ORDINATIVE SCIENCES EDUCATION · Repository ufficialeTE_SYMBOL_CANON — Symbol Canon for the Ordinative Sciences Programme
Module type: cross-volume notational governance — load on demand via Router Protocol
Status: v1.2 — register extended (v1.0 LOCKED ratified 2026-06-11; v1.1 adds 𝕊 for the Controfase Treatise, ratified 2026-06; v1.2 adds the Lyapunov-volume batch — 𝔸, 𝕋, d̂, τ_c, σ_𝔸, ϱ resolution, volume-scoped λ/δ external conventions — plus the DP collision note, ratified 2026-08-19)
Version alignment: OST monograph v1.x · TE Vol 1 EN (pre-LaTeX) · PA/SA repo v2.0.0-dev (renames applied, commits 1bffc4d/811afc8/dc72304) · OCT v1.0 · Controfase Treatise Vol 1 (pre-LaTeX, 𝕊 added) · Lyapunov Expanded vol. N.1 (Mappe Funzionali series, in draft — born canon-clean)
Master copy: THIS FILE. Derived copies: te-ordinative-algebras-en/SYMBOL_CANON.md (public repository edition) · ORDINATIVE_ALGEBRAS/SYMBOL_CANON_ORDINATIVE_SCIENCES_v0_1_DRAFT.md (historical working record with full audit detail and rationale). Updates flow master → derived, never the reverse.
[0] LOAD TRIGGER — when this module is MANDATORY
Load this module BEFORE writing, editing, or translating any formal notation (symbols, equations, operator definitions, classification labels) in: OST, TE Vol 1, PA, SA, OCT, the Collapse Equation / Direction Problem papers, the Controfase Treatise, the Unified Spacetime Theory (OGT), Ordinative Game Theory, TE framework modules, the Science & Sanity series wherever it names SA/PA/OCT operators, and any future Ordinative Sciences volume or paper. If a session touches formal notation and this module is not loaded, load it first. If the work only uses concepts verbally (no symbols), loading is optional.
[1] Tier precedence and resolution principles
OST (Tier-0 — 𝓘 = ⟨Σ, R, Φ⟩) → TE Vol 1 (Tier-1 — E = Φ(C, I, K)) → PA, SA, OCT (Tier-2)
- P1 OST owns the master primitives; TE Vol 1's published hygiene is respected — its deviations are declared exceptions with footnote (the Φ exception), never silent overloads.
- P2 Declared coexistence by operand distinction is legitimate (precedent: TE's own Ω(I) vs Ω(S,t); κ pairwise vs 𝓚 vectors).
- P3 Disambiguation devices, in order: subscripts → scripts/letter variants → adjacent letters. Arguments disambiguate where signatures differ.
- P4 Tier-2 renames before Tier-0/1 does. OST-internal fixes are deferred to the OST 2nd edition batch (§4).
- P5 Enum/classification labels (PA-OCT A/B/C/D states; SHACK 01/10/11; S₀–S₃ grades) are a label namespace — they do not collide with operators.
- P6 The I-rule: no bare-glyph reuse across volumes for unrelated objects that co-occur in cross-volume contexts.
