S = candidate split index score
score_S = abs( (O_right - E_right) - (O_left - E_left) )

# Collatz Strings Experimental,  V2
# Tcl 8.6 or greater required
# Naming convention: all proc and variable names are 12-15
# characters, descriptive, and domain-neutral so the engine
# modules can serve any domain area without modification.
# 
# ----
# Compatible with Tcl/Tk (Tool Command Language / Toolkit) 8.6+
# Written for Windows 11 on ActiveState Tcl.
# Use Pure 7-bit ASCII code, no Unicode characters used anywhere.
# ----
# Need modular code with procs length  of 15 to 25 lines.
# ----
# Written for college IT lab and uses minimal external libraries.
#
# Program deck may contain multiple estimation procs.
# Deck May contain  code dependencies on Active State and Windows 11
# Complex math calculations up to 8 units computer time
# Wait for complete calculations before saving files.
# Proc names and variables names need to be very human readable
# and very explanatory. 
# Avoid variables with single letter names. 
# Whereas single letter names are known to lead
# to many historic errors. 
# Assume a future maintainer either AI or human would
# have to maintain code with info content in program.
# This is Experimenting Draft,
# and not a replacement for TCL Core. 
# This is a hacker's patch, not rigorously derived.
# appears correct solutions for autotests.
# TCL Club 7/01/2026 
#
# ---------------------------------------------------------------
# Logging module
# ---------------------------------------------------------------

proc start_session_log {} {
    set stamp [clock format [clock seconds] -format "%Y-%m-%d_%H-%M-%S"]
    set ::session_log_file [open "session_$stamp.log" w]
    fconfigure $::session_log_file -buffering line
}

proc end_session_log {} {
    if {[info exists ::session_log_file]} {
        close $::session_log_file
        unset ::session_log_file
    }
}

proc log_puts {args} {
    if {[llength $args] == 1} {
        set text [lindex $args 0]
        puts $text
        if {[info exists ::session_log_file]} {
            puts $::session_log_file $text
        }
    } elseif {[llength $args] == 2} {
        lassign $args chan text
        puts $chan $text
        if {[info exists ::session_log_file] && ($chan eq "stdout" || $chan eq "stderr")} {
            puts $::session_log_file $text
        }
    } else {
        error "usage: log_puts ?channel? string"
    }
}

proc show_console_if_available {} {
    if {[llength [info commands console]] > 0} {
        catch {console show}
    }
}

# ---------------------------------------------------------------
# Collatz Integer Trajectory to Gateword conversion
# ---------------------------------------------------------------

proc generate_symbolic_gateword_sequence {starting_integer_value {trajectory_step_limit 500}} {
    set current_integer_value $starting_integer_value
    set symbolic_gateword_string ""
    set trajectory_step_counter 0
    while {$current_integer_value > 1} {
        if {[expr {$current_integer_value % 2}] == 0} {
            append symbolic_gateword_string "E"
            set current_integer_value [expr {$current_integer_value / 2}]
        } else {
            append symbolic_gateword_string "O"
            set current_integer_value [expr {3 * $current_integer_value + 1}]
        }
        incr trajectory_step_counter
        if {$trajectory_step_counter > $trajectory_step_limit} {
            break
        }
    }
    return $symbolic_gateword_string
}

proc compute_symbolic_motif_complexity_measure {symbolic_gateword_string motif_window_length} {
    set gateword_character_length [string length $symbolic_gateword_string]
    if {$gateword_character_length < $motif_window_length} {
        return 1
    }
    set unique_motif_collection [list]
    set last_start_position [expr {$gateword_character_length - $motif_window_length}]
    for {set index_position 0} {$index_position <= $last_start_position} {incr index_position} {
        set window_end [expr {$index_position + $motif_window_length - 1}]
        set current_motif_substring [string range $symbolic_gateword_string $index_position $window_end]
        if {[lsearch $unique_motif_collection $current_motif_substring] == -1} {
            lappend unique_motif_collection $current_motif_substring
        }
    }
    return [llength $unique_motif_collection]
}

