Chapter 4: Scattering Scale and Windowed Readout

Source Theory: euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md, §3.5-3.7

Introduction

The modular Hamiltonian $K_D$ discussed in previous chapters is a physical quantity on the geometric-information-theoretic side. But in actual measurements, we need to indirectly probe these quantities through scattering experiments.

This chapter establishes a bridge between scattering theory and modular theory:

• Birman-Krein Formula: Scattering phase $\leftrightarrow$ spectral shift function
• Wigner-Smith Formula: Group delay $\leftrightarrow$ density of states change
• Windowing Techniques: Measurement strategies under finite energy resolution
• Parity Threshold: Stability criterion for ${\mathbb{Z}}_2$ labels

Everyday Analogy:

• Geometric Side: Internal structure of a building (modular Hamiltonian)
• Scattering Side: Probing building with sound waves (scattering matrix)
• Windowing: Frequency response function of sound wave receiver
• Parity Threshold: Judging whether the “fingerprint” of detected signal is stable

1. Distributional Birman-Krein-Friedel-Lloyd Scale

1.1 Scattering Phase and Spectral Shift Function

Scattering matrix $S(E)$ (energy-dependent unitary matrix) describes particle scattering process. Define:

$$Q(E):=-\mathrm{i}{S}^{†}(E){\partial }_{E}S(E)$$

This is the group delay matrix (Wigner-Smith time delay), semi-phase defined as:

$$\phi (E):=\frac{1}{2}\arg \det S(E)$$

Relative density of states (compared to free system):

$${\rho }_{\mathrm{rel}}(E):=\frac{1}{2\pi }\mathrm{tr}Q(E)$$

Theorem F (Distributional Scale Identity):

For test function $h\in C_c^{\infty }(\mathbb{R})$ (or $h\in \mathcal{S}(\mathbb{R})$), we have

$$\int {\partial }_E\arg \det S(E)h(E)dE=\int \mathrm{tr}Q(E)h(E)dE=-2\pi \int {\xi }^{\prime }(E)h(E)dE$$

where $\xi (E)$ is the spectral shift function.

Birman-Krein Convention: $$\det S(E)=e^{-2\pi \mathrm{i}\xi (E)}$$

Mermaid Diagram: Triple Equivalence

graph TD
    A["Scattering Phase Derivative<br/>dE arg det S"] --> D["Unified Time Scale<br/>kappa(E)"]
    B["Group Delay Trace<br/>tr Q"] --> D
    C["Spectral Shift Function Derivative<br/>-2pi xi'(E)"] --> D

    D --> E["Relative Density of States<br/>rho_rel(E)"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#e1ffe1
    style E fill:#fff4e1

Physical Meaning:

• $\arg \det S(E)$: Phase accumulation in scattering process
• $\mathrm{tr}Q(E)$: Average time particles stay in scattering region (group delay)
• ${\xi }^{\prime }(E)$: Difference in density of states between perturbed and free systems

The three are strictly equal in distributional sense!

Everyday Analogy: Three equivalent methods to measure vehicle speed:

• Phase Accumulation: Timing required to pass fixed distance
• Group Delay: Measuring vehicle stay time at checkpoint
• Density Change: Statistics of increase/decrease in vehicle density on road

1.2 Branch Convention and Continuation

Technical Point: $\arg \det S$ is multi-valued (differs by $2\pi \mathbb{Z}$), need to choose continuous branch.

Branch Convention: After removing countable discrete set from energy band, take continuous branch of $\arg \det S$. Its distributional derivative ${\partial }_E\arg \det S$ does not depend on $2\pi$ jumps from branch choice, because:

1. Test function $h\in C_c^{\infty }$ “cancels” jumps
2. Matches $\mathrm{tr}Q$ through Helffer-Sjöstrand representation

Handling Threshold Singularities:

• Band edge (band threshold)
• Embedded eigenstate

Avoid by choosing $\mathrm{supp}h$, or handle as removable singularity.

1.3 Relative Aperture and Modified Determinant

Proposition F’ (Relative/Modified Aperture):

If $S_0(E)$ is reference scattering (same sheet analytic in band, no zeros/poles), and

$$U(E):=S(E)S_0(E{)}^{-1},U(E)-\mathbb{I}\in {\mathfrak{S}}_2$$

then Carleman determinant satisfies:

$$\int {\partial }_E\arg \underset{2}{\det }U(E)h(E)dE=\int \mathrm{tr}(Q(E)-Q_0(E))h(E)dE$$

where ${\det }_2$ is second-order determinant (applicable to $\mathbb{I}$ plus trace-class operators).

