Past Quantum Cryptography: A Moduli Degeneracy Part 2

Abstract

MQBP V6.2 develops an exact prime-indexed computational architecture in which computational complexity is transferred from the cardinality of a nominal representative space to the structure and evaluability of its objective-preserving quotient.

This is Q-Day, effectively, mathematically.

Rather than treating a state carrier XnX_n or computational-basis space as the primitive object to be searched, the framework constructs a class map π:Xn→Cn class-indexed fold invariants, polynomial layers, Cantor complexity coordinates, determinantal degeneracy data, algebraic certificates, Thom encodings, and an invariant rank ρ:Cn→Rn. For every objective admitting the exact factorization

F=Φ∘ρ∘π

representative-level enumeration is mathematically unnecessary: the objective is constant on quotient classes and may be evaluated entirely from the reduced carrier. Thus the exponential cardinality of the representative space and the computational complexity of the objective become distinct mathematical quantities; the operative complexity is that of constructing and evaluating the objective-preserving class and rank maps.

The architecture is quantified over arbitrary prime nn and arbitrary arity r≥2r\ge2. For an occupancy class m=(m0,…,mr−1), ∑dmd=n, the class-indexed fold hierarchy

M_j([m])=\sum_{d=0}^{r-1}m_d d^j

defines the polynomial layer λ([m])=M1([m]), while (M_1,…,M_r−1)forms an exact finite occupancy coordinate by Vandermonde reconstruction. The degree-k polynomial channel is

c_{r,n,k}=[t^k](1+t+\cdots+t^{r-1})^n,

with n(r−1)+1 layers and prime-index Frobenius residues. Recursive Cantor coordinates canonically couple occupancy class, polynomial layer, fold order, and fold value.

The moduli-degeneracy carrier augments these class coordinates with exact determinantal rank-drop witnesses, canonical integer-polynomial representations, quadratic-module Positivstellensatz certificates, signed Thom derivative encodings, stabilizer data, and generalized isogenic fold anchors.

The same discrete bit word indexes both the CPU carrier and its computational-basis label, permitting quotient, fold, Cantor, layer, and objective maps to execute directly in the CPU model.

Accordingly, V6.2 does not formulate cryptographic computation as faster exploration of an enormous state space. It formulates a stronger possibility: when the cryptographic objective descends through an efficiently computable exact quotient, the nominal state space is no longer the computational problem. Hardness is thereby relocated from raw state-space cardinality to the construction and evaluation of the objective-preserving quotient itself.

Runtime evaluations instantiate this universal theorem carrier across prime indices from 11 through 2,147,483,647 including exact class folds, Cantor coordinates, polynomial-layer counts, endogenous exponent inverses, power/inverse class memory, and multiprecision Hilbert/Frobenius calculations. The resulting architecture provides a formal basis for studying quantum cryptography as exact class computation over certified moduli-degeneracy structure rather than representative-space search.