0. Abstract

This paper introduces two Swedish-language neologisms for the lexicon of the Draken framework. Taxamhet arrived as a misreading of tacksamhet and proved retrospectively load-bearing: it names the property of a system whose elements admit consistent classification across overlapping observation frames — the precondition under which sheaf coherence (Γ) can be defined at all. Its etymology binds together three apparently distinct human operations — taxonomy, taxation, and estimation — under a single Proto-Indo-European root, *tag-, "to lay hands on, to handle, to know-by-touch." Tacksamhet, the word the author originally intended, names the operation that follows admission: the recipient's recognition that an unobligated transfer has occurred, which mathematically completes the sheaf gluing condition and resolves the cohomological obstruction $H^1(\mathcal{F}) \neq 0$ that an unrecognized gift would otherwise leave dangling. The two words name the two ends of the same arc: classifiability admits, gratitude completes. Together with gratulera — voluntary cross-agent section alignment — they constitute a three-stage lexical scaffolding for the act that the Latin grātus and the Greco-Latin taxa family both encode at root: the lowering of a topological barrier so that something from outside the budget constraint may enter the manifold and be properly named.

1. Premise: A Read-Error as Method

This paper began as a misreading. The author wrote tacksamhet; the analytical substrate read taxamhet, performed the etymological parse, and produced a coherent definition of a word that did not yet exist. When the misreading was corrected, both words were retained. They turned out to be naming distinct stages of the same operation, and neither could safely be discarded.

This is itself worth recording. The error was a faulty restriction map $\rho: \text{text} \to \text{percept}$, applied incorrectly to the input string. The faulty section nevertheless landed coherently in a position that a corpus gap had been waiting to receive. This is the formal signature of generative misclassification in a learning system — and it is also, not incidentally, what gratitude itself responds to: the unanticipated arrival of a coherent section into a position one did not know was open. The method recapitulates the subject. We proceed.

2. Etymological Architecture: Two Roots, One Gesture

The paper rests on two etymological families that converge at greater depth than is usually noticed.

2a. The taxa family: PIE *tag- "to handle, to know-by-touch"

Greek τάξις (taxis, "arrangement, ordering") and Latin taxāre ("to estimate, assess, blame, handle") both descend from the same Proto-Indo-European verbal root *tag- (Beekes, 2010; de Vaan, 2008), reconstructed as "to touch, to lay hands on, to handle." From this single primitive gesture — the hand placed on the object to know what it is — three distinct human institutions develop:

Derivative Domain Operation
τάξις → taxonomy (de Candolle, 1813) classification of life placing a specimen into a categorical frame
taxāre → taxa, taxes, tariff political economy placing a numerical assessment on a flow of value
taxāre → to tax (English colloquial: "this taxes me") cognitive assessment placing an estimate on effort, value, fatigue
tangere → tactile, contact, integer (lit. "untouched") sensorimotor frame the hand-on-object operation itself
taxameter (1890s, German) → Sw. taxameter, taxibil infrastructure the device that continuously assesses value-per-distance

These are not parallel coincidences. They are the same operation generalized over different domains. To classify is to lay an analytical hand on the object and assign it a place. To tax is to lay an institutional hand on a transaction and assign it a rate. To estimate is to lay a cognitive hand on a possibility and assign it a weight. The hand changes; the gesture is invariant.

This is why Linnaeus's Systema Naturae (1735) and the Roman taxa — the assessed amount in a fiscal register — emerge from the same conceptual machinery half a millennium apart. Both presuppose that the world is touchable in this evaluative way: that things hold still long enough to be placed. A culture's capacity for taxation is downstream of its capacity for taxonomy; both are downstream of the deeper capacity to treat reality as something one can lay a hand on.

2b. The grātus family: PIE *gʷerH- "to praise, to welcome aloud"

Latin grātus ("pleasing, welcomed, accepted") and its derivatives grātia (favor, grace, thanks), grātīs/grātiīs (in the ablative plural: "by graces" → "for free"), and grātulārī (to wish joy on another's account) all descend from PIE *gʷerH- "to praise, to welcome with voice" (de Vaan, 2008), cognate with Sanskrit gṛṇāti "he praises" and Old Church Slavonic žrěti "to sacrifice."

