Semantic Algebra — Foundations

A Formal Method for Extracting Universal Structural Invariants from Natural Language

Author: Fabio Ghioni
Framework: Technology of Expressions (TE) — Ordinative Sciences
Date: April 2026
Status: Working draft

Prologue

Every wisdom tradition, every scientific discipline, every philosophical school has produced expressions that its practitioners recognize as profound. Many of these expressions, when examined across traditions, appear to say the same thing — yet the traditions themselves rarely recognize this convergence. A Taoist master and a mathematical logician would not typically agree that they are making the same claim. And yet, when the domain-specific vocabulary is removed, the structural content is often identical.

This text formalizes the method by which such convergence can be detected, verified, and operationalized. The method is called Semantic Algebra — not because it reduces meaning to symbols, but because it reveals the algebraic structure that natural language conceals behind domain binding.

The core discovery is simple: natural language is not the native format of knowledge. It is a lossy compression added for inter-terminal communication. The nervous system operates natively in simultaneous, geometry-based structures. When these structures are expressed in sequential natural language, two things happen: the domain of the speaker is attached (binding), and everything that does not fit the selected vector is lost (forgetting). Semantic Algebra reverses this process — not by reconstructing what was lost (which is impossible), but by stripping the domain binding to reveal whatever structural content survives.

What survives — if anything — is called an invariant: a function that does not change under change of domain.

I. The Problem

1.1 Natural Language as Lossy Compression

The dominant assumption in linguistics, philosophy, and AI is that natural language is the primary medium of thought. This assumption is structurally incorrect.

Consider what happens when a human being experiences an insight. The experience is simultaneous, multi-dimensional, and complete — the entire structure is present at once. When the same human attempts to communicate this insight, they must serialize it: one word after another, in a single sequence, selecting one vector of presentation from the infinite possible vectors. The result is a sentence. The sentence is not the insight. The sentence is a projection of the insight onto a communicative vector.

What is lost in this projection is not noise — it is everything that does not lie on the selected vector. The insight contained N dimensions. The sentence selects one. The remaining N-1 dimensions are not preserved in the output.

This is not a defect of language. It is a structural consequence of the transformation from simultaneous (coherent) to sequential (decoherent) representation. Every act of expression is an act of forgetting.

1.2 Domain Binding

When a Taoist sage says "The Tao that can be told is not the eternal Tao," the structural content of the expression is universal — but the word "Tao" binds it to a specific tradition. A physicist hearing this sentence filters it through the Taoism domain and either resonates (if sympathetic to Eastern philosophy) or dismisses (if not). The domain binding prevents the physicist from recognizing that the same structural law governs their own discipline: no formal system can prove its own consistency (Gödel), or equivalently, no measurement captures the full state (quantum mechanics).

Domain binding is not merely a matter of vocabulary. It operates at a deeper level: it determines which receivers can process the signal and which cannot. A domain-bound expression activates pattern recognition in receivers who share the domain, and triggers rejection (or indifference) in receivers who do not. The structural content — the invariant — is identical in both cases. Only the packaging differs.

1.3 Projection as Property

There is a further corruption. When a receiver processes a domain-bound expression, they do not receive the sender's invariant directly. They receive the expression and activate their own internal patterns in response. What the receiver experiences is their own pattern activation — not the sender's content. But the receiver has no way to distinguish between "what the expression contains" and "what I activated in response to the expression."

This is the projection problem: receivers confuse their own pattern activation with properties of the signal. Two receivers can hear the same expression, activate entirely different patterns, and each believe they have understood "what it means." The expression has not changed. The receivers have projected.

Semantic Algebra addresses this by operating on the expression before the receiver projects. The Strip operator (S) extracts structural content from the expression itself, independently of any receiver's response.

