fix(vol4): clean inline math delimiters in 01-boundary, 08-perception, 10-intent

Author: profvjreddi

books/vol4/chapters/01-boundary/01-boundary.qmd

@@ -295,7 +295,7 @@ In computer systems engineering, developers often ask: *why cannot we run this p
 Therefore, every robust physical AI machine physically partitions its compute across heterogeneous silicon (@fig-01-dual-brain):
 
 1. **The Cognitive Proposer (Linux MPU / Edge NPU):** Operates in unprivileged user space at a deliberative cadence ($10\text{--}50\text{ Hz}$). It executes high-capacity neural policies, multi-modal perception backbones, and trajectory generators. It possesses **zero direct electrical wiring to physical actuator registers**. It can only write candidate trajectory proposals into a shared SRAM mailbox or lock-free circular ring buffer across an inter-processor communication (IPC) channel.
-2. **The Real-Time Permission Referee (Bare-Metal / RTOS MCU):** Operates on isolated, dedicated silicon at a hard real-time cadence ($1000\text{ Hz}$, $1\text{ ms}$ deadline, $$< 50\,\mu\text{s}$ jitter) with static, zero-allocation memory (`malloc` is strictly forbidden at runtime). It holds **exclusive physical authority** over the PWM compare registers, digital-to-analog converters, and gate drivers. The MCU continuously checks physical safety invariants and stopping clearance. If the proposed command satisfies all physical constraints, the MCU latches the values to hardware; if the proposal is unsafe, stalls, or panics, the MCU vetoes the command and executes a deterministic physical fallback.
+2. **The Real-Time Permission Referee (Bare-Metal / RTOS MCU):** Operates on isolated, dedicated silicon at a hard real-time cadence ($1000\text{ Hz}$, $1\text{ ms}$ deadline, $< 50\,\mu\text{s}$ jitter) with static, zero-allocation memory (`malloc` is strictly forbidden at runtime). It holds **exclusive physical authority** over the PWM compare registers, digital-to-analog converters, and gate drivers. The MCU continuously checks physical safety invariants and stopping clearance. If the proposed command satisfies all physical constraints, the MCU latches the values to hardware; if the proposal is unsafe, stalls, or panics, the MCU vetoes the command and executes a deterministic physical fallback.
 
 To bridge the asynchronous Linux OS and deterministic bare-metal firmware without priority inversion or lock contention, the MPU and MCU communicate across a shared SRAM mailbox using lock-free atomic sequence locks (*seqlocks*) or double-buffered ping-pong ring buffers.[^fn-hw-soc-mailbox] The MPU publishes candidate action chunks with a monotonically incrementing sequence version; the MCU reads the trajectory atomically without ever blocking on a mutex. If an MPU core stalls or experiences an OS scheduling delay, the sequence version fails to update, immediately signaling the MCU that proposed commands have aged beyond their freshness deadline.
 

books/vol4/chapters/08-perception/08-perception.qmd

@@ -149,12 +149,12 @@ $$\Delta x(y) = v \cdot y \cdot t_{\text{row}}$$
 
 Under rotational motion, rolling shutter distortion becomes non-linear. If the camera undergoes angular rotation at angular velocity vector $\boldsymbol{\omega} = [\omega_x, \omega_y, \omega_z]^T$ (e.g., rapid yaw or pitch during aggressive robot maneuvers), the effective camera attitude changes continuously as scanlines are read. For a scanline at vertical coordinate $y$, the differential rotation relative to the frame start is $\Delta \boldsymbol{\theta}(y) \approx \boldsymbol{\omega} \cdot y \cdot t_{\text{row}}$. This scanline-dependent rotation warps straight physical lines into curves, induces focal-length dilation (scaling distortion during pitch), and shears vertical walls into slanted surfaces.
 