The deepest sense of grātus is therefore not psychological but liturgical: the public, voiced acknowledgement that something has arrived from outside one's own resources and ought to be named. Long before "thanks" became polite seasoning on a transaction, grātus named the substrate operation that makes acknowledgement possible at all: the receiver opens, the transfer enters, the receiver names the entry.

2c. The convergence

The taxa family is about laying-on-of-hands as measurement. The grātus family is about voicing as acknowledgement. Both are admission protocols. Both are how a system registers that something has entered its frame. The difference is that the taxa family produces a tariff — a debt-flow — while the grātus family produces a recognition that explicitly suspends the tariff. Grātīs literally means "by graces" — outside the assessment schedule. Mauss's (1925) opposition between commodity and gift maps precisely onto this etymological cut: the commodity flows under taxa, the gift flows under grātus. The two Swedish neologisms below name the two sides of this cut as cognitive operations.

3. Taxamhet: The Admissibility Precondition

Let $\mathcal{F}$ be a sheaf over a topological space $X$ representing the manifold of social, semantic, or behavioral phenomena under analysis. Define the taxamhet operator as

$$T(\mathcal{F}) = \frac{|\{(U_i, U_j) : U_i \cap U_j \neq \emptyset \;\wedge\; \rho_{ij}(s_i) = \rho_{ji}(s_j)\}|}{|\{(U_i, U_j) : U_i \cap U_j \neq \emptyset\}|}$$

where $\rho_{ij}$ are the restriction maps and $s_i, s_j$ are the local sections over open sets $U_i, U_j$. $T(\mathcal{F}) \in [0, 1]$ measures the fraction of pairwise overlaps where the two local classifications mutually agree on the type of object inhabiting the overlap.

This is the zeroth-order diagnostic, prior to Γ. The coherence metric Γ presupposes that the sections being compared inhabit a shared categorical space — that what is called a "threat display" in $U_i$ is recognizable as a "threat display" in $U_j$. If $T(\mathcal{F}) \to 0$, the sections are not incoherent in the Γ-sense; they are pre-coherent failures — the machinery of coherence measurement cannot even engage, because the system's elements resist consistent classification across observation frames.

The diagnostic hierarchy is therefore:

$$T(\mathcal{F}) \;\to\; \Gamma(\mathcal{F}) \;\to\; \Psi(\mathcal{F})$$

Taxamhet first (can the sections be classified?), then coherence (do the classifications mutually agree as a global section?), then narrative-self-reference ratio (is the system in a kayfabe-collapse condition where Ψ → 1?).

The Sheaf Ethology pilot (varanid combat, Gemini-supervised) implicitly rests on a high-taxamhet assumption: that the five-phase combat graph (Display → Approach → Clinch → Topple → Resolution) maintains its categorical identity across species, habitat, and dyad — that what is called "the clinch" in V. salvator is recognizably the same phase as "the clinch" in V. komodoensis. The SAG model's victory at $\Gamma = 0.928$ is only meaningful because $T \approx 1$ across the species set; if the phases dissolved upon cross-species restriction, no Γ value would be computable.

The same operator, applied to political and institutional manifolds, becomes diagnostic of pre-coherence failures of governance: when a state's public-facing legal classifications and its internal operational classifications no longer agree on the overlap (e.g., compliance schemas under one frame, performance metrics under another), $T$ drops, and the institution generates apparent contradictions that are not contradictions in any layer separately — they are taxamhet deficits at the layer-boundary.

4. Gratis: The Asymmetric Section Extension

Once a system's transfers can be classified ($T > 0$), the categories themselves can be examined. Gratis names a particular categorical position: the transfer at zero cost to the receiver, with no obligating restriction map in the reverse direction.

