# Taciturn simplified formula from math textbooks
T ≈ 2 * pi * sqrt(L / g)

# Switching to human readable variables in Tcl
# Basic factor
set basic_formula [expr {2 * acos(-1.0) * sqrt($length / $gravity)}]

# Series correction (first two terms are usually enough)
set correction_factor_from_theta_series [expr {1 + (1.0/16) * pow($max_angle_rad, 2) \
                          + (11.0/3072) * pow($max_angle_rad, 4)}]

set accurate_period [expr {$basic_formula * $correction_factor_from_theta_series}]

# Taciturn formula from math textbooks
Momentum p = gamma * m * v, where gamma = 1 / sqrt(1 - v^2/c^2)

# Pseudocode style in plain English
# relativistic_momentum = relativistic_factor * rest_mass * velocity
# where relativistic_factor = 1 / square_root(1 - (velocity^2 / speed_of_light^2))

# Switching to human readable variables in Tcl
set rest_mass_kg          9.1093837e-31
set velocity_m_per_s      2.5e8
set speed_of_light_m_per_s 299792458.0

set relativistic_momentum_kg_m_per_s \
    [expr { (1.0 / sqrt(1.0 - pow($velocity_m_per_s, 2) / pow($speed_of_light_m_per_s, 2))) \
          * $rest_mass_kg * $velocity_m_per_s }]

   hyperellipsoid_spacetime.tcl (TCL Club, 02/19/2026)

    hyperellipsoid_spacetime.gif
    GIF89a animated GIF binary data 
    gif_pixel_width  = 320
    gif_pixel_height = 320
    n ~ 1000
   ~16 fps gives smooth-looking rotation.
   total_animation_frame_count  = 60
   definite granularity in the Animated GIF89a file

"Index","Snippet","Description","Link","Notes"
"1","Bloch Sphere","Represents a single qubit's quantum state as a point on a 3D sphere surface.","https://en.wikipedia.org/wiki/Bloch_sphere","Foundational for visualizing quantum bits; directly ties quantum mechanics to spherical geometry."
"1a","","Pure states on the surface; mixed states inside the ball.","","Relates to Pasterski's use of spheres for encoding information at boundaries."
"1b","","Measurements project along Bloch axes.","","Illustrates how geometry constrains quantum probabilities."
"2","Entanglement","Quantum correlations between particles represented as geometric constraints on multi-sphere configurations.","https://en.wikipedia.org/wiki/Quantum_entanglement","Links to ER=EPR conjecture where entanglement creates spacetime geometry."
"2a","","For two qubits: Product states vs. Bell states on CP³ manifold.","","Geometric view shows non-locality as higher-dimensional links."
"2b","","Entanglement entropy as area in holographic duals.","","Bridges QM to gravity via AdS/CFT, similar to celestial holography approaches."
"3","Stereographic Projection","Maps infinite 2D plane to finite sphere, preserving angles (conformal).","https://en.wikipedia.org/wiki/Stereographic_projection","Used in compactifying spacetime to celestial sphere in Pasterski's work."
"3a","","Adds point at infinity.","","Essential for handling asymptotic behaviors in quantum gravity."
"3b","","Conformal invariance preserved.","","Allows mapping 4D bulk to 2D boundary without losing symmetry."
"4","Hypersphere","3-sphere (S³) in 4D space for visualizing higher-dimensional objects / compactified spacetime.","https://en.wikipedia.org/wiki/N-sphere","Core to Pasterski's celestial holography: 4D spacetime compactified to S³ at infinity."
"4a","","Projections from 4D to 3D/2D.","https://wiki.tcl-lang.org/page/Snippets+Concepts+Hypersphere+Simulation","Simulates rotations and projections, mimicking gravitational scattering mappings."
"4b","","Encodes asymptotic symmetries (BMS group).","","Supertranslations and superrotations act on the hypersphere boundary."
"5","Grover's Algorithm","Quantum search via amplitude rotation in high-dimensional state space.","https://en.wikipedia.org/wiki/Grover%27s_algorithm","Demonstrates quantum evolution as geometric motion on hypersurfaces."
"5a","","Oracle and diffusion operators as rotations.","","Parallels time evolution in quantum gravity as paths on geometric manifolds."
"5b","","Optimal search in O(√N) steps.","","Geometric intuition aids in understanding quantum advantages over classical."
"6","Hyper-ellipsoid","Proposed alternative shape for mapping gravity, deformed from hypersphere due to anisotropy or mass distribution.","https://en.wikipedia.org/wiki/Ellipsoid#Generalized_ellipsoids","Standard celestial holography / Pasterski work uses round S³ hypersphere for maximal symmetry and conformal properties. Hyper-ellipsoid appears in other contexts (e.g. deformed spacetimes, anisotropic cosmologies, or effective descriptions), but is not the primary shape in celestial sphere approaches."

