = ∫_{σ(W)} λ dE(λ),

where E is a projection-valued spectral measure.

We define the scalar measure:
μ_u(B) := ⟨E(B) u, u⟩,    B ⊂ σ(W).

This is a positive Borel measure.

4. Form of the embedding via the measure
Wⁿ u = ∫ λⁿ dE(λ) u.

Thus:
‖Wⁿ u‖² = ∫ |λ|^{2n} dμ_u(λ).

5. Energy in H
‖Ψ‖_H² = ∑_{n=0}^∞ α^{-n} ∫ |λ|^{2n} dμ_u(λ).

Interchanging sum and integral (Tonelli):
= ∫ ∑_{n=0}^∞ (|λ|² / α)ⁿ dμ_u(λ).

The geometric series gives:
= ∫ 1 / (1 - |λ|² / α) dμ_u(λ),

provided |λ|² < α.

Conclusion
Ψ(W, u) ∈ H  ⟺  supp μ_u ⊂ {|λ|² < α}.

6. GNS representation
Consider the C*-algebra:
A := C*(W) ⊂ B(H).

We define the state:
ω_u(a) = ⟨a u, u⟩.

This is a positive functional of norm ‖u‖².

GNS construction
There exists a triple:
(π_u, H_u, Ω_u)

such that:
π_u : A → B(H_u),
Ω_u cyclic vector,
ω_u(a) = ⟨π_u(a) Ω_u, Ω_u⟩.

7. Identification with the spectral representation
Since W is normal:
A ≅ C(σ(W)).

The state ω_u corresponds to the measure μ_u.
The GNS representation is unitarily equivalent to:
π_u(f) = M_f   on L²(σ(W), μ_u),

and the cyclic vector is the constant function 1.

The operator W maps to the multiplication operator:
(π_u(W) φ)(λ) = λ φ(λ).

8. Embedding in GNS form
In this representation:
Wⁿ u ↔ λⁿ.

The embedding becomes the sequence of functions:
Ψ(λ) = (λⁿ)_{n≥0}.

Norm:
‖Ψ‖_H² = ∫ ∑_{n=0}^∞ α^{-n} |λ|^{2n} dμ_u(λ).

Thus the embedding is the moment functional of the measure μ_u.

9. Classification of representations
Theorem 2
Two pairs (W₁, u₁), (W₂, u₂) with normal operators are unitarily equivalent on their cyclic subspaces if and only if:
μ_{u₁} = μ_{u₂}.

Proof: classification of representations of an abelian C*-algebra by measures.

10. Case of continuous spectrum
If σ(W) contains a continuous part:
μ_u may be absolutely continuous,
the embedding is still defined,
the projection horizon depends on the boundary of the support of the measure.

Boundary convergence condition
If sup_{λ ∈ supp(μ_u)} |λ|^2 = α,
then convergence depends on the behaviour of the measure near the boundary.
More precisely,
Ψ(W,u) ∈ H  ⇔  ∫ 1 / (1 - |λ|^2 / α)  dμ_u(λ) < ∞
Thus divergence occurs whenever the measure has non-integrable mass concentration 
at the boundary, but not merely from equality of the supremum.

11. Non-normal case
If W is not normal:
there is no global spectral measure,
we apply the Dunford decomposition:
W = N + Q,
where N is normal, Q is quasinilpotent.
The GNS representation still exists for the state ω_u,
but the algebra C*(W) is non-commutative.
Classification requires the structure of the full algebra.

Even if W admits a Dunford decomposition 
W=N+Q,
the radial moments detect only the spectral radii of components contributing to u.
They do not encode:
the size of Jordan blocks,
nilpotent chains,
non-normal perturbations preserving singular values of powers.
Therefore, completeness statements are restricted to the normal case or to situations where the cyclic representation reduces to a commutative C*-algebra.

12. Crossed product in the infinite-dimensional version
If we have an automorphism:
α(a) = W a W^{-1}

on A = C*(W),

then:
A ⋊_α ℤ

acts on the GNS space via unitary implementations of the shift.

13. Limits
1. If W is unitary:
|λ| = 1.
The embedding exists only when α > 1.
2. If W has spectrum in the open disk:
the embedding always exists for sufficiently large α.
3. If the spectrum touches the boundary:
infinite horizon.

14. Final infinite-dimensional form
Ψ(W, u) is equivalent to the GNS representation for the state ω_u on C*(W).

The embedding = moment expansion of the spectral measure.

Classification of representations = classification of spectral measures.

This closes the transition to the infinite-dimensional version.


=====3=====
KK-theory (Kasparov) and noncommutative geometry as a language of invariants for the construction
A ⋊_α ℤ,   A = C*(W).

We assume:
A separable C*-algebra,
α ∈ Aut(A),
action of ℤ,
crossed product B := A ⋊_α ℤ.

1. KK-theory — core
Kasparov introduces a bifunctor:
KK(A, B)

which is an abelian group of homotopy classes of triples
(E, ϕ, F):
E — Hilbert B-module,
ϕ: A → L(E),
F ∈ L(E),
satisfying the conditions:
[F, ϕ(a)] ∈ K(E),   (F² - 1) ϕ(a) ∈ K(E),   (F - F*) ϕ(a) ∈ K(E).

2. Why is KK appropriate here?
Because:
A ⋊_α ℤ is classified by the element
[α] ∈ KK(A, A),

the dynamics is entirely contained in this class.

3. Fundamental fact
For the automorphism α:
A ⋊_α ℤ ~_{KK} Cone(1 - α_*).

More precisely: there exists the exact six-term Pimsner–Voiculescu sequence,
which in the language of KK follows from the fact:
[1 - α] ∈ KK(A, A)

is the difference element controlling the K-theory of the crossed product.

4. Classification by KK
Theorem (scheme)
If:
A is nuclear,
satisfies UCT (Universal Coefficient Theorem),

then the KK class determines the K-theory and the Morita class of the crossed product algebra.
In particular:
KK-equivalence ⇒ K-theoretic equivalence.

5. In our context
For A = C*(W):
(a) In the normal case
A ≅ C(σ(W)).

Then:
KK(A, A) ≅ KK(C(X), C(X)) ≅ K⁰(X × X).

The automorphism α corresponds to a homeomorphism of the spectrum.
Spectrum dynamics = class in K⁰.

(b) In the matrix product case
A = ∏_{n∈ℤ} M_{k_n}(ℂ).

Then:
KK(A, A) ≅ Hom(K₀(A), K₀(A)).

The shift automorphism acts as index shift.
Thus:
[α] is the shift operator in ∏ ℤ.

6. Noncommutative geometry (Connes)
In the spirit of Connes, we treat the algebra as the "algebra of functions" on a noncommutative space.
Geometric data:
C*-algebra B,
spectral triple (B, H, D).

7. Spectral triple for crossed product
For the action of ℤ:
natural candidate:
D = i d/dθ

on L²(𝕋),
if the crossed product contains the factor C(𝕋).
In the general version:
D = D_A ⊗ 1 + γ ⊗ D_ℤ.

This gives a class in:
K¹(B).

8. Cyclic cohomology
Geometric invariants:
HC*(B).

Pairing:
K_*(B) × HC*(B) → ℂ.

Gives index numbers.

9. Classification by geometric data
The crossed product algebra is classified by:
1. the KK class of the automorphism,
2. its action on K₀(A),
3. the spectral triple (if it exists),
4. element in Ext(A) (Toeplitz extension class).

10. Dynamic case
If W(t) changes continuously:
we obtain a family of automorphisms:
α_t.

The class in:
KK(A, A)

can change only discontinuously (homotopy stability).
This gives topological stability of the embedding.

11. Geometric interpretation
The embedding in H generates:
spectral space,
shift dynamics,
KK class of the element 1 - α.
This is a "noncommutative dynamical manifold".

12. Limits
If the action is minimal and aperiodic:
the crossed product is sometimes simple and often purely infinite (Kirchberg).
If the spectrum has a continuous part:
possible types II, III (in the von Neumann sense).

13. Final classificatory form
(A, α) ↦ [α] ∈ KK(A, A)

A ⋊_α ℤ classified by KK and K_*.

The primary embedding corresponds to:
GNS class + KK class of the automorphism.


=====4=====
Let:

A — separable C*-algebra,  
α ∈ Aut(A).

We want to explicitly construct the element  
[1−α] ∈ KK(A,A).

1. Element [α] ∈ KK(A,A)

An automorphism gives a natural bimodule.  
Definition of the module  
Let:  
Eα := A  
as a linear space.

Structures:  
right action: ordinary multiplication in A,  
inner product:  
⟨x,y⟩ := x* y,  
left action:  
ϕα(a) ⋅ x := α(a) x.

Then:  
(Eα, ϕα, 0)  
is a Kasparov cycle.

This defines:  
[α] ∈ KK(A,A).

For α = id we get the unit element 1A.

2. Difference 1−α

The group KK(A,A) is abelian, so:  
[1−α] := [id] − [α].

The difference in KK is realized by the direct sum and sign change (reversing the grading).

3. Explicit cycle representing 1−α

We build the direct sum of two modules:  
E := A ⊕ A.

Right action: diagonal.  
Inner product:  
⟨(x₁,x₂), (y₁,y₂)⟩ = x₁* y₁ + x₂* y₂.

Left representation:  
ϕ(a) = ( a  0  
         0  α(a) )

We assign a grading:  
γ = ( 1   0  
      0  -1 )

Operator:  
F = ( 0  1  
      1  0 )

4. Checking Kasparov conditions

We compute the commutator:  
[F, ϕ(a)] = ( 0  1 ) ( a     0    )  −  ( a     0    ) ( 0  1 )  
            ( 1  0 ) ( 0   α(a) )      ( 0   α(a) ) ( 1  0 )

After multiplication:  
= (  0          α(a)−a     )  
  (  a−α(a)       0        )

Since we act on a finitely generated module over A,  
the difference lies in the compacts K(E).

Conditions:  
F² = 1,   F = F*

Thus we obtain a KK cycle.

5. Interpretation

This element encodes the operator:  
a ↦ a − α(a).

In K-theory it acts as:  
(1−α₊) : K₊(A) → K₊(A).

6. Relation to the Pimsner–Voiculescu sequence

For the crossed product:  
B = A ⋊α ℤ,

there exists an exact sequence:

     K₀(A)  →1−α₊   K₀(A)   →   K₀(B)  
      ↑                             ↓  
     K₁(B)  ←       K₁(A)  ←1−α₊   K₁(A)

This sequence comes precisely from the class  
[1−α] ∈ KK(A,A).

7. Interpretation as a cone (mapping cone)

We consider the morphism:  
id − α : A → A.

We build the mapping cone:  
C_{1−α} = { f ∈ C([0,1], A) : f(0) = a,  f(1) = α(a) }.

The class of the cone in KK corresponds to the element 1−α.

8. Geometric interpretation

In the spirit of noncommutative geometry:  
1−α  
is the "differential in the direction of the ℤ orbit".

It is the counterpart of the operator:  
f(x) − f(Tx)  
in classical dynamics.

9. Version for A = C(X)

If:  
A = C(X),   α(f) = f ∘ T⁻¹,

then:  
[1−α] ∈ KK(C(X), C(X)) ≅ K⁰(X×X).

This is the class of the graph of the transformation:  
Γ_T ⊂ X×X.

10. Final algebraic form

[1−α] = [(A⊕A, ϕ, F)]

where:  
ϕ(a) = ( a     0    )  
       ( 0   α(a) ),      F = ( 0  1  
                                 1  0 )

This element:  
· generates the PV sequence,  
· controls the K-theory of the crossed product,  
· is the fundamental invariant of the dynamics.


=====5=====
1. Initial data

Let:  
A — separable C*-algebra,  
α ∈ Aut(A).

We define the correspondence (Hilbert bimodule)  
E := ₐA.

2. Structure of the module E

As a linear space:  
E = A.

Right action:  
x ⋅ a := x a.

Inner product:  
⟨x,y⟩ := x* y.

Left action:  
φ(a) ⋅ x := α(a) x.

This makes E a C*-Pimsner correspondence.

3. Toeplitz algebra T_E

Definition (Pimsner):  
T_E is the universal C*-algebra generated by:  
representation π : A → B(H),  
creation operator T : E → B(H),

satisfying the relations:  
T(x)* T(y) = π(⟨x,y⟩),  
π(a) T(x) = T(φ(a) x).

4. Canonical Fock model

We build the Fock space:  
F_E = A ⊕ E ⊕ E⊗² ⊕ ⋯

Since:  
E⊗ⁿ ≅ ₐⁿA,

we obtain:  
F_E ≅ ⨁_{n≥0} A.

Creation operator:  
T(a₀)(a₁, a₂, …) = (0, a₀ a₁, a₀ a₂, …)

5. Relation to the semigroup crossed product

In this representation:  
T_E ≅ A ⋊_α ℕ.

This is the Toeplitz crossed product by the semigroup ℕ.

6. Cuntz–Pimsner quotient

We define the ideal:  
J_E = closed span { T(x) T(y)* − π(θ_{x,y}) }

Cuntz–Pimsner algebra:  
O_E := T_E / J_E

7. Key theorem

If α is an automorphism, then:  

O_E ≅ A ⋊_α ℤ

8. Construction of an explicit isomorphism

Let u be a unitary implementing the action of ℤ:  
u a u* = α(a)

In O_E:  
u := image of the creation operator.

Relations:  
u* u = 1,    u u* = 1,

follow from the Cuntz relations.

We obtain:  
C*(A, u) ≅ A ⋊_α ℤ

9. Exact Toeplitz sequence

There exists a short exact sequence:  
0 ⟶ K(F_E) ⟶ T_E ⟶ A ⋊_α ℤ ⟶ 0

This is the Toeplitz extension.

10. Extension class

This extension represents the element:  
[Ext] ∈ KK¹(A ⋊_α ℤ, A)

It is precisely the class corresponding to the element:  
[1−α] ∈ KK(A,A)

11. Dynamic interpretation

T_E — algebra of "irreversible dynamics",  
O_E — full dynamics (group ℤ).  
Compacts in the kernel correspond to "loss of information when reversing time".

12. Special case A = C(X)

For:  
α(f) = f ∘ T⁻¹,

we obtain:  
O_E ≅ C(X) ⋊_T ℤ,

that is, the algebra of functions on the transformation space (X,T).

Toeplitz extension corresponds to the classical Toeplitz algebra  
for the one-sided shift.

13. Structural conclusion

A ⋊_α ℤ is the Cuntz–Pimsner algebra associated with E = ₐA.

Toeplitz algebra T_E = A ⋊_α ℕ

14. Essential in the construction

In the context of embedding:  
semigroup dynamics generates the Toeplitz algebra,  
reversibility (full spectrum) corresponds to passing to the Cuntz–Pimsner quotient,  
the class 1−α controls the entire extension.


=====6=====
analyses of pathological cases for the construction:
spectral embedding,
algebra A=C*(W),
crossed product A⋊αℤ,
identification with Toeplitz/Cuntz–Pimsner algebra.

We consider three sources of pathology:
1. unbounded operator,
2. continuous spectrum / non-point spectrum,
3. lack of separability.

Each of these cases affects a different place in the construction.

I. Unbounded operator

Let W be a densely defined operator on the Hilbert space H.

1. Basic problem

If W is unbounded, then:
W ∉ B(H),
does not generate a C*-algebra by norm closure,
C*(W) is not defined in the classical sense.