Application Scenarios:

• $S$ unitary but $S-\mathbb{I}$ not trace-class
• Relative scattering $U=S{S}_0^{-1}$ is “small perturbation”, $U-\mathbb{I}\in {\mathfrak{S}}_2$
• Provides phase-group delay consistency under “non-trace-class but relatively second-order traceable” window

Everyday Analogy: Measuring building deformation:

• Direct measurement of absolute coordinates (trace-class condition): Requires extremely high precision
• Measuring displacement relative to reference point (relative trace-class): Lower precision requirement

2. Windowing Techniques: Finite Resolution Measurement

2.1 Window Function and Scale

Actual measurements cannot be performed at single energy point, need averaging within energy window. Introduce window function $h(E)$:

$$h_{\ell }(E)={\ell }^{-1}h(E/\ell )$$

where $\ell >0$ is window scale (energy resolution), satisfying:

• ${\int }_{\mathbb{R}}h=1$ (normalization)
• $h\ge 0$ (non-negative)
• $h\in C_c^{\infty }(\mathbb{R})$ or $h\in \mathcal{S}(\mathbb{R})$ (Gaussian)

Common Window Functions:

1. Gaussian Window: $$h(E)=\frac{1}{\sqrt{2\pi }}{e}^{-E^2/2}$$

2. Kaiser-Bessel Window ($\beta \ge 6$): $$h(E)=\frac{I_0(\beta \sqrt{1-(E/W{)}^2})}{I_0(\beta )}\cdot 1_{|E|\le W}$$

where $I_0$ is zeroth-order modified Bessel function.

Windowed Phase Accumulation:

$${\Theta }_h(\gamma ):=\frac{1}{2}{\int }_{\mathcal{I}(\gamma )}\mathrm{tr}Q(E)h_{\ell }(E-E_0)dE$$

where $\mathcal{I}(\gamma )=[E_1,E_2]$ is energy region, $E_0$ is window center.

Mermaid Diagram: Windowing Process

graph LR
    A["Original Signal<br/>tr Q(E)"] -->|"Convolution"| B["Windowed Signal<br/>Q * h_ell"]
    B --> C["Integration<br/>Theta_h"]

    W["Window Function<br/>h_ell(E)"] -.->|"Scale ell"| B

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style W fill:#fff4e1
    style C fill:#e1ffe1

Everyday Analogy: Camera aperture and shutter:

• Original Signal: Instantaneous light intensity $I(t)$
• Window Function: Shutter time response $h(t)$
• Windowed Measurement: Exposure integral $\int I(t)h(t)dt$
• Scale $\ell$: Exposure time (time resolution)

2.2 Windowless Limit and Geometric Phase

Geometric Phase (windowless limit):

$${\Theta }_{\mathrm{geom}}(\gamma ):=\frac{1}{2}{\int }_{\mathcal{I}(\gamma )}\mathrm{tr}Q(E)dE={\int }_{\mathcal{I}(\gamma )}{\phi }^{\prime }(E)dE=\phi (E_2)-\phi (E_1)$$

This is total accumulation of scattering phase over energy region $\mathcal{I}(\gamma )$.

Gap Definition:

$${\delta }_{\mathrm{gap}}(\gamma ):=\mathrm{dist}({\Theta }_{\mathrm{geom}}(\gamma ),\pi \mathbb{Z})$$

This is distance from ${\Theta }_{\mathrm{geom}}$ to nearest integer multiple of $\pi$.

Physical Meaning:

• Large ${\delta }_{\mathrm{gap}}$: ${\Theta }_{\mathrm{geom}}$ far from integer multiples of $\pi$, parity label stable
• Small ${\delta }_{\mathrm{gap}}$: ${\Theta }_{\mathrm{geom}}$ close to integer multiples of $\pi$, parity label sensitive

Everyday Analogy: Judging whether weight is “significantly overweight”:

• Geometric Phase ${\Theta }_{\mathrm{geom}}$: Your actual weight
• Standard Line $\pi \mathbb{Z}$: Integer multiples of standard weight (e.g., 50kg, 100kg)
• Gap ${\delta }_{\mathrm{gap}}$: Your distance from nearest standard line
• Large distance → judgment stable; small distance → judgment sensitive (near critical point)

