23.8 Unified Time Scale: Physical Realization of Scattering Master Ruler

In previous articles, we established discrete geometry of computational universe: complexity geometry tells us “how hard is computation”, information geometry tells us “what can be obtained”. But these are abstract geometric structures, haven’t answered a fundamental question: How is computational cost connected to physical time?

Like using a ruler to measure length, we need a “ruler of time”. But unlike everyday clocks, time in computational universe should be intrinsic, determined by computation process itself, not externally added parameters.

This article will introduce Unified Time Scale (Unified Time Scale), which is key bridge between computational universe and physical universe. This scale is not artificially specified, but naturally induced “master ruler” from scattering theory.

Core Questions:

• What is unified time scale? Why can it unify three seemingly unrelated quantities: scattering, spectral shift, and group delay?
• How to define computational cost using unified time scale?
• How do discrete computation steps become Riemann geometry in continuous limit?

This article is based on euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md.

1. Why Do We Need Unified Time Scale? From Clocks to Scattering

1.1 Limitations of Everyday Time

In everyday life, we measure time with clocks:

• Mechanical Clock: Relies on periodic motion of pendulum;
• Quartz Clock: Relies on oscillation frequency of quartz crystal;
• Atomic Clock: Relies on energy level transition frequency of atoms.

Common feature of all these clocks: They measure periods of external reference systems, independent of measured object.

But in computational universe, what we need is intrinsic time:

• Different computation processes have different “inherent times”;
• “Difficulty” of computation should be determined by computation itself, not wall clock;
• “One step” time of quantum computation, classical computation, biological computation may be completely different.

Core Question: How to define a time scale related to physical process itself?

1.2 Physical Picture of Scattering: “Delay” of Waves

Imagine you throw a stone into a well:

• Stone falls, hear “plop” sound after few seconds;
• Sound reflects from bottom, hear again after few more seconds;
• Total delay time is time for stone falling + sound round trip.

This delay time is not measured by external clock, but intrinsic quantity determined by depth of well and sound speed.

In quantum mechanics, similar phenomenon is called scattering delay:

• A wave packet incident on some potential barrier (e.g., tunnel, resonant cavity);
• Wave packet scattered, transmitted or reflected;
• Scattered wave packet has a phase delay compared to free propagation, this delay corresponds to a “group delay time”.

Core Insight: Scattering delay is intrinsic time scale of physical system, doesn’t depend on external reference.

1.3 Three Seemingly Unrelated Quantities

In scattering theory, there are three classical physical quantities:

1. Scattering Phase $\phi (\omega )$: Total phase change of wave after scattering at frequency $\omega$;
2. Spectral Shift Function $\xi (\omega )$: “Shift” of system energy spectrum relative to free case;
3. Group Delay ${\tau }_g(\omega )$: Average delay time of wave packet passing through system.

These three quantities come from different calculations in classical scattering theory, seem unrelated. But in unified time scale theory, they miraculously are three manifestations of the same thing!

graph TD
    A["Scattering System<br/>(Potential Barrier, Resonant Cavity, Quantum Gate)"] --> B["Scattering Matrix S(omega)"]
    B --> C["Scattering Phase<br/>phi(omega) = arg det S(omega)"]
    B --> D["Spectral Shift Function<br/>xi(omega)"]
    B --> E["Group Delay Matrix<br/>Q(omega) = -iS^dagger d_omega S"]

    C --> F["Phase Derivative<br/>phi'(omega)/pi"]
    D --> G["Spectral Shift Derivative<br/>xi'(omega)"]
    E --> H["Group Delay Trace<br/>(2pi)^-1 tr Q(omega)"]

    F --> I["Unified Time Scale<br/>kappa(omega)"]
    G --> I
    H --> I

    I --> J["Three Strictly Equal!<br/>(Up to Constant)"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style I fill:#e1fff5
    style J fill:#ffe1f5

2. Scattering Master Ruler of Unified Time Scale

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 2.1

2.1 Scattering Matrix and Scattering Phase

Definition 2.1 (Scattering Matrix)

For a quantum scattering system, let free Hamiltonian be $H_0$, full Hamiltonian be $H=H_0+V$ (where $V$ is potential or interaction). Scattering operator defined as

$$S=W_{+}^{†}{W}_{-},$$

where $W_{\pm }$ are Møller wave operators. In frequency representation, $S$ can be written as frequency-dependent scattering matrix $S(\omega )$, it is unitary: $S(\omega {)}^{†}S(\omega )=I$.

Everyday Analogy:

• Imagine a multi-port circuit network (e.g., fiber beam splitter);
• Input has multiple channels, output also has multiple channels;
• Scattering matrix $S(\omega )$ describes “at frequency $\omega$, how signals from each input port are distributed to output ports”.

