Pairing-friendly curve cycles

Formal statement

A 2-cycle is a pair of curves with $\#E_1(\mathbb F_p)=q$ and $\#E_2(\mathbb F_q)=p$. An $m$-cycle is the analogous closed chain of length $m$.

The research questions are:

1. Do pairing-friendly 2-cycles exist outside the MNT families with embedding degrees 4 and 6?
2. Do pairing-friendly cycles of length at least 3 exist?
3. Can a cycle satisfy a clearly specified cycle-level $\rho<2$ criterion?

Target outcome

The preferred outcome is a fully verified new cycle or a non-existence proof for a stated class. The session-scale target is an exhaustive negative search over a precisely documented toy space, including near-misses and the condition that rejects each one.

Any candidate must pass independent point counting on both curves. Its exact embedding degree must be checked by proving divisibility at the claimed degree and non-divisibility at every smaller positive degree.

Scope

All experiments obey the repository-wide toy ceiling $\log_2 p\leq 60$. Negative results are claimed only for the explicitly enumerated search space.

State in five lines

EMPIRICAL: distinct primes below $2^{28}$ The exact-degree-3-through-12 primary census is candidate-complete. EMPIRICAL: same space It has 333 2-cycle hits and five directed 3-cycle hits. PROVED Every 2-cycle with both degrees in {3,4,5,6,8,10,12} is globally classified. PROVED The quartic/quartic genus-one remainder is fully closed; no arithmetic wall remains there. PROVED The remaining global frontier is degree 7, 9, or 11 and general length-at-least-3 cycles.

What is established

• PROVED For an $m$-cycle, $t_i=p_i+1-p_{i+1}$, $\sum_i t_i=m$, and exact degree is $\operatorname{ord}_{p_{i+1}}(p_i)$.
• PROVED A 2-cycle has equal Frobenius discriminants.
• EMPIRICAL: primes below $2^{28}$, degrees 3 through 12 The 333 hits are 164 degree-(6,4), 164 degree-(4,6), and five tiny exceptions.
• EMPIRICAL: three distinct primes below $2^{28}$ There are five directed hits and 61 two-of-three near-misses; no hit has a field above 43.
• EMPIRICAL: all full hits through 24 bits Every hit has explicit equations verified by BSGS plus enumeration or a prime-point Hasse certificate.

Global 2-cycle results

• PROVED Degree pairs (6,4) and (4,6) are exactly the two MNT polynomial orientations (MNT_CLASSIFICATION.md).
• PROVED If both degrees lie in {3,4,6}, no other pair occurs (QUADRATIC_DEGREE_CLASSIFICATION.md).
• PROVED If exactly one degree lies in {3,4,6} and the other in {5,8,10,12}, the unique cycle is $(7,11;10,3)$ (MIXED_DEGREE_CLASSIFICATION.md).
• PROVED If both degrees lie in {5,8,10,12}, the unique cycle is $(11,13;12,10)$ (QUARTIC_DEGREE_REDUCTION.md).
• PROVED Therefore all pairs in {3,4,5,6,8,10,12} squared are classified.

Quartic closure details

• PROVED Small gaps $2\le c\le106$ and all 34 degenerate discriminants are exhaustively audited; only $(11,13;12,10)$ occurs.
• PROVED For $c\ge108$, $|h|\le24$, leaving 750 genus-one rows.
• PROVED Congruence, real-sign, and higher-power Hensel obstructions reduce these to 47 rows on 29 normalized curves.
• EMPIRICAL: every even $108\le c\le10^7$ on 51 pre-Hensel curves An 11,333,558-candidate exact square search found no integral point.
• EMPIRICAL: exact Magma V2.29-8 computations Twenty-two curves have only small integral points, five have empty fake two-Selmer sets, and the last symmetric curve has rank zero and torsion $\mathbb Z/2$.

Length-three structure

• PROVED No consecutive MNT prime triple closes in either orientation with all exact degrees in 3 through 12 (MNT_THREE_CHAIN_OBSTRUCTION.md).
• PROVED Every equal-signed-gap degree-(4,6) path is consecutive MNT (MNT_PATH_CLASSIFICATION.md).
• EMPIRICAL: all 61 near-misses below $2^{28}$ Twenty-six are excluded MNT chains and 35 are residual; only two residuals lie above $2^{16}$.

Precisely scoped negatives

• EMPIRICAL: $5\le p<q<2^{28}$, degrees 3 through 12 No non-{(6,4),(4,6)} 2-cycle has $q\ge32$.
• EMPIRICAL: three distinct fields below $2^{28}$, same degrees No directed 3-cycle hit contains a field greater than 43.
• PROVED These finite statements do not cover fields at least $2^{28}$, degrees above 12, or global pairs involving exact degree 7, 9, or 11.

Validation

EMPIRICAL: Python 3.13.4 and SymPy 1.14.0 All 70 shared and 58 P4.2 tests pass; all P4.2 Python files compile. Magma outputs have local parser and substitution regressions.

Next action

Derive the quotient/resultant reduction for ordered degree pair (7,7). In parallel mathematically (not with agents), classify the five tiny directed 3-cycle hits by exact degree pattern and determine whether each is isolated.

Invariants - do not violate

• Keep SEARCH_SPACE.md and THREE_CYCLE_CONDITIONS.md frozen.
• Preserve all one-sided and two-of-three near-misses.
• Distinguish global proofs from the 28-bit empirical census.
• Do not rely on the anomalous empty QG012/QG013 IntegralQuarticPoints output; the independent rank-zero proof closes them.
• Cycle rho is $\max_i\log(p_i)/\log(p_{i+1})$; its geometric mean is 1.

Files that matter

RESULTS.md is the scoped write-up. QUARTIC_DEGREE_REDUCTION.md is the newest global theorem. data/magma_*_20260718.txt stores the final exact global outputs. code/search_*_targeted.py implements candidate-complete root enumeration. data/classify_three_cycle_near_misses_n61_20260718.csv is the residual ledger.

What I would tell my replacement

The quartic wall is gone: do not reopen its 750 curves. The next genuinely new algebra starts with cyclotomic degrees six and ten (exact degrees 7, 9, 11), while the length-three global question is still open beyond MNT chains.