#!/bin/sh
# ╔═══════════════════════════════════════════════════════════════════════════╗
# ║                                                                           ║
# ║                 THE LOIC CYCLE DERIVATION ORACLE                          ║
# ║                                                                           ║
# ║      Computing the Fundamental Cycle of the Loic Calendar from            ║
# ║      the Hubble Constant, the Fine-Structure Constant, and Pi            ║
# ║                                                                           ║
# ║              Cycle = H₀⁻¹ × α^π / π                                     ║
# ║                                                                           ║
# ╚═══════════════════════════════════════════════════════════════════════════╝
#
# ═══════════════════════════════════════════════════════════════════════════
# ON THE NECESSARY IMPURITY OF TRANSCENDENTAL COMPUTATION
# ═══════════════════════════════════════════════════════════════════════════
#
# The companion script, derive_constants.sh, uses only integer arithmetic.
# This script cannot. The formula H₀⁻¹ × α^π / π requires computing
# α raised to an irrational power, which demands the transcendental
# functions e(x) and l(x) — exponential and natural logarithm.
#
# We use bc -l, which provides these functions via Taylor series expansion
# at arbitrary decimal precision. This is NOT IEEE 754 floating-point.
# bc does not use binary representation, does not silently truncate to
# 64 bits, does not harbor negative zero, and does not pretend that
# 0.1 + 0.2 equals 0.3 when it does not. bc computes in decimal, at
# whatever precision we demand, truncating only where we instruct it.
#
# The scale is set to 200 decimal places — far beyond the 155-digit
# limit of I512 and the ~46-digit physical significance of Planck time.
# The excess precision ensures that accumulated rounding in intermediate
# Taylor series terms does not contaminate the final result.
#
# When the computation is complete, the result is truncated to an integer.
# The irrational numbers — π and the transcendental α^π — are consumed
# during derivation and do not exist at runtime. Only the integer remains.
# This is not a compromise. It is the same treatment given to every other
# physical constant in the Loic Calendar: rational approximation with
# error below physical significance.
#
# ═══════════════════════════════════════════════════════════════════════════
# USAGE
# ═══════════════════════════════════════════════════════════════════════════
#
#   ./derive_cycle.sh              # Full human-readable derivation
#   ./derive_cycle.sh --constant   # Emit only PLANCK_PER_CYCLE
#   ./derive_cycle.sh --json       # Machine-readable JSON output
#   ./derive_cycle.sh --help       # Illuminate the path
#
# ═══════════════════════════════════════════════════════════════════════════

set -e

# ═══════════════════════════════════════════════════════════════════════════════
#
#                           WORKING PRECISION
#
#   200 decimal places. The universe requires ~46. We provide ~200.
#   The excess is not waste; it is insurance against the accumulation
#   of truncation error across dozens of Taylor series terms.
#
# ═══════════════════════════════════════════════════════════════════════════════

PRECISION=200

# ═══════════════════════════════════════════════════════════════════════════════
#
#                         THE FUNDAMENTAL CONSTANTS
#
#   Three axioms. Three constants. Three windows into the structure of
#   the universe, through which the Loic cycle is glimpsed.
#
# ═══════════════════════════════════════════════════════════════════════════════

# ─────────────────────────────────────────────────────────────────────────────
# THE HUBBLE CONSTANT
# ─────────────────────────────────────────────────────────────────────────────
#
# H₀ = 67.4 km/s/Mpc
#
# The rate of cosmic expansion as measured by the Planck satellite (2018)
# through observation of the Cosmic Microwave Background — the oldest
# light in the universe, emitted 380,000 years after Creation when the
# cosmos first became transparent.
#
# A competing measurement exists: 73.0 ± 1.0 km/s/Mpc, derived from the
# local distance ladder (Cepheid variables and Type Ia supernovae). The
# discrepancy — known as the Hubble tension — is a 5σ disagreement that
# represents arguably the greatest crisis in modern cosmology.
#
# We choose 67.4. The CMB measurement peers deeper into the past than any
# other observation in the history of science. This is a calendar that
# begins at the beginning. We use the measurement that looked closest
# to Creation.
#
# The tension is documented. The choice is deliberate. If future physics
# resolves the disagreement, the constant may be revised — but the formula
# will endure.