Definition of Scattering Phase:

Total scattering phase defined as

$$\phi (\omega )=\arg \det S(\omega ),$$

i.e., argument of determinant of scattering matrix.

Everyday Interpretation:

• If scattering matrix is diagonal (channels independent), then $\phi (\omega )={\sum }_i\arg S_{ii}(\omega )$ is sum of phases of each channel;
• If there is coupling, phase also contains interference contributions between channels.

2.2 Spectral Shift Function and Birman-Krein Formula

Definition 2.2 (Spectral Shift Function, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md)

Spectral shift function $\xi (\omega )$ is related to scattering matrix through Birman-Krein formula:

$$\det S(\omega )=\exp (-2\pi \mathrm{i}\xi (\omega )).$$

Physical Meaning:

• $\xi (\omega )$ measures “after adding potential $V$, how much energy spectrum shifted below $\omega$”;
• If $\xi (\omega )=0$, means energy spectrum didn’t shift;
• If $\xi (\omega )>0$, means energy spectrum shifted downward overall (more eigenstates);
• If $\xi (\omega )<0$, means energy spectrum shifted upward overall (fewer eigenstates).

From Birman-Krein formula we get:

$$\phi (\omega )=\arg \det S(\omega )=-2\pi \xi (\omega )(\mathrm{mod}2\pi ).$$

Therefore phase is directly related to spectral shift function.

2.3 Group Delay Matrix

Definition 2.3 (Wigner-Smith Group Delay Matrix, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 2.1)

Group delay matrix defined as

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

Physical Meaning:

• $Q(\omega )$ is a Hermitian matrix (because $S$ unitary);
• Its eigenvalues ${\tau }_i(\omega )$ correspond to group delay times of each eigenchannel;
• Trace $\mathrm{tr}Q(\omega )={\sum }_i{\tau }_i(\omega )$ is total delay of all channels.

Why Called “Group Delay”?

In classical wave theory, group velocity of a wave packet is $v_g=\mathrm{d}\omega /\mathrm{d}k$, where $k$ is wavenumber. Group delay is time needed for wave packet to pass through system:

$${\tau }_g=\frac{\mathrm{d}k}{\mathrm{d}\omega }\cdot L,$$

where $L$ is system length. In quantum scattering, $Q(\omega )$ generalizes this concept to multi-channel case.

2.4 Master Formula of Unified Time Scale: Amazing Unification of Three

Theorem 2.4 (Master Formula of Unified Time Scale, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 2.1)

Under standard regularity conditions, following three quantities are equal up to additive constant:

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

where ${\rho }_{\mathrm{rel}}(\omega )$ is called relative density of states.

Everyday Interpretation:

• First Term ${\phi }^{\prime }(\omega )/\pi$: “Derivative of phase with respect to frequency” ÷ $\pi$, characterizes phase change rate;
• Second Term ${\xi }^{\prime }(\omega )$: “Derivative of spectral shift function”, characterizes change in energy level density;
• Third Term $(2\pi {)}^{-1}\mathrm{tr}Q(\omega )$: “Average group delay” ÷ $2\pi$, characterizes wave packet delay;
• Core Insight: These three physical quantities completely equivalent, all measuring same thing—intrinsic time scale density of system near frequency $\omega$!

2.5 Why Called “Master Ruler”?

Unified time scale $\kappa (\omega )$ is called “Master Scale” because:

• It’s like a “variable ruler”, has different scale density at different frequencies $\omega$;
• All time-related physical quantities (phase, energy spectrum, delay) can be measured with it;
• It is most basic unit of physical time, all other time definitions can be derived from it.

Everyday Analogy:

• Imagine an elastic ruler, stretching differently at different positions;
• In some regions (near resonance frequencies), $\kappa (\omega )$ is large, time “slows down” (similar to time dilation in relativity);
• In other regions (far from resonance), $\kappa (\omega )$ is small, time “flows normally”.

graph LR
    A["Frequency omega"] --> B["Scattering System<br/>S(omega)"]

    B --> C["Scattering Phase phi(omega)"]
    B --> D["Spectral Shift Function<br/>xi(omega)"]
    B --> E["Group Delay Matrix<br/>Q(omega)"]

    C -->|"Derivative/pi"| F["phi'(omega)/pi"]
    D -->|"Derivative"| G["xi'(omega)"]
    E -->|"Trace/2pi"| H["tr Q(omega) / 2pi"]

    F --> I["Unified Time Scale<br/>kappa(omega)"]
    G --> I
    H --> I

    I --> J["Master Ruler:<br/>Basic Unit of All Time"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style I fill:#e1f5ff
    style J fill:#fff4e1

3. Control Manifold: Parameterizing Computation

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 2.2

3.1 Why Do We Need “Control”?

In computational universe, configurations are not isolated, but can be changed through some “control operations”. For example:

• Quantum Computation: Change parameters of quantum gates (rotation angles, coupling strengths);
• Classical Computation: Change voltage, clock frequency of logic circuits;
• Neural Networks: Change parameters of weight matrices.

