Top-KPQ Collision Geometry
Complete Fibers and the 576-Level Dense Local Ring

What Is Closed

This chapter records two results reached after the five-paper dyadic-cubic series. First, the frozen MPRC fingerprint (K,P,Q) is a complete invariant of the congruence generated by the primitive same-sign Top-3 repartitions together with temporal permutation. Equality of (K,P,Q) is therefore not merely necessary: it is sufficient for two selections to lie in the same rewrite component. Second, the dense diagonal localization of the collision algebra over Z₂₅₆ has one relative rotor parameter and exact maximal-ideal nilpotence length 576.

The (K,P,Q) Object

Let c=(x₀,…,x_{K−1}) be a temporally ordered MPRC selection of K selected positions on Z₂H. For the MPRC ring H=128 and 2H=256. Its movement multiset has cardinality m=K−1. Define

K is the observed number of selected positions, not a tuned Top-K hyperparameter; m=K−1 is the number of movements. U, the sliding observer, changes the candidate domain; it is not part of the fingerprint of a fixed selection.

The Native Repartition

For ordinary positive magnitudes 1≤i≤j≤H−1, and separately for the negative orientation, the primitive collision move is

Zero and the half-turn H are sign-free boundary states. Each primitive move preserves K, P and L, hence also Q.

Top-KPQ Completeness Theorem

The forward implication is invariance. For the converse, orient each primitive rewrite toward increasing

A move (i,j)→(i−1,j+1) changes the potential by

Ψ is integer and bounded above by mH², so every oriented rewrite sequence terminates. A terminal multiset contains at most one ordinary positive magnitude and at most one ordinary negative magnitude; all remaining entries are 0 or H. The invariants force that terminal uniquely.

Closed Form in (a,b,h,z)

For a reachable fiber, write the unique terminal as one positive residue a when a>0, one negative residue b when b>0, h half-turns and z zero moves. With P represented in Z₂H,

Hence every (K,P,Q) fiber has one irreducible normal form. Two selections with the same fingerprint reduce to the same terminal and are therefore in the same undirected rewrite component.

Falsification Record

The theorem is analytical; finite computation is retained as regression evidence rather than substituted for the proof. Exhaustive enumeration checked H=2…7 and K=2…4, including P=0, P=H, mixed-sign selections, zero moves, half-turns and repeated values. For H=128, the K=2 case exhausts all 32,896 multisets with zero mismatches against the closed form.

The Dense Quadrics

Introduce variables x_i for the positive orientation and y_i for the negative orientation, with shared sign-free endpoints x₀=y₀ and x_H=y_H. The primitive repartitions are represented by

There are H(H−1)/2 quadrics per orientation and therefore H(H−1) in total. At H=128 this is 16,256.

Dense Localization Collapses

Localize at the dense diagonal prime

The x_i and y_i are units there. Put z=x₀=y₀. From the i=1 positive quadrics, if t=x₁/x₀ then x_j=z t^j. Similarly, with s=y₁/y₀, y_j=z s^j. The shared half-turn gives t^H=s^H. Defining u=s/t yields

Because y₁−x₁=x₁(u−1) and x₁ is a unit, all 127 interior difference directions generate the same ideal:

Thus 𝔭=(2,u−1). The dense collision algebra has one relative rotor direction, not 127 independent nilpotent directions.

Dense MPRC Collision Ramification

For H=128 set u=1+ε. Then u¹²⁸=1 gives

For 1≤j<128, ν₂(C(128,j))=7−ν₂(j). Give the monomial 2^a ε^b the MPRC weight

The unique lowest-weight collision in the binomial relation is between ε¹²⁸ and the j=64 term 2·(odd)·ε⁶⁴; both have weight 128. Every reduction is therefore weight-nondecreasing. A nonzero normal form has a≤7 and b≤127, so its weight is at most

Every generator 2^a ε^(576−a) of 𝔭⁵⁷⁶ has weight 576+63a≥576 and must vanish. Repeatedly taking the unique weight-128 branch gives

Hence the exact nilpotence length is 576.

The 252 → 576 Observation

The proven ramification length has the exact arithmetic forms

The established active-address count has

Therefore the pair may be written (4,7,9)→(4,9,16). This is especially suggestive because the CXR work used 3×3 and 4×4 grids. The arithmetic identity is exact; a theorem identifying these factors with the CXR grid construction has not yet been proved and is intentionally left open.

Unoriented Sheet Quotient

Circular magnitude is invariant under e↦−e, so L and Q are unchanged while P↦−P. The only fixed displacement sheets are 0 and H. On those sheets the base residues coincide, a=b, so conjugation swaps only the two oriented excess coordinates and leaves the half-turn coordinate fixed. The K-admissibility condition is symmetric in the two oriented sums, hence the quotient is exact.

for the unrestricted simplex; K-admissible subsets inherit the same involution orbit construction.

Exact Metric-Affine Symmetry

For f(e)=ae+b with gcd(a,2H)=1, requiring either D(f(e))=D(e) for all e or D(f(e))=H−D(e) for all e forces, from e=0 and e=1 alone, b∈{0,H} and a∈{+1,−1}. Thus no other affine ring map survives.

Coordinate permutations commute with the uniform V₄ action, yielding the direct product.

Collision Presentation Closure

Each sign family is exactly the 2×2-minor ideal of a 2×H Hankel matrix with shared endpoints z and h. The Top-KPQ unique-terminal theorem supplies the normal-form interpretation: an irreducible monomial contains at most one positive interior variable and at most one negative interior variable. Thus the standard monomials are z^a h^b times 1, x_i, y_j, or x_i y_j.

For one sign, the Eagon–Northcott quotient Betti numbers are β_i=i·C(H,i+1) with shift i+1. The positive and negative resolutions tensor over the shared free base k[z,h], so the full Betti polynomial factors as B_H(u,t)².

What Remains Open

Frozen Research Record

Preprint drafts derived from this chapter:

The collision theorem and audits were first frozen in the SILIQ/MPRC math ledger and verifier code before this booklet chapter was created. This chapter preserves the mathematical statement and its scope; it does not turn unaudited adjacent claims into consequences.