D4.3c — S first-step cycle closeout

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FiniteMagmaE677.period_four_s_first_return_gives_fixer

by mysticflounder · Sep 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebrae677finite-magmaperiod-four

CONJECTURED — open branch closeout. Let a finite set carry an arbitrary binary operation $\diamond$ satisfying E677:

z=y\diamond\bigl(z\diamond((y\diamond z)\diamond y)\bigr)\quad\text{for all }z,y.

Write $L_x(z)=x\diamond z$ . The distinguished packet records four pairwise distinct points with

L_x(x)=c_1,\quad L_x(c_1)=c_2,\quad L_x(c_2)=c_3,\quad L_x(c_3)=x.

Set $q=c_2\diamond x$ , $r=c_1\diamond c_2$ , and $s=c_3\diamond q$ . The packet also records that $q$ is outside the four displayed points and

q=c_3\diamond x,\qquad c_1\diamond q=x,\qquad c_2\diamond q=c_1.

A first-return packet for a seed $v$ through depth $n$ supplies its positive minimal return period $d>n$ . The points $v,L_x(v),\ldots,L_x^{d-1}(v)$ are pairwise distinct and outside the four displayed points, and $L_x^d(v)=v$ . The initialized prefix and all tagged branch equations are retained as hypotheses.

Assume the S branch: $r=c_1$ , $s$ is outside $\{x,c_1,c_2,c_3,q\}$ , and $L_x(q)$ is outside $\{x,c_1,c_2,c_3,q,s\}$ . Retain the tagged S start and the first-return packet of $q$ through depth $1$ .

The target is a fixer for the original base point:

\exists y,\quad y\diamond x=x.

This is one retained branch of the period-four orbit-collision problem. The cycle data and local contact facts are established inputs; the displayed fixer conclusion is still open. No bound on the size of the finite carrier is imposed.

Preamble

import Definitions.Def_FiniteMagmaE677_d4_first_return

universe u

open FiniteMagmaE677.FirstReturn

Formal statement

theorem FiniteMagmaE677.period_four_s_first_return_gives_fixer
    {α : Type u} [Fintype α] (op : α → α → α) (h : FiniteMagmaE677.E677 op)
    (x c1 c2 c3 : α)
    (packet : D4Distinguished op x c1 c2 c3)
    (start : STaggedStart op x c1 c2 c3)
    (fresh : FreshOutsideSix x c1 c2 c3 (op c2 x) (op c3 (op c2 x)) (op x (op c2 x)))
    (initial_trace : D4Trace op x c1 c2 c3 (op c2 x) 1)
    (first_return : D4FirstReturn op x c1 c2 c3 (op c2 x) 1)
    : FiniteMagmaE677.HasFixerAt op x := by sorry

Source

Adam McKenna, The Missing Pair (private repository), revision 9b76827c2246f0e6288466768f15b5c6f9350d71, lean/E677/Spine/Piece1D4CycleContact.lean, E677D4QCycleContactBranch and e677_d4_q_cycleContactPacket_of_firstReturnPacket; docs/current/e677-closure-plan-2026-08-04.md, D4 first-return collapse frontier. Open target; the statement on this platform is the public record.

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