The Geometry of Practice¶

Curious · February 2026

Developmental Trajectories as an Alternative to Grades

Abstract¶

We present an architecture in which the inverse covariance matrix \(g = \Sigma^{-1}\) of a learner's compositional history functions as a Riemannian metric tensor—a mathematical object that encodes the shape of what has been practiced, where breakthroughs have occurred, and where readiness for new learning is highest. This metric has three tested physical properties: viscosity (resistance to reorganization, measured at 0.118 ratio after 50 compositions), capacitance (stored readiness that amplifies perturbation, measured at 4.65× baseline discharge), and bimodal breakthrough (phase transitions that are discontinuous rather than gradual, gap ratio 2.86). A fourth property—the Marchenko-Pastur noise floor—provides a principled spectral threshold separating practiced dimensions from undifferentiated noise. The system produces portable developmental geometry that replaces the lossy compression of grades with a complete, observable, student-owned representation of developmental state. No optimization, no loss functions, no training. The geometry is observed, not computed.

1. Why Grades Fail¶

A grade is a scalar. It compresses a multidimensional developmental process—what was practiced, in what order, with what intensity, what broke through, what plateaued—into a single number. This compression is lossy in the information-theoretic sense: the original signal cannot be recovered from the compressed representation. A student with a B+ in biology might have deep geometric understanding of ecosystems and no grasp of molecular mechanisms, or vice versa. The grade discards the geometry.

Existing alternatives (rubrics, competency frameworks, portfolio assessment) reduce the lossiness by adding dimensions, but they are still external projections: an evaluator decides which dimensions matter, constructs a scale for each, and maps the student's work onto those scales. The student's developmental trajectory—what they actually practiced, in what temporal order, with what geometric consequences—is never directly observed. It is inferred, approximated, and projected.

We propose that the developmental trajectory itself can be the representation. Not a projection onto externally-defined dimensions. Not a rubric. The actual geometry of what was practiced, carried in a portable file, owned by the student.

2. The Architecture: From Text to Geometry¶

2.1 Compositional Extraction¶

Every piece of student work enters a linguistic extraction pipeline that produces a 17-dimensional composition vector:

5 Process-Actor dimensions (from Dowty's proto-role theory): agency, stability, influence, boundary, resonance. These capture what the actor in the student's writing does—how much it causes, persists, affects, delimits, and resonates with other entities.

12 Process-Assertion dimensions (from Bach/Vendler eventuality classification, Levin verb alternations): eventuality type, transitivity, voice, telicity, durativity, perfectivity, iterativity, dynamicity, verb class, and three constituency constraints. These capture what the student's writing asserts—its temporal structure, argument structure, and aspectual profile.

The tensor product \(5D \otimes 12D = 17D\) is the compositional space. Each text extraction produces one 17D vector. Simultaneously, a SentenceTransformer model (all-mpnet-base-v2) produces an independent 768D semantic embedding. The two representations are never mixed or optimized toward each other.

2.2 Covariance Accumulation¶

As compositions enter the field (one per text extraction), the covariance matrix \(\Sigma\) accumulates. This accumulation is append-only: no composition is ever deleted, modified, or reweighted. \(\Sigma\) only grows. Its inverse, \(g = \Sigma^{-1}\), is the precision matrix—and it functions as a Riemannian metric tensor on the learner's compositional space.

No loss function is computed. No gradient is backpropagated. The system observes the geometry that practice produces; it does not optimize toward a target.

2.3 What the Metric Means¶

\(g\) IS the learner's knowledge geometry. Where \(\Sigma\) is large (extensive practice producing high covariance), \(g\) is small (those compositions are geometrically close—finely differentiated). Where \(\Sigma\) is small (sparse practice), \(g\) is large (those compositions are geometrically far—coarsely distinguished).

The Mahalanobis distance \(d = \sqrt{(\mathbf{x}-\mathbf{y})^T g (\mathbf{x}-\mathbf{y})}\) between any two compositions is the distance through the learner's metric. Two compositions that would be far apart in Euclidean space might be close in Mahalanobis space if the learner has practiced extensively in the region between them. The metric is personal, developmental, and accumulative.