#!/usr/bin/env wish
################################################################################
# HYPERSPHERE 4D PROJECTION VISUALIZATION
# Tcl/Tk 8.6+ Implementation
# 
# PURPOSE: Visualize rotation of 4-dimensional hypersphere projected to 3D/2D
# ALGORITHM: Rotate hypersphere in 4D space, project to 3D, then render to 2D
# DISPLAY: Animated canvas showing depth through point size and color
#
# CODING STANDARD: JPL Institutional Coding Standard for C/C++ adapted to Tcl
# - No single-letter variable names (except loop indices in limited scope)
# - Defensive programming with parameter validation
# - Extensive documentation for future maintainers
# - Small, testable modules with single responsibility
# - Explicit error handling
################################################################################
# may have to check strict ASCII for Playground V9
# Compatible with Tcl/Tk 8.6+
# TCL source code follows
# Written for Windows 11 on ActiveState Tcl
# Working on TCL Playground V9, strict ASCII only
# Optimized for collegiate IT lab environments
# Working under TCL version 8.6
# Complex math calculations up to 8 units computer time
# Wait for complete calculations before saving files.
# TCL club, 02/15/2026
# This is a hacker's patch, not rigorously derived.
# appears correct solutions for autotests
# pure ASCII code - no Unicode characters used anywhere
# Approaching =>>> computer time limit on TCL laptop setup. 
################################################################################
# MODULE: MATHEMATICAL CONSTANTS
# PURPOSE: Define mathematical constants used throughout program
################################################################################
namespace eval ::constants {
    variable PI 3.14159265358979323846
    variable TWO_PI [expr {2.0 * $PI}]
    variable HALF_PI [expr {$PI / 2.0}]
}

################################################################################
# MODULE: CONFIGURATION PARAMETERS
# PURPOSE: Centralize all tunable parameters for easy maintenance
################################################################################
namespace eval ::config {
    # Display dimensions
    variable canvas_width 500
    variable canvas_height 500
    
    # Animation parameters
    variable animation_frame_delay_ms 50
    variable total_animation_frames 60
    
    # Hypersphere geometry
    variable number_of_meridians 16
    variable number_of_parallels 12
    
    # Projection parameters
    variable projection_distance 2.5
    variable screen_scale_factor 150
    
    # Visual appearance
    variable point_base_size 3
    variable point_depth_multiplier 3
    variable brightness_base 128
    variable brightness_range 127
    
    # Colors
    variable background_color "black"
    variable title_color "white"
    variable subtitle_color "yellow"
    variable info_color "cyan"
    variable point_color_base 255  ;# Blue channel always max
}

################################################################################
# MODULE: HYPERSPHERE MATHEMATICS
# PURPOSE: Core mathematical transformations for 4D hypersphere manipulation
################################################################################
namespace eval ::hypersphere {
    
    ############################################################################
    # FUNCTION: rotate_and_project_4d_to_3d
    # PURPOSE: Rotate point in 4D space and project to 3D using perspective
    # INPUTS:
    #   coord_x - X coordinate in 4D space
    #   coord_y - Y coordinate in 4D space  
    #   coord_z - Z coordinate in 4D space
    #   coord_w - W coordinate (4th dimension)
    #   rotation_angle - Angle of rotation in 4D (radians)
    # OUTPUTS: List of {x3d y3d z3d} coordinates in 3D space
    # ALGORITHM:
    #   1. Rotate in XW plane of 4D space
    #   2. Apply perspective projection based on W coordinate
    #   3. Scale remaining coordinates proportionally
    ############################################################################
    proc rotate_and_project_4d_to_3d {coord_x coord_y coord_z coord_w rotation_angle} {
        # Defensive check: ensure angle is numeric
        if {![string is double -strict $rotation_angle]} {
            error "rotation_angle must be numeric, got: $rotation_angle"
        }
        