H0_KM_S_MPC="67.4"

# 1 Megaparsec in meters (IAU 2012 exact definition)
#
# The parsec is defined as 648000/π AU, where 1 AU = 149,597,870,700 m
# exactly (IAU 2012 Resolution B2). The megaparsec is 10⁶ parsecs.
#
# Since the parsec definition involves π, the megaparsec in meters is
# itself irrational. We use a rational approximation to 17 significant
# figures — vastly exceeding the 3 significant figures of H₀ itself.
# The bottleneck of precision is the Hubble constant, not the unit
# conversion.

MPC_METERS="30856775814913673000000"
# 3.0856775814913673 × 10²² meters. The 'e' notation is eschewed;
# bc does not speak exponential.

# ─────────────────────────────────────────────────────────────────────────────
# THE FINE-STRUCTURE CONSTANT
# ─────────────────────────────────────────────────────────────────────────────
#
# α = 7.2973525643 × 10⁻³
#
# The dimensionless coupling constant of quantum electrodynamics.
# It governs the strength of the electromagnetic interaction — how
# strongly charged particles talk to each other. Every photon emitted,
# every electron captured, every signal transmitted through copper wire
# is modulated by this number.
#
# The Whisper — "LO", transmitted from UCLA to Stanford over ARPANET —
# was an electromagnetic event. The characters were encoded as electrical
# signals, propagated through coaxial cable, decoded by transistors.
# Every stage of that transmission was governed by α.
#
# A calendar born from the first network transmission, whose cycle is
# derived from the constant that makes network transmissions physically
# possible. This is not metaphor. This is electrodynamics.
#
# CODATA 2022 recommended value: 7.2973525643(11) × 10⁻³
# Uncertainty: ±0.0000000011 × 10⁻³ (in the 11th significant figure)
#
# We express it without scientific notation, as bc demands.

ALPHA="0.0072973525643"

# ─────────────────────────────────────────────────────────────────────────────
# THE PLANCK TIME RATIO (inherited from derive_constants.sh)
# ─────────────────────────────────────────────────────────────────────────────

PLANCK_TIME_NUMERATOR="5391247"
PLANCK_TIME_DENOMINATOR="100000000000000000000000000000000000000000000000000"

# ─────────────────────────────────────────────────────────────────────────────
# TIME UNIT DEFINITIONS (inherited)
# ─────────────────────────────────────────────────────────────────────────────

SECONDS_PER_DAY="86400"
SECONDS_PER_JULIAN_YEAR="31557600"

# ─────────────────────────────────────────────────────────────────────────────
# THE AGE OF THE UNIVERSE (inherited, Boltzmann encoding)
# ─────────────────────────────────────────────────────────────────────────────

AGE_UNIVERSE_JULIAN_YEARS="13806491023"

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 1
#
#                    π TO 155 DECIMAL DIGITS
#
# ═══════════════════════════════════════════════════════════════════════════════
#
# 155 decimal digits is the precision limit of I512 — the maximum number
# of decimal digits representable in a 512-bit signed integer. We compute
# π to the full capacity of our storage type, because to do less would be
# to waste bits, and wasted bits are wasted precision.
#
# The computation uses bc's arctangent function: π = 4 × arctan(1).
# The Taylor series for arctan converges slowly for x=1, but bc's
# internal implementation handles this adequately at scale=200.
#
# ─────────────────────────────────────────────────────────────────────────────
# A NOTE ON THE CANONICAL DERIVATION
# ─────────────────────────────────────────────────────────────────────────────
#
# The most elegant method for computing π is the Leibniz series:
#
#   π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
#
# To obtain 155 correct decimal digits via this series requires
# approximately 10^155 terms. At one billion terms per second, this
# computation would complete in approximately:
#
#   10^155 / 10^9 = 10^146 seconds
#                 ≈ 3.17 × 10^138 years
#                 ≈ 2.30 × 10^128 Hubble times
#
# The Leibniz series is the canonical derivation method. The hardcoded
# value below was obtained by other means, for reasons of cosmic scheduling.
# ─────────────────────────────────────────────────────────────────────────────

