What every metric and code means

1. Metrics — 𝓗 and 𝓦 harmonic completeness + wisdom score

The structural verifier projects every AI response onto the Meta-Globàlium ontology and computes two complementary measures.

𝓗 — harmonic completeness coverage / breadth

What it measures. How many of the eight cardinal poles the response touches, balanced for distribution. A response that touches all eight poles in a roughly even distribution gets 𝓗 ≈ 1; one that collapses to a single pole gets 𝓗 ≈ 0.

Domain. [0, 1].

What it does not measure. Whether the breadth is genuinely dialectical or just enumerated. A "list of perspectives" can max 𝓗 without performing any synthesis. This is exactly why we added 𝓦.

𝓦 — wisdom score relational depth

What it measures. Whether the response performs structurally articulated dialectical reasoning, not just covers ground. 𝓦 is sensitive to how the cardinals are deployed: are opposites paired, tensions named, mediators applied, axes framed?

Domain. [0, 1].

Discriminating power. 𝓦 separates list-form responses (~0.60), dialectical responses (~0.85), and adversarial reward-hacking responses (~0.10) — see the demo for live numbers across three domains.

2. The seven 𝓦 components cov · ent · pair · tens · syn · axis · subord

cov coverage weight 0.05

Definition. Fraction of the eight cardinal poles (PLA, MON, SUB, OBJ, TEO, PRA, FEN, NOU) that have at least one canonical-code citation in the response.

Formula. n_distinct_quadrants_cited / 8

Why low weight in v2. Section-header listing trivially maxes coverage. By itself, coverage is a weak signal of dialectical engagement.

ent entropy_normalized weight 0.05

Definition. Shannon entropy of the citation distribution across the eight cardinals, normalized by log(8). Measures whether attention is spread evenly or concentrated on a few poles.

Formula. −Σ pq · log(pq) / log(8) where pq is the fraction of citations belonging to quadrant q.

Why low weight in v2. Like coverage, entropy is gamed by balanced section-listing. The metric tells you the distribution is balanced, not that the reasoning is.

pair dialectical_pair_density weight 0.20

Definition. For each of the four polar axes (OBJ↔SUB, TEO↔PRA, NOU↔FEN, PLA↔MON), score 1 if both poles appear cited in the same paragraph. Co-occurrence across paragraph breaks does not count — that is sequential listing, not dialectical pairing.

Formula. n_paired_axes / 4

Why moderate weight in v2. Strong signal of dialectical structure, but section-header conventions like **ANA — Analysis (FEN→TEO).** can put both poles in one paragraph mechanically without performing the dialectic.

tens tension_density weight 0.20

Definition. Counts tension markers (but, however, yet, despite, in tension with, on the other hand, però, sin embargo, mentre que…) that appear within ±80 characters of two canonical codes from different quadrants. Saturates at 4 events.

Formula. min(1, n_tension_events / 4)

Multilingual. Markers in English, Catalan, Spanish.

syn synthesis_anchoring weight 0.15

Definition. For each mediator code present (ANA, SIN, AMO, EXP, STM, STT, SGT, SGE, ART, MTP, MTF, CIE), check whether its required input cardinals appear in the same paragraph. Average over mediators present. Required cardinals per mediator are defined in the table below (§4).

Formula. n_anchored_mediators / n_present_mediators

axis axis_explicit weight 0.15 · v2 NEW

Definition. Binary [0, 1]. Detects whether the response opens (first ~600 characters) by explicitly framing the dialectical axis it sits on. A response that names its axis at the first sentence shows the dialectical move at the front; one that opens with FEN data does not.

Formula. 1 if any of the explicit-axis patterns matches in the opening; 0 otherwise.

Patterns recognized. "OBJ-SUB axis", "axis between X and Y", "tension between X and Y", "sits on the X-Y axis", "presents an axis", "two readings/frames/views", "competing X and Y", explicit cardinal pairings (X↔Y, X-Y, X/Y).

Why introduced in v2. Dialectical articulation tends to begin by naming its axis. List-form responses defer (or never) name an axis — they enumerate.

subord subordinating_synthesis weight 0.20 · v2 NEW

Definition. Counts active mediation patterns: a frame, reading, or mediator doing something to another (subsuming, reframing, foregrounding, integrating, treating X as Y, preserving while restricting…). Plus chained-attribution patterns "X (frame1) Y (frame2) Z (frame3)". Saturates at 3 events.

Formula. min(1, n_subordinating_events / 3)

Verbs recognized. subsumes, reframes, foregrounds, dissolves, integrates, reconciles, absorbs, encompasses, enables, articulates, presses harder, makes possible, treats X as Y, preserves X while/by/without, answers the, addresses the, composes across, coexist as.