        # Calculate rotation trigonometry
        set cosine_angle [expr {cos($rotation_angle)}]
        set sine_angle [expr {sin($rotation_angle)}]
        
        # Rotate in 4D space (XW plane rotation)
        set rotated_x [expr {$coord_x * $cosine_angle - $coord_w * $sine_angle}]
        set rotated_w [expr {$coord_x * $sine_angle + $coord_w * $cosine_angle}]
        
        # Perspective projection from 4D to 3D
        # Points further in W dimension appear smaller (perspective)
        set distance $::config::projection_distance
        set perspective_scale [expr {$distance / ($distance + $rotated_w)}]
        
        # Apply perspective scaling to all 3D coordinates
        set projected_x [expr {$rotated_x * $perspective_scale}]
        set projected_y [expr {$coord_y * $perspective_scale}]
        set projected_z [expr {$coord_z * $perspective_scale}]
        
        return [list $projected_x $projected_y $projected_z]
    }
    
    ############################################################################
    # FUNCTION: generate_hypersphere_points
    # PURPOSE: Generate lattice of points on 4D hypersphere surface
    # INPUTS:
    #   rotation_angle - Current rotation angle for animation
    # OUTPUTS: List of point records, each containing:
    #   {screen_x screen_y point_size brightness_value}
    # ALGORITHM:
    #   Uses spherical coordinates extended to 4D:
    #   - Theta: longitude angle (0 to 2*PI)
    #   - Phi: latitude angle (0 to PI)  
    #   - W_angle: 4th dimension angle
    ############################################################################
    proc generate_hypersphere_points {rotation_angle} {
        set point_list {}
        
        set num_meridians $::config::number_of_meridians
        set num_parallels $::config::number_of_parallels
        
        # Iterate through meridians (longitude lines)
        for {set meridian_index 0} {$meridian_index < $num_meridians} {incr meridian_index} {
            set theta_angle [expr {$meridian_index * $::constants::TWO_PI / $num_meridians}]
            
            # Iterate through parallels (latitude lines)
            for {set parallel_index 0} {$parallel_index < $num_parallels} {incr parallel_index} {
                set phi_angle [expr {$parallel_index * $::constants::PI / $num_parallels}]
                
                # 4D hypersphere parametric equations
                # Standard sphere equations extended with W coordinate
                set coord_x [expr {cos($theta_angle) * sin($phi_angle) * cos($rotation_angle)}]
                set coord_y [expr {sin($theta_angle) * sin($phi_angle) * cos($rotation_angle)}]
                set coord_z [expr {cos($phi_angle) * cos($rotation_angle)}]
                set coord_w [expr {sin($rotation_angle)}]
                
                # Project from 4D to 3D
                lassign [rotate_and_project_4d_to_3d \
                    $coord_x $coord_y $coord_z $coord_w $rotation_angle] \
                    projected_x projected_y projected_z
                
                # Convert 3D coordinates to 2D screen coordinates
                set screen_x [expr {$::config::canvas_width / 2 + \
                    $projected_x * $::config::screen_scale_factor}]
                set screen_y [expr {$::config::canvas_height / 2 - \
                    $projected_y * $::config::screen_scale_factor}]
                
                # Point size indicates depth (Z coordinate)
                set point_size [expr {$::config::point_base_size + \
                    $projected_z * $::config::point_depth_multiplier}]
                
                # Brightness indicates depth (closer = brighter)
                set brightness_value [expr {int($::config::brightness_base + \
                    $projected_z * $::config::brightness_range)}]
                
                # Clamp brightness to valid range
                if {$brightness_value < 0} {set brightness_value 0}
                if {$brightness_value > 255} {set brightness_value 255}
                
                # Store point data
                lappend point_list [list $screen_x $screen_y $point_size $brightness_value]
            }
        }
        
        return $point_list
    }
}

################################################################################
# MODULE: GRAPHICS RENDERING
# PURPOSE: Handle all canvas drawing operations
################################################################################
namespace eval ::graphics {
    
