Chapter 41
Flat Paths in Curved Interpretation

The Flat Paths in Curved Interpretation flow implements one of the central geometric reformulations of Holographic Space-Time (HST): the proposition that physical objects, photons, and information packets do not fundamentally travel along curved trajectories. Instead, they move through a sequence of locally flat coherence transfers, with apparent curvature emerging only when these local transfers are accumulated across a non-uniform coherence field.

The flow provides a deterministic execution model that replaces traditional geodesic motion through curved spacetime with recursive alignment to local coherence gradients. Rather than assuming that space itself bends and forces objects to follow curved paths, the flow models motion as a succession of local directional updates governed by the structure of the surrounding boundary coherence field.

The result is a framework in which observed curvature becomes an emergent phenomenon rather than a fundamental property of geometry.

The first principle implemented by the flow is that every propagation step is locally flat.

The system assumes that, over sufficiently small intervals, a propagating entity experiences only its immediate coherence environment. The entity does not calculate or perceive a global trajectory.

Instead, it performs a local alignment operation:

\[ R_t \propto \nabla \rho_C \]

where:

This means the entity continuously aligns itself with the locally preferred coherence direction.

At each individual step:

Curvature(LocalSegment)≈0

which enforces the flatness invariant built into the flow.

The flow implements the HST principle that motion is generated from boundary structure rather than background geometry.

The propagation sequence begins with:

\[ B_t \rightarrow C_t \]

where:

The coherence map is then converted into a density field:$${}_{}{\mathrm{<br>}}_{}$$

\[ C_t \rightarrow \rho_C \]

and subsequently into a gradient field:

\[ \rho_C \rightarrow \nabla \rho_C \]

This chain creates the directional information that guides motion.

Thus, motion is not imposed externally.

It emerges directly from the informational structure encoded within the boundary itself.

The flow implements continuous directional correction.

Unlike a classical straight-line model in which an object receives a single initial direction, this system recalculates direction at every propagation cycle.

The update process is:

\[ \mathrm{Direction}(R_t) = \mathrm{Align} \!\left( \nabla \rho_C \right) \]

followed by:

\[ x_{t+1} = x_t + \mathrm{LocalStep} \!\left( R_t \right) \]

This means the trajectory is continuously reconstructed from local information.

The system therefore behaves as a recursive navigation process through coherence geometry.

A major purpose of the flow is to demonstrate how global curvature can emerge without local curvature.

Each propagation step remains approximately flat:

Pathlocal​≈Straight

However, because the coherence gradient changes from location to location, successive local directions are not identical.

The global trajectory becomes:

Pathglobal​=∑Pathlocal​

As these local segments accumulate, the resulting path appears curved to an external observer.

The flow therefore implements the HST principle that:

This is one of the key conceptual shifts introduced by the chapter.

The flow is equally applicable to photon propagation.

Traditional General Relativity explains gravitational lensing through curvature of spacetime.

The HST implementation instead treats light as undergoing repeated local coherence alignment.

The flow expresses this as:

Pathphoton​=∑LocalAlignment(ρC​)

Each individual transfer remains locally straight.

The apparent bending of light arises because coherence density varies around massive structures.

Thus the flow implements gravitational lensing as a coherence-navigation phenomenon rather than a spacetime-curvature phenomenon.

The flow provides an operational reinterpretation of the geodesic concept.

In General Relativity:

Geodesic=Straightest Path in Curved Space

In HST:

Geodesic=Sequence of Locally 
Optimal Coherence Transfers

The flow explicitly performs this computation through repeated gradient alignment and local propagation updates.

This reformulation preserves observed motion while replacing the underlying explanatory mechanism.

A key implementation feature is the conversion of coherence density into navigational information.

The gradient field acts as the governing structure that directs propagation.

The flow computes:

\[ \nabla \rho_C \]

and uses it to determine:

\[ R_t \]

This makes coherence gradients functionally equivalent to the directional information that conventional theories attribute to geometric curvature.

The flow therefore implements a complete motion-control system derived entirely from coherence structure.

To ensure physically meaningful execution, the flow enforces several invariants.

\[ C_t \geq C_{\min} \]

ensures that propagation occurs only within a stable coherence field.

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) \leq G_{\max} \]

prevents unphysical directional discontinuities.