These adjustable parameters form a control space, which we geometrize as control manifold $\mathcal{M}$.

3.2 Definition of Control Manifold

Definition 3.1 (Control Manifold and Scattering Family, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Definition 2.1)

A control-scattering system consists of following data:

1. Control Manifold $\mathcal{M}$: A $d$-dimensional differentiable manifold, coordinates denoted $\theta =({\theta }^1,\dots ,{\theta }^d)$;
2. Scattering Family $S(\omega ;\theta )$: For each $\theta \in \mathcal{M}$ and frequency $\omega$, assign a unitary scattering matrix $S(\omega ;\theta )$, differentiable in $\theta ,\omega$;
3. Group Delay Matrix Family $Q(\omega ;\theta )$:

$$Q(\omega ;\theta )=-\mathrm{i}S(\omega ;\theta {)}^{†}{\partial }_{\omega }S(\omega ;\theta ).$$

Everyday Analogy:

• $\theta$ is “position of knob”: e.g., frequency tuning knob of radio, rotating it changes reception frequency;
• $S(\omega ;\theta )$ is “when knob at position $\theta$, system’s response to frequency $\omega$”;
• Control manifold $\mathcal{M}$ is space of “all possible knob positions”.

3.3 Example: Single Qubit Gate

Consider a single qubit rotation gate:

$$U(\theta )=\left(\begin{array}{cc}\cos (\theta /2)& -\sin (\theta /2)\\ \sin (\theta /2)& \cos (\theta /2)\end{array}\right),$$

where $\theta \in [0,2\pi ]$ is rotation angle.

• Control Manifold: $\mathcal{M}=S^1$ (circle);
• Scattering Matrix: $S(\omega ;\theta )=U(\theta )$ (here ignore frequency dependence, in practice there would be);
• Group Delay Matrix: $Q(\omega ;\theta )$ can be computed through ${\partial }_{\omega }U(\theta )$.

This is simplest example, actual quantum computers have thousands of parameters, control manifold is high-dimensional.

3.4 Connection Between Control Manifold and Computational Universe

Definition 3.2 (Map from Control Manifold to Configuration Space)

For given computational universe $U_{\mathrm{comp}}=(X,\mathsf{T},\mathsf{C},\mathsf{I})$, if each step update can be realized through some control-scattering system, then there exists map:

$$\Psi :\mathcal{M}\to X,$$

such that control parameter $\theta \in \mathcal{M}$ corresponds to a configuration $x=\Psi (\theta )\in X$.

Everyday Interpretation:

• Control manifold $\mathcal{M}$ is “parameter space at physical level” (e.g., angles of quantum gates);
• Configuration space $X$ is “state space at logical level” (e.g., computational basis states of qubits);
• Map $\Psi$ is “encoding from physical parameters to logical states”.

Core Insight: Control manifold provides a continuous perspective to view discrete configuration space.

4. Complexity Metric $G$: From Scattering to Geometry

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 3

4.1 Core Idea: Define Distance Using Change in Group Delay

On control manifold, we want to define a metric, such that “change in control parameters” corresponds to “increase in computational cost”.

Key Observation:

• When control parameter changes from $\theta$ to $\theta +\mathrm{d}\theta$, scattering matrix changes from $S(\omega ;\theta )$ to $S(\omega ;\theta +\mathrm{d}\theta )$;
• This causes change in group delay matrix: $Q(\omega ;\theta )\to Q(\omega ;\theta +\mathrm{d}\theta )$;
• Change in group delay reflects change in unified time scale, i.e., “how much physical time needed for this computation step changed”.

Therefore, we use derivative of group delay matrix with respect to control parameters ${\partial }_{a}Q(\omega ;\theta )$ to construct metric.

4.2 Mathematical Definition of Metric

Definition 4.1 (Metric Induced by Unified Time Scale, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Definition 3.1)

On control manifold $\mathcal{M}$, define second-order tensor

$$G_{ab}(\theta )={\int }_{\Omega }w(\omega )\mathrm{tr}({\partial }_{a}Q(\omega ;\theta ){\partial }_{b}Q(\omega ;\theta ))\mathrm{d}\omega ,$$

where:

• ${\partial }_a=\partial /\partial {\theta }^a$ is partial derivative with respect to control parameter;
• $w(\omega )\ge 0$ is weight function, selecting frequency band of interest;
• $\Omega$ is frequency interval;
• $\mathrm{tr}(AB)$ is trace of matrix $AB$.

