Grid and Coordinate System

MTP (Mapping the Prompt) is a framework for shaping prompt behavior without relying on explicit instructive prose, using a dynamic model with two formal layers: Space and Intensity. Space determines axis and zone from grid position; Intensity determines strength, polarity, and tiered constraint extraction (Volcano mapping for coordinates).

Multiple tokens in one /mtp payload (including preset expansion) add ordered constraint blocks: each token is still mapped through the same Space and Intensity rules; order is not a third parameter space or dedicated transform layer.

Layer	Controls	Key concept
Space	Where — which axis/zone	19×19 grid, 3×3 macro zones, hue cycle
Intensity	How much — strength and polarity	0–100 slider, Side A / Side B duality, Volcano

This document defines the mathematical and geometric structures that make up MTP, along with the internal parameter model used to inject metadata into prompts.

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Spatial structure: 19×19 grid system

The core MTP interface uses an integer coordinate system at line intersections, similar to a Go board. The grid is 19×19.

Symmetrical three-way division model

The space is always divided into a geometrically balanced 3×3 macro-zone structure. Rather than simply splitting the number of elements into three uneven groups, MTP treats the whole surface as continuous space and places grid lines and boundaries in a fully symmetrical arrangement.

Structure from 19 grid lines

There are 19 grid lines in total. Four of them are boundaries (outer frame and between zones); inside them lie three equal-width macro zones in sequence, each containing five grid lines.

        A – F        G – L        M – S
 1–6   Yellow    |    Red      | Magenta
 7–12  Green     | Transparent | White
13–19  Cyan      |    Blue     | Purple

• Line 1: Boundary line (outer frame)
• Lines 2–6: Zone 1 (5 grid lines)
• Line 7: Boundary line (inner boundary)
• Lines 8–12: Zone 2 (5 grid lines)
• Line 13: Boundary line (inner boundary)
• Lines 14–18: Zone 3 (5 grid lines)
• Line 19: Boundary line (outer frame)

Boundary lines and zone assignment

Boundary lines are not sharp partitions; conceptually—like a gradient—they are transition regions between macro zones. The compiler assigns each coordinate (including boundary coordinates) to exactly one of the nine macro-zone cells (zone bounds are defined by the compiler) and computes intensity with the same Chebyshev distance formula as for interior coordinates.

Hue cycle and Z-order

The nine macro zones correspond to a hue cycle and also carry a generation transition order, or Z-order. Together they suggest the order in which concepts unfold and which phases processing passes through.

1. Yellow (Open: divergence, openness, margin)
2. Red (Power: force, assertion, rise)
3. Magenta (Return: return, flip, cycle)
4. Green (Grow: growth, proliferation, layering)
5. Transparent (Helix: helix, unfold, neutral structure)
6. White (Focus: focus, convergence, precision)
7. Cyan (Enter: entry, landing, structure)
8. Blue (Flow: flow, connection, rhythm)
9. Purple (Close: close, completion, wrap)

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Intensity structure: three-state model and bipolarity

Each node is not a simple on/off switch, but a continuous intensity slider from 0 to 100. Polarity—positive or negative—determines whether Side A’s behavior is active or Side B’s inverted behavior is active.

The compiler also accepts macro-zone color names as slider aliases (for example, yellow:50 ≡ open:50, yellow:-50 ≡ still:50), using the same linear intensity mapping as the paired Side A / Side B nodes.

Node symmetry (D4 symmetry) and pairing

Negative values activate Side B, while positive values activate Side A. Across the central baseline of 0, they form the following dual pairs.

Side B (-1)	Center (0)	Side A (+1)	Spatial Position
Still	Yellow	Open	Upper left
Void	Red	Power	Top
Surge	Magenta	Return	Upper right
Wither	Green	Grow	Left
Collapse	Transparent	Helix	Center
Haze	White	Focus	Right
Drift	Cyan	Enter	Lower left
Abyss	Blue	Flow	Bottom
Fade	Purple	Close	Lower right

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Coordinate-to-intensity transformation model

Coordinates such as A:1 or J:10 on the 19×19 grid (hyphen form A-1 is also accepted) are mapped by the MTP compiler to polarity (Side A / Side B) and intensity (0–100) using Chebyshev distance from the grid center J:10.

Polarity and intensity calculation algorithm (Volcano model, Chebyshev distance)

This transformation follows a Volcano model: the center is neutral, the mid-ring reaches Side A peak, and the outer frame inverts into Side B. The model uses Chebyshev distance so all macro-zone centers at the same distance share the same signed value.

Volcano model cross-section (example: row 10, from A to S):

Signed value
        Side A
 +100 |          /\                 /\
      |         /  \               /  \
      |        /    \             /    \
    0 |-------/------\-----0-----/------\-------
      |      /     Neutral Center        \
      |     /                             \
 -100 |----/                               \----
        Side B

Distance:  9      6        0        6      9
Coord:    A:10   D:10     J:10     P:10   S:10

Note: this is a one-dimensional cross-section (row 10). In full, the field is a 2D Chebyshev-distance ring around J:10—visually, a donut-shaped ring around the center.

• Center (distance 0): J:10 -> signed value 0 (neutral, no constraint emitted)
• Peak ring (distance 6): for example, D:10, D:4, P:16 -> Side A maximum (+100)
• Outer frame (distance 9): for example, A:10, A:1, S:19 -> Side B maximum (-100)

Compiler formulas:

Center (J:10)      = neutral (signed 0; no constraint)
Peak (distance 6)  = Side A maximum (signed +100)
Outer (distance 9) = Side B maximum (signed -100)

distance  = max(|x - 10|, |y - 10|)   (Chebyshev, integer 0–9)
Inner (distance ≤ 6):  signed = (distance / 6) * 100
Outer (distance > 6):  signed = 100 - 200 * (distance - 6) / 3

polarity  = +1 if signed >= 0 else -1
intensity = abs(signed), clamped to 1-100; 0 -> no constraint emitted

From center distance and macro-zone direction (axis), this mapping uniquely determines polarity and intensity; those values are injected into prompt metadata.

The grid model is visualized with scripts/mtp_grid_generator.py. Regeneration steps and SVG details are covered in Color Grid Visualization.