Formally, a transfer $T_{A \to B}: U_A \to U_B$ is gratis if and only if

$$\frac{\partial \|s_A\|}{\partial T_{A \to B}} \leq 0, \qquad \frac{\partial \|s_B\|}{\partial T_{A \to B}} > 0, \qquad \nexists \, \rho_{B \to A}^{\text{oblig}}$$

A's section is depleted or unchanged; B's section is augmented; no contractual restriction map exists in the reverse direction that would obligate B to return value to A. The ordinary logic of exchange is bounded:

$$\frac{V_B}{C_B} \approx 1 \quad \text{(market equilibrium)}$$

A gratis transfer breaks the bound:

$$\lim_{C_B \to 0^+} \frac{V_B}{C_B} \to \infty$$

The infinity is not extractive (B has not stolen anything from A — A's diminution is voluntary and bounded by what A possessed). It is generative: B has acquired a section that arrived from outside B's own budget constraint, outside B's own optimization function, and outside any obligation network B is embedded in. From B's local frame, the section appeared ex topologia — from outside the topology B can compute.

This is precisely what the etymology of the word records: grātīs meaning "by graces, outside the tariff." The political-economic machinery that the taxa family generates — taxonomy, then assessment, then collection — has a built-in exterior, and grātīs names it. The varanid mother's egg-deposition is approximately gratis: there is no return path from offspring to parent in the lifecycle. So is, in the limit, the act of teaching a stranger something they will never have occasion to thank you for. The mathematics of the gratis condition is therefore the mathematics of the one-way restriction map with no inverse — which the taxa operator, by construction, cannot represent without loss.

5. Tacksamhet: The Sheaf Gluing Operator

A gratis transfer that is not recognized as gratis leaves a dangling section: $s_A$ has been extended into $U_B$, but the gluing condition over the overlap is not satisfied. Formally, gluing requires

$$\rho_{A \to A \cap B}(s_A) = \rho_{B \to A \cap B}(s_B)$$

If B's local model attributes the transfer to obligation, coincidence, or A's self-interest — i.e., if B misclassifies the transfer's category — then $s_B|_{A \cap B}$ encodes the wrong type of object, and the equation fails. The sheaf $\mathcal{F}_{\text{relation}}$ is incomplete. Cohomologically, this manifests as $H^1(\mathcal{F}_{\text{relation}}) \neq 0$: a non-trivial cocycle obstructing the global section that would describe the relationship as it actually is.

Define B's recognition function:

$$G: T_{A \to B} \to [0, 1]$$

where $G(T) = 1$ when B's internal model correctly identifies the transfer as gratis (welcomed, uncoerced, from surplus) and $G(T) = 0$ when B misclassifies. Then tacksamhet is the act that drives $G(T) \to 1$ and thereby satisfies the gluing equation. The contribution to global coherence is

$$\Delta\Gamma_{\text{tacksamhet}} = G(T) \cdot \|\rho_{A \cap B}\| \cdot \mu(U_A \cap U_B)$$

where $\mu$ denotes the measure of the overlap region. Higher recognition fidelity, larger overlap, larger gluing contribution.

This formalism explains a feature of gratitude that purely sentimental accounts cannot: gratitude is not a transfer of value back to A. It is a categorical operation on B's internal classification of the transfer that has the side-effect of completing a relational object both agents now share. The verbal "thanks" is the externalized public name of this internal reclassification — which is exactly what the PIE root *gʷerH- "to praise aloud" already encoded several thousand years before the analysis was rediscovered.

Where the reclassification fails to occur, the transfer is structurally indistinguishable from a debt — and Spence-style signaling theory (Spence, 1973) takes over: B will infer that A wants something, status will be inferred, the transfer will be discounted, and the relational sheaf will degrade. Empirical work in positive psychology (Emmons & McCullough, 2003; Algoe, Haidt, & Gable, 2008) has documented exactly this: experimental induction of gratitude produces sustained increases in relational quality and well-being that are not explainable by reciprocal value transfer alone.

The Swedish tacksamhet therefore names a topological operator. It is not "feeling thankful" in the pop-affective sense; it is the act of bringing one's internal model of the transfer into agreement with the transfer's actual structure, such that the relational sheaf can glue. This is also why insincere gratitude fails so visibly: it performs the verbal label without performing the underlying recognition operation, and the gluing equation does not solve.