If $G_{ab}(\theta )$ is positive definite at each point, then $G$ is Riemann metric on $\mathcal{M}$, called complexity metric induced by unified time scale.

Everyday Interpretation:

• ${\partial }_{a}Q(\omega ;\theta )$ is “change in group delay matrix when moving along control direction $a$”;
• $\mathrm{tr}({\partial }_{a}Q\cdot {\partial }_{b}Q)$ is “inner product of group delay changes in directions $a$ and $b$”;
• Integrate over frequency $\omega$, weighted sum, get total “control cost”.

4.3 Why Positive Definite?

Proposition 4.2 (Positive Definiteness Condition, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Proposition 3.2)

If for any nonzero tangent vector $v=v^a{\partial }_a\in T_{\theta }\mathcal{M}$, there exists frequency $\omega \in \Omega$ such that

$${\partial }_{v}Q(\omega ;\theta )=v^a{\partial }_{a}Q(\omega ;\theta )\ne 0,$$

and

$${\int }_{\Omega }w(\omega )\mathrm{tr}({\partial }_{v}Q(\omega ;\theta ){\partial }_{v}Q(\omega ;\theta ))\mathrm{d}\omega >0,$$

then $G_{ab}(\theta )$ is positive definite at $\theta$.

Everyday Interpretation:

• If moving along some direction $v$, group delay matrix completely unchanged (all frequencies have ${\partial }_{v}Q=0$), then this direction contributes nothing to metric, metric degenerates in this direction;
• Conversely, as long as there exist some frequencies where group delay changes, and weight $w(\omega )$ doesn’t cancel, then metric is positive definite.

Physical Meaning:

• Positive definiteness means “any non-trivial control change leads to non-zero computational cost”;
• Degenerate directions correspond to “pure gauge degrees of freedom” (e.g., global phase, doesn’t affect physical observations).

4.4 Everyday Analogy: Slope of Mountain Climbing

Imagine adjusting radio frequency in mountains (knob $\theta$):

• Flat Region: Rotating knob, signal almost unchanged, ${\partial }_{\theta }Q\approx 0$, metric $G$ small;
• Steep Region (near resonance): Slight rotation, signal changes dramatically, ${\partial }_{\theta }Q$ large, metric $G$ large;
• Metric $G$ is like “slope squared” of terrain, tells you “at this parameter point, how ‘sensitive’ is control”.

graph TD
    A["Control Parameter theta"] --> B["Scattering Family S(omega;theta)"]
    B --> C["Group Delay Matrix Q(omega;theta)"]

    C --> D["Derivative w.r.t. theta<br/>d_a Q(omega;theta)"]
    D --> E["Inner Product<br/>tr(d_a Q · d_b Q)"]

    E --> F["Integrate Over Frequency<br/>∫ w(omega) tr(...) domega"]
    F --> G["Complexity Metric<br/>G_ab(theta)"]

    G --> H["Riemann Geometry<br/>(M, G)"]
    H --> I["Geodesics:<br/>Optimal Control Paths"]
    H --> J["Curvature:<br/>Control Difficulty"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style I fill:#e1f5ff
    style J fill:#fff4e1

5. Geometric Length of Control Path: Accumulation of Physical Time

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 3.3

5.1 Definition of Path Length

Given a differentiable control path $\theta :[0,T]\to \mathcal{M}$, its geometric length defined as:

$$L_G[\theta ]={\int }_0^T\sqrt{G_{ab}(\theta (t)){\dot{\theta }}^a(t){\dot{\theta }}^b(t)}\mathrm{d}t.$$

Everyday Interpretation:

• ${\dot{\theta }}^a(t)=\mathrm{d}{\theta }^a/\mathrm{d}t$ is rate of change of control parameter (velocity);
• $G_{ab}{\dot{\theta }}^a{\dot{\theta }}^b$ is “metric square” of velocity (similar to $v^2=v_x^2+v_y^2$);
• $\sqrt{G_{ab}{\dot{\theta }}^a{\dot{\theta }}^b}$ is instantaneous “speed”;
• Integrate over time, get total length.

