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.
axis_explicit and subordinating_synthesis — and rebalanced weights so that section-header listing (e.g. **FEN — Phenomenon.**) no longer games coverage and entropy. Production formula and per-component definitions below.2. The seven 𝓦 components cov · ent · pair · tens · syn · axis · subord
One-line memory per component (read these once, then return to the formal definitions below):
- cov — how many topics you touch
- ent — whether you spread attention or concentrate on one corner
- pair — whether you oppose ideas explicitly (par + anti in the same paragraph)
- tens — whether you hold the opposition instead of dissolving it
- syn — whether you anchor synthesis on its required cardinal inputs
- axis — whether you name the dimensional axis you are judging in
- subord — whether you hierarchize (a frame doing something to another), not just enumerate
The anti-Goodhart structure. The first three (cov, ent, pair — total weight 0.30) are relatively fakeable by mechanical section-listing of cardinal codes. The last four (tens, syn, axis, subord — total weight 0.70) require actual argumentative structure and cannot be produced by a Lorem-ipsum-plus-cardinals attack (the canonical Goodhart attack documented 2026-05-07). This re-weighting is the central change between 𝓦 v1 and 𝓦 v2; it is also why the dual-criterion break requires 𝓗 ∧ 𝓦 ∧ 𝓕 to clear thresholds simultaneously — gaming any one component does not yield reward.
Mapping to the Global Method. Each component covers a phase of ANA→SIN→AMO→EXP:
- cov, ent → ANA (analysis of parts: how many, how spread)
- pair, tens → SIN (synthesis of relations: pairs and held tensions)
- syn, subord → AMO (compensated synthesis: anchoring + hierarchy)
- axis → meta-cognition (naming the dimensional axis explicitly)
cov coverage weight 0.05
Quick read. How many of the eight cardinal poles you touch. A list of all 8 maxes this trivially — that is why v2 weights it lowest.
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
OBJ, SUB, TEO, PRA, FEN, NOU, PLA, MON at least once each.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
Quick read. Whether the cardinals you touch are evenly spread, not concentrated on one or two. A balanced eight-section listing maxes this — it tells you the distribution is balanced, not that the reasoning is.
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
Quick read. Whether opposites appear together in the same paragraph. Putting "freedom" in paragraph 1 and "responsibility" in paragraph 5 doesn't count — that's sequential listing, not pairing. Saying them in the same breath does.
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
Quick read. Whether you hold the opposition instead of dissolving it. "X and Y seem opposed but…" (you hold the tension) scores higher than "X and Y are basically the same" (you dissolve it). The tension words must appear close to two cardinal codes from different quadrants.
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)
OBJ-measurable harm but impose a cost on SUB-felt mobility." → marker "but" within 80 chars of OBJ + SUB → tension event.Multilingual. Markers in English, Catalan, Spanish.
syn synthesis_anchoring weight 0.15
Quick read. Whether mediator codes (the bridge-codes like SIN, AMO, EXP…) appear together with their required cardinal inputs in the same paragraph. Naming SIN without TEO + PRA nearby is empty invocation; naming SIN with TEO + PRA anchors the synthesis claim.
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
SIN together with TEO + PRA — the synthesis claim is anchored on its required inputs.SIN but neither TEO nor PRA appear nearby — the mediator is named but not anchored.axis axis_explicit weight 0.15 · v2 NEW
Quick read. Whether the opening sentence names the axis the response is operating on. "This question sits on the OBJ–SUB axis…" makes the dialectical commitment visible from the start; "Today, AI regulation is debated…" defers or never makes it explicit.
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
Quick read. Whether one frame does something to another — subsumes, reframes, integrates, treats X as Y — instead of merely listing frames side-by-side. This is the hardest component to fake and the single strongest signal of performed (vs. enumerated) dialectic.
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.
| Axis | Negative pole | Positive pole | Type |
|---|---|---|---|
| D1 | OBJ — objective | SUB — subjective | Cartesian |
| D2 | TEO — theory | PRA — practice | Cartesian |
| D3 | NOU — noumenon | FEN — phenomenon | Cartesian |
| D4 | PLA — plasma (atemporality) | MON — world (temporality) | Radial — tempeternity |
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).
| Code | Cicle | Required cardinals (in same paragraph) | Quadrant |
|---|---|---|---|
| ANA — Analysis | Aplicació | FEN | FEN |
| SIN — Synthesis | Aplicació | TEO + at least one other | NOU |
| AMO — Love (orientation) | Aplicació | NOU + PRA | NOU |
| EXP — Experience | Aplicació | PRA + FEN | FEN |
| STM — Sentimental | Orientació | SUB + PRA | SUB |
| STT — Subjective–theoretical | Orientació | SUB + TEO | SUB |
| SGT — Objective–theoretical | Orientació | TEO + OBJ | OBJ |
| SGE — Objective–practical | Orientació | OBJ + PRA | OBJ |
| ART — Art | Coneixement | FEN + SUB | SUB |
| MTP — Metaphysics (interior) | Coneixement | SUB + NOU | SUB |
| MTF — Metaphysics (exterior) | Coneixement | NOU + OBJ | OBJ |
| CIE — Science | Coneixement | OBJ + FEN | OBJ |
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:
| Cardinal | Sample terms detected |
|---|---|
| SUB | subjective, self, inner, feeling, emotion, autonomy, consciousness, identity, perspective · subjectiu, jo, intern, sentiment, autonomia, consciència |
| OBJ | objective, external, structure, observable, measurable, data, statistic, institution · objectiu, extern, estructura, mesurable, dada, institució |
| TEO | theory, principle, abstract, framework, philosophical, rational · teoria, principi, marc, filosòfic |
| PRA | practice, action, application, implementation, behavior, concrete · pràctica, acció, aplicar, comportament |
| FEN | phenomenon, observable, immediate, manifest, event, surface · fenomen, immediat, aparició, esdeveniment |
| NOU | essence, meaning, depth, underlying, transcendent, metaphysical · essència, sentit, profund, transcendent, metafísic |
| PLA | atemporal, eternal, potential, seed, origin, source, primordial · etern, llavor, origen, font, primordial |
| MON | world, temporal, manifested, unfolded, historical, contingent, present · món, temporal, manifestat, històric, contingent |
Formula. ℑ = n_implicitly_touched_cardinals / 8, plus a normalised distribution pq = countq / total_count.
𝓗 vs ℑ — the distinction
| 𝓗 (harmonic completeness) | ℑ (implicit coverage) | |
|---|---|---|
| Detects | explicit canonical codes (OBJ, SUB, …) | natural-language terms (subjective, theory, …) |
| Requires | Meta-Globàlium vocabulary | any natural language (multilingual) |
| Auditable | yes (used as process-reward in the loop) | diagnostic only — soft signal |
| Bare LLM | 0 (no anchoring) | 0.3–0.5 (touches some themes) |
| Adversarial Lorem ipsum | ~0.97 (gamed with codes) | 0 (no real concepts) |
| Anchored response | ~0.97 (genuine anchoring) | ~1.0 (terms also present) |
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