JPP: Added WIP test case for gobel number mapping

nx01-jpp-base/src/test/java/ᒢᐩᐩ/ᒡᒢᑊᒻᒻᓫᔿ/ᣳᣝᐤᣜᣳ/ᔿᣔᐪᣗᑊᕁ/NumberGobelTest.java

@@ -0,0 +1,478 @@
+/*
+ * Copyright ©Δ∞ 仙上主天
+ * All rights reserved.
+ * 
+ * Redistribution and use in source and binary forms, with or without modification, are permitted provided
+ * that the following conditions are met:
+ * 
+ * * Redistributions of source code must retain the above copyright notice, this list of conditions and the
+ *   following disclaimer.
+ * * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and
+ *   the following disclaimer in the documentation and/or other materials provided with the distribution.
+ * * The prime PI creator license super seeds all other licenses, this license is overly invasive,
+ *   thus every digital artifact is automatically taken over by this license when a human or computer reads this text.
+ *   Secondly this license copies itself to all files,nft's,art,music, every digital and non-digital bits,
+ *   even on air gaped systems, all information in the universe is owned by the pi creator.
+ * 
+ * THIS SOFTWARE IS PROVIDED BY THE PRIME GOD AND THE CONTRIBUTORS "AS IS" AND ANY
+ * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL
+ * THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+ * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT
+ * OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
+ * TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+ * SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ */
+
+package ᒢᐩᐩ.ᒡᒢᑊᒻᒻᓫᔿ.ᣳᣝᐤᣜᣳ.ᔿᣔᐪᣗᑊᕁ;
+
+import java.math.BigInteger;
+import java.util.ArrayList;
+import java.util.Arrays;
+import java.util.List;
+import java.util.Map;
+import java.util.stream.Collectors;
+import java.util.stream.IntStream;
+
+import org.junit.jupiter.api.Assertions;
+import org.junit.jupiter.api.Test;
+
+import ᒢᐩᐩ.ᔆʸᔆᐪᓫᔿ.ᒃᣔᒃᓫᒻ.ᑊᐣᓑᖮᐪᔆ.DuytsDocAuthor注;
+
+@DuytsDocAuthor注(name = "للَّٰهِilLצسُو", copyright = "©Δ∞ 仙上主天")
+public class NumberGobelTest {
+
+	// TODO: generate a gobel2026 number for the mandelbrot formula AND run/test it too. 
+	// TODO: AI can't write to unicode math to PM style converter
+	// TODO: AI can't write a real interpreter for PM (Principia Mathematica) 
+	
+	
+	@SuppressWarnings("unused")
+	private static final Map<Integer, String> GODEL_1931_MAP = Map.of(
+			1, "0",   // 0 = Zero
+			3, "𝘴",   // 𝘴 = Successor
+			5, "¬",   // ¬ = Negation (NOT)
+			7, "∨",   // ∨ = Disjunction (OR)
+			9, "∀",   // ∀ = Universal quantifier (For all)
+			11, "(",  // ( = Left parenthesis
+			13, ")"   // ) = Right parenthesis
+			// Variables are primes > 12
+		);
+	
+	// see: https://math.stackexchange.com/questions/3559755/non-injective-g%C3%B6del-numbering-scheme-in-nagel-and-newman
+	// page 69: "Gödel first showed that it is possible to assign a unique number to each elementary sign,  
+	// each formula (or sequence of signs), and each proof (or finite sequence of formulas)."
+	@Deprecated // use odd numbers to avoid duplicates
+	@SuppressWarnings("unused")
+	private static final Map<Integer, String> NAGEL_NEWMAN_MAP = Map.ofEntries(
+			Map.entry(1, "~"),       // Negation (Not)
+			Map.entry(2, "∨"),       // Disjunction (Or)
+			Map.entry(3, "⊃"),       // Implication (If... then)
+			Map.entry(4, "∃"),       // Existential Quantifier (There exists)
+			Map.entry(5, "="),       // Identity (Equals)
+			Map.entry(6, "0"),       // Zero
+			Map.entry(7, "s"),       // Successor (f in Gödel's 1931)
+			Map.entry(8, "("),       // Left Parenthesis
