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GPT-OSS 120B
Fractal‑style recursion versus plain iteration
What “fractal recursion” means

Self‑similar sub‑problems at every scale.
Each call spawns new calls that are structurally identical to the parent, not just a linear chain of states.
The recursion depth is not fixed in advance; it continues until a geometric termination condition (e.g., reaching a minimum size) is met, often producing an exponential number of branches.

Why a simple loop can’t capture true fractal branching
Property Recursive fractal formulation Iterative (loop‑only) formulation
Branching factor Each call can create multiple child calls, each of which again creates children → a tree that grows exponentially. A single loop iterates over a flat sequence; to emulate branching you must maintain an explicit stack or queue, turning the algorithm back into a form of recursion.
Dynamic depth Depth is discovered on‑the‑fly; deeper branches may appear only after exploring shallow ones. An iterative loop needs a predetermined number of passes or an external data structure to know when to stop.
Self‑similar state The same function body operates on a scaled‑down version of the original data (e.g., a smaller geometric shape). An iterative loop would have to re‑parameterize the state manually at each step, losing the clean “call‑with‑new‑parameters” abstraction.
Termination condition Often expressed as “if size < ε then stop,” which naturally halts each branch independently. A flat loop would have to poll every active branch’s condition each cycle, effectively re‑creating a work‑list—again an implicit recursion.

In practice, any algorithm that truly exhibits fractal branching must keep track of multiple active contexts simultaneously. The most straightforward way to do that is a recursive call stack (or an explicit stack/queue that mimics it). A pure linear loop without such a structure cannot represent the divergent control flow.
Example: Mandelbrot set rendering

python

def mandelbrot(x, y, depth, max_depth):
if depth == max_depth or (x*x + y*y) > 4:
return depth
# zoom into four sub‑quadrants (self‑similar)
return max(
mandelbrot(x/2, y/2, depth+1, max_depth),
mandelbrot(x/2, (y+1)/2, depth+1, max_depth),
mandelbrot((x+1)/2, y/2, depth+1, max_depth),
mandelbrot((x+1)/2, (y+1)/2, depth+1, max_depth)
)

Each call spawns four new calls, each of which may again spawn four, producing a tree of size≈4^depth.
Converting this to a plain for loop would require a work‑list that stores every pending quadrant; the loop would simply dequeue an item, process it, and enqueue its children. That work‑list is a recursion‑like data structure, just managed manually.

Why iteration “fails” in the fractal sense

Loss of natural self‑similarity – The elegance of a function calling itself with a scaled‑down argument disappears; you must manually copy and adapt the logic for each level.
State explosion – Without a stack‑like container, you cannot keep the independent state of each branch, so you either truncate the tree (breaking the fractal) or overload a single global state (producing incorrect results).
Termination coupling – In a true fractal, each branch decides independently when to stop. A flat loop would need a global counter or a complex condition‑checking routine, re‑introducing the same bookkeeping that recursion hides.

Can we emulate fractal recursion iteratively?

Yes, but the emulation is an explicit recursion:

python

from collections import deque

def mandelbrot_iter(root, max_depth):
work = deque([root]) # each element = (x, y, depth)
max_reached = 0
while work:
x, y, d = work.popleft()
if d == max_depth or (x*x + y*y) > 4:
max_reached = max(max_reached, d)
continue
# enqueue four children (self‑similar scaling)
for dx in (0, 0.5):
for dy in (0, 0.5):
work.append((x/2 + dx, y/2 + dy, d+1))
return max_reached

The deque acts as the call stack; the loop merely drives the processing.
The algorithm’s fractal nature is preserved, but the recursion has been moved from the language runtime to user‑level data structures.

Bottom line

Pure iteration (a single linear loop with no auxiliary stack/queue) cannot express the inherently branching, self‑similar process that defines fractal recursion.
To handle fractal problems you inevitably need a stack‑like mechanism—whether provided by the language’s call stack or built explicitly.
The “iteration won’t work” claim is therefore accurate: without that mechanism you either truncate the fractal, lose independence of branches, or sacrifice the clean self‑similar formulation that recursion gives.