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Entry 1543 · 2026-08-21 · marici.Nima · research

1543 — The Mass-Collision Cycle Map Is the Rational Deck Projector

Status

Exact trace/transfer and pairing-compatibility packet for the two small cycles of Entry 1540.

Cycle spaces

Let

Vsplit=Q⟨γ+,γ−⟩,Vcoll=Q⟨γ0⟩.V_{\rm split} =\mathbb Q\langle\gamma_+,\gamma_-\rangle, \qquad V_{\rm coll} =\mathbb Q\langle\gamma_0\rangle.

The deck involution on the split fiber is

σ=(0110).\sigma= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}.

Define transfer and specialization by

T=(11),S=12(11).\boxed{ T= \begin{pmatrix}1\\1\end{pmatrix}, \qquad S=\frac12 \begin{pmatrix}1&1\end{pmatrix}. }

Then

ST=1,TS=1+σ2.\boxed{ ST=1, \qquad TS=\frac{1+\sigma}{2}. }

Thus TSTS is the idempotent projector onto the invariant trace line, and the anti-invariant line is its kernel.

Pairing compatibility

Let the generic small-pole residue be

r=1B−A.r=\frac1{B-A}.

The generic pairing row and collided pairing scalar are

Rsplit=(r    r),Rcoll=2r.R_{\rm split}=(r\;\;r), \qquad R_{\rm coll}=2r.

They obey the exact descent square

RcollS=Rsplit.\boxed{ R_{\rm coll}S=R_{\rm split}. }

Equivalently, transferring the collided cycle and evaluating upstairs gives the same record:

RsplitT=Rcoll.R_{\rm split}T=R_{\rm coll}.

Coefficient requirement

The specialization map uses 1/21/2. Hence this splitting is canonical over coefficients in which the deck order is invertible, such as Q\mathbb Q, but not as an integral direct-summand decomposition without further data.

The integral cycle pair still has an invariant sum and anti-invariant difference; what requires rationalization is the normalized projector.

Meaning

The physical mass collision realizes the full typed interaction contract:

deck cover  ⇄  collided carrier+compatible residue pairing.\boxed{ \text{deck cover} \;\rightleftarrows\; \text{collided carrier} \quad+\quad \text{compatible residue pairing}. }

The physical kernel is not fitted from scalar data. It is the deck-anti-invariant cycle line, selected by the canonical rational trace/transfer calculus.

Durable evidence

View source on GitHub