    ############################################################################
    # FUNCTION: rgb_to_hex_color
    # PURPOSE: Convert RGB values to Tcl/Tk hex color format
    # INPUTS: red_value green_value blue_value (0-255 each)
    # OUTPUTS: Hex color string like "#RRGGBB"
    ############################################################################
    proc rgb_to_hex_color {red_value green_value blue_value} {
        return [format "#%02X%02X%02X" $red_value $green_value $blue_value]
    }
    
    ############################################################################
    # FUNCTION: sort_points_by_depth
    # PURPOSE: Sort points for painter's algorithm (back-to-front rendering)
    # INPUTS: point_list - List of point records
    # OUTPUTS: Sorted list with furthest points first
    # NOTE: Sorts by point_size as proxy for depth
    ############################################################################
    proc sort_points_by_depth {point_list} {
        return [lsort -real -index 2 $point_list]
    }
    
    ############################################################################
    # FUNCTION: draw_hypersphere_frame
    # PURPOSE: Render single frame of hypersphere on canvas
    # INPUTS:
    #   canvas_widget - Tk canvas widget path
    #   rotation_angle - Current rotation angle
    ############################################################################
    proc draw_hypersphere_frame {canvas_widget rotation_angle} {
        # Clear previous frame
        $canvas_widget delete all
        
        # Draw background
        $canvas_widget configure -background $::config::background_color
        
        # Draw title text
        $canvas_widget create text \
            [expr {$::config::canvas_width / 2}] 20 \
            -text "4D Hypersphere Projection" \
            -fill $::config::title_color \
            -font {Arial 20 bold} \
            -anchor n
        
        # Draw subtitle (Pasterski reference)
        $canvas_widget create text \
            [expr {$::config::canvas_width / 2}] \
            [expr {$::config::canvas_height - 50}] \
            -text "What Sabrina Pasterski is Doing (Gravity Mapping on Hypersphere?)" \
            -fill $::config::subtitle_color \
            -font {Arial 12} \
            -anchor center
        
        # Draw info line
        $canvas_widget create text \
            [expr {$::config::canvas_width / 2}] \
            [expr {$::config::canvas_height - 30}] \
            -text "Rotating in 4th dimension - size shows depth" \
            -fill $::config::info_color \
            -font {Arial 12} \
            -anchor center
        
        # Generate and sort points
        set point_list [::hypersphere::generate_hypersphere_points $rotation_angle]
        set sorted_points [sort_points_by_depth $point_list]
        
        # Draw each point
        foreach point_record $sorted_points {
            lassign $point_record screen_x screen_y point_size brightness_value
            
            # Only draw points with positive size
            if {$point_size > 0.5} {
                # Calculate point color (blue gradient based on depth)
                set point_color [rgb_to_hex_color \
                    $brightness_value \
                    $brightness_value \
                    $::config::point_color_base]
                
                # Calculate bounding box for oval
                set bbox_x1 [expr {$screen_x - $point_size}]
                set bbox_y1 [expr {$screen_y - $point_size}]
                set bbox_x2 [expr {$screen_x + $point_size}]
                set bbox_y2 [expr {$screen_y + $point_size}]
                
                # Draw point as filled oval
                $canvas_widget create oval \
                    $bbox_x1 $bbox_y1 $bbox_x2 $bbox_y2 \
                    -fill $point_color \
                    -outline $point_color
            }
        }
    }
}

################################################################################
# MODULE: ANIMATION CONTROL
# PURPOSE: Manage animation loop and frame timing
################################################################################
namespace eval ::animation {
    variable current_frame_number 0
    variable animation_running 0
    variable animation_id ""
    
    ############################################################################
    # FUNCTION: calculate_rotation_angle
    # PURPOSE: Calculate rotation angle for given frame number
    # INPUTS: frame_number - Current frame in animation sequence
    # OUTPUTS: Rotation angle in radians (0 to 2*PI)
    ############################################################################
    proc calculate_rotation_angle {frame_number} {
        set total_frames $::config::total_animation_frames
        return [expr {$frame_number * $::constants::TWO_PI / $total_frames}]
    }
    
