The paired-cube decoder does not extend to 4-dimensional ports: a Lucas-parity obstruction (no new κ)

[GamingPuzzled]

Scoping note, no new κ: one README and one standard-library checker, with no existing file changed.

Question: does the paired-cube decoder generalize from triples (3-dimensional cubes) to k-dimensional ports, with more sources per cell and a bigger deficit?

Answer: consider the star family of mod-2 decoders for k-ports:

x_T = Σ_{0<l<k} a_l Σ_{L⊆T,|L|=l} star_L + Σ_{0<j<k} b_j Σ_{|S∩T|=j} x_S

This is the family the paired-cube lemma uses: level-1 stars plus overlap-1 corrections for k = 3. A decoder in this family exists iff k is not a power of two.

The coefficient of x_S depends only on j = |S∩T|. The corrections b_j fix every 0 < j < k. The diagonal coefficient is Σ a_l C(k,l), and by Lucas' theorem every C(k,l) with 0 < l < k is even exactly when k is a power of two.

k	Decoder
3 (paired cubes)	yes
4 (hypercubes)	none
5, 6	yes; they need corrections at more overlap sizes (5) or level-2 stars (6)

python3 -B research/hypercube-decoder-parity/check.py   # <1 s; tests every coefficient vector for k = 3..6

Scope:

• This covers the star or overlap-class family only. It is not an impossibility proof for every 4-port decoder.
• It makes no claim that k = 5 or 6 beats k = 3; that needs a full construction.

Credits: icekylinx (#144), eumemic (paired-cube bit word and lemma). Prepared by Joel Pulikkan with Anthropic Claude assistance; Apache-2.0.

🤖 Generated with Claude Code