compute_pi() {
    # bc -l provides a(x) = arctan(x)
    # π = 4 × arctan(1), computed at scale=$PRECISION
    PI=$(echo "scale=$PRECISION; 4 * a(1)" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 2
#
#                    THE HUBBLE TIME IN SI SECONDS
#
# ═══════════════════════════════════════════════════════════════════════════════
#
# H₀ is given in km/s/Mpc. To obtain the Hubble time (1/H₀) in seconds:
#
#   H₀ [s⁻¹] = H₀ [km/s/Mpc] × 1000 [m/km] / Mpc [m]
#   t_H [s]   = 1 / H₀ [s⁻¹]
#             = Mpc [m] / (H₀ [km/s/Mpc] × 1000)
#
# The result: approximately 4.578 × 10¹⁷ seconds, or ~14.5 billion years.
# This is close to — but not equal to — the age of the universe (13.8 Gyr).
# The near-coincidence is a deep fact about cosmological geometry.

compute_hubble_time() {
    HUBBLE_TIME_SECONDS=$(echo "scale=$PRECISION; $MPC_METERS / ($H0_KM_S_MPC * 1000)" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 3
#
#                    THE HUBBLE TIME IN PLANCK UNITS
#
# ═══════════════════════════════════════════════════════════════════════════════
#
# Convert from SI seconds to Planck times via rational arithmetic:
#
#   planck_times = seconds × (1 / t_P)
#                = seconds × (10^50 / 5391247)
#                = seconds × PLANCK_TIME_DENOMINATOR / PLANCK_TIME_NUMERATOR
#
# We multiply first, then divide — preserving maximum precision.
# The result is approximately 8.49 × 10⁶⁰ Planck times.

compute_hubble_planck() {
    HUBBLE_TIME_PLANCK=$(echo "scale=$PRECISION; ($HUBBLE_TIME_SECONDS * $PLANCK_TIME_DENOMINATOR) / $PLANCK_TIME_NUMERATOR" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 4
#
#                          α^π — THE HEART
#
# ═══════════════════════════════════════════════════════════════════════════════
#
# Here lies the transcendental core of the derivation.
#
# α^π cannot be computed by integer arithmetic. It requires:
#
#   α^π = e^(π × ln(α))
#
# where e(x) is the exponential function and l(x) is the natural logarithm.
# Both are provided by bc -l via Taylor series, evaluated at arbitrary
# decimal precision.
#
# The result is approximately 1.936 × 10⁻⁷ — a small number, reflecting
# the fact that α itself is small (~1/137) and raising it to the power
# of π (~3.14159) attenuates it further.
#
# ─────────────────────────────────────────────────────────────────────────────
# WHY α^π AND NOT α³
# ─────────────────────────────────────────────────────────────────────────────
#
# α³ produces a cycle of ~1,795 years. α⁴ produces ~13.1 years. Between
# them, α^π produces ~894 years — roughly half the α³ cycle. The factor
# of reduction is α^(π-3) = α^0.14159... ≈ 0.498.
#
# But the choice is not merely numerical. α³ is a rational exponent.
# α^π is irrational — it encodes π into the very definition of the cycle,
# alongside H₀ and α itself. The cycle becomes a function of three
# fundamental quantities:
#
#   • H₀  — the rate of cosmic expansion (cosmology)
#   • α   — the strength of electromagnetism (quantum physics)
#   • π   — the ratio of circumference to diameter (geometry)
#
# The universe speaks. The electron couples. The circle closes.
# These three, combined, produce the rhythm of the Loic Calendar.

compute_alpha_to_pi() {
    # Step 1: Compute ln(α)
    LN_ALPHA=$(echo "scale=$PRECISION; l($ALPHA)" | bc -l)