2. Solution: passing through functional calculus

If W is:
self-adjoint,
or normal,
then we apply functional calculus:
A := C*( (W-i)⁻¹ ).

Then:
(W-i)⁻¹ ∈ B(H),
and the algebra is well-defined.

3. Spectral pathology

For an unbounded operator:
σ(W)
may be unbounded.

In the embedding:
∑ α⁻ⁿ ‖Wⁿ u‖²
often diverges for every α.

Criterion
If:
u ∈ ⋂_{n≥0} Dom(Wⁿ),
and the growth of norms is subexponential, the embedding exists.

Otherwise:
Ψ(W,u) ∉ H for every α.

4. Consequence for crossed product

Automorphism:
α(a)=W a W⁻¹
is not well-defined, because W⁻¹ may not exist.

Possible only semigroup dynamics:
αₙ(a)=Wⁿ a W⁻ⁿ
for positive n, provided the domains are stable.

Then instead of crossed product by ℤ we have a semigroup product by ℕ.

II. Continuous spectrum

We now assume W bounded, normal, but:
σ(W) contains a continuous part.

1. Spectral measure

Embedding:
‖Ψ‖² = ∫ 1/(1 - |λ|²/α) dμ_u(λ).

If:
sup |λ|² = α,
and the measure has mass at the boundary,
the integral diverges.

2. Von Neumann type

GNS representation generates a von Neumann algebra:
π(A)''.

For continuous spectrum possible are:
type I (if the representation decomposes directly),
type II (if there exists a semifinite trace),
type III (no trace).

Dynamics of crossed product may change the type (Connes–Takesaki).

3. Classification pathology

If:
the action of ℤ is minimal,
the spectrum is continuous,
then A⋊αℤ may be a simple purely infinite algebra (Kirchberg).

Then classification reduces to KK-theory (under UCT).

If lack of UCT — classification by KK may be insufficient.

III. Lack of separability

Assume:
H is not separable.

1. Problem in KK

KK-theory in the classical form assumes separability.

Without it:
no guarantee of existence of Kasparov product,
UCT may not hold.

2. Spectral problems

Spectral theorem still works,
but the measure may be supported on an uncountable set without possibility of reduction to a sum of simple representations.

3. Fock pathology

Fock space:
F_E = ⨁_{n≥0} E⊗ⁿ
may be non-separable.

Then:
K(F_E)
is not σ-unital,
and Toeplitz extension may not be a "good" extension in the sense of KK.

IV. Dynamic pathologies

1. Non-continuous automorphism

If α is not norm-continuous,
crossed product does not exist in the classical sense.

2. Lack of kernel in 1-α

If:
1-α_* is not Fredholm on K_*(A),
then PV sequence may not give finite groups.

V. Extreme cases

1. Quasinilpotent operator

σ(W)={0}.

Trivial embedding.

Crossed product degenerates to Toeplitz algebra without reversibility.

2. Full disk spectrum

σ(W)=D̅.

For α=1 embedding diverges at the boundary.

3. Unitary spectrum

If W unitary:
|λ|=1.

Embedding exists only when α>1.

Crossed product is the rotation algebra.

VI. Summary of pathologies

Pathology | Consequence
Unbounded operator | lack of C*-algebra without resolvent transformation
Continuous spectrum | possible type II/III
Lack of separability | KK may not work
Lack of UCT | incomplete classification
Irreversibility | only Toeplitz, lack of full crossed product

VII. Structural conclusion

The entire construction is stable under the conditions:
1. W bounded,
2. A separable,
3. α norm-continuous automorphism,
4. algebra satisfies UCT.

Outside these conditions:
embedding may cease to exist,
crossed product may change type,
KK classification may not suffice.


=====7=====
I proceed to the analysis in the embedding construction
Ψ(W,u)=(Wⁿ u)ₙ≥0, ‖Ψ‖_H² = ∑ₙ≥0 α^{-n} ‖Wⁿ u‖², Ψ(W,u) = (Wⁿ u)ₙ≥0, ‖Ψ‖_H² = ∑ₙ≥0 α^{-n} ‖Wⁿ u‖², Ψ(W,u)=(Wⁿ u)ₙ≥0, ‖Ψ‖_H² = ∑ₙ≥0 α^{-n} ‖Wⁿ u‖², 
and the algebra
A=C^*(W), B=A ⋊_α ℤ. A = C^*(W), B = A ⋊_α ℤ. A=C^*(W), B=A ⋊_α ℤ. 
I am not interested in abstract pathologies — only those that can actually destroy the embedding as a spectral invariant of the representation.

I. Is an unbounded operator a real threat?
1. Minimal condition for the existence of the embedding
The embedding requires:
u ∈ ⋂ₙ≥0 Dom(Wⁿ). u ∈ ⋂ₙ≥0 Dom(Wⁿ). u ∈ ⋂ₙ≥0 Dom(Wⁿ). 
If W is unbounded, this set may be:
dense,
trivial,
empty.
If it is trivial → the embedding does not exist for most vectors.

2. Does W originate from:
matrices (graph),
transition operator,
bounded operator?
If so, unboundedness does not occur.
If however W is a generator (e.g., of Laplacian type, Hamiltonian), then:
The embedding is well-defined only on the class of analytic vectors.
This does not destroy the theory, but reduces the class of allowed states.

Conclusion I
If W originates from graphs/transition matrices → the threat is practically zero.
If it is an infinite-dimensional generator → one must explicitly restrict the domain.

II. Continuous spectrum is a real threat
The embedding reduces to:
‖Ψ‖² = ∫ 1/(1 - |λ|²/α) dμ_u(λ). ‖Ψ‖² = ∫ 1/(1 - |λ|²/α) dμ_u(λ). ‖Ψ‖² = ∫ 1/(1 - |λ|²/α) dμ_u(λ). 

1. Here is the problem;
If:
sup |λ|² = α, sup |λ|² = α, sup |λ|² = α, 
and the measure has mass at the boundary, we have divergence.
This means:
the embedding depends not only on the spectrum,
but on the distribution of the mass of the spectral measure.

2. What can be spoiled in the construction
If the goal was:
spectrum as an invariant of the representation.
In the case of continuous spectrum:
two representations with the same spectrum,
but different spectral measures,
may give different embeddings.
This is not a mathematical flaw — it means that:
the invariant is μ_u, not only σ(W). the invariant is μ_u, not only σ(W). the invariant is μ_u, not only σ(W). 

Conclusion II
Continuous spectrum does not destroy the embedding,
but changes its nature:
point spectrum → sequence of moments point spectrum → sequence of moments point spectrum → sequence of moments 
The embedding ceases to be a finite-dimensional invariant.
This is the most real conceptual threat.

III. Lack of separability
In the construction:
graphs are countable,
matrices are countable,
GNS representations are separable.
Lack of separability appears only when:
the base space is uncountable,
I have an unindexable family.

What does this spoil?
KK may not work.
UCT may not hold.
But the embedding as such still exists.

Conclusion III
Lack of separability does not threaten the embedding.
It threatens only the classification program.

IV. Irreversible automorphism
If W is not invertible:
α(a)=W a W^{-1} α(a)=W a W^{-1} α(a)=W a W^{-1} 
does not exist globally.
Then:
there is no full crossed product by ℤ,
we have only the semigroup product (Toeplitz).

This does not destroy the embedding,
but changes the algebraic interpretation.

V. Only two phenomena really threaten:

1. Spectrum touching the radius α^{1/2} 1. Spectrum touching the radius α^{1/2} 1. Spectrum touching the radius α^{1/2}
This causes:
loss of convergence,
sensitivity to the boundary.
This is the projection horizon in the strict sense.

2. Different measures with the same spectrum
Two graphs may have:
identical collective spectrum,
different distributions of eigenvectors,
and then the embedding distinguishes them.
If you want to classify only by spectrum,
this is a real trap.

VI. What is NOT a real threat
Pathology
Does it destroy the embedding?
Unbounded operator (in graphs)
No
Lack of separability
No
Type II/III von Neumann
No
Lack of UCT
No
They destroy the KK classification,
not the embedding.

VII. The construction is:
algebraically stable,
topologically stable,
sensitive only to the boundary of the spectrum and the spectral measure.
This means that the real risk is not in KK,
not in Toeplitz,
not in von Neumann,
only in:
analysis of the spectral measure. analysis of the spectral measure. analysis of the spectral measure. 

VIII. Precisely...
The embedding is a complete invariant of the cyclic representation if and only if:
μ_u is uniquely recoverable from the sequence of moments μ_u is uniquely recoverable from the sequence of moments μ_u is uniquely recoverable from the sequence of moments 
that is, when the moment problem is determinate.
If the moment problem is indeterminate,
I get indistinguishable cases.
This is the only real structural threat.


=====8=====
When do two different spectral measures give the same embedding?
I am working in the normal case:
W = ∫_{σ(W)} λ dE(λ),     μ_u(B) = ⟨E(B)u, u⟩. 
W = ∫_{σ(W)} λ dE(λ),     μ_u(B) = ⟨E(B)u, u⟩. 
W = ∫_{σ(W)} λ dE(λ),     μ_u(B) = ⟨E(B)u, u⟩. 

Embedding:
‖Ψ(W,u)‖² = ∑_{n≥0} α^{-n} ‖Wⁿ u‖² = ∫ ∑_{n≥0} (|λ|²/α)^n dμ_u(λ). 
‖Ψ(W,u)‖² = ∑_{n≥0} α^{-n} ‖Wⁿ u‖² = ∫ ∑_{n≥0} (|λ|²/α)^n dμ_u(λ). 
‖Ψ(W,u)‖² = ∑_{n≥0} α^{-n} ‖Wⁿ u‖² = ∫ ∑_{n≥0} (|λ|²/α)^n dμ_u(λ). 

Key information:
m_n := ∫ |λ|^{2n} dμ_u(λ). 
m_n := ∫ |λ|^{2n} dμ_u(λ). 
m_n := ∫ |λ|^{2n} dμ_u(λ). 

The embedding depends solely on the sequence of moments
(m_n)_{n≥0}. 
(m_n)_{n≥0}. 
(m_n)_{n≥0}. 

Thus the question reduces to:
When do two different measures have the same sequence of moments?
This is the classical moment problem.

I. Reduction to the Stieltjes problem
We introduce the variable:
x = |λ|² ∈ [0,R]. 
x = |λ|² ∈ [0,R]. 
x = |λ|² ∈ [0,R]. 

We obtain the measure ν on [0,R]:
m_n = ∫ x^n dν(x). 
m_n = ∫ x^n dν(x). 
m_n = ∫ x^n dν(x). 

This is the Stieltjes moment problem (on the half-line).

II. Moment determinacy
Definition
The measure ν is moment determinate, if:
m_n(ν₁) = m_n(ν₂) ∀n ⇒ ν₁ = ν₂. 
m_n(ν₁) = m_n(ν₂) ∀n ⇒ ν₁ = ν₂. 
m_n(ν₁) = m_n(ν₂) ∀n ⇒ ν₁ = ν₂. 

Otherwise it is indeterminate.

III. When does determinacy hold?
1. Compact support
If:
supp(ν) ⊂ [0,R] 
(compact set),
then the moment problem is always determinate.
This is the Hausdorff theorem.

Key conclusion
If:
σ(W) ⊂ {|λ|² ≤ R} 
with bounded R,
then the embedding uniquely determines the measure.
There is no pathology.

2. Unbounded support
If:
supp(ν) ⊂ [0,∞),
then determinacy depends on the growth of moments.

IV. Carleman criterion
The measure is determinate, if:
∑_{n=1}^∞ m_{2n}^{-1/(2n)} = ∞. 
If the sum is finite → possible indeterminacy.

V. What this means for the embedding...
Case A: bounded operator
In graphs and matrices:
‖W‖ < ∞. 
Then:
x ∈ [0, ‖W‖²], 
that is, compact support.
The embedding uniquely determines the measure.
There are no two different measures with the same embedding.

Case B: unbounded operator
If the spectrum is unbounded,
the moment problem may be indeterminate.
There exist classical examples (e.g., the family of log-normal measures),
which have the same sequence of moments.
Then:
ν₁ ≠ ν₂,     m_n(ν₁) = m_n(ν₂). 
The embedding does not distinguish.

VI. Very important observation
The embedding uses only:
|λ|^{2n}. 
That is:
it does not see the phase of λ,
it does not see the structure on the circle,
it sees only the radius.
This means:
Two measures differing in angular distribution,
but having the same radial distribution,
give identical embedding.
This is not a moment pathology but a loss of information due to the construction.

VII. Formal theorem
Let μ₁, μ₂ be measures on ℂ.
If their images under the map:
λ ↦ |λ|² 
are equal, then the embedding is identical.
Conversely:
If the operator is bounded,
then equality of embeddings implies equality of these radial measures.

VIII. Full version (with phase)
If the embedding included moments:
∫ λ^n dμ(λ), 
the problem would be the classical complex moment problem,
and in the case of compact support also determinate.
The radial version is weaker.

IX. Final answer
Two different measures give the same embedding if and only if:
1. they have identical radial distribution, and
2. the moment problem for this radial measure is indeterminate
(which is possible only with unbounded support).

X. What really threatens your construction?
	
Situation                         		 Is it possible to have indistinguishable measures?
Bounded operator (graph, matrix)          			NO
Unbounded operator                       			YES
Different phase distribution                            		YES (always)
Compact support                                 			NO
Unbounded support                         			POSSIBLE

XI. Most important practical conclusion
For graph constructions (bounded operator):

The embedding uniquely determines the radial measure.

The only loss of information is the phase of the spectrum.
This is the real, structural limit of the method.


=====9=====
The construction must be precise: the embedding depends solely on the sequence
m_n = ∫ |λ|^{2n} dμ(λ). 
m_n = ∫ |λ|^{2n} dμ(λ). 
m_n = ∫ |λ|^{2n} dμ(λ). 

Thus two operators will be indistinguishable exactly when their radial measures are identical.
Below are two explicit types of examples.

I. Simplest example (different phase, same radius)
Space:
H = ℂ². 
I will take two different diagonal matrices:
W₁ = ( 1  0 )
     ( 0 -1 ),     W₂ = ( 1  0 )
                       ( 0  1 ). 

They are different as operators (different complex spectrum).
I will choose a cyclic vector:
u = 1/√2 (1, 1). 

1. Spectral measures
For W₁:
μ₁ = 1/2 δ₁ + 1/2 δ₋₁. 
For W₂:
μ₂ = δ₁. 

The measures are different.

2. Radial moments
Since:
|1|² = |-1|² = 1, 
for both:
m_n = ∫ |λ|^{2n} dμ = 1. 
For all n.

3. Embedding
‖Ψ‖² = ∑_{n≥0} α^{-n}. 
Identical in both cases.

Conclusion
(W₁, u) ≇ (W₂, u)    (different spectrum)
but the embedding is identical.
The information about the phase has been lost.

II. More structural example (continuous spectrum)
Let us take:
H = L²(S¹). 
Let M_z denote the multiplication operator by z.
I define:
W₁ = M_z,     W₂ = M_z̄. 

Both are unitary.

1. Spectral measures
For the constant vector u = 1:
μ₁ = Lebesgue measure on S¹, 
μ₂ = Lebesgue measure on S¹ (since reflection does not change the measure). 

Complex spectrum different as functions, but radial distribution identical:
|λ| = 1. 