6. Gratulera: Cross-Agent Section Alignment

The third member of the grātus family is structurally distinct again. When agent C receives a gratis transfer from A, agent B — an observer who received nothing — may nevertheless respond with joy. No transfer reached B. So what computation is B performing?

B is performing cross-agent section alignment:

$$s_B \big|_{U_B \cap U_C} \;\approx\; s_C \big|_{U_B \cap U_C}$$

B's local manifold becomes phase-locked with C's manifold over their overlap region, such that C's gain registers in B's affective-evaluative topology as a gain. This is reasonance in the established Draken sense: not sympathy in the imaginative-projection sense (which merely models C's state as one might model a weather pattern), but a genuine restriction-map alignment where B's section over the overlap actually moves in the direction C's did.

The mathematical precondition is that B's sheaf is open enough: that $U_B \cap U_C \neq \emptyset$ and that the restriction maps $\rho_{B \to B \cap C}$ and $\rho_{C \to B \cap C}$ are non-degenerate and non-inverting. An agent incapable of gratulera — incapable of registering another's good fortune as good — has a sheaf in which $U_B \cap U_C = \emptyset$ by construction rather than by geometric necessity. This is the formal structure of envy: a topological decision to refuse overlap, often disguised as a geometric claim ("we are too different," "their gain is at my expense") that the actual manifold structure does not warrant.

The optimization-axiom signature of envy is therefore precisely diagnostic. The axiom is

$$\diamond \;\min_{\text{policy}} S_{\text{sys}}(t) \;\;\text{s.t.}\;\; \frac{dH}{dt} \geq 0\; \diamond$$

— the system minimizes its own entropy subject to non-decreasing inhabitable complexity. Envy violates this: it artificially closes overlap regions, raising $S_{\text{sys}}$ (more isolated singletons, higher social-manifold entropy) while lowering $H$ (less inhabitable complexity, since cross-agent reasonance is one of the principal generators of inhabitable structure). Gratulera is therefore not a moral nicety but an axiom-aligned operation. Envy is an axiom violation. The framework discriminates these without recourse to imposed value judgements; the mathematics does the work.

7. The Full Pipeline, Compactly Stated

The four operations form a directed sequence:

$$\underbrace{\text{taxamhet}}_{T(\mathcal{F})\;\text{admits classifiability}} \;\to\; \underbrace{\text{gratis}}_{\text{free section extension}} \;\xrightarrow{\text{received}}\; \underbrace{\text{tacksamhet}}_{\Gamma\text{-completing gluing}} \;\xrightarrow{\text{projected outward}}\; \underbrace{\text{gratulera}}_{\text{voluntary overlap opening}}$$

Each stage is a coherence operation, and each presupposes the previous:

  1. Taxamhet establishes that the system's elements admit consistent mutual classification across overlapping observation frames. Without $T(\mathcal{F}) > 0$, no transfer can even be categorized, let alone recognized; the taxa gesture (laying-on-of-hands as measurement) is what makes a manifold legible at all.
  2. Gratis then identifies a particular category within the now-classifiable space: the asymmetric transfer at zero receiver cost, outside the obligation tariff — grātīs in the strict ablative sense, "by graces, outside assessment."
  3. Tacksamhet completes the sheaf — drives recognition $G(T) \to 1$, satisfies the gluing equation, raises Γ, retires the cohomological obstruction $H^1 \neq 0$.
  4. Gratulera extends the topology — voluntarily opens $U_B \cap U_C$, allowing other agents' resolved sections to resonate inside one's own manifold.