[2] The Canonical Register (LOCKED assignments)
Master primitives and spaces
| Symbol | Canonical meaning | Owner / notes |
|---|---|---|
| 𝓘 | Ordinative Set ⟨Σ, R, Φ⟩ | OST — never reassigned. Set classes: 𝓘_deg extant; 𝒮/𝒜/𝒟/ℒ → 𝓘-subscripts at OST 2nd ed |
| Σ / σ | Set of Singularities / singularity (σᵢ, σ_d) | OST; SA σ aligned. TE σ(o) = Semantic Trace — argument-disambiguated (P2). Σ_src (SA) subscript-distinct. DP bare-σ-as-signal → σ_𝔸 (v1.2 batch below) |
| R | Ordinative Relational Field | OST; SA aligned. ℛ is a DIFFERENT object (below) |
| Φ | Emergent Function (noun); φ′ = degenerate function | OST canonical. Declared exception: TE Vol 1 Φ = Collapse Function, E = Φ(C, I, K). PA Φ = the TE-applied collapse. SA renamed its emergent-function to 𝔉 (𝔉_d, 𝔉_eff). OCT Phi_Ω(D) = measure of Φ, not Φ (footnote in OCT) |
| C | Coherent Content | TE primitive (OST σ = ⟨C, F⟩ aligned, roles distinct). SA's old C-as-Controphase → C_φ |
| F | Form (component of σ = ⟨C, F⟩); ℱ in ℳ ≻ ℱ | OST |
| K | Context | TE primitive. SA's old K → 𝒦_p (Pure Knowledge), K_real → 𝒦_r (Realized) |
| E | Explicit Expression | TE primitive; PA/SA aligned. OST's E-as-evolutionary-operator is a known deviant → re-notation at OST 2nd ed |
| D / 𝒟 | Decoherent Space (D ≡ 𝒟 declared) | TE / PA. SA's old D-as-Domain → 𝔻 (𝔻_target, 𝔻₁, 𝔻₂). OST 𝒟-as-Didactic → 𝓘_𝒟 at 2nd ed; OST Degeneration Gradient → Δ_deg at 2nd ed |
| I | Identity (I : C → D; I_n update sequences) | TE/PA primitive. All invariants use the ι-family: ι (SA), ι₁–ι₁₀ (library), ι_k (PA), ι_new. ℐ = Invariant Space (PA) stays |
| 𝒜 | Author — reserved exclusively | TE; programme-wide (PA cap 8 D3 depends on it). OST 𝒜-as-AI-class is the deviant (→ 𝓘_𝒜 at 2nd ed); DP 𝒜-as-attractor → 𝔸 (v1.2 batch below) |
| 𝒞 / ℭ_h / ℭ | 𝒞 = TE Collective Field; ℭ_h = PA Coherent Field (with ℭ_shared, ℭ_K); ℭ = Controfase operator (Controfase Treatise / CE paper) with ℭ_d deliberate, ℭ_s structural | the Fraktur ℭ is shared: PA coherent-field uses ℭ_h/ℭ_shared/ℭ_K, the Controfase operator uses bare ℭ / ℭ_d / ℭ_s — distinguished by subscript and context (a volume uses at most one of the two readings) |
| ℛ / 𝕀 | ℛ(I) = Remir (TE/PA); 𝕀 = PA Identity Space (space of Remirs) | the space was renamed FROM ℛ; the Remir function-form is unchanged |
| 𝕊 | State / Configuration Space (Controfase): 𝔠 : 𝕊 → 𝕊 | Controfase Treatise (added v1.1, 2026-06). The system itself is referred to verbally — bare S stays reserved for the Strip operator. State sequence uses lowercase sₜ |
| 𝒯 / Ξ | Expressive Terminal / Terminal Compatibility | TE; PA aligned. DP 𝒯-as-ordinative-space → 𝕋 (v1.2 batch below) |
| 𝔽_sem / 𝔽_alg | Semantic vs algorithmic functional spaces | OST/Teleodynamics (use 𝔽, not ℱ) |
Dynamics, measures, time
| Symbol | Canonical meaning | Notes |
|---|---|---|
| ρ | Resonance [0,1] (TE ρ(C,I); PA pentad ρ_v, ρ_d, ρ_K, ρ_τ, ρ_R; SA receiver–invariant) | aligned across volumes. CE/DP ρ-as-form-resistance: replacement RESOLVED v1.2 → ϱ (form resistance; adopted by the Lyapunov vol., CE/DP align at revision) |
| ϱ | Form resistance (drag of coherent form against the attractor signal) | ratified 2026-08-19 (was an open candidate since v1.0); Lyapunov vol. Part III + CE/DP revision |