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \leq \epsilon \]

guarantees that each propagation step remains locally flat.

\[ \mathrm{Velocity}(P_t) \leq c \]

maintains compatibility with relativistic speed limits.

Together these constraints ensure that apparent curvature emerges from stable recursive updates rather than arbitrary trajectory manipulation.

The flow is not intended to deny the observational success of General Relativity.

Instead, it implements an alternative structural interpretation.

General Relativity describes motion through:

\[ G_{\mu\nu} = 8\pi T_{\mu\nu} \]

where matter generates spacetime curvature.

The HST flow replaces this with:

\[ \mathrm{Direction} \propto \nabla \rho_C \]

where coherence gradients generate directional updates.

Both frameworks aim to reproduce the same observed trajectories.

The distinction lies in what is considered fundamental.

General Relativity treats curvature as fundamental.

HST treats coherence structure as fundamental.

This flow implements a complete coherence-based mechanism for apparent curved motion without requiring the existence of a physically curved spacetime substrate.

The execution chain:

\[ B_t \rightarrow C_t \rightarrow \rho_C \rightarrow \nabla \rho_C \rightarrow R_t \rightarrow \mathrm{Path}_{\mathrm{global}} \]

demonstrates how local flat propagation can generate globally curved trajectories through recursive interaction with a structured coherence field.

Within HST, this provides a unified explanatory framework for:

The flow therefore serves as a formal computational model of the HST claim that paths are fundamentally flat at the local level, while curvature emerges as a large-scale consequence of evolving coherence geometry.

This flow describes how apparently curved trajectories emerge from a sequence of locally flat coherence transfers. Within Holographic Space-Time (HST), objects, photons, and information packets do not fundamentally travel through a curved spacetime fabric. Instead, they repeatedly align with local coherence gradients encoded within the boundary structure.

The flow should therefore be read as a recursive propagation engine that transforms boundary information into motion. Every stage represents a physical operation occurring within the coherence field, and the apparent curvature observed at large scales emerges naturally from the accumulation of many locally straight propagation steps.

The key principle throughout the flow is:

$$\text{Local Flatness}\to \text{Recursive Alignment}\to \text{Apparent Global Curvature}$$

The complete execution chain is:

\[ B_t \rightarrow C_t \rightarrow \rho_C \rightarrow \nabla \rho_C \rightarrow R_t \rightarrow x_{t+1} \rightarrow \mathrm{Path}_{\mathrm{global}} \rightarrow \kappa \]

This chain should be interpreted as a sequence of transformations.

Each stage produces the information required by the next stage.

The flow begins with a boundary state and ends with an observable trajectory.

The flow starts with:

\[ B_t \]

where:

In HST, boundaries act as the primary carriers of physical information.

Nothing moves directly because of spacetime curvature.

Instead, motion emerges from information encoded within the boundary geometry.

When reading the flow, think of $B_t$ as the complete local physical configuration available at time $t$.

The first transformation is:

\[ C_t = \mathrm{CoherenceMap}(B_t) \]

This stage converts raw boundary information into a structured coherence field.

The resulting map describes:

The coherence map is the first usable physical representation of the environment.

When reading this stage, imagine generating a navigational chart from the underlying boundary structure.

The next operation is:

\[ \rho_C = \mathrm{Density}(C_t) \]

This computes the local coherence density field.

The density field measures how strongly coherence is concentrated at each location.

High-density regions correspond to stronger structural organisation.

Lower-density regions correspond to weaker organisation.

This stage transforms the coherence map into a measurable field quantity.

Conceptually:

\[ C_t \rightarrow \rho_C \]

means:

The flow then calculates:

\[ \nabla \rho_C \]

This is one of the most important stages in the entire execution chain.

The gradient describes:

The gradient acts as the local directional guide for motion.

In the flow:

\[ \rho_C \rightarrow \nabla \rho_C \]

should be read as:

At this point the system knows nothing about a global path.

Only local information exists.

The next stage generates the propagation direction:

\[ R_t = \mathrm{Align} \!\left( P_t,\, \nabla \rho_C \right) \]

where:

This stage performs local directional alignment.

The propagating entity consults the coherence gradient and determines the most coherent local route.