2. Moments
m_n = 1. 
Embedding identical.

III. Example with unbounded support (moment indeterminacy)
Now the true pathology.
There exist classical families of different measures on [0,∞)
having identical moments (e.g., modifications of the log-normal measure).
One can construct:
ν₁ ≠ ν₂,     ∫ x^n dν₁ = ∫ x^n dν₂. 

I define multiplication operators:
(W_i f)(x) = √x f(x)    on L²(ν_i). 

Cyclic vector: u = 1.
Then:
m_n = ∫ x^n dν_i. 
Identical for both.
The operators are not unitarily equivalent,
but the embedding is identical.
This is pure moment indeterminacy.

IV. And what about graphs... because that is the goal after all, I am not deriving formalisms without purpose;
In graphs/matrices:
‖W‖ < ∞. 
Compact support ⇒ moment problem determinate.
Thus case III does not occur.
The only real mechanism of indistinguishability is:
loss of information about the phase. 

V. General classification theorem
For bounded operators:
Ψ(W₁, u₁) = Ψ(W₂, u₂) 
⇔
the radial measures are identical.

Differences in:
angular distribution,
complex conjugation,
symmetries with respect to the circle,
are not visible.

VI. Minimal constructive diagnosis
The method classifies:
the operator modulo transformations preserving |λ|. 

This is exactly the class:
λ ∼ e^{iθ} λ. 


=====10=====
I proceed to the analysis strictly from the perspective of the embedding
Ψ(W,u) = (Wⁿ u)ₙ≥0, ‖Ψ‖² = ∑ₙ≥0 α^{-n} ‖Wⁿ u‖², 
that is, the moment invariant
m_n = ∫ |λ|^{2n} dμ_u(λ). 

I am not interested in general pathologies of C*-algebras, but only those that can really:
destroy the existence of the embedding,
destroy its uniqueness,
disturb its classification stability.

I. Unbounded operator
Let W be a densely defined, closed operator.

1. Domain problem
The embedding requires:
u ∈ ⋂ₙ≥0 Dom(Wⁿ). 

For operators of type:
semigroup generator,
Laplacian,
Schrödinger operator,
this set may be:
dense (like analytic vectors),
very small,
trivial.
If it is trivial → the embedding practically does not exist.

2. Growth of moments
If the spectrum is unbounded:
m_n = ∫ xⁿ dν(x) 
may grow faster than exponentially.
Then:
∑ α^{-n} m_n 
always diverges.
The embedding does not exist for any α.

3. Moment problem
On [0,∞) the Stieltjes moment problem may be indeterminate.
There exist:
ν₁ ≠ ν₂,     m_n(ν₁) = m_n(ν₂). 

Then the embedding does not distinguish operators.

Conclusion I
Unbounded operator threatens:
the existence of the embedding,
uniqueness (moment indeterminacy),
norm stability.
This is a serious threat.

II. Continuous spectrum (bounded operator)
We assume W ∈ B(H), but:
σ(W) contains a continuous part.

1. Convergence
If:
sup |λ|² < α, 
the embedding exists.
If:
sup |λ|² = α, 
convergence depends on the behavior of the measure at the boundary.
Possible divergence.

2. Moment determinacy
Since the support is compact:
ν ⊂ [0, ‖W‖²], 
the moment problem is determinate (Hausdorff).
Thus:
m_n uniquely determine ν. 
No moment pathology.

3. Loss of angular information
The embedding depends only on:
|λ|. 
Thus:
different phase distributions,
different symmetries on the circle,
are not visible.
This is a structural loss of information, but not a pathology.

Conclusion II
Continuous spectrum does not destroy the embedding,
but:
makes it sensitive to the boundary,
reduces the information to the radial part.

III. Lack of separability
Assume H is not separable.

1. Embedding
The very definition:
Ψ(W,u) 
does not require separability.
The moment representation works.

2. KK and classification
Lack of separability spoils:
Kasparov product,
UCT,
Elliott program.
But the embedding as a sequence of moments exists.

3. Spectral measures
The measure may be supported on an uncountable set,
but if the support is compact — the moment problem is still determinate.

Conclusion III
Lack of separability does not threaten the embedding.
It threatens only the classification by KK.

IV. Most dangerous configurations
Let us collect this systematically.

Critical
1. Unbounded operator with unbounded spectrum.
2. Moment indeterminacy.
3. Vector u not belonging to all domains Wⁿ.
4. Spectrum touching the convergence boundary.

Structural, but not destructive
1. Continuous spectrum (compact support).
2. Loss of phase.
3. Different angular distributions.

Irrelevant for the embedding
1. Lack of separability.
2. Type II/III von Neumann.
3. Lack of UCT.
4. KK pathologies.

V. Key diagnosis for your graph construction
If:
W originates from a graph,
is a bounded matrix,
compact spectrum,
then:

The embedding is stable and moment determinate.

The only loss of information is the phase.

VI. The deepest point
The construction is essentially:
transformation of the operator into its radial measure. 

Therefore the exact boundary of the theory:
Two operators are indistinguishable if and only if they have identical distribution of |λ|².
All other pathologies are secondary.

Application regime (my plan)
In the intended graph-theoretic setting:
W is a bounded finite matrix,
spectrum is compact,
the moment problem is determinate,
domain issues do not arise.
Therefore, within this regime the only structural loss of information in the basic embedding is the loss of phase.
All other pathological phenomena belong to genuinely infinite-dimensional operator theory and do not affect the graph case.

=====11=====
The current embedding
Ψ(W,u)=(Wⁿ u)ₙ≥0 
sees only the moments
mₙ=∫ |λ|²ⁿ dμ_u(λ), 
that is, exclusively the radial part of the spectral measure. The phase is lost, because the norm eliminates the complex information.
I want to construct an enhancement that recovers the full spectral measure
μ_u on σ(W)⊂ℂ. 
We further assume that W is a normal operator (this is the minimal condition to even speak about a measure on ℂ).

I. What exactly needs to be recovered?
From the spectral theorem:
W=∫_{σ(W)} λ dE(λ),     μ_u(B)=⟨E(B)u,u⟩. 
The full classification of the cyclic representation reduces to recovering μ_u.
To do this, one needs to know all the complex moments:
M_{k,ℓ}=∫ λᵏ λ̄^ℓ dμ_u(λ). 
The radial embedding sees only M_{n,n}.
This is too little.

II. Minimal enhancement: complex moments
I will define a two-dimensional embedding:
Φ(W,u)=(⟨Wᵏ u, Wˡ u⟩)_{k,ℓ≥0}. 
Since:
⟨Wᵏ u, Wˡ u⟩=∫ λᵏ λ̄^ℓ dμ_u(λ), 
we obtain the full matrix of complex moments.
This is exactly the moment description of the measure on ℂ.

III. Determinacy theorem (compact support)
If:
supp(μ_u) is compact, 
then the set of all moments (M_{k,ℓ}) uniquely determines the measure.
The proof is based on:
Stone–Weierstrass theorem,
density of polynomials in C(σ(W)).
Thus:
Φ(W,u) uniquely determines μ_u.

IV. Interpretation as a reproducing kernel
The matrix
K(k,ℓ)=⟨Wᵏ u, Wˡ u⟩ 
is positive definite.
It defines a reproducing kernel space
H_Φ. 
The operator W acts as a shift operator:
S e_k = e_{k+1}. 
In this space:
S 
has exactly the same spectral measure as W in the cyclic representation generated by u.
This gives a full reconstruction.

V. Alternative: Cauchy transform
An even stronger enhancement:
Define the function
G(z)=⟨(z−W)^{-1}u,u⟩. 
This is the Cauchy transform of the measure:
G(z)=∫ 1/(z−λ) dμ_u(λ). 
From classical theory:
G uniquely determines the measure,
one can recover it via the Stieltjes formula.
This is a full, analytic spectral invariant.

VI. Minimal practical embedding
If you want a minimal modification of the original construction, it suffices to pass from norms to inner products:
Instead of:
‖Wⁿ u‖², 
take:
⟨Wᵏ u, Wˡ u⟩. 
This is exactly the Gram matrix of the orbits.

VII. Classification characteristic
For normal operators:
(W₁,u₁) ≅ (W₂,u₂) 
(unitarily equivalent cyclic representations)
⇔
Φ(W₁,u₁)=Φ(W₂,u₂). 
This is a full invariant.

VIII. This works because the algebra...
A=C*(W) ≅ C(σ(W)) 
is commutative.
The matrix of moments is exactly the values of the state
ω_u(f)=∫ f(λ) dμ_u. 
Complex polynomials are dense,
so the moments determine the state,
and the state determines the GNS representation.

IX. What does this change in the construction?
The original embedding:
Ψ(W,u) 
classifies the operator modulo phase symmetries.
The enhanced embedding:
Φ(W,u) 
classifies the full cyclic representation.
This removes the only real source of indistinguishability.

Theorem (Scope of Completeness of the Embedding)
Let W ∈ B(H), u ∈ H, and consider the embedding
Ψ(W,u) = (W^n u)_{n≥0}
with associated radial moments
m_n = ∫ |λ|^{2n} dμ_u(λ)
Then:
Normal case.
If W is normal and σ(W) is compact, then:
the enhanced embedding
Φ(W,u) = (<W^k u, W^ℓ u>)_{k,ℓ≥0}
uniquely determines the spectral measure μ_u, hence determines the cyclic representation up to unitary equivalence.
Radial embedding only.
The original embedding Ψ(W,u) determines only the pushforward measure under
λ ↦ |λ|^2
Thus it classifies the cyclic representation modulo phase symmetries:
λ ∼ e^{iθ} λ
Non-normal case.
If W is not normal, then the radial moment sequence
m_n = ||W^n u||^2
does not determine:
- the Jordan structure,
- the nilpotent part,
- the non-normal component of the operator.
In particular, two non-unitarily equivalent cyclic representations may yield identical radial embeddings.


X. Final form of the enhancement
Φ(W,u)=(⟨Wᵏ u, Wˡ u⟩)_{k,ℓ≥0} 
or equivalently
G(z)=⟨(z−W)^{-1}u,u⟩. 
Both versions uniquely determine the full complex spectrum (with weights).

The basic embedding is a radial spectral invariant;
the enhanced embedding is a complete invariant of the cyclic representation in the normal case.

=====12=====
Algebraic-operator analysis of the kernel of the integrated representation
π⋊V: A⋊_α ℤ → B(H) 

for the dynamical system (A,α), where A is a C*-algebra (separable), α∈Aut(A).

1. Dynamical system and crossed algebra
Let:
A — C*-algebra,
α: A→A — automorphism.
The algebraic crossed algebra:
C_c(ℤ,A) = { ∑_{n∈ℤ} a_n uⁿ : a_n∈A finite support }, 
with multiplication
(a uᵐ)(b uⁿ) = a αᵐ(b) u^{m+n}, 
and involution
(a uⁿ)* = α^{-n}(a*) u^{-n}. 
Its closure in the universal norm gives
A⋊_α ℤ. 

2. Covariant pair
Let π: A→B(H) be a non-degenerate representation, and
V∈U(H) a unitary satisfying the covariance condition:
V π(a) V* = π(α(a)). 
Then there exists the integrated representation
π⋊V: A⋊_α ℤ → B(H), 
given by
(π⋊V)( ∑ a_n uⁿ ) = ∑ π(a_n) Vⁿ. 

3. Kernel of the representation — elementary description
Definition:
ker(π⋊V) = {x∈A⋊_α ℤ: (π⋊V)(x)=0}. 
It is a closed and α-invariant ideal.

4. Reduction to the kernel of π
Denote:
I := ker π ⊂ A. 
Then:
I is an α-invariant ideal,
there exists the quotient algebra
A/I, 
induced automorphism
α̅(a+I)=α(a)+I. 
The representation factors through
(A/I)⋊_α̅ ℤ. 
Thus always:
I⋊_α ℤ ⊂ ker(π⋊V). 

5. Basic theorem (full case)
If:
π is faithful,
(π,V) is the regular representation (induced from π),
then
ker(π⋊V)=0. 
Proof: application of the theorem on faithfulness of the regular representation (for amenable discrete groups; ℤ is amenable).

6. Faithfulness condition (Gauge–Invariant Uniqueness)
Let there exist a circle action (gauge action):
γ_z(a)=a, γ_z(u)=z u. 
If:
1. π is faithful,
2. the representation admits a compatible gauge action,
then
π⋊V is faithful. 

7. General structure of the kernel
For any covariant pair:
ker(π⋊V) = span̅{ ∑ a_n uⁿ: ∑ π(a_n) Vⁿ=0 }. 
It is essential that:
the kernel is always an α-invariant ideal,
the intersection with A gives exactly ker π.

8. Case A=C(X)
Let:
α(f)=f∘T^{-1}. 
If π corresponds to a measure μ on X,
then:
ker π = {f: f=0 μ-a.e.}. 
Then:
ker(π⋊V) ≅ C_0(Y)⋊_T ℤ, 
where Y⊂X is the largest T-invariant set on which π vanishes.

9. Analysis via conditional expectation
There exists a conditional expectation
E: A⋊_α ℤ → A, E( ∑ a_n uⁿ )=a_0. 
If π is faithful and
(π⋊V)(x)=0, 
then:
π(E(x*x))=0. 
Hence:
E(x*x)∈ker π. 
This gives effective control over the kernel.

10. Characterization of the kernel via primitive spectrum
Kernels of crossed representations correspond to:
α-invariant ideals I⊂A 
and representations of the quotient
(A/I)⋊_α̅ ℤ. 
Thus the classification of kernels reduces to:
1. classification of α-invariant ideals,
2. analysis of quotient representations.

11. Relation to the element 1−α
In K-theory:
ker(π⋊V) 
is controlled by the action
1−α_*: K_*(A)→K_*(A). 
If:
1−α_* 
is injective,
then no additional torsion elements arise in the kernel.

12. Version in KK language
The kernel of the representation corresponds to the vanishing of the class
[π] ⊗_A [1−α] ∈ KK(A,ℂ). 
If this class is nonzero,
the representation is not faithful.

13. Structural conclusion
For the system (A,α):
ker(π⋊V) = ideal generated by ker π and covariance relations 

The representation is faithful if and only if:
1. π faithful,
2. no nonzero elements x≠0 with
∑ π(a_n) Vⁿ=0.

14. Context of the construction
In the case:
A=C*(W), 
the GNS representation (π_u,V) is faithful
exactly when:
u is cyclic,
the spectral measure does not vanish nonzero elements of C(σ(W)).
Then:
ker(π_u⋊V)=0. 


=====13=====
1. Algebraic structure of the continuous level
Let:
AR := Cb(ℝ) 
— the Banach algebra of bounded functions with the supremum norm.
We introduce a distinguished subset:
Z := { x ∈ ℝ | sin x = cos x }. 
Explicitly:
Z = { π/4 + k π/2 : k ∈ ℤ }. 
These are the intersection points of the trigonometric functions.

2. Structure of the discrete level
Let:
D := ℤ 
indexing the cells between consecutive points of Z.
We define the decomposition:
ℝ = ⨆_{k ∈ ℤ} I_k, 
where
I_k = [z_k, z_{k+1}). 
Discrete algebra:
AD := ℓ^∞(ℤ). 

3. Rounding as an algebraic morphism
We define the mapping:
R : AR → AD 
by:
(R f)(k) = arg ext_{x ∈ I_k} f(x), 
where the choice of extremum depends on the “tangent rule”:
if sin x > cos x in a neighbourhood — we choose the positive direction,
if cos x > sin x — the opposite direction.
This is a forced discretization by the gradient structure.