3. Three Tested Physical Properties¶

3.1 Viscosity: The Metric Stiffens Through Practice¶

Prediction: As compositions accumulate, each new composition produces a smaller shift in \(g\). The metric becomes progressively harder to reorganize.

Mechanism: The eigenvalues of \(\Sigma\) grow monotonically (append-only). The eigenvalues of \(g = \Sigma^{-1}\) shrink. The Frobenius norm of the metric shift per composition, \(\|\Delta g\|_F\), decreases because each rank-one update (new composition) is a smaller proportion of the accumulated covariance.

Result: After 50 compositions, the metric shift per composition was 11.8% of its initial value. Viscosity ratio = 0.118.

What it means for education: A learner who has practiced extensively in a domain has a viscous metric in that region—it resists perturbation. This is not rigidity; it is developmental maturity. The geometry has been shaped by practice and does not easily reshape. A novice has a fluid metric—every new input produces large shifts. The viscosity curve IS the developmental trajectory.

3.2 Capacitance: Stability Stores Readiness¶

Prediction: During periods of stability (consistent, low-magnitude metric shifts), the system stores readiness. When a perturbation arrives, the response is disproportionate.

Mechanism: An adaptive threshold tracks the tonic (baseline) of recent metric shifts. During stability, the threshold drops. When perturbation arrives and exceeds the lowered threshold, the stored difference between the stable tonic and the perturbation magnitude produces a discharge that exceeds what the perturbation alone would cause.

Result: Following a stability period of 20 compositions, a novel input produced a metric shift 4.65× larger than the running baseline. Capacitive discharge ratio = 4.65.

What it means for education: The most productive time to introduce challenging new material is after a period of stable practice—not during turbulence. The learner's capacitive state (observable from the tonic charge and adaptive threshold) predicts when they are maximally ready for reorganization. Counterintuitively, the learner becomes more sensitive the more stable they are. Stability is not stasis. It is stored readiness.

3.3 Bimodal Breakthrough: Phase Transitions, Not Gradual Progress¶

Prediction: When perturbations enter a viscous metric, the distribution of metric shifts is bimodal: the metric either absorbs the perturbation (small shift) or breaks through (large shift). There is no gradual middle.

Mechanism: Each new composition adds a rank-one update to \(\Sigma\). The Cauchy interlacing theorem constrains the new eigenvalues to interlace with the old. Whether the perturbation exceeds the critical threshold for phase transition (BBP transition) depends on the alignment between the new composition and the existing eigenstructure.

Result: The distribution of \(\|\Delta g\|_F\) over 50 compositions showed a bimodal structure with gap ratio 2.86 (mean of above-median shifts was 2.86× mean of below-median shifts). The metric absorbed or broke through—no intermediate responses.

What it means for education: Conceptual change is not gradual. This is consistent with decades of educational research (Posner et al., 1982; Chi, 2008) but now has a precise geometric mechanism. The learner's metric either absorbed the new input (reinforced existing structure) or broke through (reorganized—the metric shifted significantly in one or more dimensions). The gap ratio 2.86 means the breakthrough events were nearly three times the magnitude of absorptions. This is directly observable in the eigenvalue spectrum. No rubric needed. No evaluator judgment. The geometry records it.

4. Eventuality Classification: What Kind of Development Is Happening?¶

4.1 Typing Eigenvalue Trajectories¶

Each of the 17 dimensions of \(g\) has an eigenvalue that evolves over time. The trajectory of each eigenvalue is a curve. Using the Bach/Vendler eventuality classification:

Type	Curve Shape	Developmental Meaning
STATE	Flat—eigenvalue stable	This dimension is not changing. Practice hasn't reached it, or it has reached equilibrium.
ACTIVITY	Sustained drift	Ongoing development. No endpoint yet visible.
ACCOMPLISHMENT	Drift that saturated	A bounded process completed. This dimension converged.
ACHIEVEMENT	Sudden jump	Phase transition. Breakthrough. The metric reorganized.