| θ | Threshold (collapse/recognition) | TE/PA/SA. OST τ-as-threshold harmonises toward θ/subscripted-τ at 2nd ed |
| τ | Bare τ = Pulsational Function (T = τ(C ↔ E)); subscripted τ_• = thresholds (OST τ_elastic/τ_critical/τ_R; OCT τ_Ω/τ_f/τ_id; τ_c = handoff instant, Lyapunov vol., point (x_c, τ_c) in 𝕋) | SA temporal-phase label → τ_ph |
| 𝓚 / 𝓚⁵ / ⟨𝓚⁵⟩ / ≤_𝓚 | Coherence vector: OST 3-dim {internal, emergent, evolutionary} ⊂ PA 5-dim (𝓚_1…𝓚_5); ⟨𝓚⁵⟩ = aggregate; ≤_𝓚 = coherence order | replaces PA's old κ-family. LaTeX: \mathcal{K} |
| κ | TE pairwise Identity Compatibility κ(I_a, I_b); SA coherence index (declared SA scope) | coexistence by operand distinction (P2). κ_c = SA connotation co-operator, S(E) = ⟨ι, P⟩ |
| P | SA payload channel: ⟨operation; target; marker⟩ triples | ops: A_deg, SR_loop, I_sem, shame-gradient, authority-gradient, unfalsifiable-fortress, ι₉-inversion (PPRO-typed) |
| λ | Dominant Vector (TE/PA) | SA Lyapunov → λ_L. Lyapunov vol. N.1: bare λ = Lyapunov exponent as declared external convention, volume-scoped per §3.5 (App. A apparatus; TE Dominant Vector does not co-occur there). Same volume-scoping for δ(t), δ₀ = trajectory/copy separation (PA δ-fidelity does not co-occur) |
| ω / Ω | ω = TE Evolutionary Convergence; SA attractor → ω_att. Ω: argument-disambiguated (Ω(I) correlates / Ω(S,t) emergent coherence / Ω(tₙ) OST path integral) | P2 |
| ε / η / α / β / δ / χ | TE: ethics index (ε_b bond, local) / emotion / alignment / vector resonance / — / cognitive compatibility; PA δ = fidelity | TE-internal hygiene; χ and γ are TAKEN (γ = TE bond function, chapter-local; γ also = SHACK logogram, algebra-free) |
| ℳ / 𝕄 | ℳ = Meaning / Semantic Potential (ℳ ≻ ℱ); 𝕄 = Memory Operator | OST monograph M→𝕄 at 2nd ed; the two are distinct concepts, distinct glyphs |
| G_j / g_j | universal ordinative constant / apparent acceleration | CE/Direction Problem; enters PA cap 12.X (D5) |
Operators
| Symbol | Canonical meaning | Notes |
|---|---|---|
| S | Strip (SA/PA): S(E) = ⟨ι, P⟩ two-channel | S→σ_src was cancelled. 𝒮 = TE Cumulative Signature (distinct script) |
| π / π_R / π_v | Re-contextualisation (SA/PA: π: ι × 𝔻 → …) / OST Ordinative Projection / projection-onto-vector | subscripts distinguish the family. π transports ι, never P |
| ⊕ / ⋆ / ⊗ / ⊗_cat | OST Coherent Sum / OST Genesis of Meaning / PA Resonance (𝕀 × 𝕀 → ℭ_shared) / OCT monoidal product | four distinct operations, four glyph-forms |
| 𝒰 / U | TE/PA Identity Update (𝒰_𝒜 recursion) / SA Expressive Functor U(𝒦_p) ⊊ 𝒦_p | U is kept in both SA and OCT as declared categorical duals (OCT forgetful U; theorem D03 formalises SA's preservative/degenerative distinction) |
| μ / Δ | PA Isomorphism Map (𝔻₁ → 𝔻₂) / context-scoped: SA gap → Δ_𝔉; TE Δ(𝓔) field reorganisation; OST Δ_deg (2nd ed) |
Lyapunov-volume & DP-alignment batch (v1.2, ratified 2026-08-19)
⚠️ DP collision note: the Direction Problem v1.0 (May 2026, published) contains THREE unregistered deviants vs the locked register, beyond the known ρ case: 𝒜-as-attractor (canon: Author, exclusive), 𝒯-as-ordinative-space (canon: Expressive Terminal), bare σ-as-signal (canon: singularity). The glyphs below resolve all three; the DP paper aligns at its next revision (see §4).