3. Error Decomposition: EM-Poisson-Toeplitz Triangle Inequality

3.1 Total Error Budget

There is error between windowed measurement ${\Theta }_h$ and geometric limit ${\Theta }_{\mathrm{geom}}$:

$$\boxed{{\mathcal{E}}_h(\gamma ):=\underset{\text{EM Endpoint}}{\underbrace{{\int }_{\mathcal{I}}|R_{\mathrm{EM}}|dE}}+\underset{\text{Poisson Aliasing}}{\underbrace{{\int }_{\mathcal{I}}|R_{\mathrm{P}}|dE}}+\underset{\text{Toeplitz Commutator}}{\underbrace{C_{\mathrm{T}}{\ell }^{-1/2}{\int }_{\mathcal{I}}|{\partial }_{E}S{|}_{2}dE}}+\underset{\text{Out-of-Interval Tail}}{\underbrace{R_{\mathrm{tail}}(\ell ,\mathcal{I},E_0)}}}$$

where:

$$R_{\mathrm{tail}}(\ell ,\mathcal{I},E_0):={\int }_{\mathbb{R}\setminus \mathcal{I}(\gamma )}|h_{\ell }(E-E_0)|dE\in [0,1]$$

If $h\ge 0$ and ${\int }_{\mathbb{R}}h=1$, then $R_{\mathrm{tail}}=1-{\int }_{\mathcal{I}(\gamma )}{h}_{\ell }(E-E_0)dE$.

Mermaid Error Source Diagram

graph TD
    A["Windowing Error<br/>E_h(gamma)"] --> B["Euler-Maclaurin Endpoint<br/>R_EM"]
    A --> C["Poisson Aliasing<br/>R_P"]
    A --> D["Toeplitz Commutator<br/>R_T"]
    A --> E["Out-of-Interval Tail<br/>R_tail"]

    B --> B1["O(ell^-(m-1)) Decay"]
    C --> C1["O(exp(-c(2pi ell/Delta)^2))<br/>(Gaussian Window)"]
    D --> D1["O(ell^-1/2) Decay"]
    E --> E1["Support-External Integral"]

    style A fill:#ffe1e1
    style B fill:#e1f5ff
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1

Everyday Analogy: Four types of photo distortion:

• EM Endpoint: Edge blur (discrete sampling truncation)
• Poisson Aliasing: Moiré pattern (frequency aliasing)
• Toeplitz Commutator: Motion blur (temporal non-commutativity)
• Tail Leakage: Light interference from objects outside frame

3.2 Euler-Maclaurin Endpoint Remainder

Lemma P (Euler-Maclaurin):

If $h\in C_c^{2m+1}$ and endpoints have vanishing jets up to order $\le 2m$, then:

$${\int }_{\mathcal{I}}|R_{\mathrm{EM}}|dE\le C_m{\ell }^{-(m-1)}$$

Physical Meaning:

• Smoother window function (larger $m$), smaller endpoint remainder
• Larger window scale $\ell$, smaller remainder (lower energy resolution, smaller discretization error)

Corner Estimate for Kaiser-Bessel Window: Kaiser window belongs to compactly supported piecewise $C^{2m}$ window, endpoints are corners. Its EM remainder is accounted by corner version:

$${\int }_{\mathcal{I}}|R_{\mathrm{EM}}|dE\le C_{\mathrm{KB}}{\ell }^{-1}$$

Decay order reduced from $O({\ell }^{-(m-1)})$ to $O({\ell }^{-1})$.

3.3 Poisson Aliasing Term

Lemma P (Poisson Summation):

Define energy sampling step size $\Delta >0$ (lattice spacing), then:

$${\int }_{\mathcal{I}}|R_{\mathrm{P}}|dE\le C_h\sum _{|q|\ge 1}\left|\hat{h}\left(\frac{2\pi q\ell }{\Delta }\right)\right|$$

where $\hat{h}$ is Fourier transform of $h$.

Exponential Decay for Gaussian Window: If $h(E)=\frac{1}{\sqrt{2\pi }}{e}^{-E^2/2}$, then $\hat{h}(\omega )=e^{-{\omega }^2/2}$. The above sum exhibits exponential square decay:

$$\sum _{|q|\ge 1}{e}^{-c(2\pi q\ell /\Delta {)}^2}\sim e^{-c(2\pi \ell /\Delta {)}^2}$$

Rapidly approaches zero when $2\pi \ell /\Delta \gg 1$.

