# Top-KPQ Collision Geometry — Theorem Register

Muhammad Arshad · September 21, 2026

This register secures the post-five-paper MPRC collision results in the booklet repository. It is intentionally separate from `research/dyadic-cubic/release-v1/`; that release remains immutable publication provenance for the five papers.

## Frozen definitions

For a temporally ordered selection on `Z_(2H)`, with MPRC specialization `H=128`:

- `K`: number of selected positions/states in the temporally ordered selection. It is not a tuned Top-K hyperparameter.
- `m = K-1`: number of movement slots in that selection.
- `P = (x_last - x_first) mod 2H`.
- `D = min(P, 2H-P)`.
- `L = Σ d_circle(x_i,x_{i+1})`.
- `Q = (L-D)/2`.
- Primitive collision rewrites:
  - `{+i,+j} <-> {+(i-1),+(j+1)}`
  - `{-i,-j} <-> {-(i-1),-(j+1)}`
  for `1 <= i <= j <= H-1`, with `0` and `H` sign-free.

## Notation correction — gate-discovered

The frozen Top-K quantity `K` counts selected positions. A `K`-position word has `m=K-1` movements. The canonical terminal `(a,b,h,z)` counts movement slots, so its zero count is

`z = m-h-[a>0]-[b>0] = (K-1)-h-[a>0]-[b>0]`.

An earlier booklet draft wrote `K` in this slot-count formula, which was off by one. The theorem and `(a,b,h)` formulas are unchanged.

## Theorem A — Top-KPQ Completeness

Two selections are congruent under primitive same-sign repartitions and temporal permutation if and only if they have the same `(K,P,Q)`.

Equivalently,

`F(c1)=F(c2) iff c1 ~ c2`, with `F(c)=(K,P,Q)`.

### Proof spine

1. Every primitive rewrite preserves `K,P,L`, hence `Q`.
2. Orient rewrites toward increasing `Psi = Σ D(e)^2`.
3. Each forward rewrite increases `Psi` by `2(j-i+1) >= 2`.
4. `Psi` is integer and bounded above by `m H^2`, hence every oriented sequence terminates.
5. Every terminal multiset has at most one ordinary positive magnitude and at most one ordinary negative magnitude; all remaining entries are `0` or `H`.
6. The invariants determine the terminal uniquely:
   - `a=((L+P)/2) mod H`
   - `b=((L-P)/2) mod H`
   - `h=(L-a-b)/H`
   - `z=m-h-[a>0]-[b>0]=(K-1)-h-[a>0]-[b>0]`.
7. Thus each `(K,P,Q)` fiber is exactly one rewrite component.

### Regression record

Pinned implementation snapshot:

- https://github.com/muhammadarshad/siliq-rotor/blob/7c99e8f85859e3d6294f0533dbe385fed3b31e44/model/motif/collision_theorem.py
- https://github.com/muhammadarshad/siliq-rotor/blob/7c99e8f85859e3d6294f0533dbe385fed3b31e44/model/motif/audit_collision.py
- https://github.com/muhammadarshad/siliq-rotor/blob/7c99e8f85859e3d6294f0533dbe385fed3b31e44/model/motif/audit_simplex.py

Finite audit: exhaustive small `H`, edge cases by name, and all 32,896 `H=128, m=2` movement multisets (equivalently `K=3` selected-position words). These are regression evidence; the general theorem is the termination + unique-terminal proof.

## Theorem B — Dense MPRC Collision Ramification

Represent the collision quadrics by

`x_i x_j = x_(i-1) x_(j+1)`

and

`y_i y_j = y_(i-1) y_(j+1)`

with shared sign-free endpoints `x_0=y_0`, `x_H=y_H`.

At the dense diagonal prime

`p=(2, y_i-x_i)`,

all dense coordinates are units. The quadrics force

`x_j = z t^j`, `y_j = z s^j`.

With `u=s/t`, the shared half-turn gives `u^H=1`, and

`y_i-x_i = x_i(u^i-1)`.