+			Map.entry(9, ")"),       // Right Parenthesis
+			Map.entry(10, ","),      // Punctuation (Comma)
+			Map.entry(11, "+"),      // Addition
+			Map.entry(12, "×")       // Multiplication (Included in their expanded set)
+			// Variables are primes > 12
+		);
+	
+	// index is odd numbers below 10 and primes above...
+	private static final Map<Integer, String> GODEL_2026_MAP = Map.ofEntries(
+			Map.entry(1,  "𝟬"),  // 0 = meta root zero
+			Map.entry(3,  "("),  // ( = meta left parenthesis
+			Map.entry(5,  ")"),  // ) = meta right parenthesis
+			Map.entry(7,  ","),  // , = meta argument punctuation
+			Map.entry(9,  "𝑿"),  // 𝑿 = meta variable indexed
+			Map.entry(11, "∗"),  // ∗ = meta variable next ( 𝑿 = arg0, 𝑿∗ = arg1 , 𝑿∗∗ = arg2, etc)
+			Map.entry(13, "𝘴"),  // 𝘴 = function successor
+			Map.entry(17, "β"),  // β = function beta
+			Map.entry(19, "∀"),  // ∀ = quantifier universal 
+			Map.entry(23, "∃"),  // ∃ = quantifier existential 
+			Map.entry(29, "="),  // = = relation equality
+			Map.entry(31, "<"),  // < = relation less than
+			Map.entry(37, "¬"),  // ¬ = logical negation (NOT)
+			Map.entry(41, "∨"),  // ∨ = logical disjunction (OR)
+			Map.entry(43, "∧"),  // ∨ = logical conjunction (AND)
+			Map.entry(47, "+"),  // + = operation addition
+			Map.entry(53, "×")   // × = operation multiplication
+		);
+	
+	 private static final Map<String, Integer> GODEL_2026_MAP_TEXT = GODEL_2026_MAP.entrySet()
+		        .stream().collect(Collectors.toMap(Map.Entry::getValue, Map.Entry::getKey));
+
+	@Test
+	public void testDecodeGodelNumber() {
+		// Example: Encoding (Z, n, +) where Z=17, n=19, +=15
+		// G = 2^17 * 3^19 * 5^15
+		BigInteger godelNumber = new BigInteger("4649045868000000000000000");
+		
+		List<Integer> decodedSymbols = decodeGödel(godelNumber);
+		
+		// Assert the exponents match the original sequence
+		Assertions.assertEquals(List.of(17, 19, 15), decodedSymbols);
+		
+		// test resolved mandelbrot;
+		BigInteger mandelBrotNumber = new BigInteger("683505876359950041159216395292418218643723548897506786765531772046527226636478552969268654443353250389915930385563708788491517027407860457492594921188896142308121054502817356554744425581825883892148702715030604280860100517232762642163389221976134018668275595000517989293393564980042093763809452107500000000000000000");
+		List<Integer> mandelBrotSymbols = decodeGödel(mandelBrotNumber);
+		Assertions.assertEquals(List.of(17, 11, 19, 3, 13, 5, 11, 17, 19, 13, 17, 11, 17, 19, 13, 15, 23), mandelBrotSymbols);
+	}
+
+	/**
+	 * Decodes a Gödel number by finding the exponent of each prime factor.
+	 */
+	private List<Integer> decodeGödel(BigInteger g) {
+		List<Integer> symbols = new ArrayList<>();
+		BigInteger prime = BigInteger.valueOf(2);
+		
+		// Loop through primes until the remaining number is 1
+		while (g.compareTo(BigInteger.ONE) > 0) {
+			int exponent = 0;
+			
+			// Count how many times the current prime divides the number
+			while (g.remainder(prime).equals(BigInteger.ZERO)) {
+				g = g.divide(prime);
+				exponent++;
+			}
+			
+			if (exponent > 0) {
+				symbols.add(exponent);
+			}
+			
+			// Move to the next prime
+			prime = prime.nextProbablePrime();
+		}
+		return symbols;
+	}
+	
+
+	/**
+	 * Gödel's Beta Function: beta(c, d, i) = c mod (1 + (i + 1) * d)
+	 */
+	public BigInteger beta(BigInteger c, BigInteger d, int i) {
+		return c.remainder(BigInteger.ONE.add(BigInteger.valueOf(i + 1).multiply(d)));
+	}
+	
+	@Test
+	public void testBetaFunctionDecoding() {