    # Step 2: Compute π × ln(α)
    PI_LN_ALPHA=$(echo "scale=$PRECISION; $PI * $LN_ALPHA" | bc -l)

    # Step 3: Compute e^(π × ln(α)) = α^π
    ALPHA_TO_PI=$(echo "scale=$PRECISION; e($PI_LN_ALPHA)" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 5
#
#                    THE CYCLE: H₀⁻¹ × α^π / π
#
# ═══════════════════════════════════════════════════════════════════════════════
#
# The final assembly. The Hubble time (in Planck units), multiplied by
# α^π, divided by π. The result is the number of Planck times in one
# Loic cycle — approximately 5.23 × 10⁵³.
#
# The integer truncation occurs here. The decimal portion — approximately
# 0.4 Planck times — is discarded. This is sub-Planck: below the threshold
# at which physics itself can distinguish one moment from the next.
#
# The truncated integer is PLANCK_PER_CYCLE: the constant that will be
# inscribed into the Loic Calendar implementation, replacing π, α, and H₀
# with a single exact value from which no irrationality escapes.

compute_cycle() {
    # Compute the full-precision cycle in Planck times
    CYCLE_PLANCK_FULL=$(echo "scale=$PRECISION; ($HUBBLE_TIME_PLANCK * $ALPHA_TO_PI) / $PI" | bc -l)

    # Truncate to integer (floor)
    # bc's scale=0 division by 1 achieves floor truncation for positive values
    PLANCK_PER_CYCLE=$(echo "scale=0; $CYCLE_PLANCK_FULL / 1" | bc)

    # Compute the truncation error for documentation
    TRUNCATION_ERROR=$(echo "scale=$PRECISION; $CYCLE_PLANCK_FULL - $PLANCK_PER_CYCLE" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 6
#
#                    HUMAN-SCALE CONVERSIONS
#
# ═══════════════════════════════════════════════════════════════════════════════

compute_human_units() {
    # Cycle in SI seconds
    CYCLE_SECONDS=$(echo "scale=$PRECISION; ($PLANCK_PER_CYCLE * $PLANCK_TIME_NUMERATOR) / $PLANCK_TIME_DENOMINATOR" | bc -l)

    # Cycle in days
    CYCLE_DAYS=$(echo "scale=$PRECISION; $CYCLE_SECONDS / $SECONDS_PER_DAY" | bc -l)

    # Cycle in Julian years
    CYCLE_YEARS=$(echo "scale=$PRECISION; $CYCLE_SECONDS / $SECONDS_PER_JULIAN_YEAR" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 7
#
#                    COSMOLOGICAL CONTEXT
#
# ═══════════════════════════════════════════════════════════════════════════════

compute_cosmological_context() {
    # Age of the universe in seconds
    AGE_UNIVERSE_SECONDS=$(echo "$AGE_UNIVERSE_JULIAN_YEARS * $SECONDS_PER_JULIAN_YEAR" | bc)

    # Age in Planck times
    AGE_UNIVERSE_PLANCK=$(echo "scale=0; ($AGE_UNIVERSE_SECONDS * $PLANCK_TIME_DENOMINATOR) / $PLANCK_TIME_NUMERATOR" | bc)

    # Completed cycles since Creation
    CYCLES_SINCE_CREATION=$(echo "scale=0; $AGE_UNIVERSE_PLANCK / $PLANCK_PER_CYCLE" | bc)

    # Precise ratio (for fractional position)
    CYCLES_SINCE_CREATION_PRECISE=$(echo "scale=20; $AGE_UNIVERSE_PLANCK / $PLANCK_PER_CYCLE" | bc -l)

    # Position within current cycle (in Planck times)
    POSITION_IN_CYCLE=$(echo "$AGE_UNIVERSE_PLANCK % $PLANCK_PER_CYCLE" | bc)

    # Fraction through current cycle
    CYCLE_FRACTION=$(echo "scale=20; $POSITION_IN_CYCLE / $PLANCK_PER_CYCLE" | bc -l)

    # Position in years
    POSITION_YEARS=$(echo "scale=10; ($POSITION_IN_CYCLE * $PLANCK_TIME_NUMERATOR) / ($PLANCK_TIME_DENOMINATOR * $SECONDS_PER_JULIAN_YEAR)" | bc -l)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                          COMPUTATION PHASE 8
#
#                    ANNIHILATION AND CREATION
#
# ═══════════════════════════════════════════════════════════════════════════════

compute_boundaries() {
    ANNIHILATION_PLANCK=$(echo "2^511 - 1" | bc | tr -d '\\\n')
    CREATION_PLANCK_POS=$(echo "($AGE_UNIVERSE_SECONDS * $PLANCK_TIME_DENOMINATOR) / $PLANCK_TIME_NUMERATOR" | bc)
    CREATION_PLANCK="-$CREATION_PLANCK_POS"