The eventuality signature compresses the full 17-dimensional developmental state into a compact string: SSSAA|SSSSS|SSASAAA (gradient|compressed|categorical). This string IS the temporal fingerprint of what kind of development is happening across every dimension simultaneously.

4.2 The Marchenko-Pastur Developmental Noise Floor¶

Not all eigenvalue trajectories carry signal. Early in development, many dimensions are indistinguishable from what random input would produce. The Marchenko-Pastur (MP) distribution provides the principled threshold:

For \(p = 17\) dimensions and \(n\) compositions, the ratio \(\gamma = p/n\) determines the MP bulk:

\[\lambda_- = \sigma^2(1 - \sqrt{\gamma})^2 \leq \lambda \leq \sigma^2(1 + \sqrt{\gamma})^2 = \lambda_+\]

Eigenvalues within this bulk are noise — they represent dimensions where practice has not yet produced structure beyond what randomness alone would generate. Eigenvalues above \(\lambda_+\) carry genuine signal — dimensions where the learner has differentiated their geometry from noise.

What it means for education: After 10 compositions (\(\gamma = 1.7\)), the MP bulk is wide — almost nothing is distinguishable from noise. After 50 compositions (\(\gamma = 0.34\)), the bulk tightens significantly. After 200 compositions (\(\gamma = 0.085\)), the bulk is narrow — most dimensions that have been practiced are now distinguishable as signal.

The number of eigenvalues above \(\lambda_+\) is a developmental measure that no existing assessment captures: how many dimensions has this learner differentiated from randomness? A learner with 12 of 17 dimensions above the MP threshold has broadly differentiated practice. A learner with 3 of 17 above the threshold has deep but narrow development. Both might have the same grade. The geometry distinguishes them.

4.3 The Observation Vocabulary¶

The four physical properties—viscosity, capacitance, bimodal breakthrough, and the MP noise floor—together with the eventuality classification and the dual-space surplus, constitute six composable overlays on the same metric tensor \(g\). Each reads a different derivative or transform of \(g\): viscosity reads \(\|\Delta g\|_F\) over time; eventuality classifies \(d\lambda/dt\); the MP threshold filters signal from noise. Composing overlays diagonally—for example, "above MP + ACTIVITY" (genuine development) versus "within MP + ACTIVITY" (proto-development, not yet distinguishable from noise)—produces compound observables that no single overlay captures. The full mathematical treatment of the overlay surface and its composable structure appears in the companion paper (Phillips, 2026c).

5. Portable Gems: The Student Owns the Geometry¶

The developmental geometry is stored in a portable Gem file (.npz): the frozen covariance \(\Sigma\), the eigenvalue history, the eventuality signatures, the tonic state. This file is the student's. It lives on their Google Drive, in their sovereign storage. No institution controls it. No platform locks it in.

When the student moves to a new context—a new school, a new country, a new discipline—the Gem travels with them. Any system that can compute \(g = \Sigma^{-1}\) can read the student's developmental geometry. The diagonal lens \(L = \Sigma_{\text{observer}}^{-1} \cdot \Sigma_{\text{gem}}\) computes compatibility between the new context and the student's accumulated geometry. Eigenvalues of \(L\) near unity indicate resonance—the student's practice aligns with the new context's structure. Eigenvalues far from unity indicate compression or expansion—the student's geometry diverges from the new context in specific, identifiable dimensions.

5.1 What a Gem Carries¶

\(\Sigma\): the full covariance matrix — the complete record of compositional practice
Eigenvalue history: how each dimension's eigenvalue evolved — the developmental trajectory
Eventuality signatures: what kind of development happened per dimension — the temporal fingerprint
Tonic state: coherence, stability, dissonance — the metabolic readiness
Pressure map: where practice is concentrated — the dimensional density
MP profile: which dimensions are above the noise floor — the differentiation signature

A new teacher receiving this Gem sees the student's developmental geometry immediately. Not "this student got a B+ in biology." Instead: "This student has differentiated 11 of 17 dimensions. Agency and transitivity are deeply developed (high viscosity, above MP, STATE). Durativity and iterativity are actively developing (moderate viscosity, above MP, ACTIVITY). Telicity just showed a breakthrough (ACHIEVEMENT, above MP, high pressure). Perfectivity and dynamicity are undifferentiated (within MP bulk). The student's capacitive state is high — they're ready for perturbation in the developed dimensions."