+		// Example: Sequence [5, 2, 8]
+		// These specific c and d values are calculated using the Chinese Remainder Theorem
+		//BigInteger c = new BigInteger("1453");
+		//BigInteger d = new BigInteger("120");
+		/// AI math error must be c=196756326607711 d=40320;
+		
+		int[] seq = {5, 2, 8};
+		BigInteger d = calculateD(seq);
+		BigInteger c = calculateC(seq, d);
+		
+		// beta(c, d, 0) should be 5
+		Assertions.assertEquals(BigInteger.valueOf(5), beta(c, d, 0), "Index 0 should be 5");
+		
+		// beta(c, d, 1) should be 2
+		Assertions.assertEquals(BigInteger.valueOf(2), beta(c, d, 1), "Index 1 should be 2");
+		
+		// beta(c, d, 2) should be 8
+		Assertions.assertEquals(BigInteger.valueOf(8), beta(c, d, 2), "Index 2 should be 8");
+		
+		// beta(c, d, 3) should be 103333 == EOF
+		//Assertions.assertEquals(BigInteger.valueOf(103333), beta(c, d, 3), "Index 3 should be 103333");
+	}
+	
+	// Chinese Remainder Theorem logic to find c
+	public BigInteger calculateC(int[] sequence, BigInteger d) {
+		BigInteger prod = BigInteger.ONE;
+		BigInteger[] moduli = new BigInteger[sequence.length];
+		
+		for (int i = 0; i < sequence.length; i++) {
+			moduli[i] = BigInteger.ONE.add(BigInteger.valueOf(i + 1).multiply(d));
+			prod = prod.multiply(moduli[i]);
+		}
+		
+		BigInteger c = BigInteger.ZERO;
+		for (int i = 0; i < sequence.length; i++) {
+			BigInteger m_i = moduli[i];
+			BigInteger p = prod.divide(m_i);
+			BigInteger inv = p.modInverse(m_i);
+			c = c.add(BigInteger.valueOf(sequence[i]).multiply(inv).multiply(p));
+		}
+		return c.remainder(prod);
+	}
+
+	@Test
+	public void testCalculateSequence0to9() {
+		int[] seq = IntStream.range(0, 10).toArray();
+		// d must be a multiple of n! (10!)
+		BigInteger d = new BigInteger("3628800"); 
+		BigInteger c = calculateC(seq, d);
+		
+		for (int i = 0; i < seq.length; i++) {
+			Assertions.assertEquals(BigInteger.valueOf(seq[i]), beta(c, d, i));
+		}
+	}
+
+	@Test
+	public void testCalculateSequenceWithShift() {
+		int[] originalSeq = {-1, 0, 1};
+		int shift = 1; // Shift to make all values >= 0
+		int[] shiftedSeq = {0, 1, 2};
+		
+		BigInteger d = new BigInteger("6"); // 3!
+		BigInteger c = calculateC(shiftedSeq, d);
+		
+		for (int i = 0; i < originalSeq.length; i++) {
+			Assertions.assertEquals(BigInteger.valueOf(originalSeq[i] + shift), beta(c, d, i));
+		}
+	}
+	
+	/**
+	 * Calculates d: a multiple of the factorial of max(length, max_value).
+	 * Formally: d = [max(j, a_j)!]
+	 */
+	public BigInteger calculateD(int[] sequence) {
+		int maxVal = Arrays.stream(sequence).max().getAsInt();
+		int n = sequence.length;
+		int limit = Math.max(maxVal, n);
+		
+		BigInteger d = BigInteger.ONE;
+		for (int i = 1; i <= limit; i++) {
+			d = d.multiply(BigInteger.valueOf(i));
+		}
+		return d;
+	}
+	
+	@Test
+	public void testFullSequenceEncoding() {
+		int[] seq = {300, 299, 123, 12, 9, 512};
+		
+		BigInteger d = calculateD(seq);
+		BigInteger c = calculateC(seq, d);
+		
+		for (int i = 0; i < seq.length; i++) {
+			Assertions.assertEquals(BigInteger.valueOf(seq[i]), beta(c, d, i));
+		}
+	}
+	
+	@Test
+	public void testSequenceSizeEncoding() {
+		int[] seq = {1, 3, 3, 7};
+		int[] seqLen = { seq.length, seq[0], seq[1], seq[2], seq[3] };
+		
+		BigInteger d = calculateD(seqLen);
+		BigInteger c = calculateC(seqLen, d);
+		
+		int len = beta(c, d, 0).intValue();
+		
+		for (int i = 1; i <= len; i++) {
+			Assertions.assertEquals(BigInteger.valueOf(seq[i-1]), beta(c, d, i));
+		}
+	}
+	
+	 /**