The governing relationship is:

\[ \mathrm{Direction}(R_t) \propto \nabla \rho_C \]

When reading this section of the flow, imagine a navigator selecting the best next step based solely on nearby terrain information.

No global destination is required.

Only local optimisation occurs.

After determining direction, motion occurs:

\[ x_{t+1} = x_t + \mathrm{LocalStep} \!\left( R_t \right) \]

This is the actual propagation stage.

The entity moves a small distance along the chosen direction.

Importantly:

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \leq \epsilon \]

which means every individual propagation segment remains approximately flat.

This is one of the fundamental assumptions of the chapter.

The flow does not permit large local curvature.

Instead, it constructs apparent curvature through many successive flat segments.

After propagation, the process repeats.

A new boundary state is encountered.

The flow then executes again:

\[ B_{t+1} \rightarrow C_{t+1} \rightarrow \rho_{C,t+1} \rightarrow \nabla \rho_{C,t+1} \]

and a new direction is calculated.

This recursive behaviour is crucial.

The trajectory is not precomputed.

It is continuously reconstructed through local interactions with the coherence field.

The path emerges dynamically.

As the recursion continues, individual propagation segments accumulate:

\[ \mathrm{Path}_{\mathrm{global}} = \sum \mathrm{Path}_{\mathrm{local}} \]

Each local segment remains approximately straight.

However, the direction changes slightly at every update.

The accumulation of these direction changes produces the observed trajectory.

This stage should be read as:

The global path is therefore an emergent structure.

It is not an independent object within the flow.

The final stage computes:

\[ \kappa = \mathrm{Curvature} \!\left( \mathrm{Path}_{\mathrm{global}} \right) \]

This quantity represents the curvature visible to an external observer.

Notice that curvature appears only after the entire path has been assembled.

The flow never requires curvature to exist at the local propagation level.

Instead:

\[ \mathrm{Path}_{\mathrm{local}} \approx \mathrm{Straight} \]

while:

\[ \mathrm{Path}_{\mathrm{global}} \approx \mathrm{Curved} \]

This distinction is the central idea implemented by the chapter.

The invariants define conditions that must remain true during execution.

\[ C_t \geq C_{\min} \]

The coherence field must remain sufficiently organised.

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) \leq G_{\max} \]

The gradient field must not contain physically impossible jumps.

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \leq \epsilon \]

Each propagation step must remain approximately flat.

\[ \mathrm{Velocity}(P_t) \leq c \]

Propagation cannot exceed the maximum physical speed.

These invariants guarantee that the resulting trajectory emerges from coherent physical evolution.

Failure states indicate conditions under which valid propagation can no longer occur.

\[ C_t < C_{\min} \]

The coherence structure becomes too weak to support stable navigation.

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) > G_{\max} \]

The directional field becomes physically inconsistent.

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) > \epsilon \]

The local-flatness assumption is violated.

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) > \epsilon \]

The propagation exceeds allowable physical limits.

When any failure condition occurs, execution halts because the assumptions of the model are no longer satisfied.

The flow should ultimately be read as a coherence-based alternative to conventional geodesic motion.

Traditional interpretation:

Curved Space→Straight Geodesics

HST interpretation:

Straight Local Transfers→Apparent Curved Paths

The flow demonstrates how a propagating entity can continuously follow locally flat coherence gradients while still producing the large-scale curved trajectories observed in gravitational systems.

Its purpose is to formalise the HST claim that curvature is not a fundamental property of motion itself, but an emergent consequence of recursive navigation through a non-uniform coherence field encoded on the boundary structure.

\[ B_t \rightarrow C_t \rightarrow \rho_C \rightarrow \nabla \rho_C \rightarrow R_t \rightarrow x_{t+1} \rightarrow \mathrm{Path}_{\mathrm{global}} \rightarrow \kappa \]

\[ \mathrm{Direction}(R_t) \propto \nabla \rho_C \]

\[ x_{t+1} = x_t + \mathrm{LocalStep} \!\left( R_t \right) \]

\[ \mathrm{Path}_{\mathrm{global}} = \sum \mathrm{Path}_{\mathrm{local}} \]

\[ \mathrm{Path}_{\mathrm{local}} \approx \mathrm{Straight} \]

\[ \mathrm{Path}_{\mathrm{global}} \approx \mathrm{Curved} \]

\[ \kappa = \mathrm{Curvature} \!\left( \mathrm{Path}_{\mathrm{global}} \right) \]

Locally Flat Propagation⇒Recursive Gradient Alignment⇒Globally Curved Observation

This is the complete Flow → Equation → Physical Interpretation mapping for Chapter 41: Flat Paths in Curved Interpretation.

