cell-sdf-topology

Time-conditioned signed distance fields (SDFs) as continuous geometric records of living form - representing cells, their shape changes, and attached signals as unified world-tubes through time.

cell-sdf-topology explores how change can be represented not as a sequence of moments, but as a single continuous shape that exists through time.

It serves as a research sandbox for time-conditioned signed distance fields (SDFs) - a way of treating both form and evolution as a unified field, suitable for representing cells, topology-changing processes, and attached signals.

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Concept

A signed distance field describes where a surface exists in space. If we add time as another dimension and let the SDF evolve through it, the shape doesn't just move - it extrudes through time.

That extrusion forms a continuous "tube" in space-time:

• Each time slice shows the shape's boundary.
• The full tube shows how that boundary deforms, divides, merges, or dissolves.

This is the world-tube - a smooth geometric object that contains the entire history of a shape. What once was animation becomes topology.

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Dimensional Analogy

Spatial Dim	Extruded Form	Physical Meaning	Biological Analogy
0D (a point)	1D world-line	A particle’s position traced through time	A single molecule or ion’s trajectory — diffusion path of one signaling molecule
1D (a curve)	2D world-sheet	A filament’s deformation through time	A cytoskeletal filament (e.g., actin, microtubule) growing, bending, and remodeling through time
2D (a surface)	3D world-tube	A membrane or boundary evolving through time	A cell membrane undergoing division, migration, or morphogenesis — its “world-tube” represents the full motion and shape evolution of that cell
3D (a volume)	4D world hyper-tube	A volumetric body evolving through time	A full cell or multicellular organism’s 4D life history — capturing both morphology and state transitions across its lifespan
4D (a hyper-volume)	5D evolutionary manifold	The space of all possible histories — how rules of change themselves evolve	Lineage and evolutionary processes: developmental programs, gene networks, and morphogenetic constraints shifting across generations
5D (a manifold of histories)	6D entangled field-space	Interacting systems of evolving manifolds — coupling across histories	Ecosystem- or biosphere-scale coupling: co-evolving species, metabolic and environmental feedbacks, where the “field” of life’s coherence spans multiple interacting world-tubes

Each new dimension "blooms" the representation - adding another axis of experience. The tube describes not only where something is, but also where it has been and is becoming.

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Relation to Synesthetic OS

This repository is part of the broader Synesthetic OS ecosystem. It focuses on operator-level research - developing the mathematical and geometric foundations before they are promoted into production schemas.

• Use cases (controls, perceptual field mapping, topology) are defined and versioned in synesthetic-schemas.
• Platform integration occurs in sdfk, where validated operators become schema-backed controls and interactive elements.
• This repository isolates the experimental stage - exploring world-tubes, cellular packing, hyperspectral overlaps, and other constructs.

When an operator here proves stable and broadly useful, it is graduated into synesthetic-schemas and used within the Synesthetic OS runtime.

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Why This Repository Exists

Most bioimaging pipelines are voxel-centric (OME-NGFF, HDF5, segmentation masks) or mesh-centric (surface triangulations). They store massive arrays or polygon soups that are:

• Heavy to store and stream.
• Hard to compare or search.
• Poor at handling topological events (division, fusion, blebbing).

This project takes a different path:

• SDF-first - geometry defined as functions, not voxel blobs.
• Operator catalog - reusable primitives (division, fusion, ripple, bleb, neck) fit from voxel data.
• Residuals as discovery - systematic errors become signals for new operators.
• Lineage as world-tubes - continuous identity functions across time, not ad hoc IDs.
• Signals anchored - spectral or temporal curves tied directly to surfaces or regions in the SDF.

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Positioning: SDF-First vs. Voxel/Mesh

Unlike voxel or mesh storage, which freeze data as arrays or polygons, the SDF-first approach treats geometry as a function.

This enables:

• Compact storage (parameters, not pixels).
• Natural handling of topological changes (splits, merges, blebs).
• Direct attachment of signals and lineage without lossy conversions.
• Residuals that surface new biology instead of being discarded as error.

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Why This Approach

• Continuity - no discrete frames; motion and form are one object.
• Physical fidelity - time is built into the geometry, not sampled afterward.
• Analytical depth - curvature and derivatives reveal motion and growth rates.
• Extensibility - the model can "bloom" beyond time to include chemical, acoustic, or perceptual fields.

Getting Started

1. Clone the repository.
2. Explore basic SDF constructions under examples/.
3. Apply temporal extrusion to any static field.
4. Visualize slices to see the cell's evolution embedded in space-time.

Future Directions

• Coupling morphology with reaction-diffusion or signaling fields.
• Learning world-tubes from microscopy or simulation data.
• Rendering and slicing 4D+ manifolds interactively.
• Quantifying topological events: division, fusion, collapse.