4. Ideal point I
At points
x ∈ Z 
we have:
sin x = cos x, tan x = 1. 
We define a new element:
I := { x ∈ Z : |sin x| = 1 }. 
That is:
x = π/4 + 2k π. 
Then:
sin x = cos x = 1/√2. 
However, the global maximum of sin and cos does not occur simultaneously —
therefore I is not a numerical but a semantic symmetry point.
Formally:
I is a central idempotent 
in the algebra extension:
ÃD = AD ⊕ ℂ I, 
with the relation:
I² = I, I a = a I. 

5. Machine shift
Let:
ε > 0. 
Then instead of Z we have:
Z_ε = { x : sin x = cos x + ε }. 
The partition ceases to be symmetric.
Then the morphism:
R_ε 
is purely projective and:
ker R_ε 
no longer contains the class corresponding to I.
The semantics disappears.

6. Rounding as a morphism of the category of algebras
We consider the category:
Alg_ℂ. 
Rounding is a functor:
R : (AR, ‖·‖_∞) → (ÃD, ‖·‖_∞). 
Properties:
1. linearity,
2. preservation of unit,
3. is not an isomorphism,
4. is not an isometry,
5. in the ideal version possesses a central idempotent I,
6. in the machine version is a projection without a central element.

7. Operator interpretation
If we represent:
π_R : AR → B(H), 
then rounding induces:
π_D = π_R ∘ R^{-1} 
on the image.
In the ideal case:
I ↦ orthogonal projection. 
In the numerical case:
there does not exist a corresponding projection operator.

8. Key algebraic property
Ideal version:
ker R = { f : f|_Z = 0 }. 
Machine version:
ker R_ε = { f : f|_{Z_ε} = 0 }. 
Difference:
Z is symmetric, Z_ε is not. 

9. Formal conclusion
Rounding is:
a quotient morphism with respect to the trigonometric tangency relation. 
Ideal version:
contains the central idempotent I,
preserves symmetry.
Machine version:
removes the element I,
forces branch choice,
introduces bias.


=====14=====
The goal is:
to extend the continuous level ℝ,
so that the boundary classes (intersection points) are not “lost” in the quotient,
but are encoded as a central idempotent,
that is, as an irreducible decision requiring further specification.

1. Base algebra
Let
A := C_b(ℝ) 
(with the supremum norm; one can also take C_0(ℝ), it does not change the construction).
Let
Z = { π/4 + k π/2 : k ∈ ℤ } 
— the set of boundary points (intersections of sin and cos).

2. Quotient corresponding to “pure” rounding
We define the ideal:
J := { f ∈ A : f|_Z = 0 }. 
This is a closed ideal of the C*-algebra.
Pure discretization is the quotient:
A_D := A / J. 
Since Z is discrete, we have an isomorphism:
A_D ≅ ℓ^∞(Z). 
This is the algebra of values at the boundary points.
But:
in this construction, the boundary class is already resolved,
there is no place in it for “undecidedness”.

3. Extension by idempotent
I want to preserve the information:
“the value belongs to the boundary class, but has not been specified further”.
We do this as an extension of the C*-algebra by a central projective element.

Definition of the extended algebra
We define:
à := A ⊕ ℂ I 
with the norm:
‖(f, λ I)‖ = ‖f‖_∞ + |λ|. 
Multiplication:
(f, λ I) ⋅ (g, μ I) = (f g, f|_Z μ + g|_Z λ + λ μ I). 
Conditions:
I² = I,     I* = I,     I f = f I = f|_Z. 

4. Algebraic interpretation
The element I:
is a central projection,
represents the “undecided boundary state”,
is not a function on ℝ,
is a new element in the extension.
We have a short exact sequence of C*-algebras:
0 ⟶ A ⟶ Ã ⟶ ℂ ⟶ 0. 
This is an extension by one central projective element.

5. Spectrum structure
Gelfand spectrum:
Ẫ = Â ⊔ {ω_I}. 
The new point ω_I corresponds to the character:
ω_I(f, λ I) = λ. 
This is a formal semantic point.

6. Rounding as a morphism of C*-algebras
Ideal rounding:
R : Ã → A_D 
defined by:
R(f, λ I) = [f] + λ 1_Z. 
That is:
the continuous part passes to the quotient,
the part I becomes a constant function on Z.
Machine version:
R_ε 
does not have the I component,
that is, it operates on A without extension.

7. Key property
In the ideal version:
ker R = { (f, -f|_Z I) : f ∈ A }. 
This is a non-trivial ideal.
Machine version:
ker R_ε = J. 
The corrective component is missing.

8. Operator interpretation
For the representation:
π : Ã → B(H), 
we have:
π(I) = P 
where P is a central orthogonal projection.
If P ≠ 0,
there exists an irreducible space of boundary states.
In the numerical version:
P = 0. 

9. What I really did...
Instead of:
A → A/J, 
we built:
à = algebra with Busby extension 
in which the boundary class is not identified with zero,
but with a new central projection.
This is exactly:
Extension of ℝ by the condition of further specification of the value in the boundary class.

10. K-theoretic structure
Since we added a central projection:
K_0(Ã) = K_0(A) ⊕ ℤ. 
The additional generator corresponds to I.
The machine version does not have this component.


=====15=====
I'm doing a purely algebraic-operator version, in the language of this construction:
states space H,
algebra of connection operators,
crossed product,
central idempotent as rounding effect,
and interpretation of "zero divisor".

1. Reminder of the operational construction
We have:
H = ⨁̅_{n≥0} Aₙ 
and algebra of local operators:
A = ∏_{n∈ℤ} End(Aₙ). 
Shift automorphism:
α: A → A,     α(a)ₙ = a_{n-1}. 
Crossed product:
B = A ⋊_α ℤ. 
Unitary implementing the shift:
U a U* = α(a). 

2. Where does rounding come in?
In the "pure" ℝ version:
connection operates in a continuous way,
there are no distinguished boundary classes,
the spectrum of the connection operator is treated as an object in H.
Rounding introduces:
the condition of resolution when passing through the boundary class.
At the operator level, this means:
the appearance of a central projection P corresponding to the "undecided state".

3. Operator construction of the boundary element
We define the projection:
P ∈ Z(B) 
such that:
P² = P,     P* = P,     [P, a] = 0,     [P, U] = 0. 
That is:
P ∈ Z(B). 

4. Extended connection algebra
New algebra:
B̃ = B ⊕ ℂ P. 
This is a C*-extension by a central projective element.
Structure:
(a + λ P)(b + μ P) = ab + λ b P + μ a P + λ μ P. 
Since P is central:
a P = P a. 

5. Dynamic interpretation
In the crossed product:
B = span̅{ a Uᵏ }. 
Rounding causes:
at certain values (boundary classes)
the connection operator does not act unambiguously
but passes through the projection P.
Formally:
instead of
T ∈ B 
we obtain
T̃ = T (1 - P) + T P. 
The part T P represents the "boundary state".

6. Why is this a zero divisor?
Because:
P (1 - P) = 0,     P ≠ 0,     1 - P ≠ 0. 
Thus the algebra ceases to be a ring without zero divisors.
This exactly corresponds to:
the lack of multiplication by the inverse of zero in ℝ.
Instead of:
1/0 
we use:
lim_{ε→0}. 
In the language of C*-algebra:
we do not pass to an irreversible element,
but algebraically split the space by projection.

7. Infinite boundary class
If boundary classes are indexed by k ∈ ℤ,
then we have a family of projections:
{Pₖ}_{k∈ℤ} 
such that:
Pₖ Pⱼ = 0 (k ≠ j),     ∑ₖ Pₖ ≤ 1. 
Then:
Z(B̃) ⊃ ℓ^∞(ℤ). 
That is, the center becomes infinite-dimensional.

8. Rounding as a modification of the crossed product
In the pure version:
B = A ⋊_α ℤ. 
In the version with rounding:
B̃ = (A ⊕ Z₀) ⋊_α ℤ, 
where Z₀ is the central algebra generated by projections Pₖ.
The automorphism satisfies:
α(Pₖ) = P_{k+1}. 
That is, the projections shift together with the dynamics.
This is exactly:
crossed product with a central boundary fiber.

9. Spectral interpretation
If the connection operator has spectrum σ(T),
then in the version without rounding:
σ(T) 
is continuous.
After extension:
σ(T̃) = σ(T (1 - P)) ∪ σ(T P). 
A spectrum splitting arises.

10. What this means constructively
Introduction of rounding:
1. adds central projections,
2. introduces zero divisors,
3. splits the spectrum,
4. eliminates the need for "division by zero",
5. replaces irreversibility with projective splitting.

11. Key structural observation
If I do not introduce P,
then:
B 
may be simple (simplicity criterion crossed product).
After introducing P:
B̃ 
is not simple — it has a non-trivial center.
That is, rounding destroys the simplicity of the algebra.
This is very important.


=====16=====
Strictly topological and measure-theoretic version:
continuous level ℝ → topological space,
discrete level D → quotient space,
rounding = push-forward of the measure through the quotient map,
ideal version = extension of the space by boundary point(s),
machine version = no boundary point.
Without metaphor.

1. Continuous level
Let:
X=ℝ 
with the standard topology.
Let:
Z = { π/4 + k π/2 : k∈ℤ } 
— the set of boundary points.
We consider the space of Radon measures:
M(X). 

2. Rounding map as a topological mapping
We define the map:
r: X → D 
where:
D=ℤ,
r(x)=k if x∈Iₖ,
where
Iₖ=[zₖ, z_{k+1}). 
This is a Borel mapping (discontinuous at Z).

3. Rounding as push-forward
For a measure μ∈M(X) we define:
r_* μ(E) = μ(r^{-1}(E)). 
This is the standard push-forward.
Then:
r_* μ ∈ M(D). 
Since D is discrete:
r_* μ = { μ(Iₖ) }_{k∈ℤ}. 

4. What happens at boundary points?
Since:
Iₖ = [zₖ, z_{k+1}), 
the point zₖ belongs to exactly one cell.
Thus:
the mass at the boundary point is arbitrarily assigned to one cell,
this corresponds to the machine version (bias to the right/left).

5. Ideal version: space extension
I want to preserve the information:
the mass lies exactly at the boundary point.
We define the extended space:
D̃ = ℤ ⊔ { Iₖ }_{k∈ℤ}. 
Interpretation:
k = interior of the cell,
Iₖ = boundary point between I_{k-1} and Iₖ.

New map
We define:
r̃: X → D̃ 
r̃(x)= 
| k if x∈(zₖ, z_{k+1})
| Iₖ if x=zₖ 
This is a measurable mapping.

6. Ideal push-forward
For a measure μ:
r̃_* μ(k) = μ((zₖ, z_{k+1})) 
r̃_* μ(Iₖ) = μ({zₖ}). 
Now:
the boundary mass is not lost,
nor is it arbitrarily assigned.

7. C*-algebraic interpretation
Function spaces:
C₀(X) and C₀(D̃). 
Inverse map:
r̃^* : C₀(D̃) → C_b(X) 
(r̃^* f)(x)=f(r̃(x)). 
This is a *-homomorphism.
Push-forward of measures is dual to this homomorphism.

8. Connection with the central idempotent
In C₀(D̃):
the characteristic function of the set:
χ_{{Iₖ}} 
is a central projection.
Its image in the dual operator algebra is the projection Pₖ.
Thus:
central idempotent = characteristic function of the boundary class in the quotient space.

9. Zero divisor in the language of measures
If:
μ = δ_{zₖ}, 
then:
r̃_* μ = δ_{Iₖ}. 
But in the machine version:
r_* μ = δ_k or δ_{k-1}. 
That is:
ideal version preserves the boundary atom,
machine version forces a choice.
This is exactly the "ε-shift" effect.

10. Spectral interpretation
If the operator T has spectral measure μ_T,
then spectral rounding is:
μ_T ↦ r̃_* μ_T. 
That is:
compression of the spectrum to classes,
with preservation of boundary atoms.

11. Key observation
Topological rounding is:
quotient of the spectral space by the relation of belonging to the cell. 
Ideal version:
quotient with attached boundary fibers.
Machine version:
pure quotient without fibers.

12. What this means for your construction
In H:
spectral embedding operates on the measure,
rounding is the push-forward of this measure,
central projections correspond to atoms in D̃,
zero divisor = mass concentrated in the boundary class.


=====17=====
In the language of noncommutative geometry:
rounding = fiber splitting in a continuous field of C*-algebras,
and the ideal version = addition of a separate boundary fiber,
which corresponds to the central idempotent.

1. Starting point: algebra as a noncommutative space
I have the algebraic construction:
B = A ⋊_α ℤ. 
I interpret it as a noncommutative space:
X = Prim(B) 
(primitive spectrum).
In the commutative case:
C_0(X) ↔ X. 
In the noncommutative case:
B ↔ “quantum space”. 

2. Fiber spectral structure
If I have a central subalgebra:
Z(B), 
then the algebra is a continuous field of C*-algebras over
Ẑ = Spec(Z(B)). 
In the version without rounding:
Z(B) = ℂ, 
that is, the field is trivial — one fiber.

3. After introducing rounding
I added central projections:
P_k. 
Thus:
Z(B̃) ≅ ℓ^∞(ℤ). 
Spectrum of the center:
Ẑ ≅ βℤ 
(Stone–Čech).
This means:
B̃ 
is a continuous field of C*-algebras over a discrete space.

4. Fibers
For each character χ_k on the center:
B_k = B̃ / ker χ_k. 
In particular:
B_k = P_k B̃. 
This is the algebra of the boundary fiber.
Whereas the regular part:
(1 - ∑ P_k) B̃ 
is the “continuous” fiber.

5. Interpretation as splitting
Without rounding:
B 
is unsplit (one fiber).
After rounding:
B̃ = ⨁_k B_k ⊕ B_reg. 
This is exactly:
splitting of the spectral fiber into components corresponding to boundary classes.

6. Topological version in the language of Connes
In noncommutative geometry the space given by algebra A has:
algebra of functions,
Hilbert module,
Dirac operator.
Rounding causes:
1. growth of the center of the algebra,
2. appearance of new points in the spectrum of the center,
3. the structure becomes non-simple,
4. the space splits into components.
This is the counterpart of:
X → X ⊔ {boundary points}. 

7. Dirac operator and splitting
If I have a spectral triple:
(B, H, D), 
then after extension:
(B̃, H, D) 
the operator D splits:
D = D_reg ⊕ D_gran. 
The fibers have their own spectra.
The geometry becomes the sum of geometries on the fibers.

8. Geometric interpretation
In the noncommutative sense:
rounding = degeneration of continuous geometry,
appearance of singular strata,
central projections = characteristic functions of the strata.
This is the analogue of:
transition from a smooth manifold to a space with singularities.

9. What happens with the crossed product
I had:
B = A ⋊_α ℤ. 
After rounding:
B̃ = (A ⊕ Z_0) ⋊_α ℤ. 
If:
α(P_k) = P_{k+1}, 
then the dynamics shifts the fibers.
This means:
the splitting is compatible with the dynamics.
A field of fibers over the orbit of the ℤ-action arises.

10. K-theoretic consequence
Since:
B̃ = ⨁_k B_k, 
we have:
K_*(B̃) = ⨁_k K_*(B_k). 
That is, rounding changes the K-theoretic classification.