Super-Polynomial Decay for Kaiser Window: Kaiser window has known Fourier tail bounds giving exponential or super-polynomial decay (specific order depends on $\beta$ parameter).

Everyday Analogy: Sampling theorem and aliasing:

• Poisson Summation: Frequency aliasing caused by discrete sampling
• $\Delta$: Sampling interval
• $2\pi \ell /\Delta$: Dimensionless parameter (window scale/sampling interval)
• Gaussian window: Fast decay in frequency domain, aliasing nearly zero

3.4 Toeplitz Commutator Term

Lemma T (Toeplitz/Berezin Compression Error):

Let ${\mathsf{T}}_{\ell }$ be windowed compression operator on energy axis (convolution with kernel $h_{\ell }(E-E^{\prime })$), suppose ${\partial }_{E}S\in {\mathfrak{S}}_2$ and ${\int }_{\mathcal{I}}|{\partial }_{E}S{|}_{2}dE<\infty$. Then there exists constant $C_T>0$ such that:

$$\left|\mathrm{tr}\right(Q\ast h_{\ell })-\int Q(E)h_{\ell }(E-E_0)dE|\le C_T{\ell }^{-1/2}{\int }_{\mathcal{I}}|{\partial }_{E}S{|}_{2}dE$$

Proof Points:

• Compression error written as commutator $[{\mathsf{T}}_{\ell },\cdot ]$
• Average estimate for energy derivative
• Using Hilbert-Schmidt Hölder and window extension scale $\int (E-E_0{)}^2{h}_{\ell }\sim {\ell }^{-1}$ gives ${\ell }^{-1/2}$ decay

Physical Meaning:

• Group delay $Q(E)$ and window function $h_{\ell }$ do not commute
• Commutator term $[{\mathsf{T}}_{\ell },Q]$ produces $O({\ell }^{-1/2})$ error
• Larger window scale $\ell$, smaller error

Everyday Analogy: Camera shutter and moving subject:

• Static Scene: Shutter time (window function) commutes with object position (signal), no error
• Moving Scene: Non-commutative, produces motion blur (Toeplitz error)
• Longer shutter ($\ell$ larger), relatively smaller blur

3.5 Out-of-Interval Tail Leakage

$$R_{\mathrm{tail}}(\ell ,\mathcal{I},E_0):={\int }_{\mathbb{R}\setminus \mathcal{I}(\gamma )}|h_{\ell }(E-E_0)|dE$$

Physical Meaning:

• Window function support extends beyond interested energy region $\mathcal{I}(\gamma )$
• Signals outside energy region “leak” into measurement

Advantage of Compact Support Window: If window function is compactly supported (e.g., Kaiser), and $\mathrm{supp}{h}_{\ell }\subset \mathcal{I}(\gamma )$, then $R_{\mathrm{tail}}=0$.

Tail of Gaussian Window: Gaussian window has no compact support, but tail decays exponentially. If $E_0$ is at center of $\mathcal{I}(\gamma )$ and $\ell$ appropriately small, then $R_{\mathrm{tail}}\ll 1$.

4. Parity Threshold Theorem: Stability of ${\mathbb{Z}}_2$ Labels

4.1 Theorem Statement

Theorem G (Windowed Parity Threshold):

Define windowed phase accumulation:

$${\Theta }_h(\gamma ):=\frac{1}{2}{\int }_{\mathcal{I}(\gamma )}\mathrm{tr}Q(E)h_{\ell }(E-E_0)dE$$

Chain ${\mathbb{Z}}_2$ Label:

$${\nu }_{\mathrm{chain}}(\gamma ):=(-1{)}^{\lfloor {\Theta }_h(\gamma )/\pi \rfloor }$$

Define threshold parameter:

$${\delta }_{\ast }(\gamma ):=\min \{\frac{\pi }{2},{\delta }_{\mathrm{gap}}(\gamma )\}-\epsilon$$

where $\epsilon \in (0,{\delta }_{\mathrm{gap}}(\gamma ))$ is safety margin.