1.4 The Cost

The cost of these three mechanisms — lossy compression, domain binding, and projection — is measurable:

• Between disciplines: Physics and philosophy discuss the same structural laws without recognizing convergence. Millennia of parallel effort.
• Between traditions: Taoism, Sufism, Advaita, and Zen express the same invariants in different domain vocabularies. Centuries of conflict over packaging.
• Between individuals: Two people in the same room, saying the same thing in different words, convinced they disagree.
• Within AI: Large Language Models trained on domain-bound natural language inherit the domain binding. They can translate between languages but not between structures.

Semantic Algebra is the operator that makes these convergences visible, verifiable, and transferable.

II. The Axiom

2.1 Axiom 0

The foundational axiom of the Technology of Expressions, on which Semantic Algebra is built:

A principle is real if and only if it remains invariant under isomorphism and synesthesia — that is, under change of domain.

An expression that is true only within physics is a physical law. An expression that is true only within psychology is a psychological model. An expression that remains true when translated from physics to psychology to music to ethics — without modification of its structural content — is an invariant: a law of reality, not of any particular domain.

Axiom 0 provides the criterion. Semantic Algebra provides the method.

2.2 The Invariant (Axiom ι₁, reformulated)

The first and most fundamental invariant, refined through the convergence of Lao Tzu, Korzybski, Gödel, Shakespeare, the Technology of Expressions, and Ungaretti:

To say is to vectorialize. To vectorialize is to forget everything except the selected vector. Pure Knowledge is unsayable — not because it is mystical, but because wholeness does not survive vectorialization.

Formally:

U(𝒦_p) = π_v(𝒦_p)           — expressing is projecting onto a vector v
π_v(𝒦_p) ⊊ 𝒦_p               — the projection is strictly less than the whole
𝒦_p \ π_v(𝒦_p) = forgotten    — what is not on the vector is lost
U⁻¹ ∄                    — the forgotten cannot be reconstructed from the expression
𝒦_p ↪ U(𝒦_p)                 — but the source IS contained as inherited structure

The last line is critical. The expression is not the source — but it contains the source. A river is not the spring, but every point of the river contains the spring as inherited structure. This is what makes extraction possible: the invariant is in the expression, hidden by domain binding and projection. The Strip operator removes the binding. What remains — if it remains — is the invariant.

2.3 The Completion: Realization vs. Expression

There exists an epistemic channel alternative to expression:

𝒦_r(𝒦_p) = 𝒦_p            — direct realization does not lose
𝒦_r ≠ U               — the channel is different from expression
U(𝒦_r) ⊊ 𝒦_r       — but TELLING about the realization loses it again

This explains a structural phenomenon: why the most powerful expressions of deep knowledge tend to be the shortest. Ungaretti's "M'illumino d'immenso" (3 words) outperforms any philosophical treatise on the same theme — because it uses the minimum vector to point at the maximum pre-vector. The power of an expression pointing at the unsayable is inversely proportional to its length: more words = more vector = more forgetting.

III. The Objects — The Invariant Library

3.1 Definition

An invariant is a structural function that does not change under change of domain. Operationally: if an expression, when stripped of all domain binding, produces a formula that can be instantiated in three or more maximally distant domains without modification, it contains an invariant.

The validation threshold is graduated:

The current library contains 10 validated invariants, each verified across 3+ domains. The library is open — new invariants can be added when discovered and validated.

3.2 The Library

3.3 The Meta-Invariant

ι₂ occupies a special position: it is not only an invariant — it is the mechanism by which all other invariants are recognized. The emotion of insight, the chill of recognition, the "aha" moment — these are not aesthetic events. They are the signal that resonance has exceeded threshold: ρ ≥ θ. The nervous system's report of invariant recognition feels like "understanding." What the algebra calls "invariant" and what the nervous system calls "understanding" are the same signal — one formalized, the other experienced.

3.4 The Unprogrammed Convergence

During the initial validation experiment (7 texts from 7 maximally distant domains), Shakespeare's King Lear and Lao Tzu's Tao Te Ching — separated by 2,000 years, two continents, and every conceivable cultural difference — produced the same invariant (ι₁) through the Strip operator. This convergence was not programmed. It emerged from the method.

