Why this exists

TP notation is optimized for infinite-width neural limits. Epimechanics notation is optimized for representational mechanics across domains. This crosswalk lets us reuse TP results without pretending the theories are identical.

Crosswalk table

Tensor Programs concept	TP notation (typical)	Closest Epimechanics concept	Epimechanics notation	Mapping status	Notes
Representation state in model internals	activations / hidden states (`h`, `x`, layerwise vectors)	Representation state	`X \in S`	Partial	TP internal vectors are implementation-level coordinates; Epimechanics `X` can be broader and cross-domain.
Parameter update magnitude	SGD/Adam update recursions, scale exponents (`a,b,c,d`)	Kinetic-like representational change rate	`\dot{X}` and momentum `p_i = \mathcal{M}_{ij}\dot{X}^j`	Analogy only	Update norms can be empirical proxies for `\|\dot X\|`, not literal equality.
Width/depth scaling coefficients	`abc` / `abcd` parametrization exponents	Effective mass/coupling scaling constraints	`\mathcal{M}_{ij}`, `T^i{}_j`	Hypothesis-level	Good candidate bridge: exponents constrain responsiveness and cross-layer coupling. Requires explicit calibration.
Kernel regime	frozen NTK / tangent limit	Near-linear response regime	local quadratic approximation around trajectory	Structural analogy	Corresponds to low internal restructuring; useful baseline, not full equivalence.
Feature-learning regime / µP	maximal-update limits	High internal restructuring regime	higher representation-geometry evolution	Structural analogy	Good for testing when representation basis itself changes substantially.
NTK kernel	`\Theta(x,x')`	Effective local response operator	Jacobian/Hessian-derived response tensors	Partial	NTK is architecture/training specific; can be embedded as one response observable.
Master Theorem limits	NETSOR / NETSORᵀ / NE⊗ORᵀ limits	Asymptotic law under scaling	large-system limit statements	Compatible form	Great methodological import: explicit assumptions + convergent observables.
Spectral laws (TP3)	singular value/eigenvalue distributions	Stability/conditioning geometry	spectral stats of `\mathcal{M}`, Jacobian, Hessian	High utility	Strongest quantitative bridge for instrumentation.
Hyperparameter transfer invariance (TP5)	µTransfer	Control invariance under scaling	approximately conserved optimal-control settings	Empirical bridge	Treat as invariance test over resource scale, linked to representational efficiency claims.
Depth-µP scaling (TP6)	depthwise transfer laws	Hierarchical depth dynamics in meta-entities	multi-scale coupling in layered entities	Hypothesis-level	Promising for AI-organization analogies, but needs explicit ontological care.

Non-equivalences (important)

TP is not an ontological theory of entities. It is a formal scaling theory for neural computations.
Epimechanics X is not restricted to neuron activations. It includes any representable system state.
Mass tensor \mathcal{M}_{ij} is not directly “network width.” Width may influence empirical proxies, but not identity.
Auto-causality \rho_{ac} has no direct TP primitive. Must be constructed from dynamics/feedback observables.
Lagrangian postulate remains independent. TP can inform test design; it does not prove Epimechanics postulates.

Practical mapping rules for experiments

When logging TP-inspired experiments for Epiphysics, include:

tp_parametrization: standard / NTK / µP / depth-µP
epi_state_proxy: what was used as X (e.g., hidden state stack, feature map stats)
velocity_proxy: chosen \dot X proxy (e.g., per-step feature drift)
mass_proxy: chosen \mathcal{M} proxy (e.g., inverse sensitivity / conditioning metric)
coupling_proxy: chosen T proxy (cross-layer or cross-module transfer sensitivity)
regime_label: kernel-like / feature-learning-like / mixed

Mapping confidence legend

Compatible form: mathematical style aligns; direct reuse usually safe.
Partial: careful reinterpretation needed.
Structural analogy: useful conceptual bridge, not formal identity.
Hypothesis-level: candidate mapping to test, not assume.

Implementation status

Implemented baseline module:

experiments/src/tp_bridge_metrics.py

Current outputs:

representation drift,
layer update norms,
NTK drift,
spectral radius (from provided eigenspectra),
transfer regret.

Fixture + sample output:

experiments/src/tp_bridge_example_trace.json
experiments/results/tp_bridge/example_metrics.json

Next increment:

connect this script to real experiment traces (instead of synthetic fixture).

Notation Crosswalk: Tensor Programs ↔ Epimechanics

Contribute to Epiphysics