    ############################################################################
    # FUNCTION: animate_next_frame
    # PURPOSE: Draw next frame and schedule subsequent frame
    # INPUTS: canvas_widget - Tk canvas widget path
    ############################################################################
    proc animate_next_frame {canvas_widget} {
        variable current_frame_number
        variable animation_running
        variable animation_id
        
        if {!$animation_running} {
            return
        }
        
        # Calculate rotation angle for this frame
        set rotation_angle [calculate_rotation_angle $current_frame_number]
        
        # Draw frame
        ::graphics::draw_hypersphere_frame $canvas_widget $rotation_angle
        
        # Advance to next frame (loop back to 0 after last frame)
        incr current_frame_number
        if {$current_frame_number >= $::config::total_animation_frames} {
            set current_frame_number 0
        }
        
        # Schedule next frame
        set animation_id [after $::config::animation_frame_delay_ms \
            [list ::animation::animate_next_frame $canvas_widget]]
    }
    
    ############################################################################
    # FUNCTION: start_animation
    # PURPOSE: Begin animation loop
    ############################################################################
    proc start_animation {canvas_widget} {
        variable animation_running
        variable current_frame_number
        
        set animation_running 1
        set current_frame_number 0
        animate_next_frame $canvas_widget
    }
    
    ############################################################################
    # FUNCTION: stop_animation
    # PURPOSE: Halt animation loop
    ############################################################################
    proc stop_animation {} {
        variable animation_running
        variable animation_id
        
        set animation_running 0
        if {$animation_id ne ""} {
            after cancel $animation_id
            set animation_id ""
        }
    }
}

################################################################################
# MODULE: USER INTERFACE
# PURPOSE: Create and configure GUI elements
################################################################################
namespace eval ::gui {
    
    ############################################################################
    # FUNCTION: create_main_window
    # PURPOSE: Build complete user interface
    # OUTPUTS: Canvas widget path for animation
    ############################################################################
    proc create_main_window {} {
        # Configure main window
        wm title . "4D Hypersphere Projection - Tcl/Tk 8.6"
        wm geometry . "${::config::canvas_width}x[expr {$::config::canvas_height + 60}]"
        wm resizable . 0 0
        
        # Create canvas for drawing
        set canvas_widget [canvas .canvas \
            -width $::config::canvas_width \
            -height $::config::canvas_height \
            -background $::config::background_color \
            -highlightthickness 0]
        
        pack $canvas_widget -side top -fill both -expand 1
        
        # Create control frame
        set control_frame [frame .controls -relief raised -borderwidth 2]
        pack $control_frame -side bottom -fill x
        
        # Create control buttons
        button $control_frame.start \
            -text "Start Animation" \
            -command [list ::animation::start_animation $canvas_widget]
        
        button $control_frame.stop \
            -text "Stop Animation" \
            -command ::animation::stop_animation
        
        button $control_frame.quit \
            -text "Quit" \
            -command exit
        
        pack $control_frame.start $control_frame.stop $control_frame.quit \
            -side left -padx 5 -pady 5
        
        return $canvas_widget
    }
}

################################################################################
# MODULE: MAIN PROGRAM
# PURPOSE: Entry point and initialization
################################################################################

############################################################################
# FUNCTION: main
# PURPOSE: Initialize application and start event loop
############################################################################
proc main {} {
    # Create user interface
    set canvas_widget [::gui::create_main_window]
    
    # Draw initial frame (frame 0)
    ::graphics::draw_hypersphere_frame $canvas_widget 0.0
    
    # Auto-start animation
    after 500 [list ::animation::start_animation $canvas_widget]
    
    # Note: Tk event loop starts automatically with wish interpreter
}

################################################################################
# DOCUMENTATION SECTION: SNIPPETS CONCEPTS EFFECTS
################################################################################
# CONCEPT: 4D Hypersphere
#   A hypersphere is the 4-dimensional analogue of a sphere
#   Just as a sphere is the set of all points equidistant from a center in 3D,
#   a hypersphere is the set of all points equidistant from a center in 4D
#
# EFFECT: Rotation Visualization  
#   By rotating the hypersphere in 4D space and projecting to 3D, then 2D,
#   we can visualize aspects of 4-dimensional geometry
#   Points appear to expand/contract and change brightness based on their
#   position in the 4th dimension
#
# EFFECT: Perspective Projection
#   Points further away in the W dimension (4th dimension) appear smaller
#   This mimics how 3D perspective works when projecting to 2D screens
#
# EFFECT: Depth Cueing
#   Point size and brightness encode depth information
#   Larger, brighter points are "closer" to the viewer in 3D projection
#   This helps the human visual system interpret the 3D structure
################################################################################