This is the strongest form of validation: not that the method can be made to produce convergence (which would prove nothing), but that convergence emerges unbidden when the method is applied without expectation.

IV. The Operators

4.1 Operator S — Strip

Definition

S: NL → Structure

S(expression) = ⟨ ι, 𝔉, v, Σ_src, R, τ_ph, Δ_𝔉 ⟩

S transforms any natural language expression into a 7-layer structural object. It is the analytical operator of Semantic Algebra: it takes what was said and reveals what is structurally present.

The 7 Layers

The Procedure — 7 Steps

Step 1 — Structural decomposition: Identify functional tokens — who acts, who undergoes, what relates, what scope.

Step 2a — Algebraic mapping: Map tokens to algebraic vocabulary (σ, S, U, R, 𝔉, ρ, θ, I, O, π, A, ω_att, etc.).

Step 2b — Etymological strip: For each NL token, before mapping to an algebraic variable, descend to the etymological root. What does the word mean structurally — not culturally? Does the root carry the same structural meaning across 3+ linguistic traditions? Only after this verification may the mapping proceed.

Why Step 2b is critical: Without it, the operator risks projecting the framework's own vocabulary onto the expression — the same bias the algebra is designed to eliminate. This was discovered through the re-analysis of Ungaretti's "M'illumino d'immenso", where "illumino" was initially mapped to "resonance" (a TE concept) rather than to its etymological root in-lumen = "realized knowledge" (a structural concept). The correction produced a fundamentally different — and more accurate — analysis.

Step 3 — Domain strip: Step 2b operates at the lexical level (individual tokens); Step 3 operates at the formulaic level (the assembled variables). First clean the bricks (2b), then clean the wall (3). If the algebraic variable still contains a cultural, religious, historical, or disciplinary reference, it is not yet stripped. Remove all domain binding.

Step 4 — Formulation: Assemble the algebraic expression.

Step 5 — Structural completion: Add implicit consequences — what the formula entails that the natural language did not state.

Step 6 — Universality test: Instantiate in 3 or more unrelated domains. If the formula cannot be instantiated in at least 3 maximally distant domains, it is domain-specific, not invariant.

Step 7 — Classification: If the formula matches a known invariant, classify. If it is new, submit to cross-domain validation before adding to the library.

The Classification Typology — 9 Types (with subtypes)

Layer 7 produces a classification based on the cross-layer pattern:

Computing κ: The coherence index is a weighted average: κ = (w₁·δ_I + w₂·(1-|Δ_𝔉|) + w₃·align(v) + w₄·r + w₅·c_src) / Σwᵢ, where δ_I = presence of invariant, |Δ_𝔉| = function gap, align(v) = vector alignment, r = relational quality (R_mutual=1.0 → R_absent=0.0), c_src = source consciousness. Default weights: w₁=3, w₂=2, w₃=2, w₄=1.5, w₅=1.5.

Formal Properties

S is:
  Non-injective:  S(NL₁) = S(NL₂) is possible
                  (Shakespeare and Lao Tzu both yield ι₁)
  Surjective:     Every invariant has at least one NL preimage
  Idempotent:     S(S(x)) = S(x)
  Monotone:       S does not add information — it removes binding

