The probability of repeating a specific sigil and seal design depends on the vast number of possible combinations created by the system.

This number comes from multiplying the possibilities for each part: sigil lines (based on input text), archetype words, icons, and arc texts.

For an input with 4 unique consonants (like in "HELLO"), the sigil lines alone offer about 351,384 options (11⁴ × 4!, since each consonant maps to one of 11 positions and can be arranged in 24 ways).

Then, 167,960 combinations come from picking 4 out of 22 archetype words (22 × 21 × 20 × 19).

Next, 10,123,800 possibilities arise from selecting 4 icons (53 × 60 × 53 × 60) from two alternating pools.

Finally, 25,000 options stem from choosing 4 Arc texts (25 × 10 × 10 × 10) from four pools.

Multiplying these together (351,384 × 167,960 × 10,123,800 × 25,000) gives roughly 1.495 × 10²² total possibilities.

The chance of hitting the same outcome is 1 divided by this huge number, making repetition extremely unlikely with random inputs.

For inputs with more or fewer consonants, the total shifts (e.g., 10²⁰ to 10²⁴ for 3-5 consonants), but the odds remain astronomically small.

How the Sigil and Seal Are Drawn: Mathematical Process

The "Syncretica Sigil & Seal Creator" uses a combination of deterministic mapping (sigil lines) and geometric transformations to craft a unique design from user input. Here’s the math behind each step, based on the code.

1. Sigil Lines: From Input to Grid

- Input Processing (`processIntention`):

- Takes user input, converts to uppercase, removes vowels (A, E, I, O, U), and keeps unique consonants.

- Example: "HELLO" → H, L (E, O are vowels, L repeats once).

- Result: "HL".

- Prime Number Mapping (`computeModIndices`):

- Each consonant is assigned a prime number from `primeMap`:

- A: 2, B: 3, C: 5, ..., H: 19, ..., L: 37, ..., Z: 101.

- For "HL": H = 19, L = 37.

- Modulo 11 Transformation:

- Maps primes to indices using modulo 11 (since there are 11 predefined grid points):

- 19 % 11 = 8

- 37 % 11 = 4

- Result: Indices [8, 4].

- Grid Mapping:

- `blackDotPositions` defines 11 points on a 6x6 grid (scaled within a square):

- [3: [1, 3], 7: [1, 7], 15: [3, 1], 17: [3, 3], 19: [3, 5], 25: [4, 4], 31: [5, 3], 33: [5, 5], 35: [5, 7], 43: [7, 1], 47: [7, 5]].

- Index 8 → 35 [5, 7], Index 4 → 25 [4, 4].

- Result: Points [[5, 7], [4, 4]].

- Adjustments:

- Offset: Random shift (e.g., offset = 1 → [[4, 4], [5, 7]]).

- Reverse: 50% chance (e.g., stays [[5, 7], [4, 4]]).

- Line Segment:

- Connects points: [[5, 7], [4, 4]] → one segment.

- Mirroring:

- Reflects across a 6x6 grid (8x8 total canvas, but sigil uses 6x6):

- Horizontal flip (\( c' = 8 - c \)): [[5, 7], [4, 4]] → [[5, 1], [4, 4]].

- Vertical flip (\( r' = 8 - r \)): [[5, 7], [4, 4]] → [[3, 7], [4, 4]].

- Both (\( r' = 8 - r, c' = 8 - c \)): [[5, 7], [4, 4]] → [[3, 1], [4, 4]].

- Result: 4 segments (bl, h, v, hv).

- Grid-to-Pixel Conversion (`gridToBoxCoords`):

- Square size = 37% of canvas (`boxSide = baseSize * 0.37`).

- Cell size = \( boxSide / 6 \).

- For [5, 7] at \( boxLeft = centerX - boxSide/2 \), \( boxTop = centerY - boxSide/2 \):

- \( x = boxLeft + (7-1) \times cellSize \)

- \( y = boxTop + (5-1) \times cellSize \).

- If canvas = 800px, \( boxSide = 296 \), \( cellSize = 49.33 \), \( centerX = 400 \):

- \( x = 400 - 148 + 6 \times 49.33 = 252 + 296 = 548 \)

- \( y = 400 - 148 + 4 \times 49.33 = 252 + 197.32 = 449.32 \).

- Draws white lines (stroke weight 2) between points.

2. Blue Dots

- Placement: Marks start/end of each quadrant’s segment.

- For [[5, 7], [4, 4]]: Dots at [5, 7] and [4, 4] (and mirrored points).

- Total: Up to 8 dots (if unique across quadrants).

- Math:

- Same grid-to-pixel as above.

- Size = \( baseSize \times 0.033 \) (e.g., 26.4px for 800px canvas).

- Drawn as "◉" in blue (RGB: 0, 132, 255).

3. Square Outline

- Geometry:

- \( boxSide = baseSize \times 0.37 \) (e.g., 296px for 800px).

- Centered at \( (centerX, centerY) = (400, 400) \).

- Coordinates: \( (252, 252) \) to \( (548, 548) \).

- Animation: Traces perimeter (4 × 296 = 1184px) over time.

4. Archetype Words

- Placement:

- Top: \( (centerX, boxTop - boxSide \times 0.1) = (400, 222.4) \).

- Right: \( (boxLeft + boxSide + boxSide \times 0.1, centerY) = (577.6, 400) \).

- Bottom: \( (centerX, boxTop + boxSide + boxSide \times 0.1) = (400, 577.6) \).

- Left: \( (boxLeft - boxSide \times 0.1, centerY) = (222.4, 400) \).

- Text Size: \( baseSize \times 0.02 = 16px \) (for 800px).

- Rotation: 0°, 90°, 180°, -90° respectively.

5. Inner Circle

- Radius: \( innerRadius = baseSize \times 0.26 = 208px \) (for 800px).

- Diameter: 416px, centered at (400, 400).

- Math: \( x = 400 + 208 \cos(\theta), y = 400 + 208 \sin(\theta) \).

6. Arc Text and Icons

- Icons:

- Angles: 45°, 135°, 225°, 315° (π/4, 3π/4, 5π/4, 7π/4).

- Radius: \( dotRadius = baseSize \times 0.31 = 248px \).

- Positions:

- 45°: \( (400 + 248 \cos(π/4), 400 + 248 \sin(π/4)) \approx (575.36, 575.36) \).

- Size: \( baseSize \times 0.055 = 44px \).

- Arc Text:

- Angles: 90°, 180°, 270°, 0° (π/2, π, 3π/2, 0).

- Span: \( 0.03 \times \text{length} \) radians (e.g., "fræktəl" = 7 chars, span = 0.21 rad).

- Spacing: 1.1 × span = 0.231 rad.

- Start angle = center - span/2 (e.g., 90° - 0.1155 = 1.455 rad).

- Per char: \( \text{angle} = 1.455 + (i/6) \times 0.231 \).

7. Outer Circle

- Radius: \( outerRadius = baseSize \times 0.36 = 288px \).

- Diameter: 576px, centered at (400, 400).

- Pulse: Expands to \( 2.2 \times outerRadius = 633.6px \), alpha fades from 33 to 0.

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Summary

"Type 'HELLO,' and the sigil starts with H (19) and L (37), mapped via modulo 11 to grid points [5, 7] and [4, 4] on a 6x6 grid within a 296px square (for an 800px canvas). Lines connect these, mirrored four ways (e.g., [5, 1], [3, 7]). Blue dots mark ends, a 416px inner circle and 576px outer circle frame it, with icons at 248px radius (45°, 135°, etc.) and text arcing around 90°, 180°, etc. Each element scales with the canvas, rooted in precise math!"