This diagram illustrates one of the most important geometric reinterpretations within Holographic Space-Time (HST): the idea that observed curved trajectories do not require locally curved motion. Instead, the diagram demonstrates how globally curved paths emerge from the accumulation of many locally flat propagation steps guided by coherence-density gradients.

The figure is organised to show the complete transformation from boundary information to observable curvature. Beginning with the boundary state, the system constructs a coherence field, extracts directional information through coherence gradients, performs local alignment, and recursively generates a trajectory that appears curved when viewed at larger scales.

The central scientific claim visualised by the diagram is:

\[ \mathrm{Path}_{\mathrm{local}} \approx \mathrm{Straight} \]

while simultaneously:

\[ \mathrm{Path}_{\mathrm{global}} \approx \mathrm{Curved} \]

Thus, curvature becomes an emergent property of recursive propagation rather than a fundamental property of motion itself.

The upper-left panel represents the boundary state:

\[ B_t \]

depicted as an interconnected boundary lattice.

Within HST, boundaries serve as the primary information-bearing structures of reality.

The diagram illustrates that the environment does not directly impose motion through forces or curved geometry. Instead, it encodes coherence relationships that later determine propagation behaviour.

The boundary network symbolises:

The arrow leading from $B_t$ into the coherence field represents the transformation:

\[ C_t = \mathrm{CoherenceMap}(B_t) \]

which converts raw boundary information into a usable coherence geometry.

The central upper region illustrates the coherence density field:

\[ \rho_C \]

shown as a series of nested contour structures surrounding regions of elevated coherence density.

The contour lines represent gradients of coherence intensity.

Regions labelled:

\[ \mathrm{High}\,\rho_C \]

indicate locations where coherence is strongly concentrated.

These high-density regions function similarly to attractors within the coherence landscape.

The diagram intentionally resembles field contours because the coherence field serves as the navigational structure that governs propagation.

Importantly, no physical spacetime curvature is drawn.

Only variations in coherence density are shown.

This distinction reflects the HST principle that directional behaviour emerges from coherence structure rather than geometric deformation of space itself.

The upper-right panel illustrates:

\[ \nabla \rho_C \]

the coherence-density gradient field.

The arrows show the directional information available at each location.

These vectors indicate:

The gradient field functions as the directional engine of the system.

Mathematically:

\[ \mathrm{Direction}(R_t) \propto \nabla \rho_C \]

The diagram therefore visualises how directional information emerges directly from the coherence distribution.

No external force law is required.

Motion becomes a consequence of local coherence geometry.

The left-middle panel illustrates the local alignment operation:$$R_t = Align(P_t,\nabla\rho_C)$$

\[ R_t = \mathrm{Align} \!\left( P_t,\, \nabla \rho_C \right) \]

The propagating entity $P_t$ is shown examining the local gradient field before selecting its next direction.

This stage is crucial because it demonstrates that the system possesses only local information.

The propagating entity does not know the global trajectory.

It performs only a local optimisation process.

The diagram emphasises that motion arises from repeated local decisions rather than from a predetermined curved path.

This local-alignment principle forms the foundation of the HST reformulation of geodesic motion.

The centre of the figure contains the most important visual element.

A large arc-shaped trajectory connects the two high-density coherence regions.

This path is labelled:

Apparently Curved Path

The curved appearance is what an external observer would measure.

However, the diagram simultaneously shows that the path is constructed from discrete propagation points linked by local directional updates.

The trajectory does not arise because the object is intrinsically following a curved route.

Instead, each successive propagation step is generated through:

\[ R_t \rightarrow x_{t+1} \]

using the local coherence gradient available at that location.

The apparent curvature is therefore a cumulative effect.

Directly beneath the curved trajectory, the diagram presents the local segment decomposition.

The path is divided into:

and continuing propagation intervals.