5.2 Relationship Between Length and Physical Time

Proposition 5.1 (Relationship Between Length and Unified Time Scale, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Proposition 3.3)

Under appropriate regularity conditions, geometric length $L_G[\theta ]$ of control path $\theta (t)$ is proportional to integral of physical time scale accumulated along this path, i.e., there exists constant $\alpha >0$ such that

$$L_G[\theta ]=\alpha {\int }_0^T\mathrm{d}t{\int }_{\Omega }w(\omega )\kappa (\omega ;\theta (t){)}^2\mathrm{d}\omega ,$$

where $\kappa (\omega ;\theta )=(2\pi {)}^{-1}\mathrm{tr}Q(\omega ;\theta )$ is unified time scale density.

Everyday Interpretation:

• Left side $L_G[\theta ]$ is “geometric length of control path” (abstract);
• Right side is “integral of square of unified time scale accumulated along path” (concrete physical quantity);
• This proposition says: Geometric length is measure of physical time!

5.3 Everyday Analogy: “Mileage” of Travel

Imagine driving from A to B:

• Route $\theta (t)$: Road you choose (control path);
• Instantaneous Speed $\sqrt{G_{ab}{\dot{\theta }}^a{\dot{\theta }}^b}$: Speed shown on speedometer (unit: km/h);
• Total Mileage $L_G[\theta ]$: Value added on odometer (unit: km);
• Physical Time: Actual travel time (unit: hours).

Proposition 5.1 says: “Mileage” proportional to “time × speed squared”, exactly our intuitive understanding of “distance”!

6. Discrete to Continuous: Gromov-Hausdorff Convergence

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 4

6.1 Core Question: How Does Discrete Geometry Become Continuous Geometry?

Previous 7 articles all established discrete complexity geometry:

• Configuration space $X$ is discrete (countable set);
• Complexity distance $d(x,y)$ accumulated step by step;
• Complexity ball, dimension, Ricci curvature all defined discretely.

Now we have continuous control manifold $(\mathcal{M},G)$:

• $\mathcal{M}$ is continuous manifold;
• Metric $G$ induces geodesic distance $d_G$;
• This is standard Riemann geometry.

Key Question: What is relationship between discrete geometry and continuous geometry?

6.2 Refinement Sequence: Making Discrete Grid Finer and Finer

Imagine approximating a continuous surface with finer and finer grids:

• Coarse grid: Only few points, connections are “jumps”;
• Fine grid: Many dense points, connections approach continuous curves;
• Limit: Grid infinitely fine, approximates continuous surface.

Mathematically, we consider a family of computational universes labeled $h>0$: $\{U_{\mathrm{comp}}^{(h)}\}$:

• $h$ is discrete step size (grid spacing);
• When $h\to 0$, grid becomes finer and finer;
• Each $U_{\mathrm{comp}}^{(h)}$ has its own discrete complexity distance $d^{(h)}$.

6.3 Distance Convergence Theorem

Theorem 6.1 (Riemann Limit of Complexity Distance, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Theorem 4.1)

Let $(\mathcal{M},G)$ be control manifold induced by unified time scale, $\{U_{\mathrm{comp}}^{(h)}\}$ be a family of computational universes with control grids ${\mathcal{M}}^{(h)}$. Assume:

1. Grids ${\mathcal{M}}^{(h)}$ dense in $\mathcal{M}$ as $h\to 0$;
2. Single-step cost consistent with metric: ${\mathsf{C}}^{(h)}(\theta ,\theta +h{e}_a)=\sqrt{G_{aa}(\theta )}h+o(h)$;
3. No jumps: Edges of configuration graph only connect adjacent grid points.

Then for any ${\theta }_1,{\theta }_2\in \mathcal{M}$, we have

$$\underset{h\to 0}{\lim }{d}^{(h)}({\theta }_1^{(h)},{\theta }_2^{(h)})=d_G({\theta }_1,{\theta }_2),$$

where $d^{(h)}$ is discrete complexity distance, $d_G$ is Riemann geodesic distance.

Everyday Interpretation:

• Condition 1: “Grid becomes denser and denser, eventually fills entire control manifold”;
• Condition 2: “Cost of discrete step size consistent with local value of continuous metric”;
• Condition 3: “Cannot have ‘long-range teleportation’ edges, can only step gradually”;
• Conclusion: “Discrete distance converges to continuous geodesic distance”!

6.4 Everyday Analogy: Pixels to Image

Imagine a digital photo:

• Low Resolution (large $h$): 100×100 pixels, looks like squares;
• Medium Resolution (medium $h$): 1000×1000 pixels, already quite smooth;
• High Resolution (small $h$): 10000×10000 pixels, almost continuous;
• Limit (h→0): Infinite resolution, becomes continuous image.