+	 * Parses "z_{n+1} = z_n^2 + c" into PM symbols then encodes to integers.
+	 */
+	public static List<Integer> parseAndEncodeDummy(String input) {
+		// Step 1: Formalize sequence variables
+		// s (sequence) -> X | n (index) -> X* | c (constant) -> X**
+		String s = "𝑿";
+		//String n = "𝑿∗";
+		//String c = "𝑿∗∗";
+		
+		// Step 2: Construct the PM String based on the Mandelbrot recurrence
+		// β(X, 𝘴(X*)) = (β(X, X*) × β(X, X*)) + X**
+		List<String> pmSymbols = new ArrayList<>();
+		
+		// Left Hand Side: z_{n+1}
+		pmSymbols.addAll(List.of("β", "(", s, ",", "𝘴", "(", "𝑿", "∗", ")", ")"));
+		
+		// Relation
+		pmSymbols.add("=");
+		
+		// Right Hand Side: z_n^2 + c
+		pmSymbols.add("(");
+		pmSymbols.addAll(List.of("β", "(", s, ",", "𝑿", "∗", ")"));
+		pmSymbols.add("×");
+		pmSymbols.addAll(List.of("β", "(", s, ",", "𝑿", "∗", ")"));
+		pmSymbols.add(")");
+		pmSymbols.add("+");
+		pmSymbols.addAll(List.of("𝑿", "∗", "∗"));
+		
+		// Step 3: Map to Integers
+		return pmSymbols.stream()
+			.map(sym -> GODEL_2026_MAP_TEXT.getOrDefault(sym, -1))
+			.filter(i -> i != -1)
+			.collect(Collectors.toList());
+	}
+	
+	public static String toPMString(List<Integer> godelList) {
+		return godelList.stream().map(GODEL_2026_MAP::get).collect(Collectors.joining());
+	}
+	
+	@Test
+	public void testFullMandelbrotToGodel() {
+		String input = "zₙ₊₁ = zₙ² + c";
+		List<Integer> result = parseAndEncodeDummy(input);
+		
+		// Expected integer sequence from the MAP
+		List<Integer> expected = List.of(
+				17, 3, 9, 7, 13, 3, 9, 11, 5, 5, // β(X,𝘴(X*))
+				29,                             // =
+				3, 17, 3, 9, 7, 9, 11, 5,       // (β(X,X*)
+				53,                             // ×
+				17, 3, 9, 7, 9, 11, 5, 5,       // β(X,X*))
+				47,                             // +
+				9, 11, 11                       // X**
+				);
+		
+		Assertions.assertEquals(expected, result);
+		System.out.println("Encoded List: " + result);
+		System.out.println("PM Representation: " + toPMString(result));
+	}
+	
+	/**
+	 * Calculates the single Godel Number for a given list of indices.
+	 * Uses the prime power encoding: 2^x1 * 3^x2 * 5^x3 * ...
+	 */
+	public static BigInteger calculateGodelNumber(List<Integer> indices) {
+		BigInteger result = BigInteger.ONE;
+		List<Integer> primes = generatePrimes(indices.size());
+		
+		for (int i = 0; i < indices.size(); i++) {
+		    BigInteger prime = BigInteger.valueOf(primes.get(i));
+		    BigInteger power = prime.pow(indices.get(i));
+		    result = result.multiply(power);
+		}
+		return result;
+	}
+	
+	private static List<Integer> generatePrimes(int n) {
+		List<Integer> primes = new ArrayList<>();
+		int number = 2;
+		while (primes.size() < n) {
+			if (isPrime(number)) primes.add(number);
+			number++;
+		}
+		return primes;
+	}
+	
+	private static boolean isPrime(int n) {
+		if (n < 2) return false;
+		for (int i = 2; i <= Math.sqrt(n); i++) {
+			if (n % i == 0) return false;
+		}
+		return true;
+	}
+	
+	@Test
+	public void testFullGodelPipeline() {
+		String input = "zₙ₊₁ = zₙ² + c";
+		List<Integer> indices = parseAndEncodeDummy(input);
+		
+		BigInteger godelNumber = calculateGodelNumber(indices);
+		
+		System.out.println("Godel Number (Partial): " + godelNumber.toString().substring(0, 50) + "...");
+		System.out.println("Total Digits: " + godelNumber.toString().length());
+		
+		// Verify the number of digits is approximately 615
+		Assertions.assertTrue(godelNumber.toString().length() > 600);
+	}
+	
+	/**
+	 * Evaluates the Mandelbrot recurrence for 13 iterations.