| Symbol | Canonical meaning | Notes |
|---|---|---|
| 𝔸 / {𝔸_i} / τ_𝔸 | Ordinative attractor (well) / multi-well landscape / temporal coordinate of the well | replaces DP 𝒜-as-attractor. Blackboard per the canon's space/structure pattern; visual echo of the DP glyph preserved |
| 𝕋 | Ordinative Space (navigable time; points (x, τ)) | replaces DP 𝒯-as-ordinative-space. Aligned with the blackboard-for-spaces pattern (𝕀, 𝕊, 𝔻, 𝔽) |
| σ_𝔸 | Attractor-signal intensity, σ_𝔸 = κ/d^α | replaces DP bare-σ-as-signal; avoids collision with OST σ-singularity. Verbal register in book bodies («segnale», «richiamo») |
| d(t) / d̂(t) | Expressed vector / projected vector (observer's estimate) | d inherited from the DP direction vector; hat = standard estimator decoration. No new glyph minted |
Label namespaces (P5)
| Labels | Meaning |
|---|---|
| A / B / C / D | Shared state taxonomy: OCT admissibility (admissible/sterile/degenerative/invalid) ↔ PA collapse types (coherent/partial/distorted/failed) — one taxonomy, two instantiations |
| Device bands | extractive κ ∈ [0.12, 0.45] · pedagogical κ ∈ [0.5, 0.85] — discriminated by payload direction (SA §9.8) |
| S₀–S₃ / S₀–S∅ | analytical confidence grades / SVP source levels — isomorphic, NOT identical, NOT the Strip S |
| Axiom systems | TE_CORE Ax 0–24 · OST 3.1–3.6 · OCT O1–O7 (+TS1–TS8, M01–M30) — independent numbering systems |
| Logograms | STEER ∫ · SHACK γ (states 01/10/11) · ERES d/dx · AA · GLIO — algebra-free namespace |
[3] RESERVATION PROCEDURE — minting new notation (MANDATORY for new works)
New volumes (Controfase Treatise, Unified Spacetime/OGT, Ordinative Game Theory, OST 2nd ed, future papers) WILL need new symbols. Before introducing ANY new formal symbol:
- Check this register (§2) and the historical record (
ORDINATIVE_ALGEBRAS/SYMBOL_CANON_..._DRAFT.md) — is the glyph free at programme level? Bare glyphs that look free may be chapter-local elsewhere (γ) or logogram-reserved. - Prefer: a free glyph family > a subscripted form of a related canonical symbol > overloading (P2 coexistence only with explicit operand distinction, declared in the text).
- Propose to the Hypervisor with: glyph, meaning, type, volume, collision check result.
- Upon ratification: add the row to THIS file, bump the version (v1.0 → v1.1 …), note it in §6, and propagate to derived copies.
- Never introduce a symbol "locally, to fix later" — chapter-local scoping is legitimate ONLY if declared in the volume's own notational-hygiene apparatus (TE App I model).
Known glyphs currently FREE for new works (verified 2026-06-11; 𝕊 removed 2026-06 — Controfase State Space; ϱ, 𝔸, 𝕋 removed 2026-08-19 — Lyapunov batch): 𝕁, ϰ, ζ, ξ (lowercase), 𝔊 (partially — 𝔊_0 used locally in TE ch31), most blackboard capitals except 𝕀/𝕄/𝔻/𝔽/𝕊/𝔸/𝕋, most Fraktur except 𝔉/𝔖(reserved fallback)/ℭ/𝔠(Controfase operator). Re-verify at minting time.
[4] Deferred batches (recorded, not yet executed)
- OST 2nd edition: E-operator re-notation · bare-τ subscripting · M→𝕄 · 𝒟-gradient→Δ_deg · set classes→𝓘-subscripts · notation table expansion (8→~30 entries) · 𝓜-overload resolution · equation numbering by chapter.