Theorem: If there exist $\ell >0,\Delta >0,m\in \mathbb{N}$ such that:

$${\mathcal{E}}_h(\gamma )\le {\delta }_{\ast }(\gamma )$$

then for any window center $E_0$ satisfying window quality conditions, we have:

$${\nu }_{\mathrm{chain}}(\gamma )=(-1{)}^{\lfloor {\Theta }_h(\gamma )/\pi \rfloor }=(-1{)}^{\lfloor {\Theta }_{\mathrm{geom}}(\gamma )/\pi \rfloor }$$

That is, parity label of windowed measurement agrees with geometric limit.

Mermaid Logic Diagram

graph TD
    A["Geometric Phase<br/>Theta_geom"] --> B["Compute Gap<br/>delta_gap = dist(Theta_geom, pi Z)"]
    B --> C["Set Threshold<br/>delta_* = min(pi/2, delta_gap) - epsilon"]

    D["Windowed Measurement<br/>Theta_h"] --> E["Estimate Error<br/>E_h"]
    E --> F{"E_h <= delta_* ?"}

    F -->|"Yes"| G["Parity Stable<br/>nu_chain = (-1)^floor(Theta_h/pi)"]
    F -->|"No"| H["Parity Unstable<br/>Need to Improve Parameters"]

    C -.->|"Threshold"| F

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#ffe1f5
    style F fill:#ffcccc
    style G fill:#e1ffe1
    style H fill:#ffaaaa

Physical Meaning:

• $\pi /2$ Buffer: Fluctuations of ${\Theta }_h$ within $\pm \pi /2$ range do not change parity of $\lfloor {\Theta }_h/\pi \rfloor$
• Gap Threshold: If ${\delta }_{\mathrm{gap}}<\pi /2$, further tighten requirement to ${\delta }_{\mathrm{gap}}-\epsilon$
• Error Control: Adjust $\ell ,\Delta ,m$ to make total error ${\mathcal{E}}_h$ satisfy threshold

Everyday Analogy: Stability of scale reading:

• True Weight ${\Theta }_{\mathrm{geom}}$: 70.3 kg
• Standard Line $\pi \mathbb{Z}$: 50 kg, 100 kg
• Gap ${\delta }_{\mathrm{gap}}$: 20.3 kg (from 50kg) or 29.7 kg (from 100kg), take smaller 29.7
• Threshold ${\delta }_{\ast }$: $\min \{25,29.7\}-0.5=24.5$ kg
• Measurement Error ${\mathcal{E}}_h$: Scale precision $\pm 0.2$ kg
• If ${\mathcal{E}}_h=0.2<24.5$, then judgment “not overweight” (parity label stable)

4.2 Origin of $\pi /2$ Buffer

Note ($\pi /2$ Buffer):

In parity determination, $(-1{)}^{\lfloor \Theta /\pi \rfloor }$ flips only when $\Theta$ crosses odd number of $\pi$.

• If $\Theta$ fluctuates near some $k\pi$ by $<\pi /2$, then:

  • $\lfloor (\Theta -\pi /2)/\pi \rfloor =k-1$ or $k$
  • $\lfloor (\Theta +\pi /2)/\pi \rfloor =k$ or $k+1$
  • Parity remains unchanged

• Converging total perturbation to $<\pi /2$ guarantees not crossing nearest integer multiple of $\pi$

Taking ${\delta }_{\ast }(\gamma )=\min \{\pi /2,{\delta }_{\mathrm{gap}}(\gamma )\}-\epsilon$ is explicit formulation of this buffer.

Geometric Intuition:

  ───────────┼─────────────┼─────────────┼─────────────
            0            π           2π           3π
             └──┬──┘        └──┬──┘        └──┬──┘
            π/2 Buffer    π/2 Buffer    π/2 Buffer

If $\Theta$ is within $\pm \pi /2$ range near some $k\pi$, $\lfloor \Theta /\pi \rfloor$ remains $k$ or $k-1$, parity unchanged.

4.3 Transition for Non-Smooth Windows

Technical Extension: If window $h\in C_c^0$ and piecewise $C^{2m}$ within support (endpoints allow corners), can transition through smoothing:

1. Take standard smoothing kernel ${\rho }_{\delta }$
2. Define $h_{\ell ,\delta }:=h_{\ell }\ast {\rho }_{\delta }$
3. For fixed $\ell >0$, have $\Vert h_{\ell ,\delta }-h_{\ell }{\Vert }_{L^1}=O(\delta )$
4. Incorporate smoothing error $R_{\mathrm{smooth}}(\delta ):={\int }_{\mathcal{I}}|h_{\ell ,\delta }-h_{\ell }|dE$ into total error budget

Choose $\delta =\delta (\ell ,m)$ such that $R_{\mathrm{smooth}}(\delta )\le \frac{1}{2}{\delta }_{\ast }(\gamma )$, preserving same parity threshold conclusion.