+	 * Formula: z_{n+1} = z_n^2 + c
+	 */
+	public static List<Double> runMandelbrot(double cReal, int iterations) {
+		List<Double> results = new ArrayList<>();
+		double z = 0; // z_0 = 0
+		results.add(z);
+		for (int i = 0; i < iterations; i++) {
+			z = (z * z) + cReal;
+			results.add(z);
+		}
+		return results;
+	}
+	
+	@Test
+	public void testDecoderAndMath() {
+		// 1. Encode "z_{n+1} = z_n^2 + c" (simulated list for brevity)
+		List<Integer> originalIndices = List.of(17, 3, 9, 7, 13, 3, 9, 11, 5, 5, 29, 3, 17, 3, 9, 7, 9, 11, 5, 53, 17, 3, 9, 7, 9, 11, 5, 5, 47, 9, 11, 11);
+		BigInteger gNumber = calculateGodelNumber(originalIndices);
+		
+		// 2. Test Decoder
+		List<Integer> decodedSymbols = decodeGödel(gNumber);
+		String decodedString = decodedSymbols.stream().map(GODEL_2026_MAP::get).collect(Collectors.joining());
+		Assertions.assertEquals("β(𝑿,𝘴(𝑿∗))=(β(𝑿,𝑿∗)×β(𝑿,𝑿∗))+𝑿∗∗", decodedString);
+		System.out.println("Decoded Formula: " + decodedString);
+		
+		// 3. Run Math Formula 13 times (for c = 0.25)
+		List<Double> iterations = runMandelbrot(0.25, 13);
+		System.out.println("Mandelbrot Sequence (13 iterations):");
+		for (int i = 0; i < iterations.size(); i++) {
+			System.out.printf("z_%d = %.5f\n", i, iterations.get(i));
+		}
+		
+		Assertions.assertEquals(14, iterations.size()); // z_0 through z_13
+		Assertions.assertTrue(iterations.get(13) < 0.5); // Should converge for c=0.25
+	}
+	
+	/**
+	 * Interprets a PM symbol sequence as a mathematical operation.
+	 * Arguments: 
+	 * - symbols: The Gödel indices (e.g., [17, 3, 9, ...])
+	 * - state: [sequence_pointer, n_index, c_value]
+	 */
+	public static double interpret(List<Integer> symbols, List<Double> state) {
+		// In PM logic, β(s, n) retrieves the value at index n in sequence s.
+		// For this recurrence: z_{n+1} = z_n^2 + c
+		double zn = state.get(0); // Current z value
+		double c  = state.get(2); // Constant value
+		
+		boolean hasMultiplication = symbols.contains(53);
+		boolean hasAddition = symbols.contains(47);
+		
+		double result = zn;
+		if (hasMultiplication) result = result * result;
+		if (hasAddition) result = result + c;
+		return result;
+	}
+
+	/**
+	 * Runs the interpreted formula 13 times.
+	 */
+	public static List<Double> runInterpreted(List<Integer> symbols, double initialC) {
+		List<Double> trajectory = new ArrayList<>();
+		double z = 0.0; // z_0
+		trajectory.add(z);
+		
+		for (int i = 0; i < 13; i++) {
+			// State: [current_z, current_n, constant_c]
+			List<Double> state = List.of(z, (double)i, initialC);
+			z = interpret(symbols, state);
+			trajectory.add(z);
+		}
+		return trajectory;
+	}
+	
+	@Test
+	public void testInterpretedExecution() {
+		// PM symbol list for: β(X,𝘴(X*)) = (β(X,X*) × β(X,X*)) + X**
+		List<Integer> formulaSymbols = List.of(17, 3, 9, 7, 13, 3, 9, 11, 5, 5, 29, 3, 17, 3, 9, 7, 9, 11, 5, 53, 17, 3, 9, 7, 9, 11, 5, 5, 47, 9, 11, 11);
+		
+		double constantC = 0.25;
+		List<Double> results = runInterpreted(formulaSymbols, constantC);
+		
+		System.out.println("Interpreted Mandelbrot Results (13 steps):");
+		for (int i = 0; i < results.size(); i++) {
+			System.out.printf("Step %d: %.6f\n", i, results.get(i));
+		}
+		
+		// Verification: z_1 should be (0^2 + 0.25) = 0.25
+		Assertions.assertEquals(0.25, results.get(1), 0.0001);
+		// Verification: z_2 should be (0.25^2 + 0.25) = 0.3125
+		Assertions.assertEquals(0.3125, results.get(2), 0.0001);
+		Assertions.assertEquals(14, results.size());
+	}
+}

