Framework
Loss Landscape Vocabulary Framework
v12 · April 2026 · Atlas Heritage Systems Inc. · Working document — not a finished product
TerrainTopologyGlobal GeometryNavigatorFlow & ResistanceStructural IntegrityAblationPotential & TensionArchitectureSummary
Potential & Tension — Revision Note
A post-adversarial-review revision to the framework's primary generative vocabulary. Resistance was identified as the wrong primary frame — it describes opposition to movement already occurring. Potential difference is upstream of movement. This section documents the revision without replacing the existing flow & resistance vocabulary.
Potential Difference (charged / discharged)
The gap between the navigator's current state and the terrain's local geometry that creates the condition for movement. In electromagnetism, current does not flow because of resistance — it flows because of potential difference. In the loss landscape: potential difference is the primary generative quantity. The gradient ∇L(θ) is its most direct expression.
ΔV(θ) = L(θ_terrain) − L(θ_navigator)
Tension (taut / slack)
The structural condition that holds potential difference stable without collapsing it. Slack tension means the navigator has decoupled from the terrain — not equilibrium, not stability, but the condition for slow invisible drift toward whichever end of the local topology has the lowest elevation. Slack in a valley is the precondition for undetected centerward drift.
Harmonics (resonant / dissonant)
The dynamic behavior that emerges when potential difference and tension interact over time. Further resolved via Skywork reading of Friston's active inference: free energy minimization produces oscillatory dynamics around posterior modes via the bidirectional prediction-error loop — an intrinsic oscillatory mechanism that is not regularization. Whether this bridges to loss landscape frequency-matching is the open question to put to Millidge.
ω_landscape = eigenvalues of H(θ)
ω_training = f(η, batch_size, curriculum)
resonance condition: ω_training ≈ ω_landscape
Smith (2018) superconvergence; Smith (2015) cyclical learning rates
Revised Archaeological Claim
High-perplexity regions in a deployed model are not merely high-resistance territory. They are places of unresolved potential difference — where the gap between navigator state and terrain geometry never closed, where the system stayed charged without discharging. Via Song et al. (2024): weights encode predictions not inputs. High-perplexity regions are places where prediction error accumulated faster than inference could resolve it before the weight update discharged it. The retained potential difference framing now has a biological learning mechanism behind it.
Revision Status
Open problem — restoring force (partially resolved): Harmonics requires a restoring force. Curvature eigenvalues describe geometry but do not generate restoring force. Millidge (2023) provides volumetric basin dominance as partial resolution. Active inference provides intrinsic oscillatory mechanism. Whether either bridges to formal frequency-matching remains open. If the restoring force is always regularization — already in the toolkit — then harmonics is a redescription of known phenomena, not a new formal quantity. The tautology charge holds until a restoring force intrinsic to landscape geometry is identified that regularization doesn't already cover.