11. Key structural observation
Without rounding:
the algebra may be simple,
the geometry is unsplit,
there are no central idempotents.
With rounding:
the algebra is not simple,
the noncommutative space has many fibers,
central components arise.
This is exactly:
fiber splitting in the sense of noncommutative geometry.


=====18=====
Criterion for the triviality of fiber splitting.
It concerns the case in which adding central projections Pₖ (rounding) does not change the structure of the algebra, neither geometrically, nor in K-theory, nor in representations.

1. Formal setting
We have:
B = A ⋊_α ℤ 
and extension:
B̃ = B ⊕ Z₀, Z₀ = C*({Pₖ}) ⊂ Z(B̃). 
With dynamics:
α(Pₖ) = P_{k+1}. 
The splitting is trivial if and only if
B̃ ≅ B. 

2. Algebraic condition
The splitting is trivial ⇔
Pₖ = 0 for all k. 
This is an elementary condition, but too weak structurally.
I need to move to representations and spectral measures.

3. Measure-theoretic criterion (most important)
Let T ∈ B have spectral measure μ_T.
Rounding generates additional fibers exactly when:
μ_T({zₖ}) ≠ 0 
for some boundary point zₖ.
Thus:
The splitting is trivial ⟺ μ_T(Z) = 0 
for all operators considered in the construction.
In other words:
lack of spectral atoms in boundary classes.

4. Representation criterion
Let
π: B̃ → B(H) 
be a GNS representation.
The splitting is trivial ⇔
π(Pₖ) = 0 ∀ k. 
Then the representation factors through:
B. 
That is, the boundary fibers are “invisible”.

5. Criterion in the center
The splitting is trivial ⇔
the center after extension does not increase:
Z(B̃) = ℂ. 
This occurs when:
the base algebra is simple,
and the extension by Pₖ degenerates to zero.

6. Dynamic criterion (crossed product)
If the action
α: A → A 
is minimal (no non-trivial α-invariant ideals),
then the crossed product is simple.
Then every central extension requires the existence of
α-invariant boundary mass.
If:
α(Z) ∩ Z = ∅ 
in the spectral sense (no return to boundary classes),
then:
Pₖ = 0. 
The splitting disappears.

7. K-theoretic criterion
The splitting is trivial ⇔
the new projections are trivial in K₀:
[Pₖ] = 0. 
If
K₀(B̃) = K₀(B), 
then the extension does not contribute a new class.

8. Geometric interpretation
In the language of noncommutative geometry:
The splitting is trivial ⇔
the spectrum of the Dirac operator does not possess points with mass concentrated in boundary classes.
That is:
lack of singular strata,
the space remains unsplit.

9. Intuitive operational interpretation
Rounding changes nothing when:
the signal does not hit exactly the intersection points,
there is no ideal symmetry sin = cos,
all transitions are “transversal”.
Then:
push-forward of the measure 
does not generate boundary atoms.

10. Strongest theorem
Theorem (triviality of splitting).
Let B = A ⋊_α ℤ.
The extension by central projections Pₖ is trivial if and only if:
∀ π ∈ Rep(B), π(χ_Z(T)) = 0, 
that is, the characteristic function of the boundary set acts as zero in every representation.

11. Conclusion for the construction
The splitting is NOT trivial only in the case of:
ideal symmetry,
atomic measure in the boundary class,
exact hitting of the intersection point.
In every numerical model:
μ(Z) = 0 
with probability 1.
That is:
in practice, rounding geometrically changes nothing,
but algebraically remains a potential singularity.


=====19=====
I am proceeding to a full, strict analysis of cases in which two different graphs have identical radial distribution, even though their (complex) spectrum is different, that is, exactly the situation that limits the original radial spectral embedding. I will operate in the language of operator algebra and spectral measures in the space H.

1. Formal setting
Let G₁, G₂ be two finite graphs (or infinite, but with bounded degrees), and let W₁, W₂ be their adjacency operators/matrix of connection on the Hilbert space H.
For a cyclic vector u ∈ H we define the radial spectral measure:
μᵤʳᵃᵈ(B) := μᵤ({λ ∈ σ(W) : |λ| ∈ B}), 
that is, push-forward of the measure μᵤ through the map λ ↦ |λ|.
Radial embedding:
Ψʳᵃᵈ(W,u) = (⟨Wⁿ u, Wⁿ u⟩)ₙ≥0 = (mₙ)ₙ≥0, mₙ = ∫ |λ|²ⁿ dμᵤ(λ). 
Two graphs are indistinguishable in this embedding if and only if:
μᵤ₁ʳᵃᵈ = μᵤ₂ʳᵃᵈ. 

2. Moment problem
The radial embedding sees only the moments:
mₙ = ∫₀^∞ r²ⁿ dν(r), ν = μᵤʳᵃᵈ. 
Two different graphs may have different spectrum σ(W₁) ≠ σ(W₂), but the sequences of radial moments may coincide if their radial measures ν₁ = ν₂.

3. Case analysis
3.1. Graphs with different phases of eigenvalues
Minimal example: 2×2 matrices
W₁ = (1 0 
      0 -1),   W₂ = (1 0 
                     0 1). 
For a cyclic vector u = 1/√2 (1,1)ᵀ:
|λ₁| = |λ₂| = 1 ⟹ ν₁ = ν₂ = δ₁. 
Different spectrum (1,-1 vs 1,1),
radially identical.
This is a classic case of different phase, the same modulus.

3.2. Graphs with different geometry, but the same degree distribution
For regular graphs (degree d):
each ...(truncated 242 characters)...,0,-2}{2,0,0,−2},
Spectrum K₂,₂: {2,0,0,−2} — here the example is the same; if we add orientation or weight, phases may differ, but |λ| will be identical.

3.3. Infinite graphs, with continuous spectrum
For infinite regular graphs, e.g., two-dimensional lattice grid:
spectrum of the adjacency operator: continuous band [-4,4],
for different operators (e.g., phase shift or different weights), σ changes,
but the radial distribution may be identical if the moduli of eigenvalues are the same.

4. Necessary and sufficient conditions
Let W₁, W₂ have spectral distributions μ₁, μ₂ and radial push-forward ν₁, ν₂.
Theorem (radial equivalence condition)
Ψʳᵃᵈ(W₁,u₁) = Ψʳᵃᵈ(W₂,u₂) ⟺ ν₁ = ν₂. 
Necessary: if the radial embedding is identical, then ν₁ = ν₂.
Sufficient: if ν₁ = ν₂, all radial moments coincide, so the embedding is identical.

5. Mechanism of formation of indistinguishable graphs
1. Different phases of eigenvalues → |λ| the same, λ ≠ λ'.
2. Different geometric structures → symmetry in the spectral modulus preserves the radial distribution.
3. Atomic vs. continuous spectrum → continuous spectrum may “mask” phase differences if the push-forward preserves identical radial distribution.

6. Boundary class
For finite graphs:
the radial distribution is finite,
number of atoms = number of vertices,
different graphs have the same radial embedding if their histograms |λᵢ| are identical.
For infinite graphs:
the Stieltjes moment problem for the radial distribution must be deterministic for the embedding to be unique.

7. Conclusion
Critical summary:
Two graphs may be indistinguishable in the radial embedding if and only if they have identical radial measure of their adjacency operators.
The spectrum may be different (different phase λᵢ), different structures, different weights, but the moments |λ|²ⁿ are identical.
Real cases: small graphs with different eigenvalues of the same modulus, regular graphs with different symmetric structures, graphs with continuous spectrum of identical radial distribution.


=====20=====
Properties of the projection operator applied to data; In the context of projection degeneration;
I formalize this in three layers:
1. data model through graph and connection matrix,
2. definition of projection |π|,
3. definition of projection degeneration as loss of structural distinguishability.

1. Algebraic model of data
Let:
G = (V, E) — graph (finite or locally finite),//for infinite we will pass in the next chapter and locally finite will return;
W — connection matrix (adjacency operator or generalized),
H = ℓ²(V).
Data:
x ∈ H. 
Cascade dynamics:
xₙ = Wⁿ x. 
Data cascade:
K(x) = {xₙ}ₙ≥₀. 

2. Data projection
Let:
π : H → K 
be a linear operator (usually irreversible).
In practice:
π(x) = P x 
for some matrix P (projection, compression, radial embedding, etc.).
Observed signal:
yₙ = π(Wⁿ x). 

3. Definition of projection degeneration
Definition (projection degeneration with respect to W).
The projection π is degenerate with respect to W, if there exist two different states x ≠ x' such that:
∀n≥0: π(Wⁿ x) = π(Wⁿ x'). 
Equivalently:
∃v≠0 such that ∀n: π(Wⁿ v)=0. 
That is:
v ∈ ⋂_{n≥0} ker(π Wⁿ). 
This is the cascade kernel of the projection.
We denote:
N_π := ⋂_{n≥0} ker(π Wⁿ). 

The projection is non-degenerate ⇔
N_π = {0}. 

4. Projection horizon
I define the horizon as the moment from which the signal vanishes.
Definition (projection horizon).
For a given x I define:
N(x) = inf{ N : ∀n≥N, π(Wⁿ x)=0 }. 
If such N exists, we say that:
the projection has a finite horizon for x.
Global horizon:
N_π = sup_{x≠0} N(x). 

5. Redshift (formally)
If:
‖π(Wⁿ x)‖ → 0 
monotonically or exponentially, despite ‖Wⁿ x‖ not decreasing, then we have:
definition (projection redshift)
∃λ < ρ(W) such that ‖π(Wⁿ x)‖ ~ λⁿ. 
That is, the projection compresses the spectrum:
σ(W) ↦ σ(π W π*), 
and the spectral radius decreases.
This is purely spectral degeneration.

6. Spectral degeneration
Let:
W = ∫ λ dE(λ). 
The projection generates the operator:
W_π = π W π*. 
Spectral degeneration ⇔
σ(W_π) ⊂ σ(W). 
In particular:
ρ(W_π) < ρ(W). 
This is the formal counterpart of “fading”.

7. Multifractality (inter-projection degeneration)
Let there be two projections:
π₁, π₂. 
Each generates a kernel:
N_π₁, N_π₂. 
If:
N_π₁ ≠ N_π₂, 
then the state space decomposes into:
H = (N_π₁ ∩ N_π₂) ⊕ other components. 
If:
N_π₁ + N_π₂ ≠ H, 
then none of the projections reconstructs the full structure.
This is formally:
bifractality of the space with respect to the set of projections.
For a family {π_i}_{i=1}^k:
N_tot = ⋂_i N_π_i. 
If:
dim N_tot > 0, 
I have multifractality. It was supposed to be.

8. This is NOT a singularity of the graph
The graph is determined by W.
If:
W 
is uniquely determined in H,
then degeneration occurs solely because:
π 
is not a full observation.
Formally:
ker π ≠ {0}. 
Thus:
degeneration = loss of information by the observer,
not a defect of the graph structure.

9. Impossibility of reconstruction
Reconstruction of the graph from the projection is possible ⇔
\overline{span}{ Wⁿ ker(π)^⊥ : n≥0 } = H. 
If not:
the graph is not reconstructible from the given projection.
This is the classic observability problem (control theory).

10. Definitional summary
Degeneration of the projection with respect to data through graph and connection matrix is:
1. existence of a non-zero cascade kernel N_π,
2. spectral compression ρ(W_π) < ρ(W),
3. existence of a finite projection horizon,
4. lack of reconstruction of the graph from projection data.
These are four equivalent levels of description of the same structure.


=====21=====
I formalize the mechanism of “fading”, “redshift” and “horizon” as a purely operator effect of projection on an infinite graph.
Metaphors and physics are omitted. I assume that it will be easier for the reader to navigate the reasoning taking into account such nomenclature. It is not related to physics here, but to an algebraic projection phenomenon equivalent to the discussed one.

1. Setting
Let:
G = (V, E) — infinite graph, locally finite,
W — adjacency operator (or connection matrix) acting on
H = ℓ²(V). 
We assume:
W : H → H 
bounded (e.g., graph with bounded degree).
We choose a base vertex v₀ ∈ V.
Let:
e_{v₀} ∈ H 
— base vector (delta at v₀).

2. Exploratory cascade
We define:
xₙ := Wⁿ e_{v₀}. 
The vector xₙ has support exactly in the set:
Sₙ = { v : dist(v, v₀) = n }. 
Ball of radius n:
Bₙ = ⋃_{k=0}^n Sₖ. 
This is purely graph-based layered exploration.

3. Local projection
Let:
π_N : H → H 
be the orthogonal projection onto:
H_N := ℓ²(B_N). 
That is, we observe only the ball of radius N.
Observed signal:
yₙ = π_N Wⁿ e_{v₀}. 

4. Projection horizon
Definition (horizon).
The horizon for e_{v₀} with respect to π_N is:
n > N ⟹ yₙ = 0. 
That is:
∀n > N: π_N Wⁿ e_{v₀} = 0. 
This always holds, because Wⁿ e_{v₀} has support outside B_N.
Conclusion:
The projection generates an artificial horizon at distance N.
The graph remains infinite, but the projection makes it locally finite.

5. Redshift as norm damping
We consider the observed norm:
‖yₙ‖² = ‖π_N Wⁿ e_{v₀}‖². 
For a regular graph of degree d:
‖Wⁿ e_{v₀}‖ ~ d^{n/2}. 
Whereas:
‖yₙ‖ = { ‖Wⁿ e_{v₀}‖    if n ≤ N  
          0               if n > N  }

This is an extreme case.

6. Generalization: weighted projection with damping
Instead of sharp cutoff, we take a weighted projection:
(π_α f)(v) = α^{dist(v,v₀)} f(v),   0 < α < 1.   
This is not an orthogonal projection, but a compression operator.
Then:
yₙ = π_α Wⁿ e_{v₀}. 
Norm:
‖yₙ‖² = ∑_{v∈Sₙ} α^{2n}. 
If the number of vertices in the layer grows as:
|Sₙ| ~ dⁿ, 
then:
‖yₙ‖² ~ (d α²)ⁿ. 

7. Critical parameter (asymptotic horizon)
If:
d α² < 1, 
then:
‖yₙ‖ → 0. 
This is the formal “redshift”.
If:
d α² = 1, 
critical state.
If:
d α² > 1, 
the projection does not damp the growth.

8. Spectral approach
Effective operator:
W_α := π_α W π_α^{-1}. 
Spectral radius:
ρ(W_α) = α ρ(W). 
Thus:
ρ(W_α) < ρ(W) 
for α < 1.
This is strict spectral compression.

9. Local closure of the graph
We define the subspace:
H_α = \overline{span} { π_α Wⁿ e_{v₀} : n ≥ 0 }. 
If:
d α² < 1, 
then:
H_α 
is separable and closed, even though the graph is infinite.
Further increase of n does not contribute norm energy.
This is the formal “pointlessness of further collection”.

10. Projection limit
We consider the limit:
lim_{N→∞} π_N Wⁿ e_{v₀}. 
If:
∑_n ‖π_α Wⁿ e_{v₀}‖² < ∞, 
then:
e_{v₀} 
is in the domain of the resolvent:
(I − α W)^{-1}. 
This is algebraic closure.

11. Structural conclusion
For an infinite graph:
local projection generates a finite horizon,
weighted projection generates an asymptotic horizon,
both are artifacts of the projection operator,
the graph remains infinite and irreducible.
Formally:
\overline{⋃_n B_n} = V, 
but:
\overline{span} H_α ⊂ H. 