-As an analytical example, consider a ground robot traversing an aisle at $v = 3.0\text{ m/s}$ equipped with a rolling shutter camera operating at $30\text{ Hz}$ with vertical resolution $H = 1080\text{ rows}$. If the active readout time across all rows is $t_{\text{readout}} = 33.3\text{ ms}$, the inter-row readout interval is $t_{\text{row}} = 33.3\text{ ms} / 1080 \approx $30.8\,\mu\text{s}$$. A vertical pallet edge spanning the full height of the frame does not appear vertical in the captured array. The bottom scanline ($y = 1079$) is sampled $33.3\text{ ms}$ after the top scanline ($y = 0$), producing a total physical spatial skew of:
+As an analytical example, consider a ground robot traversing an aisle at $v = 3.0\text{ m/s}$ equipped with a rolling shutter camera operating at $30\text{ Hz}$ with vertical resolution $H = 1080\text{ rows}$. If the active readout time across all rows is $t_{\text{readout}} = 33.3\text{ ms}$, the inter-row readout interval is $t_{\text{row}} = 33.3\text{ ms} / 1080 \approx 30.8\,\mu\text{s}$. A vertical pallet edge spanning the full height of the frame does not appear vertical in the captured array. The bottom scanline ($y = 1079$) is sampled $33.3\text{ ms}$ after the top scanline ($y = 0$), producing a total physical spatial skew of:
 $$\Delta x = 3.0\text{ m/s} \times 0.0333\text{ s} \approx 0.10\text{ m} = 100\text{ mm}$$
 With an exposure duration of $t_{\text{exp}} = 10.0\text{ ms}$, each individual scanline also suffers a motion blur width of $\Delta x_{\text{blur}} = 3.0\text{ m/s} \times 0.010\text{ s} = 0.03\text{ m} = 30\text{ mm}$. If downstream geometry reconstruction treats this image as an instantaneous rigid projection, the perceived $100\text{ mm}$ tilt causes the motion planner to identify an artificial intrusion into the free corridor. Correcting this distortion requires **rolling-shutter motion compensation**, the continuous-time geometric unwarping algorithm that utilizes high-rate ($1000\text{ Hz}$) inertial angular velocities ($\boldsymbol{\omega}$) and linear velocities ($\mathbf{v}$) to interpolate the instantaneous 6-DoF camera pose $\mathbf{T}_{wb}(t(y))$ for each individual scanline $y$, restoring rigid epipolar geometry prior to 3D triangulation.
 
 ::: {.callout-case-study title="Mobile Robot Rolling-Shutter Image Shear Fracture"}
-**System:** Autonomous mobile warehouse transport robot equipped with a forward-facing rolling-shutter CMOS camera ($1080\text{p}$ at $30\text{ Hz}$, row readout time $t_{\text{row}} = $30.8\,\mu\text{s}$$, total frame readout $t_{\text{readout}} = 33.3\text{ ms}$) and an onboard 6-DoF Visual-Inertial Odometry (VIO) pipeline running a sliding-window factor graph optimizer.
+**System:** Autonomous mobile warehouse transport robot equipped with a forward-facing rolling-shutter CMOS camera ($1080\text{p}$ at $30\text{ Hz}$, row readout time $t_{\text{row}} = 30.8\,\mu\text{s}$, total frame readout $t_{\text{readout}} = 33.3\text{ ms}$) and an onboard 6-DoF Visual-Inertial Odometry (VIO) pipeline running a sliding-window factor graph optimizer.
 
 **Incident:** While executing a rapid $90^\circ$ turn at an angular velocity of $\omega_z = 2.4\text{ rad/s}$ ($137.5^\circ/\text{s}$) alongside an industrial storage rack, the robot's state estimator suffered an instantaneous divergence fracture ($>0.52\text{ m}$ lateral position jump). The motion planner commanded a corrective steer directly into the steel racking upright at $2.2\text{ m/s}$, shearing the camera mount and causing structural chassis deformation.
 