################################################################################
# DOCUMENTATION SECTION: SNIPPETS CONCEPTS HYPERSPHERE SIMULATION  
################################################################################
# SIMULATION ALGORITHM:
#   1. Generate points on 4D hypersphere using parametric equations
#   2. Rotate points in 4D space (XW plane rotation)
#   3. Project from 4D to 3D using perspective division
#   4. Map 3D coordinates to 2D screen coordinates
#   5. Render points with size/color encoding depth
#   6. Animate by incrementing rotation angle each frame
#
# MATHEMATICAL FOUNDATION:
#   4D Hypersphere Equation: x^2 + y^2 + z^2 + w^2 = r^2
#   Parametric Form (unit sphere):
#     x = cos(theta) * sin(phi) * cos(psi)
#     y = sin(theta) * sin(phi) * cos(psi)  
#     z = cos(phi) * cos(psi)
#     w = sin(psi)
#   Where:
#     theta = longitude (0 to 2*PI)
#     phi = latitude (0 to PI)
#     psi = 4th dimension angle
#
# PROJECTION MATHEMATICS:
#   4D Rotation (XW plane):
#     x' = x * cos(angle) - w * sin(angle)
#     w' = x * sin(angle) + w * cos(angle)
#   
#   Perspective Projection (4D to 3D):
#     scale = distance / (distance + w')
#     x3d = x' * scale
#     y3d = y * scale
#     z3d = z * scale
#
# REFERENCE: Sabrina Pasterski
#   Theoretical physicist working on celestial holography and quantum gravity
#   Hypersphere geometries relevant to asymptotic symmetry analysis
#   Connection to conformal field theory on celestial sphere
################################################################################

# Execute main program
main

# End of hypersphere_4d_projection.tcl

# Relativistic momentum calculation V2
# may have to check strict ASCII for Playground V9
# Compatible with Tcl/Tk 8.6+
# TCL source code follows
# Written for Windows 11 on ActiveState Tcl
# Working on TCL Playground V9, strict ASCII only
# Optimized for collegiate IT lab environments
# Working under TCL version 8.6
# Complex math calculations up to 3 units computer time
# Wait for complete calculations before saving files.
# TCL club, 02/24/2026
# This is a hacker's patch, not rigorously derived.
# appears correct solutions 
# pure ASCII code - no Unicode characters used anywhere
# May Approach =>>> computer time limit on TCL laptop setup. 

# Taciturn formula from math textbooks
# Momentum p = gamma * m * v, where gamma = 1 / sqrt(1 - v^2/c^2)

# Pseudocode style in plain English
# relativistic_momentum = relativistic_factor * rest_mass * velocity
# where relativistic_factor = 1 / square_root(1 - (velocity^2 / speed_of_light^2))
# Relativistic momentum calculation

# Switching to human readable variables in Tcl
# All speeds must be in the same units (e.g. m/s); speed_of_light ≈ 299792458 m/s

set rest_mass_kg          9.1093837e-31     ;# example: electron rest mass
set velocity_m_per_s      2.5e8              ;# example: 0.833 c
set speed_of_light_m_per_s 299792458.0

# Step 1: Compute the relativistic factor (gamma)
set velocity_squared_over_c_squared [expr {pow($velocity_m_per_s, 2) / pow($speed_of_light_m_per_s, 2)}]
set relativistic_factor [expr {1.0 / sqrt(1.0 - $velocity_squared_over_c_squared)}]