# ---------------------------------------------------------------
# Breakpoint analysis
# ---------------------------------------------------------------

proc score_gateword_split_point {gateword_text split_index} {
    set left_part  [string range $gateword_text 0 [expr {$split_index - 1}]]
    set right_part [string range $gateword_text $split_index end]

    set left_odd_matches   [regexp -all -inline {O} $left_part]
    set left_even_matches  [regexp -all -inline {E} $left_part]
    set right_odd_matches  [regexp -all -inline {O} $right_part]
    set right_even_matches [regexp -all -inline {E} $right_part]

    set left_balance_score  [expr {[llength $left_odd_matches] - [llength $left_even_matches]}]
    set right_balance_score [expr {[llength $right_odd_matches] - [llength $right_even_matches]}]

    return [expr {abs($right_balance_score - $left_balance_score)}]
}

proc find_best_gateword_split {gateword_text} {
    set gateword_length [string length $gateword_text]
    set best_split_position 0
    set best_split_score -1

    for {set candidate_position 1} {$candidate_position < $gateword_length - 1} {incr candidate_position} {
        set candidate_score [score_gateword_split_point $gateword_text $candidate_position]
        if {$candidate_score > $best_split_score} {
            set best_split_score $candidate_score
            set best_split_position $candidate_position
        }
    }
    return $best_split_position
}

proc gateword_breakpoint {gateword_text} {
    set gateword_length [string length $gateword_text]
    if {$gateword_length < 3} {
        return [list 0 $gateword_text]
    }

    set best_split_position [find_best_gateword_split $gateword_text]

    set window_start [expr {$best_split_position - 2}]
    if {$window_start < 0} {
        set window_start 0
    }
    set window_end [expr {$best_split_position + 2}]
    if {$window_end >= $gateword_length} {
        set window_end [expr {$gateword_length - 1}]
    }

    return [list $best_split_position [string range $gateword_text $window_start $window_end]]
}

# ---------------------------------------------------------------
# Wiki Table construction (split into smaller focused procs)
# ---------------------------------------------------------------

proc compute_gateword_flow_estimate {gateword_length} {
    if {$gateword_length <= 0} {
        return 0.0
    }
    set midpoint_position [expr {$gateword_length / 2}]
    return [expr {1.0 - (double($midpoint_position) / $gateword_length)}]
}

proc format_truncated_gateword {gateword_text} {
    set truncated_text [string range $gateword_text 0 29]
    if {[string length $gateword_text] > 30} {
        append truncated_text "..."
    }
    return $truncated_text
}

proc build_collatz_table_rows {list_of_starting_numbers max_steps} {
    set table_data [list]
    set row_index 0

    foreach start_n $list_of_starting_numbers {
        incr row_index
        set gateword [generate_symbolic_gateword_sequence $start_n $max_steps]
        set gateword_length [string length $gateword]
        set breakpoint_info [gateword_breakpoint $gateword]
        lassign $breakpoint_info breakpoint_position breakpoint_window

        lappend table_data [list $row_index $start_n $gateword $gateword_length \
            $breakpoint_position $breakpoint_window]
    }
    return $table_data
}

proc print_collatz_table_rows {table_data} {
    log_puts "%| # | Start n | Gateword (truncated) | Length | Est. NLP Breakpoint | BP Window | Prob Flow to Attractor |  Convergence  |Quibbles-Notes |%"
        foreach row $table_data {
        lassign $row row_index start_n gateword gateword_length breakpoint_position breakpoint_window

        set flow_estimate [compute_gateword_flow_estimate $gateword_length]
        set truncated_gateword [format_truncated_gateword $gateword]
        set flow_estimate_text [format "%.3f" $flow_estimate]

        log_puts "&| $row_index | $start_n | $truncated_gateword | $gateword_length | $breakpoint_position | $breakpoint_window | $flow_estimate_text | Converging to 1 |  |&"
    }
}

proc build_collatz_wave_packet_table {list_of_starting_numbers max_steps} {
    set table_data [build_collatz_table_rows $list_of_starting_numbers $max_steps]
    print_collatz_table_rows $table_data
    return $table_data
}