@@ -560,7 +560,7 @@ An observation that arrives with zero sensor noise, perfect spatial resolution,
 
 *The scenario.* An autonomous vehicle transitions from direct sunlight ($100{,}000\text{ lux}$) into a dark loading bay ($20\text{ lux}$) at $v = 6.0\text{ m/s}$. The illumination change triggers four cameras to adjust exposure gains simultaneously, bursting raw pixel telemetry across the shared SoC interconnect at $4.8\text{ GB/s}$ against a memory controller limit of $5.2\text{ GB/s}$ ($92\%$ bus utilization).
 
-*The systems dilemma.* Real-time safety monitors reading high-frequency IMU and wheel encoder telemetry ($128\text{ kB/s}$ at $1000\text{ Hz}$) experience head-of-line blocking in the shared memory controller, inflating access latency from $$15.0\,\mu\text{s}$$ to $2.8\text{ ms}$ ($P_{99.9}$) and starving the closed-loop traction controller.
+*The systems dilemma.* Real-time safety monitors reading high-frequency IMU and wheel encoder telemetry ($128\text{ kB/s}$ at $1000\text{ Hz}$) experience head-of-line blocking in the shared memory controller, inflating access latency from $15.0\,\mu\text{s}$ to $2.8\text{ ms}$ ($P_{99.9}$) and starving the closed-loop traction controller.
 
 *The systems verdict: Interconnect Quality-of-Service Law.* Perceptual data ingestion must never starve real-time safety reflexes. The SoC architecture must enforce hardware AXI QoS priority registers, isolating real-time safety channels on dedicated SRAM or high-priority lanes while rate-limiting or dropping perception frames during transient bus saturation.
 <!-- END -->

books/vol4/chapters/10-intent/10-intent.qmd

@@ -210,7 +210,7 @@ In an analytical example of an articulated manipulator picking components from a
 $$\tau \le \frac{15\text{ mm} - 3\text{ mm}}{200\text{ mm/s}} = \frac{12\text{ mm}}{200\text{ mm/s}} = 0.060\text{ s} = 60\text{ ms}$$
 As plotted along the dynamic operating curve in @fig-10-lease-dynamics-tradeoff (Panel A), a lease duration longer than $60\text{ ms}$ permits the component to drift outside the gripper's physical capture envelope while the planner is still driving the arm toward the initial coordinate. If the conveyor slows to $v_{\text{drift}} = 0.02\text{ m/s}$ ($20\text{ mm/s}$), the allowable validity window expands to $600\text{ ms}$. Because physical drift rates vary with workspace dynamics, the safe lease duration is a property of the environment and the tooling, not an arbitrary parameter tuned to software convenience.
 
