Growth & Repair
A body-plan grows to a target shape, then, cut it, and watch it regrow the missing part.
Layer 1 of 2 · the accepted science
The classical picture
How does a cell in a growing embryo know where it is, and so what to become? The standard answer in developmental biology is positional information, the idea Lewis Wolpert set out in 1969 with his French flag model: a group of cells can form a pattern if each one reads a signal that varies with position and responds according to its local value. The signal is a morphogen, Alan Turing's 1952 word for a substance that shapes form, released from a small source region often called an organizer (after Spemann and Mangold's 1924 transplant experiments in amphibian embryos).
The textbook model of such a gradient is a morphogen made at a source, spreading by diffusion, and breaking down as it goes:
Why these terms
- Diffusion, D ∂²c/∂x². Molecules spread from high concentration to low (Fick's law). On its own, diffusion from a steady source would eventually fill the whole tissue and carry no positional information.
- Degradation, k c. Molecules are removed at a rate proportional to how many there are. Balanced against diffusion, this gives a stable profile that falls off over a characteristic length λ, so concentration really does encode distance from the source.
- A threshold, θ. Cells above the threshold respond one way and cells below it another. The boundary between them sits at a definite distance, and because that distance depends on the logarithm of source strength, doubling the source moves the boundary outward by only about 0.7 λ. The best-studied real case is the Bicoid gradient that patterns the head-to-tail axis of the fruit-fly embryo.
Regeneration
Some animals rebuild lost parts by re-reading the same kind of positional system. Cut a planarian flatworm into pieces and each piece can regrow a whole worm; cut the head off a Hydra and a new head organizer forms and a new head grows. In these animals the tissue that remains re-establishes its signalling sources and then regrows toward the form those signals specify, which is why regeneration is described as a return to a target shape.
Where the classical picture stops
The panel below keeps the logic and drops the chemistry. There is no diffusing morphogen. Each step the rule reads every cell's straight-line distance from the organizer directly, and uses a second test in place of a signal that has to travel: a cell counts as reached only if a path of living tissue connects it to the organizer. Tissue grows into empty space within the target radius and dies if it lies beyond the radius or is cut off from the organizer. The organizer sits at the centre and can be cut out, but nothing in the rule rebuilds it. That is the largest difference from a regenerating animal such as Hydra, where a lost organizer can form again.
What you are looking at
A lattice seen from above. The yellow cell at the centre is the organizer, green is living tissue, and black is empty space. Target radius sets how far the form should reach, growth rate sets how readily tissue spreads into empty space within it, Small wound and Amputate half cut tissue away and always leave the organizer, Cut out the organizer removes it, and clicking on the canvas cuts wherever you click, organizer included. Match to target is the fraction of the target disk that is filled.
The Body-Plan
Duncan(√5). No sovereignty verdict is claimed, that is gated.
Layer 2 of 2 · the framework's reading
The wrapped reading
Everything below is written for readers working through Principia Attractum. If you came for the developmental biology, you have already had it.
The molecular and population panels asked whether a flow could hold a
basin. This one asks a different question: can a shape hold itself? A form grows
outward from an organizer until it matches a target body-plan, then stops. The real test is what
happens when you damage it: a self-holding form treats its shape as an
attractor, disturb it and it returns. This is morphological regeneration,
the planaria trick, made runnable, and it maps to BEP (§3.4): identity sustained
while the boundary is restored.
The target body-plan is supplied by the modeler: an organizer emits a signal, and
cells grow or die to match how far that signal says the boundary should reach. That authored
target is the supplied structure, shown plainly. What is worth watching is not that it reaches
the shape, it is whether it returns to the shape after damage, and what
damage makes the form fail to come back (a rupture, β).
Click on the canvas to cut away tissue, or use the buttons. Any cut that leaves the organizer heals, however much it takes. A cut that takes the organizer does not: the tissue withers and the form cannot come back. No sovereignty verdict is claimed, only the observable: did the form regenerate, or rupture?
The Shape of the Basin
At the default settings (target radius 26, growth rate 0.60) the home is the full target disk: 2,121 tissue cells filling 100% of it, reached from the organizer alone in 18 to 22 steps. Once there, the form holds it exactly.
Every damaged form tested that kept its organizer fell home. Twelve settled forms each took five kinds of damage that leave the organizer in place, and all sixty returned to the full disk:
| damage | match right after | back to 100% |
|---|---|---|
| Small wound | 95% | 12 of 12, in 5 to 9 steps |
| Amputate half | 49% | 12 of 12, in 16 to 20 steps |
| a ring cut around the organizer, severing the whole outer body | 97% | 12 of 12, in 17 to 25 steps |
| 90% of all tissue deleted at random | 12% | 12 of 12, in 16 to 23 steps |
| all tissue deleted, organizer left | 0% | 12 of 12, in 19 to 23 steps |
Even a form reduced to the organizer alone regrows completely. How much tissue is lost does not decide the outcome.