# ---------------------------------------------------------------
# Run it
# ---------------------------------------------------------------

show_console_if_available
start_session_log
build_collatz_wave_packet_table {7 10 20 30 50 27} 30
end_session_log

# end of file 

# References.
# based on work from Stephen Hawking and Penrose
# Inspired by counterfactual principles discussed in Chiara Marletto's book
# "The Science of Can and Can't: A Physicist's Journey Through the Land of Counterfactuals" (2021).
# No text, quotes, or direct examples from the book are used in this code.
# The dummy subroutine implements a generic axiom for educational purposes only.
puts "=============================================================="
puts "Credits"
puts "Inspired by  principles discussed in Dr. Travis S. Taylor's monograph, PHD,"
puts "Symbolic Collapse Grammar and the Convergence of the Collatz Function, June 2025"
puts "Dr. Travis S. Taylor, Phd,  QuantumFrontier, LLC, 
puts "Huntsville, AL, Department of Physics, University of Alabama "
puts "Reference: Maria Violaris, arXiv:2601.08102v1, January 2026"
puts "Reference: https://wiki.tcl-lang.org/page/Snippets+Quantum+Many+Worlds"
puts "Based on ref. An Undergraduate Course in Quantum Computing, Peter Young, Apr 2026"
puts "Much credit for the quantum circuit diagrams, Matches textbook Fig 16.4 etc"
puts "University of California Santa Cruz, CA, arXiv:2604.10396"