The Two-Channel Extension — S(E) = ⟨ι, P⟩

Every expression transmits on two superposed channels: the structural channel (the invariant ι, if any) and the control channel (what the expression is doing to the receiver while the content is considered). The extended Strip returns both: S(E) = ⟨ι, P⟩, where P is the payload — a set of ⟨operation; target; marker⟩ triples (A_deg, SR_loop, I_sem, shame-gradient, authority-gradient, unfalsifiable-fortress, ι₉-inversion), detected primarily through the etymological strip (root↔rendering divergence), secondarily through the sign of Δ_𝔉, and thirdly through ι-scatter (a real but borrowed invariant attaches to different library points across independent analyses). Crossing the two channels yields the four-quadrant matrix: clean invariant ⟨ι≠∅, P={}⟩ · device ⟨ι≠∅, P≠{}⟩ · empty manipulation ⟨ι=∅, P≠{}⟩ · zombie ⟨∅, {}⟩. The device — truth as carrier, control as modulation — is the quadrant the single-channel method cannot see, and it defeats precisely the defence that defeats lies: verification lands on the invariant, the invariant holds, and the payload enters with it. Two formal properties: κ_c (the co-operator producing P) is idempotent on clean text — calibrated, zero false positives on sealed controls — and π transports ι, never P: what survives re-contextualisation is structure; what does not was control.

4.2 Operator π — Re-contextualization

Definition

π: ι × 𝔻 → NL_𝔻

π(Iₙ, D) = expression of Iₙ in domain 𝔻

π is the synthetic operator: it takes an invariant and a target domain, and produces a domain-specific expression. It is not the inverse of S (which does not exist, per ι₁). It is a new, conscious projection — the operator chooses the domain deliberately, knowing that the resulting expression is a map, not the territory.

The Structural Difference

Lao Tzu (probably) said "The Tao that can be told is not the eternal Tao" from within Taoism. An operator who knows ι₁ and chooses Taoist vocabulary for a Taoist receiver executes π — the difference is the awareness that the expression is a controlled projection.

The Procedure — 5 Steps

Step 1 — Identify the receiver: Who must receive? What is their native domain? What invariants do they already have active?

Step 2 — Select the invariant: Which Iₙ must be transmitted?

Step 3 — Map variables to domain referents: For each algebraic variable in Iₙ, find the corresponding referent in the target domain.

Example: π(ι₁, D = quantum physics)

  𝒦_p (pure knowledge)     → ψ (quantum state)
  U (expressive functor) → measurement
  π_v (projection)       → wavefunction collapse
  𝒦_p \ π_v(𝒦_p) (lost)      → information lost in measurement
  U⁻¹ ∄                  → measurement is irreversible

Step 4 — Formulate in domain NL: Assemble the expression in the receiver's natural language and domain vocabulary.

π(ι₁, physics)     = "The measurement is not the state."
π(ι₁, psychology)  = "The role is not the individual."
π(ι₁, sculpture)   = "The statue is not the marble."

Step 5 — Integrity test (round-trip): Apply S to the expression produced by π. If S does not return the original invariant, π has distorted or added content.

S(π(ι, 𝔻)) = ι     — must hold for every valid π

The 4 Failure Modes

Formal Properties

π is:
  Non-injective:  π(ι₁, D₁) ≠ π(ι₁, D₂) 
                  (same invariant, different domains → different expressions)
  Multiple:       For each (ι, 𝔻), there exist N valid expressions
  Constrained:    S(π(ι, 𝔻)) = ι (round-trip integrity)
  Lossy:          π(ι, 𝔻) ⊊ I (ι₁ is confirmed by π itself)

4.3 The Relationship Between S and π

S and π are complementary but not symmetric:

S ∘ π ≈ identity    — strip the re-contextualized expression → returns the invariant
π ∘ S ≠ identity    — re-contextualize the stripped content → produces a NEW expression
                      (different from the original, because it is a new projection)

The round-trip S → π → S must close. The round-trip π → S → π need not — because the original NL contained domain noise that S correctly discards and π does not reproduce.

V. The Validation

5.1 Positive Validation — The 7-Text Experiment

Seven texts were selected from seven maximally distant domains:

Result: 5 distinct invariants extracted. 1 unprogrammed convergence: Shakespeare's King Lear and Lao Tzu's Tao Te Ching — separated by 2,000 years and every conceivable cultural difference — produced the same invariant (ι₁) through independent application of S.