12. Key theorem
Let G be a graph with exponential growth of layers Sₙ ~ dⁿ.
For radially damping projection with parameter α:
the projection has an asymptotic horizon ⇔ d α² ≤ 1. 
Then:
‖π_α Wⁿ e_{v₀}‖ → 0. 

13. Intuitions
This is not a singularity of the graph.
This is:
effect of spectral compression,
reduction of spectral radius,
loss of norm energy in layers,
local closure despite open edges.
This is a purely operator mechanism.


=====22=====
Assuming:
G = (V, E) — infinite graph, locally finite,
W — connection operator on H = ℓ²(V),
for a vertex v ∈ V we consider the local projection π_v satisfying the condition of local closure (e.g., radial damping with parameter α_v such that the criterion from the previous step holds).
For simplicity we write:
H_v := \overline{span} { π_v Wⁿ e_v : n ≥ 0 } 
— the projection space generated from node v.
This is a closed subspace of H.
Local closure means:
H_v ⊊ H. 

1. Definition of inter-projection multifractality
Let v₁, v₂ ∈ V and the corresponding spaces:
H₁ := H_{v₁},   H₂ := H_{v₂}. 
Definition (bifractality).
The system (π_{v₁}, π_{v₂}) is bifractal if:
H₁ ≠ H₂. 
Stronger form (non-trivial bifractality):
H₁ ⊈ H₂ and H₂ ⊈ H₁. 
This means lack of mutual reconstructibility.

2. Case I — disjoint boundaries
Assume:
supp(H₁) ∩ supp(H₂) = ∅. 
Formally:
H₁ ⊥ H₂. 
Then:
H = H₁ ⊕ H₂ ⊕ H_res. 
This is the case of maximal bifractality.
Purely algebraic interpretation:
projections generate disjoint cyclic subspaces,
no state generated from v₁ is visible from v₂.
Sufficient condition:
dist(v₁, v₂) > N₁ + N₂ 
for finite horizons.

3. Case II — partially common boundaries, but without base nodes
We assume:
H₁ ∩ H₂ ≠ {0}, 
but:
e_{v₁} ∉ H₂,   e_{v₂} ∉ H₁. 
That is:
the spaces have a common part,
but no base node belongs to the space of the other.
Formally:
H₁ = K ⊕ K₁,   H₂ = K ⊕ K₂, 
where K = H₁ ∩ H₂.
This is bifractality with a common part.
Structurally:
H₁ + H₂ = K ⊕ K₁ ⊕ K₂. 
Reconstruction of the full geometry from one projection impossible.

4. Case III — “smooth”
We assume:
e_{v₁} ∈ H₂,   e_{v₂} ∈ H₁. 
But:
H₁ ≠ H₂. 
That is:
each sees the node of the other,
the spaces partially overlap,
but are not identical.
Formally:
H₁ ∩ H₂ ≠ {0},   H₁ + H₂ ⊊ H. 
This is mild bifractality.
The difference arises from:
π_{v₁} ≠ π_{v₂} 
(e.g., different damping parameters α₁ ≠ α₂).

5. Asymmetric visibility
Now the unequal case:
e_{v₂} ∈ H₁,   e_{v₁} ∉ H₂. 
That is:
H₂ ⊊ H₁. 
This is one-sided projection inclusion.
Formally:
H₂ ⊂ H₁. 
Typical condition:
α₁ > α₂, 
that is, the projection v₁ has a larger spectral range.

6. Operator classification of bifractality
For two projections we define:
N_i = ⋂_{n≥0} ker(π_{v_i} Wⁿ). 
Then:
disjointness ⇔ H₁ ⊥ H₂,
partial commonality ⇔ N₁ ≠ N₂,
asymmetry ⇔ N₂ ⊂ N₁.

7. Spectral approach
Let:
μ_{v_i} 
— spectral measures generated by e_{v_i}.
The projection spaces correspond to spectrum lifts:
H_i = L²(σ(W), μ_{v_i}^{(α_i)}). 
Bifractality ⇔
μ_{v₁}^{(α₁)} ≠ μ_{v₂}^{(α₂)}. 

8. Fundamental property
Multifractality is exclusively a property of the pair:
(π_{v_i}, W), 
not of the graph itself.
The graph may be homogeneous, and yet:
H₁ ≠ H₂. 

9. Generalization to n projections
Family {π_{v_i}}_{i=1}^k.
We define:
H_i = cyclic space generated by π_{v_i}. 
Multifractality ⇔
{H_i} does not form one common subspace cover of H. 
Extreme case:
⊕_i H_i ⊊ H. 

10. Formal summary
Inter-projection bifractality is a relation between closed subspaces H_i generated by different local projections.
Classification:
1. H₁ ⊥ H₂ — disjoint,
2. H₁ ∩ H₂ ≠ {0}, lack of mutual bases,
3. mutual visibility, but different spaces,
4. one-sided inclusion H₂ ⊊ H₁.
This is the complete classification of the two-projection case.


=====23=====
I assume exactly the same model as above:
H = ℓ²(V),
W — self-adjoint connection operator,
for node v we have the cyclic space
H_v := \overline{span} { Wⁿ e_v : n ≥ 0 }, 
(possibly with damping by the radial function g_v(W), which spectrally gives a modification of the measure).
From the spectral theorem:
H_v ≅ L²(σ(W), μ_v), 
where
μ_v(B) = ⟨E(B) e_v, e_v⟩ 
is the spectral measure of the vector e_v.

1. Reduction of the problem
The condition of asymmetric inclusion:
H₂ ⊊ H₁ 
is equivalent to the existence of an isometric embedding
L²(σ(W), μ_{v₂}) ↪ L²(σ(W), μ_{v₁}) 
as a closed subspace.
From classical theory of L² spaces:
L²(μ₂) ⊂ L²(μ₁)  ⇔  μ₂ ≪ μ₁. 
That is:

Theorem (absolute continuity criterion)
H₂ ⊂ H₁  ⇔  μ_{v₂} ≪ μ_{v₁}. 
The inclusion is proper (⊊), when additionally:
μ_{v₁} ≪̸ μ_{v₂}. 

2. Interpretation through Lebesgue decomposition
We decompose:
μ_{v₂} = f μ_{v₁} + μ_{v₂}^⊥. 
Then:
H₂ ⊂ H₁  ⇔  μ_{v₂}^⊥ = 0,
the inclusion is proper ⇔ f is not invertible in L∞(μ_{v₁}).
More precisely:
H₂ = { f^{1/2} ϕ : ϕ ∈ L²(μ_{v₁}) }. 
If f > 0  μ_{v₁}-almost everywhere → the spaces are equal.
If f = 0 on a set of positive μ_{v₁}-measure → the inclusion is proper.

3. Support criterion
Necessary condition:
supp(μ_{v₂}) ⊂ supp(μ_{v₁}). 
If the inclusion of supports is proper and there is no singular part — we obtain H₂ ⊊ H₁.

4. Point / continuous decomposition
We write:
μ_v = μ_v^{pp} + μ_v^{ac} + μ_v^{sc}. 
Conditions for asymmetry:
(A) Point
If
supp(μ_{v₂}^{pp}) ⊂ supp(μ_{v₁}^{pp}) 
and lack of additional masses outside this range — the point part gives inclusion.

(B) Absolutely continuous
If
dμ_{v₂}^{ac} / dμ_{v₁}^{ac} ∈ L∞, 
then the continuous part is embedded.

(C) Singular
The occurrence of a singular component μ_{v₂}^{sc} absent in μ_{v₁}
immediately destroys the inclusion.

5. Functional criterion through the resolvent
The spectral measure is given by the Stieltjes transform:
m_v(z) = ⟨ (W − z)^{-1} e_v, e_v ⟩. 
Condition:
H₂ ⊂ H₁ 
⇔
Im m_{v₂}(λ + i0) = f(λ) Im m_{v₁}(λ + i0) 
for a function f ≥ 0.

6. Approach through graph transfer functions
If the graph is homogeneous, and the difference arises from damping α₁, α₂,
then spectrally:
μ_{v_i}^{(α_i)} = |g_{α_i}(λ)|² μ_v(λ). 
Then:
H₂ ⊂ H₁ 
⇔
|g_{α₂}|² = f |g_{α₁}|² 
with f ∈ L∞(μ_v).
In practice:
|g_{α₂}(λ)| ≤ C |g_{α₁}(λ)|. 
If additionally the inequality is strict on a set of positive measure → the inclusion is proper.

7. Minimal algebraic criterion
Equivalently:
e_{v₂} ∈ H₁. 
Because:
H₂ = \overline{span} { Wⁿ e_{v₂} } 
and if the generator lies in H₁, then the whole does too.
Thus:

Elementary criterion
H₂ ⊂ H₁  ⇔  e_{v₂} ∈ H₁. 
Spectrally this is exactly the condition μ_{v₂} ≪ μ_{v₁}.

8. Geometric interpretation in the graph
Asymmetry occurs when:
1. All spectral paths from v₂ pass through the “visibility zone” of v₁,
2. but not conversely.
Which in the model with damping corresponds to:
α₁ > α₂. 

9. Necessary and sufficient condition (final form)
H₂ ⊊ H₁ 
if and only if:
1. μ_{v₂} ≪ μ_{v₁},
2. there exists a set A such that
   μ_{v₁}(A) > 0,   μ_{v₂}(A) = 0. 
This is the complete spectral criterion.


=====24=====
I am transitioning from spectral language to purely combinatorial.
We have:
infinite locally finite graph G = (V, E),
connection matrix A (adjacency / connection matrix),
H = ℓ²(V),
for v ∈ V
H_v = closed span { Aⁿ e_v : n ≥ 0 }. 
We want a purely graph-theoretic condition when:
H_{v₂} is a proper subspace of H_{v₁}. 

1. Elementary algebraic criterion
From the previous step:
H_{v₂} ⊂ H_{v₁} ⇔ e_{v₂} ∈ H_{v₁}. 
And by definition:
H_{v₁} = closed span { Aⁿ e_{v₁} }. 
Thus we need:
e_{v₂} = limit_{k→∞} ∑_{n=0}^{N_k} c_{n,k} Aⁿ e_{v₁}. 

2. Interpretation through paths
The u-coordinate of the vector Aⁿ e_{v₁} is the number of paths of length n from v₁ to u.
Thus:
e_{v₂} ∈ H_{v₁} 
means that the delta function at v₂ can be approximated by combinations of path counts outgoing from v₁.
This is possible only when:

3. Necessary condition (reachability)
v₂ belongs to the closure ∪_{n≥0} S_n(v₁), 
where S_n(v₁) is the graph ball of radius n.
That is:
there must exist a path from v₁ to v₂.
Lack of path ⇒ immediate lack of inclusion.

4. Stronger condition — local cyclicity
The existence of a path is not sufficient.
We need the functions:
u ↦ number of paths of length n from v₁ to u 
to distinguish v₂ from the rest of the graph.
Formally:
for every u ≠ v₂ there exists n such that: (Aⁿ)_{v₁,u} ≠ (Aⁿ)_{v₁,v₂}. 
If there exists a vertex u ≠ v₂ such that:
(Aⁿ)_{v₁,u} = (Aⁿ)_{v₁,v₂} for all n, 
then v₁ does not spectrally distinguish v₂ from u.
Then:
e_{v₂} ∉ H_{v₁}. 

5. Combinatorial definition (radial indistinguishability)
We define the relation:
Let ~_{v₁} be the equivalence relation given by the formula:
u ~_{v₁} w ⇔ for every n: (Aⁿ)_{v₁,u} = (Aⁿ)_{v₁,w}. 
This is a purely graph-theoretic equivalence relation.
Then:

Theorem (combinatorial criterion)
H_{v₂} ⊂ H_{v₁} 
if and only if:
1. v₂ is reachable from v₁,
2. the equivalence class of v₂ with respect to ~_{v₁} is singleton.

6. When is the inclusion proper?
For:
H_{v₂} to be a proper subspace of H_{v₁}, 
there must exist a vertex u such that:
u reachable from v₁,
but u is not radially distinguishable from v₂ (that is, u ≁_{v₂} u' for some u').
Formally:
there exists u such that: u does not belong to the same class ~_{v₂} as some u', 
where these classes are distinguishable from v₁, but indistinguishable from v₂.

7. Structural interpretation
The inclusion holds exactly when:
the structure of the subgraph seen from v₂ is a combinatorial “subpattern” of the structure seen from v₁.
This means:
all local neighborhood types of v₂ also occur in the cascade of v₁,
but not conversely.

8. Classical cases
(A) Regular tree
Every vertex has identical radial expansion.
Then:
H_{v₁} = H_{v₂} 
for all nodes.
Lack of asymmetry.

(B) Graph with one “dense island”
If:
v₁ lies in a strongly branched part,
v₂ in a simple chain outgoing from it,
then:
H_{v₂} ⊂ H_{v₁} (properly). 
Because all paths from v₂ are a subset of those from v₁,
but v₁ sees additional branchings.

(C) Graph with an automorphism swapping vertices
If there exists an automorphism g such that:
g(v₁) = v₂, 
then:
H_{v₁} = H_{v₂}. 
Lack of asymmetry.

9. Final purely matrix form
H_{v₂} ⊂ H_{v₁} 
if and only if:
1. for every n the support of row v₂ in Aⁿ is contained in the support of row v₁ in Aⁿ,
2. and the vector e_{v₂} is the limit of linear combinations of vectors Aⁿ e_{v₁} (n ≥ 0). 
Condition (1) is purely combinatorial,
condition (2) reduces to radial indistinguishability.

10. Final intuition
Inclusion asymmetry occurs when:
the cascade from v₂ does not generate any new type of paths,
and the cascade from v₁ generates more structures than the cascade from v₂.
This is strictly a property of the distribution of path counts in successive powers of matrix A,
and not a property of the global topology of the graph.


=====25=====
I am now combining three elements into one purely combinatorial structure:
A — connection matrix of the infinite graph G = (V, E),
projection from node v as the cascade {Aⁿ e_v}_{n≥0},
local closure (horizon) + fading (redshift).
I am working without spectral language — exclusively at the level of powers of A.

1. Projection cascade as radial structure
For v ∈ V we define graph balls:
S_n(v) = {u ∈ V: dist(v, u) = n}. 
and the path-counting function:
P_n(v, u) := (Aⁿ)_{v, u}. 
This is the number of paths of length n from v to u.
The projection cascade is the family of functions:
K_v := {P_n(v, ·)}_{n≥0}. 

2. Introduction of damping (combinatorial redshift)
Previously damping was a spectral function. Combinatorially we introduce a weight:
w(n) ↓ 0. 
We define the weighted cascade:
R_v(u) = ∑_{n=0}^∞ w(n) P_n(v, u). 
This is the sum of all paths with damping dependent on length.
Convergence condition:
∑_n w(n) P_n(v, u) < ∞. 
Then the projection is locally finite.

3. Definition of horizon (combinatorial)
For a threshold λ > 0 we define:
H_v(λ) := {u ∈ V: R_v(u) ≥ λ}. 
Interpretation:
vertices with too small contribution from the damped sum are no longer distinguishable,
the graph “closes locally”.
The horizon is the boundary set:
∂H_v := {u: R_v(u) = λ}. 
If there exists N such that:
w(n) P_n(v, u) < ε ∀n > N, 
then further levels do not change the projection — local closure occurs.