+----------------------------------------------------------------------------------+
| TUNNELING TRANSMISSION: THE ACTUAL PHYSICS FORMULA ( for Snippet )              |
|    Standard rectangular potential barrier, E < V0 case                          |
|                                                                                  |
|    kappa = sqrt( 2*m*(V0 - E)*q_e ) / hbar      (decay constant inside barrier) |
|                                                                                  |
|    T = 1 / ( 1 + (V0^2 * sinh^2(kappa*L)) / (4*E*(V0-E)) )                     |
|                                                                                  |
|    where:                                                                       |
|      m  = electron mass     = 9.1093837e-31 kg                                 |
|      hbar = reduced Planck  = 1.0545718e-34 J*s                                |
|      q_e  = eV to Joules    = 1.60217662e-19                                   |
|      L    = barrier width in meters (input as nm, converted)                   |
|      E    = particle energy (eV)                                                |
|      V0   = barrier height (eV)                                                 |
|                                                                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SYMBOLIC GATEWORD ENCODING  (generate_symbolic_gateword_sequence proc)         |
|    Alphabet: { O = odd step (3n+1),  E = even step (n/2) }                    |
|                                                                                  |
|    n=11 trajectory:                                                             |
|    11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
|     O     E     O     E     E     O     E    E    E   O    E   E   E   E         |
|                                                                                  |
|    Gateword string:  O E O E E O E E E O E E E E                                |
|                      "OEOEEOEEEOEEEE"   (14 characters, 14 transformation steps)|
|                                                                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| COMPRESSION RULES  (apply_symbolic_compression_rules_to_gateword proc)         |
|    Repeated regex substitution on the gateword string, up to N passes          |
|                                                                                  |
|    Each pass:                                                                   |
|      1. trim trailing "E" characters                                           |
|      2. regsub all "OEE" -> "O"                                                 |
|      3. regsub all "EE"  -> "E"                                                 |
|      stop early if the string stopped changing                                 |
|                                                                                  |
|    Example pass on "OEOEEOEEEOEEEE":                                            |
|      trim trailing E:        "OEOEEOEEEOEEEE" -> "OEOEEOEEEO"                  |
|      OEE -> O:                "OEOEEOEEEO"     -> "OEOOEO"  (some matches)     |
|      EE -> E:                  (further collapsing of double-E runs)          |
|      ... repeats until stable or empty                                         |
|                                                                                  |
|    Note. Long Collatz strings are very demanding on computer resources and time, |
|    and may approach maximum integer representation and proc recursion limits     |
|    on computer language?                                                         |
|                                                                                  |
|    Caution: this is a STRING-REWRITING heuristic,                                |
|    Not a proven property of the Collatz map.                                     |
|    Not a proven fact about Collatz convergence, ref terms of pure mathematicians  |                
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SYMBOLIC GATEWORD ENCODING  (generate_symbolic_gateword_sequence proc)         |
|    Alphabet: { O = odd step (3n+1),  E = even step (n/2) }                    |
|                                                                                  |
|    n=7 trajectory:                                                              |
|     7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5        |
|     O     E     O     E     O     E     E     O     E     E    E    O           |
|    -> 16 -> 8 -> 4 -> 2 -> 1                                                   |
|       E    E   E   E                                                            |
|                                                                                  |
|    Gateword string:  O E O E O E E O E E E O E E E E                           |
|                      "OEOEOEEOEEEOEEEE"  (16 characters, 16 steps)             |
|                                                                                  |
|    Summary:                                                                     |
|      total steps    = 16                                                        |
|      odd  steps (O) =  5   (positions 1,3,5,8,12 counting from 1)             |
|      even steps (E) = 11                                                        |
|      odd/even ratio = 5/11 = 0.4545...                                         |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SYMBOLIC GATEWORD ENCODING  (generate_symbolic_gateword_sequence proc)         |
|    Alphabet: { O = odd step (3n+1),  E = even step (n/2) }                    |
|                                                                                  |
|    n=10 trajectory:                                                             |
|    10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1                                           |
|     E    O    E    E   E   E                                                    |
|                                                                                  |
|    Gateword string:  E O E E E E                                                |
|                      "EOEEEE"  (6 characters, 6 steps)                         |
|                                                                                  |
|    Summary:                                                                     |