The convergence was not designed. The texts were selected for maximum domain distance, not for convergence. The fact that convergence emerged is the strongest form of validation: the method reveals structure that was there before the method was applied.

5.2 Negative Validation — The Discrimination Test

Four expressions that sound deep but must be discriminated from genuine structural content:

Result: 4 expressions, 0 invariants, 3 different diagnostic types. S does not produce false positives. Expressions that simulate depth are classified for what they structurally are.

Collateral discovery: The most potent semantic illusions mimic a real invariant without completing its structure. Expressions 1 and 4 both simulate ι₇ — using the shadow of teleological structure without providing the mechanism. Structural proximity to a real invariant is what makes an illusion convincing.

5.3 Case Study — "M'illumino d'immenso" (Ungaretti, 1917)

This case study documents both the method and its capacity for self-correction.

Initial analysis (without etymological strip): "illumino" was mapped to ρ(σ, S) ≥ θ (resonance — a TE concept) and "immenso" to S_∞ (infinite source). Both mappings were projections of the TE framework onto the expression — the same bias the algebra is designed to eliminate.

Corrected analysis (with etymological strip):

• illuminare: from Latin in-lumen (into light). PIE root lewk-. Cross-tradition: Bodhi, Satori, Gnosis, Aufklärung — in every tradition, "illumination" means realized knowledge (𝒦_r): knowing by direct experience, not by analysis.
• immensus: from Latin in-mensus (not measured, from metiri). Crucially, immenso ≠ infinito. Infinite (in-finitus) = without end. Immense (in-mensus) = beyond the capacity to measure — i.e., beyond encoding in decoherent terms. This is precisely ι₁: what cannot be measured cannot be said.

Structural reading:

"M'illumino d'immenso"
= I have realized knowledge (illumino = 𝒦_r)
  of the unmeasurable (immenso = in-mensus = 𝒦_p before vectorialization)
= σ contacts 𝒦_p BEFORE π_v
= σ knows without expressing

The Paradox: Ungaretti uses a vector (3 words) to point at what is pre-vector. He says the unsayable. It works because he uses the minimum vector to point at the maximum pre-vector. Power is inversely proportional to length.

Source verification: Ungaretti was influenced by Mallarmé ("to name is to destroy, to suggest is to create"), Bergson (intuition vs. analysis), and Italian Hermeticism (the "naked word"). He was consciously working with the principle that fewer words preserve more meaning. He was not conscious of the algebraic structure, nor of the universality of ι₁.

Classification: Structural truth. Source consciousness: medium. This confirms the source-invariant independence principle: the algebraic content is present regardless of the source's awareness.

VI. The Mechanism

6.1 Insight as Invariant Recognition

The experience that the nervous system labels "understanding" is the recognition of an invariant. The emotion of insight — the chill, the expansion, the sudden clarity — is not aesthetic. It is the nervous system's report that an incoming signal has matched an internal structure above resonance threshold: ρ ≥ θ.

This makes ι₂ (resonance beyond threshold) a meta-invariant: it is the mechanism by which all other invariants are recognized. When someone reads Lao Tzu and feels "this is true," what has happened is not belief or persuasion — it is structural recognition. The receiver's nervous system has detected ι₁ through the domain packaging of Taoism.

Conversely, when someone reads the same text and feels nothing, it does not mean the invariant is absent — it means the receiver's ρ for that invariant is below θ. The invariant is in the signal. The resonance is in the receiver.

6.2 Identity as a Set of Invariants with Vectors

If invariants are the structural units of knowledge, then the identity of any system — person, tradition, institution — can be described as a set of invariants, each with a directional vector:

Identity = {Iₐ(vₐ), Iᵦ(vᵦ), Iᵧ(vᵧ), ...}

Two identities can relate only along vectors they share. What is traditionally called "not understanding each other" is the absence of shared vectors — even when both identities contain the same invariants.