5. Robustness Under Weak Non-Unitary Perturbations

5.1 Non-Unitary Deviation

Actual scattering processes may have dissipation (energy loss), scattering matrix $S(E)$ no longer strictly unitary. Define non-unitary deviation:

$${\Delta }_{\mathrm{nonU}}(E):=|S^{†}(E)S(E)-\mathbb{I}{|}_1$$

This is trace norm measure of $S$ deviating from unitarity.

Corollary G (Weak Non-Unitary Stability):

If:

$${\int }_{\mathcal{I}(\gamma )}{\Delta }_{\mathrm{nonU}}(E)dE\le \epsilon$$

and:

$${\mathcal{E}}_h(\gamma )\le {\delta }_{\ast }(\gamma ):=\min \{\frac{\pi }{2},{\delta }_{\mathrm{gap}}(\gamma )\}-\epsilon$$

then ${\nu }_{\mathrm{chain}}(\gamma )=(-1{)}^{\lfloor {\Theta }_h(\gamma )/\pi \rfloor }$ unchanged, and agrees with windowless limit.

Physical Meaning:

• As long as energy integral of non-unitary deviation $\le \epsilon$
• And total error budget satisfies threshold ${\delta }_{\ast }-\epsilon$
• Parity label remains stable

Lemma N (Weak Non-Unitary Phase Difference Bound):

Write polar decomposition $S=U(\mathbb{I}-A)$, $U$ unitary, $A\ge 0$. If ${\int }_{\mathcal{I}}|S^{†}S-\mathbb{I}{|}_{1}dE\le \epsilon$, then:

$$|{\int }_{\mathcal{I}}\mathrm{tr}Q(S)h_{\ell }-{\int }_{\mathcal{I}}\mathrm{tr}Q(U)h_{\ell }|\le C_N\epsilon$$

Proof Points:

• $Q(S)=\mathrm{Im}\mathrm{tr}(S^{-1}{\partial }_{E}S)$
• Near unitarity $\Vert S^{-1}\Vert \le (1-\Vert A\Vert {)}^{-1}$
• Control difference using $\Vert {\partial }_{E}S{\Vert }_1\le \Vert {\partial }_{E}U{\Vert }_1+\Vert {\partial }_{E}A{\Vert }_1$ and $\Vert A{\Vert }_1\lesssim \Vert S^{†}S-\mathbb{I}{\Vert }_1$

Everyday Analogy: Effect of tire leak on speed measurement:

• Ideal Tire: Unitary scattering $S^{†}S=\mathbb{I}$
• Leaky Tire: Non-unitary ${\Delta }_{\mathrm{nonU}}>0$
• As long as leak not too large ($\int {\Delta }_{\mathrm{nonU}}\le \epsilon$)
• Speed measurement still reliable (parity label stable)

6. Recommended Parameters and Engineering Thresholds

6.1 Parameter Table (Satisfying Theorem G Threshold)

Window Family:

• Gaussian window: $h(E)=\frac{1}{\sqrt{2\pi }}{e}^{-E^2/2}$
• Kaiser window: $\beta \ge 6$

Smoothness Order/EM Endpoint Remainder:

• If $h\in C_c^{\infty }$ or $h\in \mathcal{S}$: Take $m\ge 6$, use ${\int }_{\mathcal{I}}|R_{\mathrm{EM}}|\le C_m{\ell }^{-(m-1)}$
• If using Kaiser window: Use corner estimate ${\int }_{\mathcal{I}}|R_{\mathrm{EM}}|\le C_{\mathrm{KB}}{\ell }^{-1}$

Step Size and Bandwidth:

• Take $\Delta \le \ell /4$, and make $2\pi \ell /\Delta \ge 5$
• Poisson aliasing: $R_{\mathrm{P}}\le C_h{\sum }_{|q|\ge 1}\left|\hat{h}\left(2\pi q\ell /\Delta \right)\right|$
  • Gaussian Window: Exponential square decay $\sim e^{-c(2\pi \ell /\Delta {)}^2}$
  • Kaiser Window: Exponential or super-polynomial decay (specific depends on $\beta$)