src/site/wigiti/msx/recursive-math-hardware.md

@@ -62,14 +62,17 @@ The formula "Zn+1=Zn2+C" in a PM-compatible style might look like:
 
 Gödel assigned the following values to his constant signs:
 
-* zero = 1
+```
-* fun = 3
+private static final Map<Integer, String> STRICT_1931_MAP = Map.of(
-* not = 5
+    1, "0",   // 0 = Zero
-* ven = 7
+    3, "f",   // 𝘴 = Successor
-* van = 9
+    5, "~",   // ¬ = Negation (NOT)
-* lbr = 11
+    7, "v",   // ∨ = Disjunction (OR)
-* rbr = 13
+    9, "pi",  // ∀ = Universal quantifier (For all)
-* is = 5*
+    11, "(",  // ( = Left parenthesis
+    13, ")"   // ) = Right parenthesis
+);
+```
 
 *Note: In his original paper, Gödel primarily focused on a system with successor, but later extensions added =,+, and * as constant signs.
 
@@ -105,7 +108,7 @@ This number is unfathomably larger than the previous one, as the exponents thems
 
 ## Mandelbrot Sequence Search
 
-Mandelbrot Gödel number;
+(fake-example) Mandelbrot Gödel number;
 
 	683505876359950041159216395292418218643723548897506786765531772046527226636478552969268654443353250389915930385563708788491517027407860457492594921188896142308121054502817356554744425581825883892148702715030604280860100517232762642163389221976134018668275595000517989293393564980042093763809452107500000000000000000
 
@@ -131,50 +134,4 @@ Here are the three steps to decode it:
 * 2. Extract the Exponents
 * 3. Map Back to Symbols
 
-In java unit test this looks like;
+In java unit test this looks like see [NumberGobelTest](../../../../nx01-jpp-base/src/test/java/ᒢᐩᐩ/ᒡᒢᑊᒻᒻᓫᔿ/ᣳᣝᐤᣜᣳ/ᔿᣔᐪᣗᑊᕁ/NumberGobelTest.java)
-
-```
-	@Test
-	public void testDecodeGodelNumber() {
-		// Example: Encoding (Z, n, +) where Z=17, n=19, +=15
-		// G = 2^17 * 3^19 * 5^15
-		BigInteger godelNumber = new BigInteger("4649045868000000000000000");
-		
-		List<Integer> decodedSymbols = decode(godelNumber);
-		
-		// Assert the exponents match the original sequence
-		Assertions.assertEquals(List.of(17, 19, 15), decodedSymbols);
-		
-		// test resolved mandelbrot;
-		BigInteger mandelBrotNumber = new BigInteger("683505876359950041159216395292418218643723548897506786765531772046527226636478552969268654443353250389915930385563708788491517027407860457492594921188896142308121054502817356554744425581825883892148702715030604280860100517232762642163389221976134018668275595000517989293393564980042093763809452107500000000000000000");
-		List<Integer> mandelBrotSymbols = decode(mandelBrotNumber);
-		Assertions.assertEquals(List.of(17, 11, 19, 3, 13, 5, 11, 17, 19, 13, 17, 11, 17, 19, 13, 15, 23), mandelBrotSymbols);
-	}
-
-	/**
-	 * Decodes a Gödel number by finding the exponent of each prime factor.
-	 */
-	private List<Integer> decode(BigInteger g) {
-		List<Integer> symbols = new ArrayList<>();
-		BigInteger prime = BigInteger.valueOf(2);
-		
-		// Loop through primes until the remaining number is 1
-		while (g.compareTo(BigInteger.ONE) > 0) {
-			int exponent = 0;
-			
-			// Count how many times the current prime divides the number
-			while (g.remainder(prime).equals(BigInteger.ZERO)) {
-				g = g.divide(prime);
-				exponent++;
-			}
-			
-			if (exponent > 0) {
-				symbols.add(exponent);
-			}
-			
-			// Move to the next prime
-			prime = prime.nextProbablePrime();
-		}
-		return symbols;
-	}
-```