Each segment is drawn as approximately straight.

This section visualises the key HST statement:

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \approx 0 \]

while:

\[ \mathrm{Curvature} \!\left( \mathrm{Path}_{\mathrm{global}} \right) > 0 \]

The figure therefore distinguishes clearly between:

\[ \mathrm{Path}_{\mathrm{local}} \approx \mathrm{Straight} \]

and

\[ \mathrm{Path}_{\mathrm{global}} \approx \mathrm{Curved} \]

This distinction forms the core argument of the chapter.

The right-middle panel introduces:

\[ \kappa = \mathrm{Curvature} \!\left( \mathrm{Path}_{\mathrm{global}} \right) \]

A conventional coordinate graph is shown containing a curved trajectory.

This panel represents the perspective of an observer measuring the complete path.

The diagram demonstrates that curvature appears only after accumulation of many local propagation events.

In HST, curvature is therefore considered an observational quantity rather than a locally fundamental one.

The observer sees:

\[ \kappa \]

but the underlying mechanism remains recursive local alignment.

The lower portion of the diagram displays the complete flow chain:

\[ B_t \rightarrow C_t \rightarrow \rho_C \rightarrow \nabla \rho_C \rightarrow R_t \rightarrow x_{t+1} \rightarrow \mathrm{Path}_{\mathrm{global}} \rightarrow \kappa \]

This chain provides the causal execution sequence of the entire model.

Each stage generates the information required for the next stage.

The flow demonstrates how observable curvature emerges from a series of deterministic transformations originating in the boundary state.

The progression moves systematically from:

Information
→
Coherence
→
Direction
→
Motion
→
Observation

The small comparison panel labelled:

summarises the entire chapter visually.

The left side depicts a straight segment.

The right side depicts a curved trajectory.

The purpose of this panel is to reinforce the distinction:

\[ \mathrm{Path}_{\mathrm{local}} \neq \mathrm{Path}_{\mathrm{global}} \]

in geometric appearance.

The local propagation law remains flat.

The global observation becomes curved.

This relationship is the central geometric insight of the chapter.

The lower-right section lists the conditions required for valid propagation.

\[ C_t \geq C_{\min} \]

The coherence field must remain sufficiently organised.

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) \leq G_{\max} \]

The gradient field must remain physically continuous.

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \leq \epsilon \]

Every propagation step must remain approximately flat.

\[ \mathrm{Velocity}(P_t) \leq c \]

Propagation must respect the maximum physical speed.

Together these constraints ensure that apparent curvature emerges from physically valid recursive evolution.

The diagram provides a complete visual representation of the HST reinterpretation of curved motion.

Traditional geometric theories describe motion through curved spacetime:

\[ \mathrm{Curved\ Space} \rightarrow \mathrm{Geodesic\ Motion} \]

The HST framework illustrated here replaces that description with:

Boundary Structure→Coherence Field→Gradient Alignment→Local Flat Transfers→Apparent Curvature

The figure therefore demonstrates how gravitational trajectories, photon deflection, and geodesic behaviour can be reformulated as recursive navigation through coherence geometry rather than motion through a fundamentally curved spacetime substrate.

Its central conclusion is that curvature is not generated locally but emerges globally from the accumulation of coherence-guided flat propagation steps across an evolving boundary field.

The Flat Paths in Curved Interpretation flow executes deterministically whenever a valid boundary state produces a continuous coherence field, a physically consistent coherence-density gradient, and locally flat propagation segments.

Execution is guaranteed because every stage of the flow produces a uniquely defined successor state derived from the preceding state without requiring probabilistic branching, hidden variables, or global trajectory knowledge.

The execution chain

\[ B_t \rightarrow C_t \rightarrow \rho_C \rightarrow \nabla \rho_C \rightarrow R_t \rightarrow x_{t+1} \rightarrow \mathrm{Path}_{\mathrm{global}} \rightarrow \kappa \]

contains no undefined transitions.

Each transformation is fully constrained by the local coherence structure encoded within the boundary state.

For every valid boundary configuration:

Bt​

the coherence mapping function produces a valid coherence representation:

\[ C_t = \mathrm{CoherenceMap}(B_t) \]

provided:$$C_t \ge C_{min}$$

\[ C_t \geq C_{\min} \]

Under this condition, coherence information remains continuously available throughout execution.