-::: {#fig-10-lease-dynamics-tradeoff fig-env="figure" fig-pos="t" fig-cap="**Physics-Derived Lease Validity Horizon and Ingestion Rejection Gateways**: (Panel A) Maximum safe lease duration $\tau$ as a function of scene drift velocity $v_{\text{drift}}$ across distinct task tolerance radii ($r_{\text{tol}} = 5\text{ mm}, 15\text{ mm}, 30\text{ mm}$) derived from $\tau \le (r_{\text{tol}} - \sigma_{\text{sensor}}) / v_{\text{drift}}$. The green shaded region denotes the admissible safe execution manifold, while the red shaded region marks the stale-goal collision hazard zone. Coupling lease duration to reasoning model inference cadence ($P_{99} = 650\text{ ms}$) violates the physical bound under dynamic drift. (Panel B) The 4-stage early rejection pipeline: (1) monotonic sequence and parent perception hash verification ($<1\ \mu\text{s}$), (2) $O(1)$ analytical kinematic workspace manifold $\mathcal{W}$ and static occupancy check ($<5\ \mu\text{s}$), (3) dynamic reachability filter ($t_{\min} = 2\sqrt{d/a_{\max}} \le \tau_{\text{rem}}$), and (4) semantic covariance ambiguity gate ($\lambda_{\max}(\mathbf{\Sigma}) \le \sigma^2_{\max}$). Inadmissible proposals are rejected in $$42\,\mu\text{s}$$ ($P_{99}$) with structured diagnostic feedback, eliminating $15\text{–}80\text{ ms}$ trajectory optimizer stalls and preventing planning queue starvation." fig-alt="Physics-Derived Lease Validity Horizon and Ingestion Rejection Gateways. (Panel A) Maximum safe lease duration \tau as a function of scene drift velocity v{\text{drift}} across distinct task tolerance radii (r{\text{tol}} = 5\text{ mm}, 15\text{ mm}, 30\text{ mm}) derived from \tau \le (r{\text{tol}} - \sigma{\text{sensor}}) / v{\text{drift}}. The green shaded region denotes the admissible safe execution manifold, while the red shaded region marks the stale-goal collision hazard zone. Coupling lease duration to reasoning model inference cadence (P{99} = 650\text{ ms}) violates the physical bound under dynamic drift. (Panel B) The 4-stage early rejection pipeline: (1) monotonic sequence and parent perception hash verification ($<1\,\mu\text{s}$), (2) O(1) analytical kinematic workspace manifold \mathcal{W} and static occupancy check ($<5\,\mu\text{s}$), (3) dynamic reachability filter (t{\min} = 2\sqrt{d/a{\max}} \le \tau{\text{rem}}), and (4) semantic covariance ambiguity gate (\lambda{\max}(\mathbf{\Sigma}) \le \sigma^2{\max}). Inadmissible proposals are rejected in $42\,\mu\text{s}$ (P{99}) with structured diagnostic feedback, eliminating 15\text{–}80\text{ ms} trajectory optimizer stalls and preventing planning queue starvation."}
+::: {#fig-10-lease-dynamics-tradeoff fig-env="figure" fig-pos="t" fig-cap="**Physics-Derived Lease Validity Horizon and Ingestion Rejection Gateways**: (Panel A) Maximum safe lease duration $\tau$ as a function of scene drift velocity $v_{\text{drift}}$ across distinct task tolerance radii ($r_{\text{tol}} = 5\text{ mm}, 15\text{ mm}, 30\text{ mm}$) derived from $\tau \le (r_{\text{tol}} - \sigma_{\text{sensor}}) / v_{\text{drift}}$. The green shaded region denotes the admissible safe execution manifold, while the red shaded region marks the stale-goal collision hazard zone. Coupling lease duration to reasoning model inference cadence ($P_{99} = 650\text{ ms}$) violates the physical bound under dynamic drift. (Panel B) The 4-stage early rejection pipeline: (1) monotonic sequence and parent perception hash verification ($<1\ \mu\text{s}$), (2) $O(1)$ analytical kinematic workspace manifold $\mathcal{W}$ and static occupancy check ($<5\ \mu\text{s}$), (3) dynamic reachability filter ($t_{\min} = 2\sqrt{d/a_{\max}} \le \tau_{\text{rem}}$), and (4) semantic covariance ambiguity gate ($\lambda_{\max}(\mathbf{\Sigma}) \le \sigma^2_{\max}$). Inadmissible proposals are rejected in $42\,\mu\text{s}$ ($P_{99}$) with structured diagnostic feedback, eliminating $15\text{–}80\text{ ms}$ trajectory optimizer stalls and preventing planning queue starvation." fig-alt="Physics-Derived Lease Validity Horizon and Ingestion Rejection Gateways. (Panel A) Maximum safe lease duration \tau as a function of scene drift velocity v{\text{drift}} across distinct task tolerance radii (r{\text{tol}} = 5\text{ mm}, 15\text{ mm}, 30\text{ mm}) derived from \tau \le (r{\text{tol}} - \sigma{\text{sensor}}) / v{\text{drift}}. The green shaded region denotes the admissible safe execution manifold, while the red shaded region marks the stale-goal collision hazard zone. Coupling lease duration to reasoning model inference cadence (P{99} = 650\text{ ms}) violates the physical bound under dynamic drift. (Panel B) The 4-stage early rejection pipeline: (1) monotonic sequence and parent perception hash verification ($<1\,\mu\text{s}$), (2) O(1) analytical kinematic workspace manifold \mathcal{W} and static occupancy check ($<5\,\mu\text{s}$), (3) dynamic reachability filter (t{\min} = 2\sqrt{d/a{\max}} \le \tau{\text{rem}}), and (4) semantic covariance ambiguity gate (\lambda{\max}(\mathbf{\Sigma}) \le \sigma^2{\max}). Inadmissible proposals are rejected in $42\,\mu\text{s}$ (P{99}) with structured diagnostic feedback, eliminating 15\text{–}80\text{ ms} trajectory optimizer stalls and preventing planning queue starvation."}
 ![](figures/fig10_lease_dynamics_tradeoff.svg){width="100%"}
 :::
 