Because the `i=1` difference is a unit multiple of `u-1`,

`p=(2,u-1)`.

For `H=128`, set `u=1+epsilon`. Then

`(1+epsilon)^128=1`.

Using `v2(C(128,j))=7-v2(j)` and the weight

`W(2^a epsilon^b)=64a+b`,

the unique lowest-weight collision is between `epsilon^128` and the `j=64` term. Reduction never lowers `W`. A nonzero normal form has maximum weight

`7*64+127=575`.

Therefore

`p^576=0`.

Following the unique weight-128 branch seven times gives

`epsilon^575 = 128 epsilon^127 != 0`.

Hence

`p^575 != 0`, `p^576=0`,

and the terminal layer is generated by `128 epsilon^127`.

## Exact arithmetic observation retained without promotion

`252 = 7*36 = 4*7*9`

`576 = 9*64 = 4*9*16 = 24^2`.

The CXR work used `3x3` and `4x4` grids, making the transition `(4,7,9)->(4,9,16)` structurally suggestive. No theorem identifying the factors with CXR grid semantics is claimed yet.

## Open items

- Associated graded structure `gr_p` of the 576-level dense local ring.
- Exact CXR structural map behind `252=4*7*9` and `576=4*9*16`.

## Booklet

See [Chapter 26 — Top-KPQ Collision Geometry](../../chapter-26.html).


## Theorem C — Self-conjugate sheet quotient

On `P=0` or `P=H`, charge conjugation `C(e)=-e` stays in the same `(P,Q)`
fiber and acts by

`(eta+,eta-,eta_H) -> (eta-,eta+,eta_H)`.

Hence the unoriented sheet is exactly `Delta_T/Z2`.

For the unrestricted simplex the orbit count is

`( C(T+2,2) + floor(T/2) + 1 ) / 2`.

The K-admissible simplex is invariant under the same involution.

## Theorem D — Exact metric-affine Q-space symmetry

For a uniform affine automorphism `f(e)=ae+b` of `Z_(2H)`, require globally either

`D(f(e))=D(e)`

or

`D(f(e))=H-D(e)`.

Then the only possibilities are

`I, C:e->-e, A:e->e+H, AC`.

Thus the ring group is exactly `V4`. K-coordinate permutations commute with it, so the
stated metric-affine group is exactly

`S_m x V4`.

The induced aggregate action is

- `C: P->-P, L->L, Lambda->Lambda`;
- `A: P->P+mH, L->mH-L, Lambda->-Lambda`;

where `Lambda=2L-mH`.

Verifier:
https://github.com/muhammadarshad/siliq-rotor/blob/master/model/motif/audit_qspace_symmetry.py


## Theorem E — Collision presentation closure

Let

`S=k[z,h,x_1,...,x_(H-1),y_1,...,y_(H-1)]`

with shared endpoints `x_0=y_0=z`, `x_H=y_H=h`.

The primitive collision quadrics form the two Hankel-minor families

`x_i x_j - x_(i-1)x_(j+1)`

and

`y_i y_j - y_(i-1)y_(j+1)`.

Each sign family is the 2x2-minor ideal of a 2xH Hankel matrix. The Top-KPQ
unique-terminal theorem orients these into a confluent quadratic Gröbner system, giving

`in(I_H)=(x_1,...,x_(H-1))^2 + (y_1,...,y_(H-1))^2`.

Consequences:

- `Hilb(t)=((1+(H-1)t)^2)/(1-t)^2`.
- Degree-K class count `N_m=H[H(m-1)+2]`.
- Krull dimension `2`.
- Cohen-Macaulay.
- Projective dimension `2H-2`.
- Koszul.
- One-sign quotient Betti numbers `beta_i=i*C(H,i+1)`, shift `i+1`.
- Full Betti polynomial `B_H(u,t)^2`, from tensoring the two Eagon-Northcott
  resolutions over the shared free base `k[z,h]`.