6. Implications¶

6.1 For Assessment¶

The developmental geometry is not an assessment in the evaluative sense. It does not judge. It does not rank. It does not compare the student to a standard. It observes the shape of what was practiced and reports that shape in its own vocabulary. The question shifts from "how well did the student perform?" to "what geometry did the student's practice produce?"

This does not eliminate the need for evaluative judgment—there are contexts where evaluation is appropriate and necessary. But it separates observation from evaluation. The geometry is the observation. What to do with it is the evaluative act. Separating the two is itself a contribution: current assessment conflates measurement with judgment, and the conflation produces the known distortions (teaching to the test, grade inflation, construct underrepresentation, the compression of multidimensional development into scalar scores).

6.2 For Instruction¶

The capacitive discharge result (4.65×) has direct instructional implications. If stability produces heightened sensitivity—if the learner becomes more responsive after a period of consistent, low-turbulence practice—then the optimal instructional sequence is: stabilize, then perturb. Build a period of consistent practice in a dimension, then introduce the challenging new material. The learner's geometry is ready for it. The metric will discharge into the perturbation and produce a disproportionate reorganization.

This is testable. The prediction is specific: a learner whose tonic state shows high coherence, low dissonance, and high stability will produce a larger metric shift in response to a perturbation than a learner whose tonic state shows low coherence, high dissonance, and low stability—even if the perturbation is identical.

6.3 For Equity and Portability¶

Grades are local. They depend on the institution, the instructor, the curve, the cohort. A grade earned at one institution does not transfer to another with full fidelity. The developmental geometry is portable because it is mathematical, not institutional. The eigenvalue spectrum of \(g\) does not depend on who assigned the grade. It depends on what was practiced.

A Gem file from a rural school in one country and a Gem file from an urban school in another country describe the same mathematical object in the same vocabulary. The diagonal lens computes their compatibility directly. No translation. No equivalency table. No accreditation chain. The geometry speaks for itself.

A caveat: the 768D semantic embeddings are produced by a multilingual SentenceTransformer model, which handles cross-lingual semantic proximity natively. The 17D compositional extraction, however, depends on spaCy's language-specific parsing models, which vary in quality across languages. The geometric portability of Gems across languages requires that the extraction pipeline produce comparable 17D compositions from comparable text in different languages. This has not yet been validated empirically. The claim of cross-linguistic portability is architecturally grounded (the geometry is language-independent) but pipeline-dependent (the extraction that produces the geometry is not). Validating extraction fidelity across spaCy language models is a priority for future work.

6.4 What We Are Proposing—and What We Are Not¶

We are not proposing that developmental geometry should replace grades tomorrow. We are proposing that the following claims are now testable and should be tested:

The metric tensor \(g = \Sigma^{-1}\) has physical properties (viscosity 0.118, capacitance 4.65×, bimodal breakthrough 2.86) that are testable in learner data.
Practice produces observable pressure profiles — learners in different domains develop different dimensional density patterns.
Conceptual change is directly observable as bimodal \(\|\Delta g\|_F\) — phase transitions in the metric, not inferred from performance.
Readiness for instruction is a measurable capacitive state — stable, high-coherence learners are maximally ready for perturbation.
The eventuality classification system types developmental trajectories using linguistic temporal logic, producing readable developmental signatures.
The Marchenko-Pastur distribution provides a principled noise floor — distinguishing practiced dimensions from undifferentiated noise, computable from existing data.

If these claims survive testing in educational settings—with real students, real curricula, real longitudinal data—then the geometry of practice becomes a viable complement to, and potentially a replacement for, the lossy scalar compression of grades.

References¶

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Phillips, P. (2026b). The field observes without an LLM. Habitat Documentation. https://docs.habitat.ooo/news/first-observation/

Phillips, P. (2026c). The precision matrix as developmental metric. Habitat Documentation. https://docs.habitat.ooo/news/precision-matrix-foundations/

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DOI: zenodo.org/records/18704763