# Dimensional slices, shadows and regular generators

## Bounded question

The release asks what an observer can infer about a finite set `M subset Z^4`
from complete labelled 3D slices, coordinate-axis line-sum projections, and a
scalar parameter indexing changing 4D geometry. `w` and `lambda` are
mathematical labels, not inferred physical axes.

## Complete labelled slices

For fixed slice axis `w`, define `M_c={(x,y,z):(x,y,z,c) in M}`. Then exactly

`M = union_c (M_c x {c})`.

Every point has one `w` label and occurs in one labelled slice; every
reconstructed point came from `M`. The engine applies this identity to a
137-voxel discrete 4-ball. Its slice counts are `7,33,57,33,7`, and the exact
reconstruction has symmetric difference zero. Losing labels, registration or
slice content loses information.

## Four-axis projection alias

Let `E={v in {0,1}^4:sum(v) even}` and `O={v in {0,1}^4:sum(v) odd}`. They are
disjoint eight-point sets. Drop any coordinate and fix the remaining three
bits. Exactly one missing bit makes the total even and exactly one makes it
odd. Consequently every coordinate-parallel line sum equals one for both sets,
along all four axes. Hence `E != O` but all four axis X-rays agree. Their
labelled `w` slices differ and reconstruct each set separately.

This exact counterexample does not say that complete tomography is ambiguous;
uniqueness depends on the object class and directions. One 3D “edge” slice is
weaker still: two 4D sets can share it and differ at unobserved `w`.

## A fifth coordinate as generator index

A scalar `lambda` can index `M_lambda`; the 5-coordinate worldvolume is
`W={(p,lambda):p in M_lambda}`. That representation does not decide whether
`lambda` is space, time, temperature, a control setting or bookkeeping.

The claim “one scalar can only trace a curve” requires regularity:

- one integer can enumerate any finite catalog (the engine maps `0..48` onto
  all 49 pairs in a 7 by 7 box family);
- a merely continuous scalar map can be space-filling (Peano);
- for a `C1` map `g:R->R^2`, its 2 by 1 derivative has rank at most one; on a
  compact interval it is locally Lipschitz, so its image has 2D measure zero
  and contains no 2D open set.

The diagnostic path `g(lambda)=(1+lambda,1+lambda^2)` has rank one at 101
sample points. The general result comes from the written theorem, not those
samples. The declared finite path `a=b` visits seven of 49 distinct 4D boxes
and misses 42; an arbitrary scalar code can nevertheless enumerate all 49.

## Evidence classification and next gate

Slice reconstruction and the parity alias are written mathematics plus exact
finite computation. The figure is synthetic. No physical measurements are
used, so no fourth or fifth spatial dimension is identified.

Next, freeze two dynamical bridge models—ordinary 3D plus internal state versus
an explicit extra spatial coordinate—that predict different calibrated outcome
distributions under one intervention. Only unused measured data can distinguish
them; observational equivalence must remain ambiguity.