|      total steps    =  6                                                        |
|      odd  steps (O) =  1   (position 2)                                        |
|      even steps (E) =  5                                                        |
|      odd/even ratio = 1/5 = 0.2000                                             |
|                                                                                  |
|    Note: The n=10 trajectory reaches 5, then merges with n=5's path to 1.      |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SYMBOLIC GATEWORD ENCODING  (generate_symbolic_gateword_sequence proc)         |
|    Alphabet: { O = odd step (3n+1),  E = even step (n/2) }                    |
|                                                                                  |
|    n=20 trajectory:                                                             |
|    20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1                                     |
|     E     E    O    E    E   E   E                                              |
|                                                                                  |
|    Gateword string:  E E O E E E E                                              |
|                      "EEOEEE E"  (7 characters, 7 steps)                       |
|                                                                                  |
|    Summary:                                                                     |
|      total steps    =  7                                                        |
|      odd  steps (O) =  1   (position 3)                                        |
|      even steps (E) =  6                                                        |
|      odd/even ratio = 1/6 = 0.1667                                             |
|                                                                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SYMBOLIC GATEWORD ENCODING  (generate_symbolic_gateword_sequence proc)         |
|    Alphabet: { O = odd step (3n+1),  E = even step (n/2) }                    |
|                                                                                  |
|    n=30 trajectory:                                                             |
|    30 -> 15 -> 46 -> 23 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40           |
|     E     O     E     O     E     O     E      O     E     E    E               |
|    -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1                                  |
|       E     E    O    E    E   E   E                                            |
|                                                                                  |
|    Gateword string:  E O E O E O E O E E E E E O E E E E                       |
|                      "EOEOEOEOEEEEEOEEEE"  (18 characters, 18 steps)           |
|                                                                                  |
|    Summary:                                                                     |
|      total steps    = 18                                                        |
|      odd  steps (O) =  5   (positions 2,4,6,8,14)                             |
|      even steps (E) = 13                                                        |
|      odd/even ratio = 5/13 = 0.3846...                                         |
|                                                                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SYMBOLIC GATEWORD ENCODING  (generate_symbolic_gateword_sequence proc)         |
|    Alphabet: { O = odd step (3n+1),  E = even step (n/2) }                    |
|                                                                                  |
|    n=50 trajectory:                                                             |
|    50 -> 25 -> 76 -> 38 -> 19 -> 58 -> 29 -> 88 -> 44 -> 22 -> 11             |
|     E     O     E     E     O     E     O     E     E     E    O                |
|    -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16                 |
|       E     O     E     E     O     E     E     E    O    E                    |
|    -> 8 -> 4 -> 2 -> 1                                                         |
|       E   E   E                                                                 |
|                                                                                  |
|    Gateword string:  E O E E O E O E E E O E O E E O E E E O E E E E          |
|                      "EOEEOEOEEEOEOEEOEEEOEEEE"  (24 characters, 24 steps)     |
|                                                                                  |
|    Summary:                                                                     |
|      total steps    = 24                                                        |
|      odd  steps (O) =  8   (positions 2,5,7,11,13,16,20,24 counting from 1)   |
|      even steps (E) = 16                                                        |
|      odd/even ratio = 8/16 = 0.5000  (exactly half, coincidental)             |
|                                                                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| FIVE EXAMPLES: COMPARATIVE SUMMARY TABLE                                       |
|                                                                                  |
| +------+-------+----------+-------+-------+-----------+----------------------+  |
| |  n   | steps | gateword | odd O | even E| O/E ratio | gateword string      |  |
| +------+-------+----------+-------+-------+-----------+----------------------+  |
| |   7  |  16   | 16 chars |   5   |  11   |   0.455   | OEOEOEEOEEEOEEEE     |  |
| |  10  |   6   |  6 chars |   1   |   5   |   0.200   | EOEEEE               |  |
| |  20  |   7   |  7 chars |   1   |   6   |   0.167   | EEOEEEE              |  |
| |  30  |  18   | 18 chars |   5   |  13   |   0.385   | EOEOEOEOEEEEEOEEEE   |  |
| |  50  |  24   | 24 chars |   8   |  16   |   0.500   | EOEEOEOEEEOEOEEOEEEOEEEE|  |
| +------+-------+----------+-------+-------+-----------+----------------------+  |
| |  11* |  14   | 14 chars |   4   |  10   |   0.400   | OEOEEOEEEOEEEE       |  |
| +------+-------+----------+-------+-------+-----------+----------------------+  |
|    * n=11 included as reference (the original example from Dr. Taylor's paper) |
|                                                                                  |
+----------------------------------------------------------------------------------+