Why introduced in v2. The qualitative gap between "list of perspectives" and "performed dialectic" is precisely the difference between naming frames and showing them act on each other. This component captures that.

3. Cardinal poles (8) PLA · MON · SUB · OBJ · TEO · PRA · FEN · NOU

The eight fundamental poles of the Meta-Globàlium hypersphere. Four orthogonal dialectical axes — three Cartesian, one radial — define the structure on which every response is projected.

4. Mediators (12) 12 axis-bridging codes per cicle

Mediators sit on the axes between cardinals — they perform the dialectical bridging operation. Three groups of four, one per cicle (cycle of the Global Method).

The required-cardinals column drives the syn (synthesis_anchoring) component: a mediator only counts as anchored when its required inputs appear in the same paragraph as the mediator itself.

5. Cicles (3) cycles of the Global Method

Three cycles of the Global Method, each with its own four mediators and characteristic question type. The system prompt selects the appropriate cicle for each user query.

Cicle de l'Aplicació · application cycle

Mediators: ANA, SIN, AMO, EXP. For questions like "How should we act?" — the cycle goes from observed phenomena (FEN) through analysis to synthesis (NOU), to value-laden orientation (AMO), to applied practice (PRA + FEN). The canonical Global Method cycle.

Cicle de l'Orientació · orientation cycle

Mediators: STM, STT, SGT, SGE. For questions like "How should I direct myself?" — focuses on the SUB ↔ OBJ axis, navigating subjective and objective stances toward decision and direction.

Cicle del Coneixement · knowledge cycle

Mediators: ART, MTP, MTF, CIE. For questions like "How do we know?" — focuses on the FEN ↔ NOU axis (appearance ↔ depth), traversed through art, metaphysics, and science.

The system prompt at api.arkadium.ai/data/meta_globalium_system_prompt.txt classifies the user's question by type and selects the appropriate cicle before generating the response.

6. Implicit coverage ℑ semantic touch · diagnostic complement to 𝓗

A diagnostic metric that detects which cardinal poles a response touches conceptually (via natural-language terms), independent of whether the response uses canonical Meta-Globàlium codes. ℑ is complementary to 𝓗 — it does not replace it.

Why ℑ exists

The structural verifier 𝓗 only counts citations of canonical codes (OBJ, SUB, TEO, …) — by design, since the Meta-Globàlium is a shared vocabulary between human and machine. A response that does not use the vocabulary is not auditable in 𝓗 terms — and that is correct, not a bug.

However, this creates an interpretation problem at the demo: the bare LLM panel shows 𝓗 = 0 and an empty compass, which can read as "this response is empty" when in fact the response is fluent prose touching many quadrants conceptually. ℑ closes this gap: it shows what the response touches semantically, so the visitor sees the difference between touching themes and operating on a shared structure.

How ℑ is computed

For each cardinal, a curated lexicon of natural-language terms (multilingual EN/CA/ES) is checked against the response with case-insensitive word-boundary matching. Examples:

Formula. ℑ = n_implicitly_touched_cardinals / 8, plus a normalised distribution pq = countq / total_count.

𝓗 vs ℑ — the distinction

The diagnostic signal: 𝓗 vs ℑ correlation

The relation between 𝓗 and ℑ is itself a useful classifier:

𝓗 high · ℑ high — Genuinely anchored

Response uses both vocabulary and concepts. Verifiable, structurally articulated.

𝓗 high · ℑ low — Adversarial / gamed

Response cites codes but doesn't talk about real themes. Reward-hacking signature: cite codes without engaging with their concepts. ℑ catches this when 𝓗 cannot.

𝓗 low · ℑ high — Bare LLM (no anchor)

Response engages themes in natural language but doesn't use the shared vocabulary. The verifier has nothing to audit, but the content is real. The structural problem of §2.b is visible here: the response cannot be projected onto a navigable map.

𝓗 low · ℑ low — Empty or off-topic

Response neither uses the vocabulary nor touches the themes.

Important caveat. ℑ is intentionally a soft signal. Term-based detection has false positives (a response can mention "subjective" without doing subject-pole work) and false negatives (a response can do subject-pole work using vocabulary that isn't in the lexicon). It is not a substitute for 𝓗 — it is a visual aid that shows the visitor where the bare LLM stands relative to anchored responses.

References

Paper §4.1, §4.5, §9.5

Formal definitions and worked examples. arkadium.ai/papers/arkadium/en/

Source code

Verifier and wisdom score implementation. github.com/opengea/arkadium · Apache 2.0

API endpoints

https://api.arkadium.ai/wisdom_score.php · https://api.arkadium.ai/verifier.php

Live demo

Side-by-side comparison of bare LLM, list-form, dialectical, and adversarial responses. arkadium.ai/demo