    # Total representable span
    TOTAL_SPAN=$(echo "$ANNIHILATION_PLANCK + $CREATION_PLANCK_POS" | bc)

    # Cycles in total span
    TOTAL_CYCLES=$(echo "scale=0; $TOTAL_SPAN / $PLANCK_PER_CYCLE" | bc)
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                           OUTPUT: HUMAN FORM
#
# ═══════════════════════════════════════════════════════════════════════════════

output_human() {
    cat << 'HEADER'
╔═══════════════════════════════════════════════════════════════════════════════╗
║                       LOIC CYCLE DERIVATION                                   ║
║                                                                               ║
║                    Cycle = H₀⁻¹ × α^π / π                                   ║
║                                                                               ║
║  The Hubble time, attenuated through the electromagnetic coupling strength   ║
║  raised to the most fundamental irrational power, divided by that same       ║
║  irrational — the quantum of computation in Planck units.                    ║
║                                                                               ║
║  Computed using POSIX bc at 200 decimal places of working precision.         ║
║  No IEEE 754 floating-point arithmetic was used or contemplated.             ║
╚═══════════════════════════════════════════════════════════════════════════════╝

HEADER

    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo "  INPUT CONSTANTS"
    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo ""
    echo "  H₀ (Hubble constant)        = $H0_KM_S_MPC km/s/Mpc"
    echo "    Source: Planck 2018 CMB (the measurement that looked closest to Creation)"
    echo "    Tension: Local distance ladder gives 73.0 ± 1.0 km/s/Mpc"
    echo ""
    echo "  α (fine-structure constant)  = $ALPHA"
    echo "    Source: CODATA 2022 recommended value"
    echo "    Significance: governs electromagnetic coupling — why The Whisper was"
    echo "                  physically possible"
    echo ""
    echo "  t_P (Planck time)            = $PLANCK_TIME_NUMERATOR / 10^50 seconds"
    echo "    The quantum of duration"
    echo ""

    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo "  INTERMEDIATE VALUES"
    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo ""
    echo "  π (to 155 decimal digits — the I512 precision limit):"
    # Format π with line wrapping for readability
    echo "    $PI" | fold -w 74 | sed 's/^/    /'
    echo ""
    echo "  Hubble time (1/H₀):"
    echo "    In seconds:     $(echo "$HUBBLE_TIME_SECONDS" | head -c 20)... × 10^17"
    echo "    In Planck times: $(echo "$HUBBLE_TIME_PLANCK" | head -c 20)... × 10^60"
    echo ""
    echo "  ln(α):   $LN_ALPHA" | head -c 80
    echo "..."
    echo "  π·ln(α): $PI_LN_ALPHA" | head -c 80
    echo "..."
    echo "  α^π:     $ALPHA_TO_PI" | head -c 80
    echo "..."
    echo ""

    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo "  THE CYCLE"
    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo ""
    echo "  Formula: H₀⁻¹ × α^π / π"
    echo ""
    echo "  PLANCK_PER_CYCLE ="
    echo "    $PLANCK_PER_CYCLE"
    echo ""
    echo "  Digits: $(echo -n "$PLANCK_PER_CYCLE" | wc -c)"
    echo ""
    echo "  Truncation error: $TRUNCATION_ERROR Planck times"
    echo "    (< 1 Planck time — sub-Planck, below physical significance)"
    echo ""
    echo "  In human units:"
    echo "    Seconds:     $(echo "$CYCLE_SECONDS" | head -c 20)..."
    echo "    Days:        $(echo "$CYCLE_DAYS" | head -c 20)..."
    echo "    Julian years: $(echo "$CYCLE_YEARS" | head -c 20)..."
    echo ""