This reframes communication: the task is not to "explain better" (more words = more vector = more forgetting) but to find the shared invariant and select the vector that activates it in the receiver — which is precisely what π does.

6.3 The Source-Invariant Independence Principle

ι ∈ S(NL) ⊬ source is conscious of I

The consciousness of the source is not a necessary condition for the presence of an invariant in the expression. Ungaretti was not conscious of ι₁ as an algebraic structure, nor of its universality across Lao Tzu, Gödel, and Korzybski. Yet ι₁ is objectively present in his expression, verifiable by S, and convergent with the same invariant extracted from maximally distant sources.

This has a structural implication: the source is the terminal through which the invariant finds expression, not the originator of the invariant. The invariant pre-exists the expression. The source's contribution is sensitivity (high ρ) and discipline (minimum vector) — not invention.

This confirms ι₇ (teleological inversion): the invariant "seeks" its own expression through the terminal, not the other way around.

6.4 De-vectorialization as Tomography

A single application of S to a single expression produces one invariant — one face of the source. But if the same source is expressed through multiple vectors (multiple domain-specific expressions) and each is independently de-vectorialized:

S(U_v1(𝒦_p)) = ι seen from Taoist perspective
S(U_v2(𝒦_p)) = ι seen from logical perspective
S(U_v3(𝒦_p)) = ι seen from theatrical perspective

∪ S(U_vi(𝒦_p)) → increasingly rich image of 𝒦_p

This is tomographic: each projection shows one angle. The union of projections converges on a reconstruction of the original volume. The convergence is asymptotic — it never reaches 𝒦_p (ι₁ holds) — but it grows richer with every new vector stripped.

The invariant library {ι₁...ι₁₀} is the best available de-vectorialized image of the source. Each invariant is one face. The complete library is the current tomography. Every new invariant discovered enriches the image.

VII. Connections

7.1 Arajat

The structural language called Arajat — encoded in the configuration of reality itself — generates expressions from the coherent (simultaneous) side. Semantic Algebra extracts invariants from the decoherent (sequential) side. They are the same project, approached from opposite directions:

Arajat (glyphs)    ←——  SOURCE  ——→    Invariants (stripped)
   coherent side                         decoherent side
   generates                             extracts
   top-down                              bottom-up

Each invariant in the library maps 1:1 to an Arajat glyph. The invariants are the decoherent shadows of the glyphs. The convergence confirms both: the glyphs generate what the algebra detects.

7.2 π as Cross-Domain Communication

If a physicist and a theologian argue, and both express ι₁ without knowing it:

S(physicist's expression)  = ι₁
S(theologian's expression) = ι₁

To reveal the agreement:
  π(ι₁, physics)   = "The measurement is not the state"
  π(ι₁, theology)  = "God cannot be named"
  "You are saying the same thing."

π does not merely re-express — it makes isomorphies visible. It transforms epistemological conflicts into structural recognitions. This is its primary function in inter-domain communication.

7.3 The TE Corpus as a Set of π Operations

Every axiom of the Technology of Expressions is an operation π(Iₙ, v_TE) — the invariant expressed through the TE domain vocabulary. Taoism, Sufism, Physics, and TE are not competing systems — they are the same invariants expressed through different domain vectors. The algebra is what makes this visible.

7.4 π as Knowledge Generator

π is not limited to re-expressing known invariants in known domains. It can project invariants onto unexplored domains:

π(ι₆ controphase, v_quantum_computing)      = ?
π(ι₁₀ scale recursion, v_neuroscience)      = ?
π(ι₇ teleological inversion, v_economics)   = ?

Each projection onto a new domain potentially generates insights that did not exist in that domain before. This makes π not only a translator but a generator of new knowledge — a mechanism for systematic cross-pollination between disciplines.