Inspiration

This project grew from thinking about how living forms persist and change - not as snapshots or keyframes, but as continuous processes that leave a trace through time. In biology and physics alike, every object has a world-line: a record of where it has been. Here, that idea is extended to complex spatial fields - giving each cell, surface, or volume a world-tube, a tangible geometric history.

It is a way to see structure and motion as one: to study evolution, interaction, and morphogenesis through a single, unbroken field.

Philosophical Grounding: The Geometry of Perception

This project begins from a simple but far-reaching intuition: that time-conditioned signed distance fields (SDFs) may not only describe how living forms evolve, but also reflect the deeper geometry through which perception itself is organized.

While the immediate application is biological modeling, the broader hypothesis is that the brain represents the world - and generates experience - through a similar field-based formalism. This perspective connects computational geometry with long-standing questions in neuroscience and the philosophy of mind:

• Continuity of Experience - Perception feels continuous, not frame-based. The world-tube formalism mirrors this: a single manifold where motion and form are unified aspects of one continuous field.

• The Binding Problem - How disparate sensory inputs become a coherent percept. In this model, binding is intrinsic: each sensory modality ($color(x,t)$, $sound(x,t)$) is simply a field anchored to the same manifold $f(x,t)=0$. The object is the geometry itself; its attributes are functions defined upon it.

• The "Thick" Present - Our sense of now includes traces of the past and anticipations of the near future. The temporal derivatives of $f$ encode these dynamics directly: $\partial f / \partial t$ corresponds to perceived motion; higher-order terms represent momentum, persistence, and prediction.

• Predictive Processing Substrate - Instead of storing voxel snapshots, a perceptual system could maintain compact parameters of a world-tube, updating it continuously by minimizing prediction error.

Formal Analogy

SDF World-Tube Construct	Proposed Perceptual Analog
The manifold $f(x, y, z, t) = 0$	The unified field of awareness - the stage of experience
Spatial gradients $\nabla_{xyz} f$	Perception of static form and structure
Temporal gradient $\partial f / \partial t$	Direct perception of motion and flow
Higher-order derivatives $\partial^2 f / \partial t^2, \ldots$	Felt momentum, anticipation, surprise
Anchored signal fields (e.g. $color(x,t)$)	Sensory qualities - color, sound, texture
Topological events (splits, merges)	Perceived discrete events: creation, fusion, division

In this view, perception is not a sequence of frames but a continuous manifold - a standing geometric object through which consciousness moves. The apparent "flow of time" is the traversal of that manifold's temporal axis.

This section defines the conceptual horizon for the project: to develop a formal geometry capable of describing not only how cells exist and transform in time, but how perception itself might arise from such transformations.

Limits of Representation: The Observation Dimension

Every system that seeks to represent its own evolution encounters a structural boundary. In logic, Godel's incompleteness theorems show that any consistent formal system rich enough to describe arithmetic contains true statements that cannot be proven within it. By analogy, a geometric system that encodes its own temporal evolution faces a comparable limit.

A time-conditioned signed distance field $f(\mathbf{x}, t)$ that captures all change necessarily embeds its own dynamics. Such a field cannot simultaneously contain the perspective from which it is observed - that perspective lies in an additional, unmodeled dimension we call the Observation Dimension.

Element	Logical Analog	Meaning
The SDF manifold $f(\mathbf{x}, t)=0$	Formal system	The complete set of describable states
Residuals or unfit data	Godel sentences	Truths the current basis cannot express
New operators or fields	Axiomatic extensions	Expanding expressivity without closure

Completeness in this context is asymptotic: each new operator expands what can be represented, but a remainder always persists - phenomena just beyond the expressible basis. Those residuals are not errors; they mark the discovery frontier - the surfaces where new operators and new couplings emerge.

Recognizing this limit keeps the framework honest: the SDF is not the world itself, but the best self-consistent slice of it that can be held inside geometry.

Repository Structure

• docs/ - SSOT design: specifications, operator catalog, audit methods, and open questions.
• docs/primer.md - Primer: a plain-language bridge between cell biology terms and SDF operators/workflows.
• meta/prompts/ - Provenance: init/, emit/, and audit/ prompts with heredocs.
• src/ - Future code: stubs for fitting, reconstruction, and APIs.
• tests/ - Test harness: placeholders targeting Python >=3.11 with pytest.
• meta/outputs/ - Audit outputs: reports and verification artifacts.

Status

• Initialization, emission, and audit loops complete.
• Documentation passes audit.
• Ready for public release as a reference repository.

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License & Citation

• License: MIT.
• Cite via CITATION.cff (GitHub will generate BibTeX/APA automatically).

For related work on time-conditioned SDFs via neural representations, see Wiesner et al., 2023.