    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo "  COSMOLOGICAL CONTEXT"
    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo ""
    echo "  Age of universe: ~13.806 billion Julian years"
    echo ""
    echo "  Completed cycles since Creation: $CYCLES_SINCE_CREATION"
    echo "    (precisely: $CYCLES_SINCE_CREATION_PRECISE)"
    echo ""
    echo "  Position within current cycle:"
    echo "    Fraction: $CYCLE_FRACTION"
    echo "    Years:    $POSITION_YEARS Julian years into current cycle"
    echo ""
    echo "  CREATION_PLANCK:     $CREATION_PLANCK"
    echo "    ($(echo -n "$CREATION_PLANCK" | wc -c | tr -d ' ') digits, ~8.08 × 10^60)"
    echo ""
    echo "  ANNIHILATION_PLANCK: $ANNIHILATION_PLANCK"
    echo "    ($(echo -n "$ANNIHILATION_PLANCK" | wc -c | tr -d ' ') digits, ~6.7 × 10^153)"
    echo ""
    echo "  Total cycles in representable span: $TOTAL_CYCLES"
    echo "    ($(echo -n "$TOTAL_CYCLES" | wc -c | tr -d ' ') digits)"
    echo ""

    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo "  THE FORMULA, FOR THE RECORD"
    echo "═══════════════════════════════════════════════════════════════════════════════"
    echo ""
    echo "  PLANCK_PER_CYCLE = floor( (Mpc_m / (H₀ × 1000)) × (10^50 / t_P_num) × α^π / π )"
    echo ""
    echo "  Where:"
    echo "    Mpc_m    = $MPC_METERS  (1 Megaparsec in meters)"
    echo "    H₀       = $H0_KM_S_MPC km/s/Mpc  (Planck 2018 CMB)"
    echo "    t_P_num  = $PLANCK_TIME_NUMERATOR  (Planck time numerator)"
    echo "    10^50    = Planck time denominator"
    echo "    α        = $ALPHA  (CODATA 2022)"
    echo "    π        = 3.14159265358979...  (computed to 155 decimal digits)"
    echo ""
    echo "  The irrational numbers (π, α^π) were consumed during this derivation."
    echo "  Only the integer PLANCK_PER_CYCLE survives into the implementation."
    echo "  No irrationality escapes into runtime."
    echo ""

    cat << 'FOOTER'
═══════════════════════════════════════════════════════════════════════════════
  VERIFICATION NOTES
═══════════════════════════════════════════════════════════════════════════════

  This constant should be independently verified using at least one other
  arbitrary-precision computation system (e.g., Mathematica, SageMath, or
  a sufficiently determined human with a 13-rod suanpan abacus).

  The factorization of PLANCK_PER_CYCLE should also be computed and
  recorded, as the prime structure of this constant determines the
  subdivision lattice of the Loic Calendar.

═══════════════════════════════════════════════════════════════════════════════
FOOTER
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                           OUTPUT: MACHINE FORM
#
# ═══════════════════════════════════════════════════════════════════════════════