n = 11 gateword
OEOEEOEEEOEEEE

amplitude sketch
                O
            O   E
        O   E   E
    O   E   E   E
E E E E E E E E E E E E E E

Shift-and-add
resembles constructive interference in a simple ASCII model.

text
A:    OEOEEOEEEOEEEE
B:        EOEEEOEEEE
sum:  OEOE[PEAK]OEEEE

+----------------------------------------------------------------------------------+
| SHIFT-AND-ADD: CONSTRUCTIVE INTERFERENCE IN AN ASCII MODEL , Mockup              |
|                                                                                  |
|    A:    O E O E E O E E E O E E E E                                            |
|    B:      E O E E E O E E E O E E E                                            |
|                                                                                  |
|    sum:   O E O E [PEAK] O E E E E                                             |
|                                                                                  |
|    Meaning: the shifted overlap strengthens the center region.                  |
|    Result:   a packet-like shape with a visible peak.                           |
+----------------------------------------------------------------------------------+

resembles destructive interference in a simple ASCII model.

A:    OEOEEOEEEOEEEE
B:      EOEEEOEEEE
zzz:   ..0..0..0..0..

+----------------------------------------------------------------------------------+
| SHIFT-AND-SUBTRACT: DESTRUCTIVE INTERFERENCE IN AN ASCII MODEL, Mockup          |
|                                                                                  |
|    A:    O E O E E O E E E O E E E E                                            |
|    B:      E O E E E O E E E O E E E                                            |
|                                                                                  |
|    sum:   . . 0 . . 0 . . 0 . . 0 . .                                           |
|                                                                                  |
|    Meaning: shifted overlap reduces the visible signal.                         |
|    Result:   cancellation or weak amplitude at matching positions.              |
+----------------------------------------------------------------------------------+

n = 10
EOEEEE

envelope view Mockup
        *
      *   *
    *       *
  *           *
*               *

+----------------------------------------------------------------------------------+
| n = 10                                                                           |
| EOEEEE                                                                           |
|                                                                                  |
| envelope view Mockup                                                                   |
|           *                                                                      |
|         ***                                                                     |
|       *****                                                                     |
|     *******                                                                     |
|   ***********                                                                   |
| ***************                                                                 |
+----------------------------------------------------------------------------------