# Step 2: Compute relativistic momentum
set relativistic_momentum_kg_m_per_s [expr {$relativistic_factor * $rest_mass_kg * $velocity_m_per_s}]

puts "Rest mass:                  [format %.3e $rest_mass_kg] kg"
puts "Velocity:                   [format %.3e $velocity_m_per_s] m/s"
puts "Relativistic factor (gamma): [format %.4f $relativistic_factor]"
puts "Relativistic momentum:      [format %.4e $relativistic_momentum_kg_m_per_s] kg m/s"

Rest mass:                  9.109e-31 kg
Velocity:                   2.500e+08 m/s
Relativistic factor (gamma): 1.8119
Relativistic momentum:      4.1264e-22 kg m/s
(bin) 1 % 

(tcl) 8 % # Switching to human readable variables in Tcl
(tcl) 9 % # All speeds must be in the same units (e.g. m/s); speed_of_light ≈ 299792458 m/s
(tcl) 10 % 
(tcl) 10 % set rest_mass_kg          9.1093837e-31     ;# example: electron rest mass
9.1093837e-31
(tcl) 11 % set velocity_m_per_s      2.5e8              ;# example: 0.833 c
2.5e8
(tcl) 12 % set speed_of_light_m_per_s 299792458.0
299792458.0
(tcl) 13 % 
(tcl) 13 % # Step 1: Compute the relativistic factor (gamma)
(tcl) 14 % set velocity_squared_over_c_squared [expr {pow($velocity_m_per_s, 2) / pow($speed_of_light_m_per_s, 2)}]
0.6954062850335115
(tcl) 15 % set relativistic_factor [expr {1.0 / sqrt(1.0 - $velocity_squared_over_c_squared)}]
1.8119221375144938
(tcl) 16 % 
(tcl) 16 % # Step 2: Compute relativistic momentum
(tcl) 17 % set relativistic_momentum_kg_m_per_s [expr {$relativistic_factor * $rest_mass_kg * $velocity_m_per_s}]
4.1263734962859223e-22
(tcl) 18 % 
(tcl) 18 % puts "Rest mass:                  [format %.3e $rest_mass_kg] kg"
Rest mass:                  9.109e-31 kg
(tcl) 19 % puts "Velocity:                   [format %.3e $velocity_m_per_s] m/s"
Velocity:                   2.500e+08 m/s
(tcl) 20 % puts "Relativistic factor (gamma): [format %.4f $relativistic_factor]"
Relativistic factor (gamma): 1.8119
(tcl) 21 % puts "Relativistic momentum:      [format %.4e $relativistic_momentum_kg_m_per_s] kg m/s"
Relativistic momentum:      4.1264e-22 kg m/s
(tcl) 22 % 
 

gravity_sim_pseudocode.csv:

text
index_number,Python style pseudocode,Tcl style equivalent,pseudocode_comment,quibble_notes
1,"# Gravitational wave perturbation (+ polarization)","# Gravitational wave perturbation (+ polarization)","Header comment for GW + polarization case","Purely descriptive, no executable effect"
2,"t = np.linspace(0, 10, 1000)","set tL [list]; set t0 0.0; set t1 10.0; set Nt 1000","Declare time array from 0 to 10 with Nt samples","Tcl uses explicit loop or helper proc to fill tL"
3,"# time array","# time array","Comment that t is the simulation time grid","Could store dt instead of full list for some uses"
4,"h_plus = 1e-4 * np.sin(2 * np.pi * 0.5 * t)","set hL [list]","Define GW strain h_plus as small sinusoid","Need later loop to append per-time h values"
5,"# strain amplitude","# strain amplitude","Note that h_plus is dimensionless strain","Amplitude 1e-4 is exaggerated vs. realistic LIGO"
6,"for each t in time array:","for {set i 0} {$i < $Nt} {incr i} {","Loop over time grid indices","Pseudocode loop; body filled in next rows"
7,"    h = 1e-4 * sin(2*pi*0.5*t)","  set ti [expr {$t0 + double($i)*($t1-$t0)/($Nt-1)}]","Compute current time ti for index i","Linear spacing between t0 and t1"
8,"    h_plus[i] = h","  set h [expr {1.0e-4 * sin(2.0*acos(-1)*0.5*$ti)}]","Evaluate sinusoidal strain at ti","acos(-1) used as pi in Tcl expr"
9,"    # store h_plus[i]","  lappend tL $ti; lappend hL $h","Append time and strain to lists","Mimics NumPy vectorization with manual list build"
10,"end for","}","End time loop over GW strain","Closing brace only, no extra action"
11,"a_t = 1.0 + h_plus","# inside loop: set a_t [expr {1.0 + $h}]","x1 semi-axis stretches with the wave","In Tcl, a_t is instantaneous value, not full array"
12,"b_t = 1.0 - h_plus","# inside loop: set b_t [expr {1.0 - $h}]","x2 semi-axis compresses with opposite sign","Same instantaneous semantics as a_t"
13,"# x3, x4 axes unaffected","# x3, x4 axes unaffected by + polarization","Comment: other axes unchanged for + mode","Cross and higher modes omitted here"
14,"a_t_list.append(a_t)","  lappend aL $a_t","Store time series of a_t semi-axis","Requires aL initialized before loop"
15,"b_t_list.append(b_t)","  lappend bL $b_t","Store time series of b_t semi-axis","Requires bL initialized before loop"
16,"# This is LIGO mechanism","# This is LIGO mechanism","Comment: differential arm stretching is detector signal","Extremely simplified description of interferometer readout"
17,"# metric oscillates, axes stretch/contract","# metric oscillates, axes stretch/contract","Summarizes spacetime interpretation of GW strain","Leaves out tensor notation and full GR details"