Losing the organizer is what ruptures the form. Cut out the organizer and the form ruptures. Every piece of tissue is then disconnected, all of it withers in the next step, and with no organizer nothing grows again: 12 of 12 boards were empty one step after the organizer was removed and still empty 300 steps later. Of 900 random canvas cuts made inside the target disk, the form ruptured exactly when the cut covered the organizer and recovered every other time, with no exceptions.
That irreversibility is authored, not discovered: the rules contain nothing that can make a new organizer, so a board without one can never grow a form again. What the runs measured is the other side of the line. Every tissue configuration tested, from a small wound to every cell deleted but the organizer, came back.
So the basin is very large, and one variable decides membership. Across everything tested, the states that return to the target disk are the states in which the organizer survives, whatever else has been lost, and the states that do not return are the states in which it has not. How much tissue a cut removes does not decide whether the form comes back; which cell it removes does. A cut that takes every cell of the body except the organizer stays in the basin, and a cut that takes the organizer and little else leaves it.
The Morph of the Basin
The sliders change the rules, and with them the target and the speed of return. They never change the condition for return. Starting from the organizer alone, twelve forms were grown at each setting. Every one of the 240 reached a full disk:
| target radius | growth 0.10 | growth 0.30 | growth 0.60 | growth 1.00 |
|---|---|---|---|---|
| 12 | 93 to 131 steps | 25 to 39 | 11 to 15 | 6 |
| 20 | 141 to 193 | 37 to 60 | 15 to 22 | 10 |
| 26 | 173 to 220 | 45 to 52 | 18 to 22 | 13 |
| 32 | 197 to 227 | 55 to 71 | 22 to 26 | 16 |
| 38 | 232 to 270 | 60 to 72 | 25 to 30 | 19 |
Changing the radius imposes a new attractor, and the form converges to it. The radius slider is not a push on the form; it rewrites the rule the form obeys, so each setting has its own target disk and its own basin. Moving the target from 26 to 38 on a settled form grew it to the new disk in 12 to 18 steps, and moving it from 26 to 12 trimmed it in 6 to 9, 12 of 12 each way. From the smallest to the largest setting and back, the form re-settled exactly every time.
Slow growth stretches the return, it does not break it. At the slowest and largest setting (growth 0.10, radius 38), forms stripped to the organizer took 229 to 276 steps to come back, and all twelve did.
The condition for return does not change. At every slider setting a form with its organizer returns and a form without it does not. The only thing the radius changes is how easy the organizer is to hit with a random cut: 62 of 300 random cuts took it at radius 12, 17 of 300 at radius 26 and 6 of 300 at radius 38, simply because a bigger body is more tissue around the same single cell.
What the morph can and cannot say here. This is the clearest case in the gallery of a basin held up from outside. The target shape is written into the panel as a radius, the positional signal is read off that radius directly each step, and the whole basin exists only while one authored cell survives. Everything the tissue does, it does in service of a shape it was handed. No maintenance cost is represented anywhere in the model, the organizer and the growth rule are supplied from outside, and the tissue cannot rebuild the one cell the shape depends on, which is what the disclosure card below records as condition 4 supplied.
BEP §3.4). An authored design claim, not a measurement.
SOV §7.2), what this
panel supplies, not how it scored- exercised 1 · Recursion lock. Morphology is a maintained attractor (BEP §3.4): the body grows to a target shape and, when a chunk is cut away, the still-connected tissue regrows to target while an isolated fragment withers, you can watch identity restored or lost.
- exercised 2 · Internal recurcline persistence. The match-to-target recovers after any damage that leaves the organizer, and when a cut takes the organizer the whole form withers and cannot regenerate. Caveat: The update runs synchronously at u=1, and the morphogen field is computed globally each step, so part of both the temporal and positional order is supplied.
- exercised 3 · Boundary retention. Unlike the other panels, a boundary IS modeled here: the connected-tissue rule maintains a form against damage. Tissue disconnected from the organizer dies, so the body's distinction from empty space is actively restored.
- supplied 4 · Maintenance-bearing continuation. The organizer cell is the free signal source and nothing in the structure can rebuild it, and growth into the target radius is at an authored rate with no modeled inflow, so the maintenance of form is bought with an authored organizer, not paid by the structure.
No sovereignty verdict is claimed here or anywhere in this gallery. The Sovereign column is empty, and that emptiness is the honest reading: no panel here has been shown to produce its own order.