Magnification Function¶

Allocation model — instantiates Neighborhood-Preserving Substrate Mapping

Sets how much scarce substrate each source region receives as a function of its importance, so high-relevance regions are magnified with fine resolution and low-relevance regions are compressed.

Magnification Function is the rule that converts a region's importance into an amount of substrate — a function from source density or task-relevance to local magnification (substrate per unit of source). Its defining move is being the explicit allocation law: rather than tiling uniformly, it states, as a tunable function, exactly how resolution is traded across the space — how much map an important region earns versus the periphery. It decides how much resolution each region gets; other mechanisms then carry that allocation out. It is a design object you can write down, inspect, and argue about, not an emergent side effect of training.

Example¶

A VR headset cannot render its entire wide field at full resolution at frame rate, so it spends pixels the way the eye does. A magnification function ties rendering resolution to gaze and eccentricity: the ~2° foveal patch the user is looking at is magnified — rendered at full detail — while resolution falls off steeply toward the periphery, mirroring the eye's own cortical magnification, where the fovea claims a hugely disproportionate share of visual cortex. The function is the trade curve: steepen its falloff and you save more GPU but risk the user catching blur when they glance sideways; flatten it and the image is uniformly crisp but the frame budget blows. It sets the allocation; the renderer and the eye-tracker execute it.

How it works¶

Take a relevance or sensitivity profile over the source — density, task-error cost, attention or gaze — and define a monotonic function mapping relevance to local magnification, subject to the constraint that the allocation integrates to the total substrate capacity (spend more here, you must spend less there). The output is a per-region resolution target. What makes it this mechanism is that it is the allocation law itself — a stated function — not the dynamic re-tiling that re-applies it and not the grid of receptive fields that fills it in.

Tuning parameters¶

Falloff steepness — how sharply magnification drops from important to unimportant regions; steep saves more substrate but concentrates distortion at the periphery.
Relevance measure — what drives magnification (source density, task-error sensitivity, attention); changing it reshapes the entire map.
Budget / normalization — the total substrate the function must sum to; a tighter budget forces sharper trade-offs.
Floor allocation — a minimum resolution guaranteed to low-relevance regions, so the periphery never goes fully blind.
Update coupling — whether the function is fixed or re-evaluated as relevance shifts, in which case it hands off to remeshing.

When it helps, and when it misleads¶

Its strength is making the resolution trade-off explicit and tunable instead of implicit, letting a finite substrate buy accuracy exactly where it pays. Its failure modes trace to the relevance signal and the steepness. It is only as good as its relevance measure — magnify by the wrong signal and you lavish resolution where it does not matter while starving where it does — and a steep function creates strong distortion gradients the periphery's users feel. The classic misuse is setting magnification to favor a region for political or vanity reasons — the loudest stakeholder's area — while dressing it up as an objective allocation; this is exactly what a Fairness and Salience Review exists to catch. The discipline is to derive magnification from a defensible, auditable relevance profile and to keep a floor for the periphery.^[1]

How it implements the components¶

It owns the allocation policy and the relevance profile that drives it; it does not execute or fill the allocation:

magnification_allocation_policy — it is the policy: the function that allocates substrate by importance.
task_relevance_and_sensitivity_profile — it reads and encodes the relevance/sensitivity profile that the allocation is a function of.

It does not dynamically re-apply the allocation as relevance drifts — that is Adaptive Remeshing and Reallocation, which consumes this function — and it does not lay down the receptive-field grid it sizes, which is the Receptive-Field Tiling Grid.

Instantiates: Neighborhood-Preserving Substrate Mapping — it is the archetype's law for magnifying resolution where it matters.
Sibling mechanisms: Adaptive Remeshing and Reallocation · Receptive-Field Tiling Grid · Adjacency-Matrix Preservation Test · Boundary and Seam Regression Test · Calibration Anchor Stimuli · Canary Region Probe · Coverage-Hole Heatmap · Elastic-Net Embedding · Lateral-Interaction Smoothing · Local Ablation or Lesion Probe · Map Fold and Collision Scan · Map Registration and Alignment · Multidimensional Scaling Layout · Neighborhood Trustworthiness and Continuity Metric · Perturbation-Response Map · Self-Organizing Map Training · Topographic Error Measure · Versioned Coordinate Atlas

Notes¶

The magnification function is where the archetype's efficiency goal meets a fairness question: "important to whom, measured how?" Because magnifying one region necessarily compresses another, the allocation is a distributive choice as much as a technical one — which is why it pairs naturally with a fairness/salience review of the relevance measure, rather than being treated as a neutral optimization.

References¶

[1] Cortical magnification — the disproportionately large area of sensory cortex devoted to high-acuity regions such as the fovea — is the real, well-documented phenomenon this mechanism generalizes. Cited as a correctly-used anchor. ↩