publishing/quarto/contents/vol4/chapters/01-boundary/01-boundary.qmd

@@ -295,7 +295,7 @@ In computer systems engineering, developers often ask: *why cannot we run this p
 Therefore, every robust physical AI machine physically partitions its compute across heterogeneous silicon (@fig-01-dual-brain):
 
 1. **The Cognitive Proposer (Linux MPU / Edge NPU):** Operates in unprivileged user space at a deliberative cadence ($10\text{--}50\text{ Hz}$). It executes high-capacity neural policies, multi-modal perception backbones, and trajectory generators. It possesses **zero direct electrical wiring to physical actuator registers**. It can only write candidate trajectory proposals into a shared SRAM mailbox or lock-free circular ring buffer across an inter-processor communication (IPC) channel.
-2. **The Real-Time Permission Referee (Bare-Metal / RTOS MCU):** Operates on isolated, dedicated silicon at a hard real-time cadence ($1000\text{ Hz}$, $1\text{ ms}$ deadline, $$< 50\,\mu\text{s}$ jitter) with static, zero-allocation memory (`malloc` is strictly forbidden at runtime). It holds **exclusive physical authority** over the PWM compare registers, digital-to-analog converters, and gate drivers. The MCU continuously checks physical safety invariants and stopping clearance. If the proposed command satisfies all physical constraints, the MCU latches the values to hardware; if the proposal is unsafe, stalls, or panics, the MCU vetoes the command and executes a deterministic physical fallback.
+2. **The Real-Time Permission Referee (Bare-Metal / RTOS MCU):** Operates on isolated, dedicated silicon at a hard real-time cadence ($1000\text{ Hz}$, $1\text{ ms}$ deadline, $< 50\,\mu\text{s}$ jitter) with static, zero-allocation memory (`malloc` is strictly forbidden at runtime). It holds **exclusive physical authority** over the PWM compare registers, digital-to-analog converters, and gate drivers. The MCU continuously checks physical safety invariants and stopping clearance. If the proposed command satisfies all physical constraints, the MCU latches the values to hardware; if the proposal is unsafe, stalls, or panics, the MCU vetoes the command and executes a deterministic physical fallback.
 
 To bridge the asynchronous Linux OS and deterministic bare-metal firmware without priority inversion or lock contention, the MPU and MCU communicate across a shared SRAM mailbox using lock-free atomic sequence locks (*seqlocks*) or double-buffered ping-pong ring buffers.[^fn-hw-soc-mailbox] The MPU publishes candidate action chunks with a monotonically incrementing sequence version; the MCU reads the trajectory atomically without ever blocking on a mutex. If an MPU core stalls or experiences an OS scheduling delay, the sequence version fails to update, immediately signaling the MCU that proposed commands have aged beyond their freshness deadline.