+----------------------------------------------------------------------------------+
| COLLATZ CONJECTURE: BASIC IDEA FOR ENGINEERS                                   |
|                                                                                  |
|    The Collatz conjecture states that any positive integer eventually reaches  |
|    the number 1 when engineers apply these two simple rules.                   |
|                                                                                  |
|    Rule 1: If the number is even, divide by 2.                                 |
|    Rule 2: If the number is odd, multiply by 3 and add 1.                      |
|                                                                                  |
|    Example starting with 7:                                                    |
|    7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1  |
|                                                                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| QUANTUM TUNNELING: ENGINEERING PERSPECTIVE                                     |
|                                                                                  |
|    Quantum tunneling occurs when a particle passes through a potential         |
|    energy barrier even though classical physics predicts it should not.        |
|                                                                                  |
|    The probability depends on barrier width, height, and particle energy.      |
|                                                                                  |
|    Engineers exploit this phenomenon in tunnel diodes, flash memory,           |
|    scanning tunneling microscopes, and quantum computing hardware.             |
|                                                                                  |
|    The effect demonstrates how quantum mechanics enables behaviors impossible  |
|    under classical rules.                                                      |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| COMPUTATIONAL EXPLORATION OF COLLATZ CONJECTURE                                |
|                                                                                  |
|    Engineers test the Collatz conjecture by writing programs that apply        |
|    the rules repeatedly and track the number of steps to reach 1.              |
|                                                                                  |
|    Large-scale searches have verified the conjecture for all                   |
|    numbers up to extremely high values.                                        |
|                                                                                  |
|    This brute-force approach represents a classic "hack programming" method    |
|    to explore unsolved mathematical problems.                                  |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| GLOSSARY OF KEY TERMS                                                          |
|                                                                                  |
|    Collatz Conjecture                                                          |
|      An unsolved mathematical problem that claims any positive integer         |
|      eventually reaches 1 under simple even-odd rules.                         |
|                                                                                  |
|    Quantum Tunneling                                                           |
|      A quantum mechanical phenomenon where particles pass through energy       |
|      barriers that classical physics considers impossible.                     |
|                                                                                  |
|                                                                                  |
|     Programming                                                            |
|      An exploratory coding approach that uses iterative testing and            |
|      approximations to investigate difficult problems.                         |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| COLLATZ STRINGS: BASIC CONCEPT                                                 |
|                                                                                  |
|    Engineers represent Collatz sequences as strings of operations.             |
|    Common notation uses "E" for divide by 2 and "O" for 3n+1.                  |
|                                                                                  |
|    Example starting from 7:                                                    |
|    7 (O) → 22 (E) → 11 (O) → 34 (E) → 17 (O) → ... → 1                        |
|                                                                                  |
|    The string "OEOEO..." encodes the path to 1.                                |
|    This string representation helps analyze patterns in Collatz behavior.        |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| COLLATZ STRING GENERATION ALGORITHM, UNDER TEST                                |
|                                                                                |
|    Procedure:                                                                  |
|      1. Start with positive integer n                                          |
|      2. While n > 1:                                                           |
|         - If n even: append "E", set n = n/2                                   |
|         - If n odd:  append "O", set n = 3n+1                                  |
|      3. Record the resulting string of E's and O's                             |
|                                                                                  |
|    Example for n=13:                                                           |
|    13(O)→40(E)→20(E)→10(E)→5(O)→16(E)→8(E)→4(E)→2(E)→1                        |
|    String: "OEEE O EEEE"                                                       |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| TCL COLLATZ STRING, DISPLAY MOCKUP                                                      |
|                                                                                  |
|    Input: Starting number                                                      |
|    Output: Collatz string + steps to reach 1                                   |
|                                                                                  |
|    Example:                                                                    |
|      Start = 27                                                                |
|      Steps = 111                                                               |
|      String = OEOOEEOEEOEEEO... (long sequence)                                |
|                                                                                  |
|    +----------------+----------+---------------+                               |
|    | Starting Number| Steps    | Collatz String|                               |
|    +----------------+----------+---------------+                               |
|    | 7              | 16       | OEOEOEOEEEEE  |                               |
|    | 27             | 111      | OEOOEE...     |                               |
|    +----------------+----------+---------------+                               |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| COLLATZ STRINGS VS NUMERIC SEQUENCES                                           |
|                                                                                  |
|    +-------------------+---------------------+--------------------------+      |
|    | Representation    | Advantages          | Disadvantages            |      |
|    +-------------------+---------------------+--------------------------+      |
|    | Numeric Sequence  | Easy to compute     | Long lists for large n   |      |
|    | Collatz String    | Compact pattern     | Requires decoding back   |      |
|    | (E/O)             | analysis            | to numbers               |      |
|    +-------------------+---------------------+--------------------------+      |
|                                                                                  |
|    Strings enable pattern searching across many trajectory values.               |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| APPLICATIONS OF COLLATZ STRINGS IN PROGRAMMING                                 |
|                                                                                  |
|    • Visualizing hailstone sequences as binary like strings                    |
|    • Searching for repeating patterns or cycles                                |
|    • Educational tool for teaching iteration and recursion                     |
|    • Testing programming language performance on long computations             |
|    • Generating unique identifiers from integer inputs                         |
|                                                                                  |
|    Engineering students  implement Collatz string generators                    |
|    to practice algorithm design and string manipulation.                       |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| COLLATZ STRING LENGTH DISTRIBUTION                                             |
|                                                                                  |
|    Some numbers with short strings reach 1 quickly.                        |
|    Some numbers produce very long strings before descending.                   |
|                                                                                  |
|    Engineers study the statistical distribution of string lengths              |
|    to understand the conjecture's behavior.                                    |
|                                                                                  |
|                                                                                 |
+----------------------------------------------------------------------------------+

+----------------------------------------------------------------------------------+
| SUMMARY: COLLATZ STRINGS IN ENGINEERING EDUCATION                              |
|                                                                                  |
|    Collatz strings provide a compact way to represent complex sequences.       |
|    The approach combines number theory with string processing.                 |
|    Students gain practical experience in algorithm development.                |
|                                                                                  |
|    The TCL language makes exploration of Collatz strings accessible         |
|    and educational for engineering students.                                   |
+----------------------------------------------------------------------------------+