Therefore the flow cannot lose directional information while propagation remains within a coherent region.

The coherence field generates a unique density distribution:

\[ \rho_C = \mathrm{Density}(C_t) \]

for every valid coherence map.

Since density is derived directly from the coherence structure, the propagation engine always possesses a measurable local coherence landscape.

This guarantees that navigational information can be constructed at every execution step.

Given a continuous coherence density field:

\[ \rho_C \]

the gradient operation:

\[ \nabla \rho_C = \mathrm{Gradient} \!\left( \rho_C \right) \]

produces a valid directional field whenever:

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) \leq G_{\max} \]

The local direction field therefore remains physically continuous and mathematically well-defined throughout propagation.

No undefined directional states can occur while the gradient continuity invariant remains satisfied.

The alignment operation:

\[ R_t = \mathrm{Align} \!\left( P_t,\, \nabla \rho_C \right) \]

always produces a unique local direction because the gradient field uniquely specifies the preferred coherence-transfer orientation.

Therefore:

\[ \mathrm{Direction}(R_t) \propto \nabla \rho_C \]

remains deterministically computable at every propagation cycle.

No ambiguity exists regarding the next locally optimal transfer direction.

The central requirement of the chapter is:

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \leq \epsilon \]

When this invariant is satisfied, every propagation update remains approximately flat.

The execution engine therefore guarantees that local motion never requires intrinsic curvature.

All local transfers remain valid coherence-navigation steps.

Thus:

\[ \mathrm{Path}_{\mathrm{local}} \approx \mathrm{Straight} \]

throughout execution.

Position updates occur through:

\[ x_{t+1} = x_t + \mathrm{LocalStep} \!\left( R_t \right) \]

Because each direction vector is uniquely determined and each step size remains bounded, the next position is always computable.

This guarantees uninterrupted recursive propagation whenever coherence continuity is maintained.

The trajectory evolves through successive local updates without requiring a precomputed global path.

The global trajectory is constructed through accumulation:

\[ \mathrm{Path}_{\mathrm{global}} = \sum \mathrm{Path}_{\mathrm{local}} \]

Since every local segment is valid and every propagation step is deterministic, the global path is guaranteed to exist.

The accumulated trajectory therefore emerges naturally from recursive execution rather than from imposed global geometry.

The observable curvature quantity:

\[ \kappa = \mathrm{Curvature} \!\left( \mathrm{Path}_{\mathrm{global}} \right) \]

is guaranteed to be computable whenever a valid global path exists.

Importantly, curvature appears only after trajectory accumulation.

The flow therefore guarantees:

\[ \mathrm{Path}_{\mathrm{local}} \approx \mathrm{Straight} \]

while simultaneously permitting:

\[ \mathrm{Path}_{\mathrm{global}} \approx \mathrm{Curved} \]

This property is a direct consequence of recursive gradient alignment.

Execution remains physically valid while the following conditions hold:

\[ C_t \geq C_{\min} \]

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) \leq G_{\max} \]

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) \leq \epsilon \]

Velocity(Pt​)≤c

These constraints ensure:

Execution is guaranteed to halt if any invariant is violated.

Specifically:

\[ C_t < C_{\min} \]

\[ \mathrm{Discontinuity} \!\left( \nabla \rho_C \right) > G_{\max} \]

\[ \mathrm{Curvature} \!\left( \mathrm{LocalSegment} \right) > \epsilon \]

\[ \mathrm{Velocity}(P_t) > c \]

In such cases:

\[ \mathrm{Execution}(P_t) = \mathrm{FAIL} \]

This prevents physically inconsistent trajectories from being generated.

For every valid boundary state satisfying the coherence, continuity, flatness, and velocity constraints:

\[ B_t \rightarrow C_t \rightarrow \rho_C \rightarrow \nabla \rho_C \rightarrow R_t \rightarrow x_{t+1} \rightarrow \mathrm{Path}_{\mathrm{global}} \rightarrow \kappa \]

executes to completion with a uniquely determined trajectory.

Therefore:

\[ \mathrm{Execution}(B_t) = \mathrm{PROPAGATE} \]

is guaranteed whenever all invariants remain satisfied.