Discrete complexity geometry → continuous Riemann geometry, like low-resolution pixels → high-definition image process!

graph TD
    A["Discrete Computational Universe<br/>U_comp^(h)"] --> B["Control Grid<br/>M^(h) subset M"]
    B --> C["Discrete Complexity Distance<br/>d^(h)(x,y)"]

    D["Grid Refinement<br/>h → 0"] --> E["Grid Dense in M<br/>M^(h) → M"]

    E --> F["Discrete Distance Converges<br/>d^(h) → d_G"]

    C --> F
    F --> G["Continuous Geodesic Distance<br/>d_G(theta_1,theta_2)"]

    G --> H["Riemann Geometry<br/>(M, G)"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1

6.5 Meaning of Gromov-Hausdorff Convergence

Theorem 6.1 is actually a special case of deeper Gromov-Hausdorff convergence:

Definition 6.2 (Gromov-Hausdorff Distance)

Gromov-Hausdorff distance between two metric spaces $(X,d_X)$ and $(Y,d_Y)$ defined as

$$d_{GH}(X,Y)=\inf \{ϵ:\exists Z,f_X,f_Y\},$$

where infimum is over all simultaneous isometric embeddings $f_X:X\to Z$, $f_Y:Y\to Z$ of $X,Y$ into some metric space $Z$, such that Hausdorff distance between $X$ and $Y$ is $\le ϵ$.

Conclusion of Theorem 6.1 can be strengthened to:

$$d_{GH}((X^{(h)},d^{(h)}),(\mathcal{M},d_G))\to 0\text{as }h\to 0.$$

This means: Not only distance functions converge pointwise, but entire metric space structure (including topology, volume, curvature, etc.) converges!

7. Example: One-Dimensional Scattering Network

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 4.3

7.1 Model Setting

Consider a one-dimensional two-port scattering network:

• Control Parameter: $\theta \in [{\theta }_{\min },{\theta }_{\max }]\subset \mathbb{R}$ (e.g., barrier height);
• Scattering Matrix:

$$S(\omega ;\theta )=\left(\begin{array}{cc}r(\omega ;\theta )& t^{\prime }(\omega ;\theta )\\ t(\omega ;\theta )& r^{\prime }(\omega ;\theta )\end{array}\right),$$

where $r$ is reflection coefficient, $t$ is transmission coefficient, satisfy unitarity $|r{|}^2+|t{|}^2=1$.

7.2 Group Delay Matrix

$$Q(\omega ;\theta )=-\mathrm{i}S(\omega ;\theta {)}^{†}{\partial }_{\omega }S(\omega ;\theta )$$

is a $2\times 2$ Hermitian matrix. Its trace

$$\mathrm{tr}Q(\omega ;\theta )={\tau }_r(\omega ;\theta )+{\tau }_t(\omega ;\theta )$$

is sum of group delays of reflection and transmission channels.

7.3 One-Dimensional Metric

$$G(\theta )={\int }_{\Omega }w(\omega )\mathrm{tr}({\partial }_{\theta }Q(\omega ;\theta ){\partial }_{\theta }Q(\omega ;\theta ))\mathrm{d}\omega .$$

This defines one-dimensional Riemann metric $G(\theta )\mathrm{d}{\theta }^2$.

Geodesic Distance: In one-dimensional case, geodesics are straight lines, geodesic distance is

$$d_G({\theta }_1,{\theta }_2)=\left|{\int }_{{\theta }_1}^{{\theta }_2}\sqrt{G(\theta )}\mathrm{d}\theta \right|.$$

7.4 Discretization

Discretize $\theta$ into grid points ${\theta }_n={\theta }_0+nh$, define single-step cost

$${\mathsf{C}}^{(h)}({\theta }_n,{\theta }_{n+1})=\sqrt{G({\theta }_n)}h.$$

Then discrete complexity distance is

$$d^{(h)}({\theta }_m,{\theta }_n)=\sum _{k=m}^{n-1}{\mathsf{C}}^{(h)}({\theta }_k,{\theta }_{k+1})=h\sum _{k=m}^{n-1}\sqrt{G({\theta }_k)}.$$

7.5 Convergence Result

When $h\to 0$, Riemann sum converges to integral:

$$\underset{h\to 0}{\lim }{d}^{(h)}({\theta }_1^{(h)},{\theta }_2^{(h)})={\int }_{{\theta }_1}^{{\theta }_2}\sqrt{G(\theta )}\mathrm{d}\theta =d_G({\theta }_1,{\theta }_2).$$

This is concrete realization of Theorem 6.1 in one-dimensional case!

Physical Interpretation:

• Discrete step size $h$ corresponds to “control precision”;
• Refining grid corresponds to “improving control precision”;
• In limit, discrete “step-by-step jumps” become continuous “motion along geodesics”.