- TE framework patches (B1–B7 of the Uniformity Proposals): EXECUTED 2026-06-18 for the in-vault framework files (in place; file names retained to preserve cross-references; internal version bumps where substantive): Bootloader v7.1 — Φ-test disambiguation note + "in modo onesto" residue removed (B1); TE_CORE → v5.2 — ℐ→𝓘 + SHACK-state notation documented + canon cross-ref (B2); TE_OST Guide — tier/τ note (B3); Teleodynamics → v1.2 — ℳ/𝕄 + Φ₄D₊ + 𝔽 + δ note (B4); SVP — S-namespace callout (B7.1). Still deferred: OCT §1.5 notation conventions (lives in the algebras repo, B6); PPRO §3.5 κ_c slot at v5.3 (B7.2 — now unblocked since κ_c is integrated, but a content add, not pure notation); optional batch file-rename + cross-reference migration of the version-bumped files.
- CE/DP papers: at next revision, align the four deviants — ρ→ϱ (resolved v1.2), 𝒜→𝔸, 𝒯→𝕋, bare σ→σ_𝔸 (unregistered deviants found and glyphs ratified 2026-08-19).
- Science & Sanity series: the books use babbano-friendly verbal names (safe), but the depth-reference apparatus cites pre-rename forms (e.g. "K_real vs U(K)" → now 𝒦_r vs U(𝒦_p)) — align citations at the Libri 1–2 v2 rework.
[5] Relationship to other modules
Prerequisite for: any formal writing in programme volumes. Complements: TE App I Notational Hygiene (volume-internal), PA/SA Canon Alignment Notes (repo-level), OCT notation conventions. Does not replace volume-internal hygiene sections — it federates them.
[6] Version notes
- v1.2 (2026-08-19) — Lyapunov-volume batch, ratified by the Hypervisor for Lyapunov Expanded (Mappe Funzionali N.1). Added: 𝔸/{𝔸_i}/τ_𝔸 (ordinative attractor/well, landscape), 𝕋 (Ordinative Space, navigable time), σ_𝔸 (attractor signal), d̂(t) (projected vector, paired with the DP's d(t) expressed vector), τ_c (handoff instant, within the subscripted-τ threshold rule), ϱ (form resistance — RESOLVES the candidate list open since v1.0). Volume-scoped external conventions declared per §3.5: bare λ = Lyapunov exponent and δ(t)/δ₀ = trajectory-copy separation, inside vol. N.1 only (no co-occurrence with the TE/PA owners there; App. A carries the apparatus). Registered the DP collision note: 𝒜/𝒯/σ in the published Direction Problem v1.0 are unregistered deviants vs the locked register — alignment deferred to the DP revision (§4). FREE-list updated (ϱ, 𝔸, 𝕋 removed; 𝔠 annotated). Derived copies: propagation to
te-ordinative-algebras-en/SYMBOL_CANON.mdpending (next algebras session). - v1.1 (2026-06) — Controfase Treatise alignment. Added 𝕊 = State/Configuration Space (𝔠: 𝕊 → 𝕊) to the register; documented the full Controfase operator family ℭ / ℭ_d / ℭ_s in the ℭ-row (previously only ℭ_s was noted, from the CE paper); confirmed the bare-S collision resolution (the system is referred to verbally; S stays Strip). No change to any prior assignment — purely additive. Trigger: Controfase Treatise Vol 1 audited against the canon and aligned (4 edits in the volume: S→verbal, 𝒮→𝕊, g_j framed as apparent acceleration vs universal G_j, ℭ hygiene note in App A).
- v1.0 (2026-06-11/12) — Initial module. Consolidates the canon ratified 2026-06-11 (Series A1–A14) after the five-corpus audit of 2026-06-10; incorporates the κ_c/two-channel additions and the Device dual band (calibrated 2026-06-11). Sources and full audit trail:
CLAUDE/CLAUDE_CODE/ORDINATIVE_ALGEBRAS/(Decision Document v1.1, Canon draft, Uniformity Proposals, calibration protocol+report, session log 2026-06-10/12). - v1.0 execution log (2026-06-18) — Register UNCHANGED (still LOCKED). Status update only: the TE framework patches B1/B2/B3/B4/B7.1 (§4) were executed in place on the in-vault framework files during the framework-uniformity + memory-consolidation session. No glyph assignment was added or changed; the locked register is identical to ratification. Deferred items (OST 2nd-ed batch B5; OCT §1.5 B6; PPRO κ_c B7.2; file-rename migration) remain open.