DMI · Foundational Concepts
Invariants, Degrees of Freedom, and the Three-Axis Classification
The foundational layer of Dimensional & Metric Intelligence: what survives transformation (invariants), what is free to change (degrees of freedom), and why every engine carries three independent classifications — a generative class, a metric class, and an invariant class.
The ladder is fundamentally geometric: each rung is an orthogonal axis, and orthogonality is a pure structural fact. The metric is what you read off the ladder, not what it is made of. Each step introduces one new dimensional distinction, independent of any domain.
- Step 1
Presence
There is a difference — otherness vs none.
- Step 2
Direction of variation
A line along which states vary.
- Step 3
Independence
A second variation that does not blend with the first (orthogonality).
- Step 4
Order
Before/after on a rung, without distance.
- Step 5
Comparability
Positions read against each other — the birth of the relational reading.
- Step 6
Magnitude
Comparison becomes graded; units become possible as a derived reading.
- Step 7
Curvature
The rung's direction itself varies — the invariant of curved metrics.
- Step 8
Composition
Superposition of rungs in fixed proportion (diagonality).
- Step 9
Periodicity
Variation that returns; frequency as its own dimension.
- Step 10
Phase
The relative setting of two periodicities — a relation between relations.
- Step 11
Coherence
Variations that hold together across a whole configuration.
- Step 12
Stability
Persistence under change of reading.
- Step 13
Recursion
Dimensional structure becomes the content of a dimension.
Each step is orthogonal to all below it; the ladder is uniform across all domains — they differ only in which rung they emphasise and how they read magnitude.
Every engine holds one structural fact fixed and varies exactly one thing. The invariant is what survives all allowed transformations; the degree of freedom is what the transformation is allowed to change. For the alternation family: the invariant is the figure under rotation, the degree of freedom is the phase.
The kinematic core of the engine suite reduces to a single principle: half-cycle phase alternation of an n-fold rotational framework. Each framework steps a figure through a rotation of 180°/n — half the fundamental sector — so the two phases are antipodal within the fundamental domain. Square↔Diamond is the minimal manifestation: two phases, one 45° step, 4-fold symmetry. A triangle cannot alternate at all — rotated 120° it lands indistinguishably on itself.
The whole suite reduces to the more general rule: hold one structural fact fixed, vary exactly one thing, read what survives. Configurational readers (DCO, DII in Orthogonal Diagonality mode) hold the object fixed and vary the reading axis within it. Relational readers (Alignment, AHA Series Comparator, DII in Curved Multi-Dimensionality mode) hold the objects fixed and vary the relation between them. Half-cycle n-fold alternation is the purest, most minimal instance of this rule.
Orientation becomes an invariant exactly when the transformation group available to the reader does not contain the operation that would flip it. Chirality is the first example of this class: in the pentagon, clockwise and counter-clockwise phase sequences are related only by reflection, and reflection is not in the rotation group. Orientation has crossed over from the phase block into the invariant block.
Chirality (pentagon)
Clockwise and counter-clockwise phase sequences are related only by reflection — and reflection is not in the rotation group. The first occurrence of the class in the polygon family.
Parity of permutations
Even vs odd orderings: no allowed swap sequence turns one into the other. Orientation as invariant of sequence.
Handedness of spirals and helices
Left- vs right-handed: no rotation superimposes them. Chirality lifted from polygons to periodic structures.
Oriented cycles
Directed loops where traversal direction is structural; reversing it produces a different reading, not the same one re-phased.
Chromatic direction
The 0–180° half-cycle has a direction of hue travel; two series shifting in opposite directions are mirror readings, not phase-shifted twins.
Consequence for RPG and SC: the Reference Record needs a general orientation flag, not a pentagon special case. SC's rule generalises cleanly — phase is always ignorable; orientation is ignorable only when the framework's group contains the flip.
Every engine carries three independent classifications. Generative class describes construction — can the figure be generated by a smaller rotation framework? Metric class describes reading — into how many independent parameter sets does the figure's measurement resolve? Invariant class describes what survives transformation and comparison — when are two readings the same object? ADI needs all three: its classification act is fundamentally about identity under transformation, and neither the generative nor the metric axis can answer that alone.
Generative Class
How the framework is generated
Defined by its symmetry group and containment relationships.