4. Redshift as a purely path effect
Combinatorial redshift is the situation when for increasing n:
P_{n+1}(v, u)/P_n(v, u) → ρ(u), 
but simultaneously:
w(n) → 0 
so quickly that:
w(n) P_n(v, u) → 0. 
Then:
the graph structure may densify,
but projectively the contribution decreases.
This is purely an effect of multiplication:
local growth × global damping. 

5. Connection with the inclusion H_{v₂} ⊂ H_{v₁}
We recall the combinatorial criterion:
H_{v₂} ⊂ H_{v₁} 
⇔
e_{v₂} ∈ closed span {Aⁿ e_{v₁}}. 
Now with weight:
R_{v_i}(u) = ∑_n w_i(n) P_n(v_i, u). 
Inclusion asymmetry occurs when:
∀u, n: P_n(v₂, u) > 0 ⇒ ∃m: P_m(v₁, u) > 0, 
and additionally:
w₂(n) P_n(v₂, u) ≤ C w₁(m) P_m(v₁, u). 
That is:
every path accessible from v₂ has a counterpart in the cascade of v₁,
and is not more strongly faded.

6. Horizon and inclusion
Assume that for some λ:
H_{v₂}(λ) ⊂ H_{v₁}(λ). 
This means:
everything that the projection v₂ “sees” above the threshold,
is contained in the visibility zone of v₁.
If the inclusion is proper:
H_{v₂}(λ) ⊊ H_{v₁}(λ), 
then:
v₁ has a more distant horizon.
This is the exact combinatorial counterpart of the earlier spectral asymmetry.

7. Condition on the damping function
Let:
β_v(n) := max_{u ∈ S_n(v)} P_n(v, u). 
If:
∑_n w_v(n) β_v(n) 
converges faster for v₂ than for v₁,
then the horizon of v₂ is closer.
Then:
H_{v₂} ⊊ H_{v₁}. 

8. Classes of cases (combinatorially)
(I) Disjoint horizons
If:
dist(v₁, v₂) > rad(H_{v₁}) + rad(H_{v₂}), 
then:
H_{v₁} ∩ H_{v₂} = ∅. 

(II) Partial coverage
If graph balls partially intersect,
but profiles P_n(v₁, ·) and P_n(v₂, ·) are different,
then we have partial commonality.

(III) Horizon asymmetry
If for all n:
S_n(v₂) ⊂ S_{n+k}(v₁) 
for some fixed shift k,
then:
H_{v₂} ⊂ H_{v₁}. 
This is a purely geometric criterion.

9. Interpretation of “black sky”
If there exists N such that:
∑_{n > N} w(n) P_n(v, u) < ε ∀u, 
then further layers are projectively indistinguishable.
The graph is infinite,
but the projection is locally finite.
This is exactly:
lim_{n → ∞} w(n) ‖Aⁿ e_v‖ = 0. 

10. Final synthesis
Inclusion asymmetry in the language of horizon and redshift:
H_{v₂} ⊊ H_{v₁} 
occurs when:
1. every path accessible from v₂ is contained in the expansion of v₁,
2. the damping function for v₂ causes faster fading,
3. there exist paths from v₁ that v₂ never generates.
This is 100% combinatorial:
exclusively a property of the sets S_n(v), path counts P_n and the function w(n).


=====26=====
I am transitioning to the full formalism of multiple projections, exclusively combinatorially.
We assume:
G = (V, E) infinite graph, locally finite,
A connection matrix,
family of projections {π_i}_{i∈I},
each projection defined by:
base node v_i ∈ V,
damping function w_i : ℕ → ℝ₊, decreasing.
We work under local closure (horizon).

1. Projection as a radial operator
For i ∈ I we define the influence function:
R_i(u) = ∑_{n=0}^∞ w_i(n) P_n(v_i, u),   where P_n(v_i, u) = (Aⁿ)_{v_i, u}. 
We define the resolution threshold λ_i > 0.
Projection zone:
H_i = {u ∈ V : R_i(u) ≥ λ_i}. 
We assume that for each i there exists an effective radius N_i such that:
w_i(n) P_n(v_i, u) < λ_i   ∀n > N_i. 
This is local closure.

2. Definition of multifractality (general)
Let the family of projections {H_i}_{i∈I} generate a multifractal structure when the sets H_i are not all identical.
Formally:
∃i ≠ j : H_i ≠ H_j. 

3. Structure of partial order
We define the relation:
π_i ⪯ π_j ⇔ H_i ⊂ H_j. 
This is a partial order on the set of projections.
Multifractality corresponds to the set {π_i} not being a total chain.

4. Types of multifractality
(A) Disjoint
H_i ∩ H_j = ∅. 
(B) Partially common
H_i ∩ H_j ≠ ∅,   H_i ⊈ H_j,   H_j ⊈ H_i. 
(C) Asymmetric
H_i ⊊ H_j. 
(D) Cyclic multifractality
There exists a cycle of indices:
π_{i₁} ⪯ π_{i₂} ⪯ ⋯ ⪯ π_{iₖ} ⪯ π_{i₁} 
without equality of all spaces.
This is possible only with different thresholds λ_i.

5. Horizons as radial functions
We define the effective radius:
ρ_i = max {n : ∃u ∈ S_n(v_i),   R_i(u) ≥ λ_i}. 
Multifractality may arise from:
1. different centers v_i,
2. different functions w_i,
3. different thresholds λ_i.

6. Generalization to graphs of arbitrary growth dimension
Let the growth function:
γ_v(n) = |S_n(v)|. 
If the graph has growth:
polynomial (e.g. ℤ^d),
exponential (regular tree),
mixed,
then the closure condition:
∑_n w_i(n) γ_{v_i}(n) < ∞ 
guarantees the existence of a horizon.
Multifractality appears when:
γ_{v_i}(n) ≁ γ_{v_j}(n). 

7. Multifractality in the lattice sense
The set {H_i} with operations:
H_i ∧ H_j = H_i ∩ H_j,   H_i ∨ H_j = H_i ∪ H_j 
forms a lattice of subsets of V.
The multifractal structure is non-trivial when the lattice does not reduce to a single element nor a chain.

8. General criterion (n projections)
The family {π_i}_{i∈I} is multifractal if and only if:
∃u ∈ V 
and indices i, j such that:
R_i(u) ≥ λ_i,   R_j(u) < λ_j. 
This is the minimal distinguishability criterion.

9. Spectral counterpart (without using spectrum)
Instead of spectral measures we have path profiles:
Φ_i(n) = {u : P_n(v_i, u) > 0}. 
Multifractality ⇔
∃n, i, j : Φ_i(n) ⊈ Φ_j(n). 

10. Final definition
Definition (projection multifractality of an infinite graph).
Graph G with a family of locally closed projections {π_i} is multifractal if the partial order ({π_i}, ⪯) is non-trivial, i.e.:
1. there exist disjoint projections,
2. or partially common,
3. or asymmetrically inclusive,
4. or form a non-trivial lattice of subsets.
Multifractality is exclusively a property of:
path distributions P_n,
damping functions w_i,
thresholds λ_i.
It is not a property of the graph itself.


=====27=====
I am transitioning from a pure graph to the model:
G = (V, E) infinite, locally finite,
on V there is given an n-metric d (metric or hypermetric; not ultrametric),
projection from vertex v collects vertices in a “cascade” consistent with d,
damping depends on this metric.

1. Horizon as a level of the influence function in n-metric space
Let:
R_v(u) = ∑_{n≥0} w(d(v, u)) P_n(v, u), 
where:
d is the given n-metric,
w monotonically decreases,
P_n as before.
Horizon:
H_v(λ) = {u : R_v(u) ≥ λ}. 
In the case of Euclidean metric the balls:
B_r(v) = {u : d(v, u) ≤ r} 
are spherical.

2. Deformation of the horizon
Assume that:
the real local structure generates Finsler metric F_x(ξ),
the projection assumes globally Riemannian metric g_x(ξ, ξ).
Locally:
d_F(v, u) = inf_γ ∫ F_{γ(t)}(γ̇(t)) dt. 
Projectively we assume:
d_g(v, u) = inf_γ ∫ √{g_{γ(t)}(γ̇, γ̇)} dt. 
If F ≠ √g, then balls:
B_r^F(v) ≠ B_r^g(v). 
This is deformation of the horizon.

3. Formal definition of deformation
Let:
ρ_v^{(F)}(λ) = sup {d_F(v, u) : R_v(u) ≥ λ}. 
Analogously ρ_v^{(g)}(λ).
Deformation of the horizon:
Δ_v(λ) = ρ_v^{(g)}(λ) - ρ_v^{(F)}(λ). 
If Δ_v ≠ 0, the projection assumes incorrect geometry of collection.

4. Lack of information about “how far one can see”
Important:
From the level of R_v(u) one cannot recover the real d_F, because:
R_v(u) = ∑ w(d_g(v, u)) P_n(v, u) 
contains only distorted information.
Different metrics d_F can give identical profiles R_v with appropriate rescaling of w.
This is degeneration of the inverse:
(d_F, w) ↦ R_v 
is not injective.

5. Appearance of radiality
Each projection generates level sets:
{u : R_v(u) = c}. 
They are isotropic with respect to the assumed metric.
That is, the projection always “sees” the space as radial
in the sense of the assumed norm.
If the real metric is Finslerian with norm:
F_x(ξ) = √{ξ^T A(x) ξ} + anisotropic component, 
then level sets are ellipsoids.
Riemannian projection replaces them with spheres.//and since Finslerian is a generalization of Riemannian, this computational trouble is guaranteed;

6. Accumulation of error with cone deviation
Let:
F_x(ξ) = √{ξ^T A(x) ξ} 
where A(x) positive definite, but directionally variable.
If the projection assumes:
g_x = I, 
then the real radius in direction θ:
r_F(θ) = r_g / √{θ^T A(x) θ}. 
Deviation of cones:
κ(θ) = √{θ^T A(x) θ}. 
Radial error:
δ r(θ) = r_g (1 - 1/κ(θ)). 
The greater the anisotropy of A(x),
the greater the distortion of the horizon.

7. Influence on redshift
Damping depends on d_g, and the real number of paths depends on d_F.
If in a certain direction the graph grows faster, then:
P_n(v, u) ~ γ(θ)^n. 
But damping uses incorrect radius.
Effect:
w(d_g) γ(θ)^n 
does not correctly compensate the growth.
This generates:
local accelerated fading,
apparent “emptiness” between axes.

8. Degeneration of projection axes
Assume two axes θ₁, θ₂.
If:
κ(θ₁) ≠ κ(θ₂), 
then effective horizon satisfies:
ρ(θ₁) ≠ ρ(θ₂). 
Riemannian projection forces:
ρ(θ₁) = ρ(θ₂). 
Thus part of the space gets “compressed”,
another “stretched”.
Between axes an artificial gap appears:
B_r^g \ B_r^F ≠ ∅. 

9. Impossibility of graph reconstruction
Because the projection aggregates data to:
R_v(u), 
and many different local structures A(x)
give identical radial profiles after smoothing,
reconstruction of the full graph structure is impossible.
This is degeneration:
(graph, metric F) ⟶ (radial profile) 
is not invertible.

10. Synthesis
Deformation of the horizon in n-metric space consists in that:
1. Projection assumes global norm (e.g. Riemannian),
2. Real local geometry is anisotropic (Finsler),
3. Damping depends on incorrect metric,
4. Level sets of influence are forced spheres,
5. Real propagation cones are ellipsoids,
6. Systematic radial deformation arises,
7. Redshift becomes directionally dependent,
8. In effect, an appearance of excess “emptiness” appears.
Correct approach:
hypermetric projection on Finsler, treated globally as Riemann, generates an increasing computational error proportional to the anisotropy of the directional norm.


=====28=====
I am transitioning to a purely operator model of “metric perception error”.
Assumptions:
G = (V, E) infinite graph, locally finite.
The graph itself has no metric — the metric appears exclusively in the projection procedure.
Projection from node v uses locally correct Riemannian metric g.
Globally the real propagation geometry has Finslerian character F.
The error arises from using g to model the dynamics, which actually obeys F.
I want to construct:
operator of accumulating error as the difference of weighted Laplacians.

1. Weighted Laplacian dependent on the projection metric
Let for each edge (x,y) ∈ E be given the projection length:
ℓ_g(x,y) 
determined by the local Riemannian metric g.
We define the weight:
w_g(x,y) = ϕ(ℓ_g(x,y)), 
where ϕ decreasing (e.g. e^{-αℓ}).
Weighted Laplacian:
(L_g f)(x) = ∑_{y∼x} w_g(x,y)(f(x)-f(y)). 

2. Real propagation operator (Finsler)
Analogously:
ℓ_F(x,y) 
from the Finsler norm F_x(ξ).
Weight:
w_F(x,y)=ϕ(ℓ_F(x,y)). 
Real Laplacian:
(L_F f)(x) = ∑_{y∼x} w_F(x,y)(f(x)-f(y)). 

3. Metric error operator
We define:
E := L_g - L_F. 
It acts:
(E f)(x) = ∑_{y∼x} (w_g(x,y)-w_F(x,y))(f(x)-f(y)). 
This is the exact operator of perception error.

4. Accumulating character
We consider the propagation semigroup:
T_g(t)=e^{-t L_g}, T_F(t)=e^{-t L_F}. 
Projection error after time t:
Δ(t) = T_g(t)-T_F(t). 
For small perturbation:
T_g(t) = T_F(t) + ∫_0^t T_F(t-s) E T_g(s) ds. 
Operator norm:
‖Δ(t)‖ ≤ t ‖E‖ e^{t‖L_g‖}. 
The error grows exponentially with the “projection radius” (diffusion time).

5. Anisotropy as a source of error
Let locally:
ℓ_F(x,y) = ℓ_g(x,y) · κ(x,y), 
where κ measures the deviation of the projection cone from spherical.
Then:
w_F(x,y) = ϕ(ℓ_g(x,y) κ(x,y)). 
For ϕ(r)=e^{-α r}:
w_F = e^{-α κ ℓ_g} = w_g^κ. 
Difference of weights:
w_g - w_F = e^{-α ℓ_g} - e^{-α κ ℓ_g}. 
For κ>1 damping stronger,
for κ<1 weaker.
This is directional deformation of the horizon.

6. Deformation of diffusion radius
Effective radius:
r_g(t) ~ √t 
(graph diffusion heuristic).
With Finsler:
r_F(t,θ) ~ √t / κ(θ). 
Thus projection horizon:
B_{r_g(t)}^g 
does not cover the real diffusion cone.
The error operator causes:
underestimation in some directions,
overestimation in others.

7. Degeneration of information
Since we observe only:
T_g(t) δ_v, 
and we do not know L_F,
different pairs (F,ϕ) can give identical T_g with appropriate matching.
The mapping:
F ↦ L_g 
is not injective.
Information about the real geometry is lost in the kernel of the error operator.

8. Operator of accumulating redshift
If the spectrum of L_F and L_g differs:
σ(L_F) ≠ σ(L_g), 
then for eigenfunctions:
L_g ψ_k = λ_k^{(g)} ψ_k, L_F ψ_k = λ_k^{(F)} ψ_k. 
Then propagation phase:
e^{-t λ_k^{(g)}} - e^{-t λ_k^{(F)}} 
causes systematic shift of the fading rate.
This is the exact counterpart of redshift in operator language.