text
PSEUDOCODE:
START
  GET user input
  IF user input is valid
    CALL process user input
    DISPLAY processed user input
  ELSE
    DISPLAY invalid input message
  END IF
END

"1","import numpy as np","package require Tcl","Initialize environment and imports.","Standard setup; Tcl uses lists instead of numpy arrays."
"2","t = np.linspace(0,10,1000)","set tL {};set t0 0.0;set t1 10.0;set Nt 1000;for {set i 0} {$i<$Nt} {incr i} {set ti [expr {$t0+double($i)($t1-$t0)/($Nt-1)}];lappend tL $ti}","Create evenly spaced time array over [0,10] with 1000 points.","Tcl has no built-in linspace; explicit loop is standard."
"3","h_plus = 1e-4np.sin(2np.pi0.5t)","set pi [expr {acos(-1.0)}];set hL {};foreach ti $tL {lappend hL [expr {1e-4sin(2*$pi0.5$ti)}]}","Define + polarization strain h₊(t): amplitude 10⁻⁴, frequency 0.5 Hz.","Toy frequency; real GW events often 10–1000 Hz range."
"4","a_t = 1.0 + h_plus","set aL {};foreach h $hL {lappend aL [expr {1.0+$h}]}","Perturb semi-axis a: stretch in + polarization direction.","Quadrupole pattern: one axis expands when orthogonal compresses."
"5","b_t = 1.0 - h_plus","set bL {};foreach h $hL {lappend bL [expr {1.0-$h}]}","Perturb semi-axis b: compress orthogonally.","Opposite sign reflects + polarization signature."
"6","","set c_t 1.0;set d_t 1.0","x3 and x4 axes remain unperturbed in + polarization.","Cross (+) polarization affects only two orthogonal axes."
"7","# Example: differential strain da = a_t - 1.0","# set daL {};foreach a $aL {lappend daL [expr {$a-1.0}]}","Track observable strain difference da(t).","Mimics LIGO arm-length change detection."
"8","# Optional: plot or export (t, h_plus, da)","# Use canvas or write to file for visualization","Visualize perturbation over time.","Useful for debugging or educational demo in TCL."
"9","# Project perturbed 4D ellipsoid to 2D celestial boundary","# Normalize (a_t,b_t,1.0,1.0) → spherical angles θ,φ","Map GW-distorted hyper-ellipsoid to celestial sphere.","Links to Pasterski celestial holography: GW memory imprints boundary."
"10","# Count/measure “pores” or features on projected 2D rind","# After projection, tally distortion nodes or curvature gradients","Quantify boundary information change due to GW.","Tangerine analog: more/stretched pores signal anomaly approach."

two separate bugs fixed on 1-2 iterations, but solution looks rough here:

The oracle is flipping the wrong component.

The diffusion step is mathematically fine, but with the wrong oracle it cancels out and leaves the state uniform.