Example: the pentagon is rotationally irreducible because 5-fold symmetry contains no smaller rotational subgroup.
Metric Class
How the framework can be measured
Defined by the independent parameter systems through which it can be read.
Example: the pentagon decomposes into orthogonal, diagonal, radial and harmonic parameter sets.
Invariant Class
What survives transformation
Defined by the properties that remain unchanged under the framework's allowed transformation group.
Example: 5-fold symmetry remains invariant across phase shifts, while orientation becomes invariant once reflection is excluded.
DMI Principle
Parameters are dimensional symbols.
Dimensional operations produce metrics.
Metrics reveal invariants.
Invariants define classes.
The pentagon is rotationally irreducible but metrically decomposable.
The pentagon is rotationally irreducible — yet metrically decomposable into four orthogonal parameter sets. Each set is complete (any one reconstructs the whole figure) and independent (no metric in one set is derivable from another without leaving that set's operations):
Orthogonal set
Vertex coordinates in (x, y): five points on a circle, fully specified by R and two axis projections. The figure lives in a 2-D orthogonal frame like any other.
Diagonal set
The pentagram: diagonal ÷ side = φ, and each diagonal intersection divides in φ again. A self-contained ratio system — the pentagon's characteristic metric content, invisible to the orthogonal set.
Radial set
Outer radius R, inner radius r = R·cos 36°, the 72° sector, five spokes. The figure as a polar construction, independent of both Cartesian axes and diagonals.
Harmonic set
The pentagon as one cycle of a 5-periodic structure: fundamental frequency 1/5 of a turn, phase positions k·72°, and the half-cycle step 36° as a beat against the fundamental. The set the alternation engine actually runs on.
| Framework | Invariant | Degree of freedom | Generative class | Metric class | Invariant class | Orientation |
|---|---|---|---|---|---|---|
| Square ↔ Diamond4-fold | Four-fold symmetry; diagonal ÷ side = √2; inner ÷ outer radius = cos 45° ≈ 0.7071 | 45° half-cycle step — edge-aligned square ↔ vertex-aligned diamond | Reducible — contains the 2-fold sub-rotation (point symmetry) | Poor — √2 is its only characteristic ratio | Phase-free — both phases are the same object; orientation flips inside the group | Ignorable — the group contains the flip |
| Pentagon5-fold | Five-fold symmetry; diagonal ÷ side = φ ≈ 1.618; inner ÷ outer = cos 36° ≈ 0.8090 | 36° half-cycle step — point-up ↔ point-down (mirror pair) | Irreducible — 5 is prime; no sub-framework, no point symmetry | Richest of the family — φ saturates the diagonal set (pentagram) | Orientation-bearing — mirror pairs are different objects; chirality survives every allowed transformation | Load-bearing — chirality is part of the invariant block |
| Hexagon6-fold | Six-fold symmetry; edge = outer radius (ratio 1); inner ÷ outer = √3/2 ≈ 0.8660 | 30° half-cycle step — flat-top ↔ pointy-top | Reducible — contains 2-fold and 3-fold sub-rotations | Moderate — ratio 1 and √3/2; plane-tiling ability | Phase-free — both phases are the same object; tiling identity survives alternation | Ignorable — the group contains the flip |
| Octagon8-fold | Eight-fold symmetry; silver ratio 1+√2 ≈ 2.414; inner ÷ outer = cos 22.5° ≈ 0.9239 | 22.5° half-cycle step — two Square↔Diamond pairs folded into one | Reducible — contains the four-fold framework (the prototype) | Rich — silver ratio plus the embedded square's √2 | Phase-free — but containment is not identity: an octagon reading never classifies as a square | Ignorable — the group contains the flip |
Irreducibility and decomposability do not even correlate: the square is rotationally reducible and metrically poor; the octagon is reducible yet rich; the pentagon — the engine you cannot simplify by rotation — is precisely the one that most rewards being decomposed by measurement.
RPG primarily visualises invariant classes — it constructs figures from their group and animates the phase so you can see what stays the same. SC primarily compares invariant classes — it normalises every reading back to phase 0, then compares only the invariant block (fold order, inner ÷ outer radius, characteristic ratio, flags) within tolerance. ADI classifies on all three axes: the generative class tells it how a figure is built, the metric class how it is measured, and the invariant class when two readings are the same object. The three-axis classification is what keeps their jobs apart.