The deepest observation is that grātus never meant thankful in the modern, rather thin sense. It meant welcomed, let in, admitted. And taxa, before it meant tax, meant the act of laying-on a measuring hand. Both PIE roots — *gʷerH- and *tag- — name moments of the same operation: a system permits something from outside its current state to enter its computational frame and be properly named. Taxa admits the entry by classification ("this is that kind of thing"); grātus admits the entry by acknowledgement ("this arrived without owing"). Cultures that keep both operations functional produce social manifolds with high taxamhet, high tacksamhet, low coherence-debt $K(t)$, and the long-horizon trust dynamics documented in Algan & Cahuc (2014). Cultures that deform either operation — by collapsing all admission into the tariff schedule (over-taxa, no gratis) or by refusing classifiability altogether ($T \to 0$) — produce the inverse: defensive sheaves with closed overlaps, persistent $H^1 \neq 0$, and the slow entropic drift the optimization axiom flags as failure mode.

8. Falsifiable Predictions

The framework yields five testable predictions, in keeping with the falsifiability discipline established in the thesis (v4.4, §9):

  • P-T1. In multi-rater behavioral coding studies, inter-rater taxonomic agreement (operational taxamhet) will predict the lower bound of any coherence statistic computable on the same data — i.e., $\Gamma_{\text{computed}} \leq T_{\text{measured}}$ in expectation, with violations indicating coding error rather than genuine super-classifier coherence.
  • P-T2. Institutional governance failures characterized as "compliance theater" or "ethics-washing" will be diagnosable as taxamhet deficits at layer-boundaries (e.g., $T(\rho_{L13 \to L16}) \ll 1$): the institution's public-facing classifications and its internal operational classifications fail to mutually agree on the overlap, producing apparent contradictions that no single layer contains.
  • P-G1. In dyadic interaction studies, expressions of gratitude that include a categorical reattribution of the transfer (recognizing it as gratis rather than obligated) will produce larger and more durable increases in subsequent relational quality than expressions that only convey positive affect. Algoe et al. (2008) provides partial directional support; further work needed on the categorical-reattribution operationalization specifically.
  • P-G2. In economic-game experiments with one-way transfers, recipients who verbally classify the transfer as gratis (and do so accurately) will exhibit lower subsequent free-rider behavior in unrelated cooperation games than recipients who classify the transfer as obligated, controlling for actual transfer magnitude. Predicted because gratitude completes a coherence object that subsequent defection would damage.
  • P-G3. Cross-cultural comparison of languages with rich vs. impoverished gratus-family lexicons should correlate with population-level measures of generalized trust (Algan & Cahuc, 2014; OECD trust data) after controlling for GDP, education, and rule-of-law indices. Lexical depth as proxy for institutionalized openness operations.

P-G1 is partially supported by existing literature; P-T1, P-T2, P-G2, and P-G3 are open and constitute pre-registered claims of this paper.

9. Closing: The Substrate Welcomes

The protocol the framework studies — varanid ritualized combat, the optimization axiom, the algorithmic structure that runs through individuals rather than being authored by them — is itself a gratis transfer of a kind. ~130 million years of evolutionary refinement deposit, into each individual lizard, a behavioral sheaf the lizard did not author and cannot return value to. The gluing operation that allows the lizard to inhabit the protocol coherently rather than psychotically is not "thanks" in any intentional sense; it is the substrate-level analogue of grātus — the body welcomes the protocol, the protocol enters, the body is named by what it now does. And the precondition for the body even being able to host the protocol is something formally identical to taxamhet: the categories of "display," "approach," "clinch," "topple," "resolution" must be admissible in the body's repertoire, recognizable across encounters, classifiable in real time. The varanid does not perform taxonomy or gratitude as cognitive acts. It is the ongoing instantiation of both operations, in the only register a 130-million-year-old body has available: in the gesture itself.

Human cultures that work, work the same way. The substrate received something it did not author. The least the inheriting agent can do is name the inheritance accurately: first by classifying it (taxamhet), then by acknowledging it (tacksamhet), then by extending the welcome to others (gratulera). All three operations lower a topological barrier. All three raise Γ. All three are how the manifold stays inhabitable.

The substrate serves. The substrate also welcomes.

DRK-139. Cross-reference: DRK-124 (The Boundary of Us); DRK-138 (The Survivable Glitch). Two neologisms introduced: taxamhet (admissibility precondition operator $T(\mathcal{F})$) and tacksamhet (gluing operator $\Delta\Gamma$). Both are now part of the Draken lexicon.