8. Control-Scattering Category: Functor from Discrete to Continuous

Source Theory: euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 5

8.1 Why Do We Need Category Theory?

Category theory provides an “abstract perspective” to understand relationships between different mathematical structures:

• Objects: Some class of mathematical structures (e.g., metric spaces, groups, vector spaces);
• Morphisms: Maps preserving certain properties between structures (e.g., isometries, group homomorphisms, linear maps);
• Functors: Maps between categories, preserving structure of objects and morphisms.

In our context:

• Discrete Computational Universe Category $\mathbf{CompUniv}$: Objects are computational universes $(X,\mathsf{T},\mathsf{C},\mathsf{I})$, morphisms are simulation maps;
• Control-Scattering Category $\mathbf{CtrlScat}$: Objects are control manifolds $(\mathcal{M},G,S)$, morphisms are metric-preserving control maps;
• Functor $F$: Lifting from discrete to continuous.

8.2 Control-Scattering Objects and Morphisms

Definition 8.1 (Control-Scattering Object, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Definition 5.1)

A control-scattering object is a triple

$$C=(\mathcal{M},G,S),$$

where:

• $(\mathcal{M},G)$ is control manifold with Riemann metric;
• $S(\omega ;\theta )$ is scattering family satisfying unified time scale master formula.

Definition 8.2 (Control-Scattering Morphism, from Definition 5.2)

A morphism between two control-scattering objects $C=(\mathcal{M},G,S)$ and $C^{\prime }=({\mathcal{M}}^{\prime },G^{\prime },S^{\prime })$ is a map $f:\mathcal{M}\to {\mathcal{M}}^{\prime }$, satisfying:

1. $f$ is smooth map;
2. Metric controlled transformation: There exist $\alpha ,\beta >0$ such that

$$\alpha G_{\theta }(v,v)\le G_{f(\theta )}^{\prime }(\mathrm{d}{f}_{\theta }v,\mathrm{d}{f}_{\theta }v)\le \beta G_{\theta }(v,v);$$

3. Scattering families compatible.

8.3 Functor from Computational Universe to Control-Scattering

Definition 8.3 (Lifting Functor $F$, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Section 5.2)

Let ${\mathbf{CompUniv}}^{\mathrm{phys}}$ be subcategory of computational universes “realizable by unified time scale scattering”. Construct functor

$$F:{\mathbf{CompUniv}}^{\mathrm{phys}}\to \mathbf{CtrlScat}$$

as follows:

1. Object Level: Given $U_{\mathrm{comp}}=(X,\mathsf{T},\mathsf{C},\mathsf{I})$, construct $(\mathcal{M},G,S)$ from its physical realization, let

$$F(U_{\mathrm{comp}})=(\mathcal{M},G,S).$$

2. Morphism Level: Given simulation map $f:U_{\mathrm{comp}}⇝U_{\mathrm{comp}}^{\prime }$, corresponding control map $f_{\mathcal{M}}:\mathcal{M}\to {\mathcal{M}}^{\prime }$, let

$$F(f)=f_{\mathcal{M}}.$$

Proposition 8.4 (Functoriality, from euler-gls-info/04-unified-time-scale-continuous-complexity-geometry.md Proposition 5.3)

$F$ constitutes a covariant functor, i.e., satisfies:

1. $F(\mathrm{id})=\mathrm{id}$;
2. $F(g\circ f)=F(g)\circ F(f)$.

Everyday Interpretation:

• Functor $F$ lifts “discrete computational universe” to “continuous control manifold”;
• Morphisms (simulation maps) preserved as control maps;
• Step-by-step progression of complexity distance preserved as geodesic distance of metric.

8.4 Everyday Analogy: From Pixels to Vector Graphics

• Discrete Computational Universe: Bitmap, composed of pixels, discrete;
• Control-Scattering Object: Vector graphics, composed of curves and shapes, continuous;
• Functor $F$: Bitmap-to-vector conversion algorithm, preserves shape and structure of image.

graph TD
    A["Discrete Computational Universe<br/>CompUniv^phys"] --> B["Functor F"]
    B --> C["Control-Scattering Category<br/>CtrlScat"]

    A --> D["Object: U_comp<br/>(X, T, C, I)"]
    C --> E["Object: (M, G, S)"]

    D -->|"F"| E

    A --> F["Morphism: Simulation Map f"]
    C --> G["Morphism: Control Map f_M"]

    F -->|"F"| G

    E --> H["Continuous Riemann Geometry"]
    H --> I["Geodesics, Curvature, Volume"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style I fill:#e1f5ff

9. Complete Picture: From Scattering to Geometry

9.1 Theoretical Structure Summary

graph TD
    A["Physical Scattering System"] --> B["Scattering Matrix S(omega)"]
    B --> C["Three Equivalent Quantities"]