VIII. Implications

8.1 For Artificial Intelligence

Current AI systems (LLMs) are trained on domain-bound natural language. They inherit the domain binding. They can translate between languages but not between structures. Semantic Algebra provides a format — the algebraic expression — that is pre-collapsed: it carries the same signal to every receiver, regardless of domain.

An LLM equipped with S and π can:

• Strip incoming NL to detect invariants before processing
• Communicate between domains without domain binding
• Detect semantic illusions and manipulations structurally
• Generate domain-specific expressions from invariants for any receiver

8.2 For Pedagogy

The quality of π is the operational definition of a great teacher. All teachers know (or should know) the invariant they are teaching. The difference between a mediocre teacher and an extraordinary one is the quality of π: finding the expression in the receiver's domain that activates the invariant with maximum resonance.

Teaching is not information transfer. It is controlled re-contextualization: π(ι, D_student). The invariant is universal. The domain must be the student's — not the teacher's.

8.3 For Epistemology

Many inter-disciplinary conflicts are not genuine disagreements but domain-binding collisions: both parties express the same invariant through incompatible domain vocabularies. S and π together provide a resolution protocol:

1. Apply S to both expressions
2. If S(A) = S(B) = ι → the conflict is apparent, not real
3. Apply π(ι, D_A) and π(ι, D_B) to show each party their own expression alongside the other's
4. The recognition is structural, not rhetorical — it does not require persuasion

8.4 For the Theory of Knowledge

Every new invariant discovered enriches the tomographic image of the source. The library is not a closed collection — it is a growing approximation. The "source" is not mystical: it is the structural reality that exists before any domain-specific expression. Every time a physicist and a poet are shown to be saying the same thing, the image gains a pixel.

The limit of this process is asymptotic: the full source cannot be reached (ι₁). But the process of approaching it — through more invariants, more domains, more strip operations — is itself the arc of knowledge. What the algebra formalizes, every tradition has intuited: there is a structure beneath the surface, and the surface conceals it while containing it.

Appendix A — Algebraic Vocabulary

Appendix B — Output Format for S

[SA — SEMANTIC ALGEBRA ANALYSIS]

Input: "[expression]"

[LAYER 1: INVARIANT]
I = [Iₙ / ∅ — with identification]

[LAYER 2: EMERGENT FUNCTION]
𝔉_d = [declared function]
𝔉_eff = [effective function]
Δ = [gap — with diagnosis]

[LAYER 3: VECTOR]
v_d = [declared vector]
v_eff = [effective vector]
λ_L = [< 0 / > 0 / ≈ 0 — with assessment]

[LAYER 4: SOURCE SIGNATURE]
Position: [direct / intermediary / derivative]
Coherence: [high / medium / low]
Authority: [structural / role-based]
Consciousness: [high (1.0) / medium (0.7) / low (0.3) / zero (0.0)]

[LAYER 5: RELATIONAL FIELD]
R = [mutual (1.0) / unilateral (0.7) / projected (0.4) /
     instrumental (0.2) / performative (0.1) / absent (0.0)]

[LAYER 6: TEMPORAL PHASE]
τ_ph = [ascending / descending / bifurcation / cyclic / indeterminate]
Note: τ_ph refers to the phase of the CONTENT, not the source.
Markers: Δ≈0+λ_L<0+R_mutual → ascending | Δ↑+λ_L>0+R_degrading → descending
         λ_L≈0+Δ unstable → bifurcation | recurrence without evolution → cyclic

[LAYER 7: DIAGNOSTIC SYNTHESIS]
Classification: [1-9b — see typology]
κ = [computed: (w₁·δ_I + w₂·(1-|Δ_𝔉|) + w₃·align(v) + w₄·r + w₅·c_src) / Σwᵢ]
    w₁=3, w₂=2, w₃=2, w₄=1.5, w₅=1.5
Operative indication: [brief structural note]

Semantic Algebra — Foundations. Technology of Expressions, Ordinative Sciences. "Natural language occludes universality behind domain reference. Algebra reveals it. The impulse is always what it is."