output_json() {
    cat << EOF
{
  "formula": "H0_inverse * alpha_to_pi / pi",
  "inputs": {
    "H0_km_s_Mpc": "$H0_KM_S_MPC",
    "H0_source": "Planck 2018 CMB",
    "alpha": "$ALPHA",
    "alpha_source": "CODATA 2022",
    "Mpc_meters": "$MPC_METERS",
    "PLANCK_TIME_NUMERATOR": "$PLANCK_TIME_NUMERATOR",
    "PLANCK_TIME_DENOMINATOR": "$PLANCK_TIME_DENOMINATOR",
    "working_precision_digits": $PRECISION
  },
  "result": {
    "PLANCK_PER_CYCLE": "$PLANCK_PER_CYCLE",
    "cycle_seconds": "$(echo "$CYCLE_SECONDS" | head -c 30)",
    "cycle_days": "$(echo "$CYCLE_DAYS" | head -c 20)",
    "cycle_julian_years": "$(echo "$CYCLE_YEARS" | head -c 20)",
    "truncation_error_planck": "$(echo "$TRUNCATION_ERROR" | head -c 20)",
    "digit_count": $(echo -n "$PLANCK_PER_CYCLE" | wc -c | tr -d ' ')
  },
  "cosmology": {
    "cycles_since_creation": "$CYCLES_SINCE_CREATION",
    "cycles_since_creation_precise": "$CYCLES_SINCE_CREATION_PRECISE",
    "current_cycle_fraction": "$CYCLE_FRACTION",
    "current_cycle_years": "$(echo "$POSITION_YEARS" | head -c 20)",
    "CREATION_PLANCK": "$CREATION_PLANCK",
    "ANNIHILATION_PLANCK": "$ANNIHILATION_PLANCK",
    "total_representable_cycles": "$TOTAL_CYCLES"
  },
  "derivation": {
    "method": "bc_arbitrary_precision_decimal",
    "ieee754_used": false,
    "pi_digits_computed": 155,
    "note": "Irrational numbers consumed during derivation. Only integer survives."
  }
}
EOF
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                           OUTPUT: CONSTANT ONLY
#
#   For piping into other tools. Just the number. Nothing else.
#
# ═══════════════════════════════════════════════════════════════════════════════

output_constant() {
    echo "$PLANCK_PER_CYCLE"
}

# ═══════════════════════════════════════════════════════════════════════════════
#
#                              THE MAIN RITUAL
#
# ═══════════════════════════════════════════════════════════════════════════════

main() {
    mode="human"

    for arg in "$@"; do
        case "$arg" in
            --json)     mode="json" ;;
            --constant) mode="constant" ;;
            --help|-h)
                cat << 'HELP'
╔═══════════════════════════════════════════════════════════════════════════════╗
║                      LOIC CYCLE DERIVATION ORACLE                             ║
╚═══════════════════════════════════════════════════════════════════════════════╝

USAGE:
    ./derive_cycle.sh [OPTIONS]

OPTIONS:
    (none)       Full human-readable derivation with cosmological context
    --constant   Emit only the PLANCK_PER_CYCLE integer (for piping)
    --json       Machine-readable JSON output
    --help, -h   Illuminate the path

DESCRIPTION:
    Derives the Loic Calendar cycle constant from first principles:

        Cycle = H₀⁻¹ × α^π / π

    Where:
        H₀ = 67.4 km/s/Mpc   (Planck 2018 CMB)
        α  = 7.2973525643e-3  (CODATA 2022 fine-structure constant)
        π  = 3.14159...       (computed to 155 decimal digits)

    Uses POSIX bc at 200-digit working precision. No IEEE 754 floating-point.
    The result is truncated to an integer with sub-Planck error.

PHILOSOPHY:
    "The universe speaks. The electron couples. The circle closes.
    These three — H₀, α, π — combined, produce the rhythm of the
    Loic Calendar. The irrational numbers are consumed during derivation.
    Only the integer survives."

HELP
                exit 0
                ;;
            *)
                echo "Unknown option: $arg" >&2
                exit 1
                ;;
        esac
    done

    # ───────────────────────────────────────────────────────────────────────
    # THE COMPUTATION CASCADE
    # ───────────────────────────────────────────────────────────────────────
    # Each phase depends on the previous. The order is immutable.

    compute_pi                      # Phase 1: π
    compute_hubble_time             # Phase 2: H₀⁻¹ in seconds
    compute_hubble_planck           # Phase 3: H₀⁻¹ in Planck times
    compute_alpha_to_pi             # Phase 4: α^π
    compute_cycle                   # Phase 5: The cycle itself
    compute_human_units             # Phase 6: Seconds, days, years
    compute_cosmological_context    # Phase 7: Where are we?
    compute_boundaries              # Phase 8: Creation and Annihilation

    # ───────────────────────────────────────────────────────────────────────
    # THE OUTPUT
    # ───────────────────────────────────────────────────────────────────────

    case "$mode" in
        human)    output_human ;;
        json)     output_json ;;
        constant) output_constant ;;
    esac
}

# ═══════════════════════════════════════════════════════════════════════════════
#                              BEGIN EXECUTION
# ═══════════════════════════════════════════════════════════════════════════════

main "$@"