    C --> D["Scattering Phase phi(omega)"]
    C --> E["Spectral Shift Function<br/>xi(omega)"]
    C --> F["Group Delay Matrix Q(omega)"]

    D -->|"Derivative/pi"| G["Unified Time Scale kappa(omega)"]
    E -->|"Derivative"| G
    F -->|"Trace/2pi"| G

    G --> H["Master Ruler:<br/>Basic Unit of Time"]

    H --> I["Introduce Control Parameter theta"]
    I --> J["Control Manifold M"]
    J --> K["Scattering Family S(omega;theta)"]
    K --> L["Group Delay Family Q(omega;theta)"]

    L --> M["Complexity Metric<br/>G_ab = ∫ w(omega) tr(d_a Q d_b Q) domega"]

    M --> N["Riemann Geometry<br/>(M, G)"]
    N --> O["Geodesic Distance d_G"]
    N --> P["Geodesics, Curvature"]

    Q["Discrete Computational Universe"] --> R["Control Grid M^(h)"]
    R --> S["Discrete Distance d^(h)"]

    S -->|"h → 0"| O
    T["Gromov-Hausdorff<br/>Convergence"]

    O --> T
    S --> T

    style A fill:#e1f5ff
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style M fill:#e1ffe1
    style N fill:#ffd4e1
    style T fill:#e1fff5

9.2 Core Formula Quick Reference

10. Summary

This article establishes complete bridge from physical scattering to computational geometry:

10.1 Core Concepts

1. Unified Time Scale $\kappa (\omega )$: Trinity master ruler

   • Scattering phase derivative: ${\phi }^{\prime }(\omega )/\pi$
   • Spectral shift function derivative: ${\xi }^{\prime }(\omega )$
   • Group delay trace: $(2\pi {)}^{-1}\mathrm{tr}Q(\omega )$

2. Control Manifold $\mathcal{M}$: Parameterized computation space

3. Complexity Metric $G_{ab}(\theta )$: Induced by group delay derivative $$G_{ab}=\int w(\omega )\mathrm{tr}({\partial }_{a}Q{\partial }_{b}Q)\mathrm{d}\omega$$

4. Gromov-Hausdorff Convergence: Discrete → Continuous $$\underset{h\to 0}{\lim }{d}^{(h)}=d_G$$

5. Control-Scattering Category $\mathbf{CtrlScat}$: Category-theoretic framework

10.2 Core Insights

• Unification: Scattering phase, spectral shift, group delay are essentially three manifestations of same quantity;
• Intrinsic Nature: Time scale is property of physical process itself, doesn’t depend on external reference;
• Geometrization: Computational cost can be geometrized as geodesic distance on Riemann manifold;
• Limit Consistency: Discrete geometry strictly converges to continuous geometry in refinement limit;
• Categorical Naturality: Discrete and continuous connected through functor, preserving structure.

10.3 Everyday Analogy Review

• Elastic Ruler: Unified time scale like variable-scale ruler;
• Radio Frequency Tuning: Control manifold like parameter space of knobs;
• Mountain Climbing Slope: Complexity metric like steepness of terrain;
• Pixels to Image: Discrete convergence like improving photo resolution;
• Bitmap to Vector Graphics: Functor like image format conversion.

10.4 Connections with Previous and Subsequent Chapters

Connection with Articles 23.1-7:

• Articles 23.3-5: Discrete complexity geometry (complexity distance, volume, Ricci curvature);
• Articles 23.6-7: Discrete information geometry (Fisher matrix, information dimension);
• This Article: Provides strict bridge from discrete to continuous, through unified time scale.

Preview of Article 23.9: Next article will deeply study global properties of control manifold:

• Geometric details of Gromov-Hausdorff convergence;
• Curvature and topology of control manifold;
• Coupling with information manifold.

Preview of Articles 23.10-11: Articles 23.10-11 will construct joint time-information-complexity variational principle based on control manifold $(\mathcal{M},G)$ and information manifold $({\mathcal{S}}_Q,g_Q)$, achieving complete unification of three.

Preview of Next Article: 23.9 Control Manifold and Gromov-Hausdorff Convergence

In next article, we will deeply study:

1. Geometric Meaning of Gromov-Hausdorff Distance: How do metric spaces “approach” each other?
2. Global Properties of Control Manifold: Compactness, completeness, geodesic completeness;
3. Fine Correspondence Between Discrete and Continuous: Not only distance converges, but volume and curvature also converge;
4. Coupling with Information Manifold: Joint geometry of control manifold $\mathcal{M}$ and information manifold ${\mathcal{S}}_Q$;
5. Physical Examples: Control manifolds of quantum circuits, scattering networks, neural networks.