9. Interpretation
The graph does not possess an inherent metric.
The metric is introduced by:
choice of weights,
choice of damping function,
construction of the Laplacian.
If the assumed metric does not correspond to the real directional structure of propagation:
level sets of influence are artificially spherical,
real ones are elliptical,
error grows with radius,
degeneration of projection axes arises.

10. Final operator structure
Full system:
E = L_g - L_F 
Δ(t) = e^{-tL_g}-e^{-tL_F} 
‖Δ(t)‖ ≤ t ‖E‖ e^{t‖L_g‖} 
This formalizes:
deformation of the horizon,
accumulating redshift,
impossibility of graph reconstruction,
degeneration between projection axes.


=====29=====
Phenomenon of projection degeneration caused by extreme local density and anisotropic damping, leading to:
one-directional inclusion asymmetry,
loss of distinguishability from outside,
lack of possibility to define a common internal projection,
emergence of illusory “merger” only at the level of weights.
Formalization of the puzzle.

1. Graph construction (abstract model)
Let:
G = (V, E) infinite graph,
there exists a subset B ⊂ V,
B = ⨆_{i=1}^n B_i,
each B_i is strongly connected and has very high local density.
Formally:
deg_{B_i}(x) ≫ deg_{V \ B}(x) ∀x ∈ B_i. 
Additionally:
between different B_i the number of edges is small,
from outside to B there are few edges.

2. Weighted Laplacian operator
Let L_g be the perceptual operator with damping ϕ.
We assume:
w(x,y) = ϕ(ℓ_g(x,y)). 
Within B_i:
w(x,y) ≈ w_int, 
outside:
w(x,y) ≈ w_ext, w_ext ≪ w_int. 

3. External reduction (effective contraction)
For an observer from vertex v ∉ B:
the solution
u(t) = e^{-t L_g} δ_v 
after time t ≫ 1 undergoes homogenization on B:
u(t)|_{B_i} ≈ c_i(t) 1_{B_i}. 
And since couplings between B_i are weak, additionally:
c_1(t) ≈ ⋯ ≈ c_n(t). 
In the projection limit:
B ~ one vertex with weight ∑_i |B_i|. 
This is formal contraction:
G → G/B 
in the Mosco–Gromov sense for diffusion operators.

4. One-directional asymmetry
I take projection from x ∈ B_1.
Since:
deg_{B_1}(x) ≫ deg_ext(x), 
we have:
H_x ⊂ B_1 
(effective horizon remains in B_1).
Whereas for external observer v:
H_v ⊃ B. 
Thus:
H_x ⊊ H_v. 
But:
H_v ⊈ H_x. 
This is one-sided projection inclusion.

5. Lack of common internal projection
Take x ∈ B_1, y ∈ B_2.
Assume weak coupling between B_1 and B_2.
Then:
H_x ∩ H_y ≈ ∅ 
(at appropriate threshold).
That is:
internally lack of common projection position,
externally full contraction.
This is classical degeneration of equivalence relation.

6. Formal lack of decidability “who fell into whom”
We define the relation:
π_i ⪯ π_j ⟺ H_i ⊂ H_j. 
In our case:
π_ext ⪰ π_{B_i},
π_{B_i} mutually incomparable.
There does not exist a common minimal element in the partial order.
//therefore there does not exist a reversible description of the order of data merging;
Therefore it makes no sense to speak of internal “merger” —
because the projection lattice has no common element for B_1, B_2.

7. Operator description of degeneration
We decompose the Laplacian blockwise:
L = (L_ext ε; ε^T L_B), 
where:
L_B = ⨁_{i=1}^n L_{B_i}. 
If:
‖L_{B_i}‖ ≫ ‖ε‖, 
then the spectrum has the form:
small eigenvalues correspond to global mass B,
large eigenvalues correspond to internal dynamics.
External projection sees only the first.
Internal sees only the second.
Effective spectra do not overlap.

8. Accumulating undecidability
If the number of B_i grows (successive “mergers”):
|B| → ∞, 
then external Laplacian reduces to:
L_eff ~ one-dimensional mass operator. 
All complex dynamics reduces to one eigenvalue.
Information about the structure:
ker(external projection) 
grows with the number of substructures.

9. Phenomenon upon approaching the projection
If the external observer moves to B:
local metric changes the proportion of weights:
w_ext / w_int ↑ 
Then contraction ceases to work.
The horizon ceases to be single-point.
A phase transition occurs in the projection structure:
contraction → internal multifractality. 

10. Theorem (formal version of the puzzle)
Let B = ⨆ B_i ⊂ V satisfy:
1. min_{x ∈ B_i} deg_{B_i}(x) → ∞,
2. the number of edges between different B_i is bounded,
3. the number of edges from B to outside is bounded.
Then there exists a threshold λ such that:
for every v ∉ B:
H_v(λ) ⊃ B, 
for every x ∈ B_i:
H_x(λ) ⊂ B_i. 
Consequently:
1. The relation ⪯ is one-sided.
2. There does not exist a common projection connecting B_i.
3. External reduction identifies B with one vertex.
4. Reconstruction of the structure of B from outside is impossible.

11. Reasoning;
global projection gives a smooth, single-point object,
local projections remain multifractal and disjoint,
it is impossible to decide the direction of “merger”,
degeneration grows with local density,
upon crossing the horizon the structure explodes multidimensionally.
This is a purely operator consequence of strongly separated block spectrum and anisotropic damping.


=====30=====
I. Minimal density conditions for irreversible contraction

Reminder of the model:
G = (V, E) infinite,
B ⊂ V   subgraph “locally dense”,
projection from outside v ∉ B,
horizon:
Hᵥ(λ) = {u : Rᵥ(u) ≥ λ}

Irreversible contraction means:
from the point of view of v the entire B reduces to one effective vertex
and there does not exist a way to reconstruct the internal structure from projection data.

1. Reduction to the spectral problem

We decompose the Laplacian blockwise:

L = \begin{pmatrix} L_{ext} & ε \\ εᵀ & L_B \end{pmatrix}

Irreversible contraction occurs when:

λ₁(L_B) ≫ ‖ε‖

where:
λ₁(L_B)  — the first non-zero eigenvalue of the Laplacian on B (i.e., spectral “stiffness”),
ε        — coupling with the exterior.

2. Interpretation through Cheeger's constant

Let:

h(B) = min_{S ⊂ B}  |∂S| / min(|S|, |B \ S|)

From Cheeger's inequality:

λ₁(L_B) ≥ h(B)² / (2 Δ_B),

where Δ_B — maximum degree in B.

Thus the minimal contraction condition:

h(B)² / (2 Δ_B) ≫ deg_{ext}(B)

This is the first pure density condition.

3. Minimal combinatorial condition

Let:

d_{int} = min_{x ∈ B} deg_B(x),
d_{ext} = max_{x ∈ B} deg_{V \ B}(x)

Sufficient condition:

d_{int} ≫ d_{ext}

More precisely:

d_{int} / d_{ext} ≥ C(λ),

where C(λ) depends on the horizon threshold.

4. Dependence on λ (“tower height”)

Yes — it depends.

The threshold λ determines which eigenvalues are visible.

If:

e^{-t λ₁(L_B)} ≪ λ,

then all internal modes decay below resolution.

The higher the “tower” (greater damping, greater t), the smaller density difference suffices for contraction.

Formally:

contraction when  λ₁(L_B) · t ≫ 1

That is:
with large damping even moderate density suffices,
with small damping extreme density is needed.

5. Minimal condition for irreversibility

Irreversibility means that the mapping:

L ↦ L_{eff}

has non-zero kernel (loss of dimension).

Minimal condition:

rank(L_{eff}) < rank(L)

This occurs when:

λ_k(L_B) > λ_{cut}   ∀ k ≥ 2

Then only the mass mode remains visible.

II. “Black projection box”

As a purely mathematical concept.

We have the mapping:

P : G ⟶ Rᵥ(·)

where:
input is the graph,
output is the projection profile from one point.

Black box means:
there exist two different graphs G₁ ≠ G₂ such that

P(G₁) = P(G₂)

That is, the mapping is not injective.

In the context of dense subgraphs:
many internal structures B,
the same mass profile from outside.

The box is “black”, because:
we know input and output,
but it is impossible to reconstruct the middle.

III. Synthetically...

Minimal conditions depend on λ (i.e., the level of damping / tower height).

The minimal density condition causing irreversible contraction is:

h(B)² / Δ_B ≫ d_{ext}

or more simply:

d_{int} ≫ d_{ext}

at a level exceeding the spectral resolution determined by λ.


=====31=====
If I use purely scalar conditions of the type
d_int ≫ d_ext 
then I indeed enter the regime of isotropic averages: Cheeger, Poincaré, concentration (Azuma–Hoeffding for random walks). This gives global control, but kills directionality.
If I want to describe one-sided closure of the view and deformation of the horizon by the graph structure, I need an anisotropic version of the operator — not scalar, but directional.

1. Instead of scalar Laplacian — Laplacian with weight tensor
Standard Laplacian:
L = D - A 
is isotropic (counts only sums of weights).
Anisotropic version:
L_Θ f(x) = ∑_{y∼x} θ_{xy} (f(x) - f(y)), 
where
θ_{xy} = ϕ(ℓ(x,y)) ⋅ κ_{xy}. 
ϕ — damping “from the tower” (height),
κ_{xy} — directional metric (locally Riemannian),
lack of symmetry: κ_{xy} ≠ κ_{yx}.
This is no longer a classical Laplacian — this is a Finslerian operator on the graph.

2. Horizon as isoline of Green's function
Let:
G_v = L_Θ^{-1} δ_v. 
Horizon:
H_v(λ) = {x: G_v(x) ≥ λ}. 
If in B we have:
θ_{xy} ≫ θ_{xz} (y ∈ B, z ∉ B), 
then level sets of G_v deform:
from the side of weak weights – they expand,
from the side of strong weights – they collapse.
This gives “lensing”.

3. Anisotropic contraction condition
It is not sufficient:
d_int ≫ d_ext. 
You need:
min_{x ∈ B} (∑_{y ∈ B} θ_{xy}) / (∑_{z ∉ B} θ_{xz}) ≫ e^{λ_{cut} t}. 
Here the tower height appears: t.
This is a local directional condition — dependent on the vertex.

4. One-sided singularity
Let the singularity exist only from one side;
formally it means:
λ_1(L_Θ|_B) ≫ sup_{x ∈ B} ∑_{z ∉ B} θ_{xz} 
but only in one directional cone.
That is:
for trajectories from a certain area of the graph — contraction occurs,
for others — not.
This gives one-sided horizon inclusion:
H_{v_1} ⊃ B, H_{v_2} ⊅ B. 

5. Lensing as perturbation of harmonic function
I consider the solution:
L_Θ u = 0. 
In the presence of density B:
the gradient of u undergoes concentration on the boundary ∂B.
An effect analogous to:
conductivity in a heterogeneous medium,
refraction in Finsler metric arises.
This deforms:
search probability distribution,
random walk trajectories,
centrality ranking.

6. Artifacts in graph exploration
If the search algorithm is based on:
BFS,
PageRank,
random walk,
then density B causes:
1. accumulation of probability mass,
2. apparent shortening of internal distances,
3. damping of influence of side branches.
Formally:
π(x) ∝ deg_Θ(x) 
concentrates on B.
Other branches become “dark”.

7. This can be done purely on the connection matrix;
I take the adjacency matrix A.
I define the perceptual operator:
A_Θ = W A 
where W is a diagonal matrix of directional weights dependent on the local metric.
Contraction occurs when:
ρ(A_Θ|_B) ≫ ρ(A_Θ|_{V \ B}), 
where ρ is the spectral radius.
This is a purely combinatorial condition.

8. Analogous interpretation
I have:
closed density,
projections that close it only from one side,
deformation of the horizon,
damping of alternative paths,
informational artifacts.
This is exactly anisotropic degeneration of the operator.
Isotropic
Anisotropic
Cheeger
local weight cones
Poincaré
directional inequalities
concentration
spectral refraction
global contraction
conical contraction

========================
Appendix "Methodological remarks and interpretive scope"
A.1. Status of the construction
The tower space together with its associated radial functional were constructed as a tool for the analysis of discrete data, and not as an ontological model.
The starting point was the analysis of coherent, coupled data represented in the form of a graph, available only in discrete (measurement) form.

The construction served to reconstruct the local significance of data, to assess the dominance of components under projection from a chosen vertex, and to analyze the distribution of “representational energy” within the coupled structure.
The space here is not a primary entity. It is a consequence of the applied projection functional.

A.2. Limitation to empirical data
The construction works correctly only when the data are real measurement results, the graph structure reflects actual relations between observations, and no artificial algebraic tables unrelated to empiricism are introduced.

If abstract data generated purely formally are introduced into the model - symmetric algebraic constructions not anchored in observation - then the functional loses its analytical interpretation and reduces to a purely formal transformation.
In particular, the Construction is not intended for arbitrary exploration of algebraic spaces independent of data.

A.3. Risk of overinterpretation (curve fitting)
From the tower functional one can derive cosmological models, hierarchical scale models, global geometric interpretations.

I am aware that such applications can lead to curve-fitting type fits, overinterpretation of radial minima, identification of the functional structure with actual physical structure.

The construction was not created as a cosmological theory. Its original purpose was the analysis of coupled data structure. Although of course I was inspired by something - one has to start somewhere. And of course I first checked whether it would work.
And it does work — fit a redshift ladder to the tower height and you get everything from Planck to horizon with temperature. That is curve fitting.

A.4. Status of space
In the presented approach, topology is not assumed, space is not given a priori, geometry emerges as an effect of energetic projection.

This reverses the standard order where the classical approach is:
space → metric → dynamics
And I did:
functional → metric → geometric structure

This does not mean an ontological primacy of the functional, but the adoption of an operational construction.

A.5. Remarks concerning the algebraic base
The historical choice of algebraic bases (ℝ, ℂ, ℍ, classical normed algebras) has a pragmatic character, deeply rooted in analysis, widely useful in physics and engineering.
At the same time, there exist many quotient algebras and extensions that are observationally indistinguishable from ℝ, yet possess different structural properties, generate different asymmetries under projection.

The question of the adequacy of the algebraic base becomes relevant only when persistent asymmetries appear in the empirical data and classical symmetry ceases to be descriptively neutral.

This is not a criticism of standard constructions, but an indication of a possible source of systematic biases. The very biases we keep running into when someone fits data into an equation and everything beautifully snaps together, making them think they have discovered something.
What they discovered were harmonics in the second derivative, symmetries, commutativity and associativity baked into the Archimedean ℝ construction.

A.6. Constructive limitations
It should be emphasized that the exponential hierarchy of damping is a constructive choice — other weighting profiles lead to different behaviors, the functional is not the only possible realization of this idea.
Alternative versions of the tower with similar formal properties can be generated.
This fact limits the possibility of treating the construction as a unique structure.

A.7. Summary of scope
The presented structure is a coherent tool for the analysis of discrete data, possesses a well-defined hierarchical functional, generates a secondary geometric structure,
is not an ontological theory of space, does not constitute an independent cosmological model, does not prejudge the fundamental nature of the algebraic base.

Interpretations going beyond data analysis should be treated as hypotheses requiring independent empirical verification.
Just as we have been refining equations to fit new data for centuries, in the future we will most likely be fitting algebras, category theory, and overly abstract solutions that do not fit the axis along which the educational process runs.
Which means for most people it will simply be incomprehensible.