Toeplitz Commutator Term: Control quantity ${\ell }^{-1/2}{\int }_{\mathcal{I}}|{\partial }_{E}S{|}_2$

Non-Unitary Tolerance: $${\int }_{\mathcal{I}}{\Delta }_{\mathrm{nonU}}\le \epsilon$$

Gap Pre-Check: Compute ${\delta }_{\mathrm{gap}}(\gamma )=\mathrm{dist}({\Theta }_{\mathrm{geom}}(\gamma ),\pi \mathbb{Z})$

Total Error Budget Formula:

$$\boxed{\int |R_{\mathrm{EM}}|+\int |R_{\mathrm{P}}|+C_{\mathrm{T}}{\ell }^{-1/2}\int |{\partial }_{E}S{|}_2+R_{\mathrm{tail}}\le {\delta }_{\ast }(\gamma )}$$

where ${\delta }_{\ast }(\gamma )=\min \{\frac{\pi }{2},{\delta }_{\mathrm{gap}}(\gamma )\}-\epsilon$.

Mermaid Parameter Adjustment Flow

graph TD
    A["Input Energy Region I"] --> B["Compute Theta_geom"]
    B --> C["Compute delta_gap"]
    C --> D["Set delta_*"]

    D --> E["Choose Window Function<br/>(Gaussian/Kaiser)"]
    E --> F["Initial Parameters<br/>ell, Delta, m"]

    F --> G["Estimate Four Error Terms<br/>R_EM, R_P, R_T, R_tail"]
    G --> H{"E_h <= delta_* ?"}

    H -->|"No"| I["Adjust Parameters<br/>Increase ell or Delta"]
    I --> G

    H -->|"Yes"| J["Parameters Qualified<br/>Execute Measurement"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5
    style G fill:#f5e1ff
    style H fill:#ffcccc
    style I fill:#ffaaaa
    style J fill:#aaffaa

6.2 Numerical Examples

Single-Channel Resonance:

$$\delta (E)=\arctan \frac{\Gamma }{E-E_0}$$

where $\Gamma$ is resonance width. Compute:

$${\Theta }_{\mathrm{geom}}=\delta (E_2)-\delta (E_1)$$

Estimate difference between ${\Theta }_h$ and actual $\int (2\pi {)}^{-1}\mathrm{tr}Q$, and mark crossing points of $\pi$ flips.

Multi-Channel Near-Unitary:

$$S(E)=U\mathrm{diag}(e^{2i{\delta }_1(E)},e^{-2i{\delta }_1(E)})U^{†}$$

Examine ${ϵ}_i$ flips and chain sign response.

Expected Results:

• When $\ell ,\Delta$ satisfy threshold inequality, parity label of windowed measurement agrees with geometric limit
• When parameters don’t satisfy, “spurious flip” appears

7. Connection with Unified Time Scale

7.1 Review of Unified Time Scale

In previous chapters (20-experimental-tests/01-unified-time-measurement.md), we established unified time scale:

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

Triple Equivalence:

• ${\phi }^{\prime }(\omega )/\pi$: Scattering phase derivative
• ${\rho }_{\mathrm{rel}}(\omega )$: Relative spectral density
• $(2\pi {)}^{-1}\mathrm{tr}Q(\omega )$: Wigner-Smith group delay

Theorem F of this chapter is exactly the distributional version of this unified scale!

7.2 Connection with Modular Hamiltonian

First-Order Variation Relation:

$$\delta K_D=2\pi \delta S_D$$

where $S_D$ is entanglement entropy. Combined with QNEC vacuum saturation:

$$⟨T_{vv}⟩=\frac{1}{2\pi }\frac{{\partial }^{2}S}{\partial v^2}$$

obtains indirect connection between modular Hamiltonian and scattering phase:

$$K_D\sim \int {\phi }^{\prime }(\omega )d\omega =\pi \int \kappa (\omega )d\omega$$

Everyday Analogy:

• Geometric Side: Internal stress distribution of building (modular Hamiltonian $K_D$)
• Scattering Side: Sound wave reflection phase ($\phi (\omega )$)
• Unified Scale: Inferring stress through acoustic probing ($\kappa (\omega )$)

7.3 Application of Windowing Techniques in Experiments

Experimental schemes in 20-experimental-tests chapter rely on windowing techniques of this chapter:

1. PSWF/DPSS Spectral Windowing (02-spectral-windowing-technique.md)

   • Window function selection consistent with recommendations of this chapter
   • Shannon number $N_0=2TW$ corresponds to energy-time window parameters $\ell ,\Delta$

2. FRB Observation Application (05-frb-observation-application.md)

   • FRB pulse as natural “window function”
   • Interstellar scattering introduces non-unitary effects, need Corollary G for evaluation

3. Topological Fingerprint Optical Implementation (03-topological-fingerprint-optics.md)

   • Measurement of ${\mathbb{Z}}_2$ parity labels depends on stability criterion of Theorem G
   • Experimental parameter design must satisfy error budget ${\mathcal{E}}_h\le {\delta }_{\ast }$

8. Chapter Summary

8.1 Core Formulas

Birman-Krein-Friedel-Lloyd-Wigner-Smith Scale Identity: $${\partial }_E\arg \det S=\mathrm{tr}Q=-2\pi {\xi }^{\prime }$$

Windowed Phase Accumulation: $${\Theta }_h(\gamma )=\frac{1}{2}{\int }_{\mathcal{I}(\gamma )}\mathrm{tr}Q(E)h_{\ell }(E-E_0)dE$$

Parity Threshold Theorem: If ${\mathcal{E}}_h(\gamma )\le {\delta }_{\ast }(\gamma )$, then $${\nu }_{\mathrm{chain}}(\gamma )=(-1{)}^{\lfloor {\Theta }_h(\gamma )/\pi \rfloor }=(-1{)}^{\lfloor {\Theta }_{\mathrm{geom}}(\gamma )/\pi \rfloor }$$

Total Error Budget: $${\mathcal{E}}_h=\int |R_{\mathrm{EM}}|+\int |R_{\mathrm{P}}|+C_{\mathrm{T}}{\ell }^{-1/2}\int |{\partial }_{E}S{|}_2+R_{\mathrm{tail}}$$

Threshold Parameter: $${\delta }_{\ast }(\gamma )=\min \{\frac{\pi }{2},{\delta }_{\mathrm{gap}}(\gamma )\}-\epsilon$$

8.2 Physical Picture

Mermaid Summary Diagram

graph TD
    A["Scattering Matrix<br/>S(E)"] --> B["Group Delay Matrix<br/>Q(E)"]
    B --> C["Distributional Scale<br/>tr Q = dE arg det S"]

    C --> D["Windowed Measurement<br/>Theta_h(gamma)"]
    D --> E["Error Decomposition<br/>E_h"]

    E --> F["EM Endpoint<br/>O(ell^-(m-1))"]
    E --> G["Poisson Aliasing<br/>O(exp(-c(2pi ell/Delta)^2))"]
    E --> H["Toeplitz Commutator<br/>O(ell^-1/2)"]
    E --> I["Tail Leakage<br/>R_tail"]

    D --> J["Parity Label<br/>nu_chain"]
    E --> K{"E_h <= delta_* ?"}
    K -->|"Yes"| L["Label Stable"]
    K -->|"No"| M["Label Unstable"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#ffe1f5
    style J fill:#e1ffe1
    style K fill:#ffcccc
    style L fill:#aaffaa
    style M fill:#ffaaaa

8.3 Key Insights

1. Triple Unified Scale:

   • Scattering phase $\leftrightarrow$ group delay $\leftrightarrow$ spectral shift function
   • Strictly equivalent in distributional sense
   • Provides multiple paths for experimental measurement

2. Necessity of Windowing Techniques:

   • Actual measurements have finite energy resolution
   • Window functions introduce systematic errors
   • Errors can be controlled through parameter adjustment

3. Robustness of Parity Threshold:

   • $\pi /2$ buffer mechanism
   • Gap ${\delta }_{\mathrm{gap}}$ determines stability
   • Weak non-unitary perturbations tolerable

4. Engineering Parameter Design:

   • Gaussian window: Exponential decay, Poisson aliasing minimal
   • Kaiser window: Compact support, tail leakage zero
   • Parameter selection needs to balance four error sources

8.4 Preview of Next Chapter

Next chapter (05-causal-diamond-summary.md) will:

• Synthesize entire chapter content
• Interface with experimental schemes
• Discuss open problems and future directions

End of Chapter

Source Theory: euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md, §3.5-3.7