[NOTICE — see description] Sequence-Dependent Cryptography (The Magmoidal Cipher): Verifiable Delay Functions, V31-QKD, and Asymmetric Sequential Locks via Non-Associative Fano-Geometry

PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested.

NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form.

The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other.

The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true".

What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing.

Additional note. This record proposes cryptographic constructions without a security proof, an attack analysis, or a reduction to a standard hardness assumption. Nothing here should be used, or relied upon as secure, in any setting. The non-associative structure it is built on is addressed above. A cryptographic proposal requires at minimum a stated threat model, a reduction to a well-studied problem, and analysis against known attacks on similar constructions; none is present.

The foundational security of modern cryptographic protocols relies heavily on the computational intractability of associative mathematics (e.g., factoring primes, elliptic curve point addition). A critical feature of associative algebra is that operations commute and associate: $(AB)C = A(BC)$. Consequently, algebraic evaluation can be arbitrarily chunked and parallelized across massive hardware clusters, accelerating both legitimate computation and brute-force attacks.

In this paper, we introduce the Magmoidal Cipher, a fundamentally novel cryptographic protocol that operates over non-associative magmas—specifically, the incidence geometry of the Octonionic Fano Plane ($PG(2,2)$). By encoding a cryptographic task as a strictly right-leaning sequence of non-associative Cayley-Dickson doublings, we intentionally violate the Mac Lane Pentagon identity. Because evaluating the state out of its strict topological sequence yields catastrophic geometric phase errors, the algorithm mathematically forbids parallel sub-chunking.

We demonstrate two distinct commercial applications for this geometry:

1. Classical Silicon: The protocol serves as an optimal Verifiable Delay Function (VDF), guaranteeing a deterministic, un-acceleratable sequential time-lock resistant to ASIC and parallel-GPU speedups.

2. Quantum Hardware (V31-QKD): When deployed on a 7-qubit Quoct substrate, guessing the incorrect sequential basis forces a measurement in an invalid geometric orientation. This triggers a topological Associator Penalty, converting adversarial eavesdropping directly into a heralded quantum erasure, fundamentally upgrading the security bounds of standard BB84 Quantum Key Distribution.