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bibkey: cfmp2026fibonacciobstruction authors: trureturing contributors year: 2026 title: CFMP Fibonacci return observation obstruction doi: null url: https://github.com/the-omega-institute/trureturing/pull/11418 claim: A scoped framed-return congruence obstruction with exact positive-modulus kernel normal form and image condition. license: citation-only triage: anchor strata_touched: []

The cyclic CFMP return observation on integer pairs is

$$C(x,y)=(2x-y,x+2y).$$

These are the additive low and high return parameters of the cyclic construction; they are not geometric edge lengths. The full discussion appears in Sections 133–139. The elementary coordinate calculations below explain the observation and its modular information loss.

Write

$$J(x,y)=(y,x),F(x,y)=(y,x+y),D(x,y)=(-x+2y,2x+y),W=J\circ F\circ J.$$

Expanding both coordinates gives

$$D^2(x,y)=(5x,5y),C=D\circ J,C\circ W=F\circ C.$$

Thus $C$ first swaps the coordinates and then applies $D$. In the coefficient basis $(1,\theta )$ of $\mathbb{Z}[\theta ]/({\theta }^2-\theta -1)$, $D$ is multiplication by $2\theta -1$, whose square is five. The substitution matrix $F$ is the ordinary Fibonacci matrix; identifying this particular readout does not identify the whole tetrahedron geometry with a Fibonacci evolution.

For $k,t\in \mathbb{N}$, set $s(k,t)=(kt,2kt)$. Direct substitution gives the two separate coordinate identities

$$C(z+s(k,t){)}_1=C(z{)}_1,C(z+s(k,t){)}_2=C(z{)}_2+5kt.$$

If $a\equiv b(\mathrm{mod}m)$ means $m\mid b-a$, these identities show that both observed coordinates of $z+s(k,t)$ and $z$ agree modulo $5k$. The complete integer description of this congruence kernel is

$$C(x,y)\equiv (0,0)(\mathrm{mod}5k)⟺\exists t,s\in \mathbb{Z}:x=kt,y=2kt+5ks.$$

For necessity, write $-2x+y=5ka$ and $-x-2y=5kb$. Then $x=-k(2a+b)$ and $y=2k[-(2a+b)]+5ka$, so take $t=-(2a+b)$ and $s=a$. For sufficiency, the displayed input has output $(-5ks,5k(t+2s))$.

When $k>0$, reducing this normal form modulo $5k$ gives precisely

$$\ker C_{5k}=\{(kt,2kt):t=0,1,2,3,4\}.$$

The first coordinates are distinct modulo $5k$: if $5k\mid k(t-t^{\prime })$, cancellation of the nonzero integer $k$ gives $5\mid t-t^{\prime }$, which forces $t=t^{\prime }$ in the stated range. Every nonempty observation fibre is a translate of this kernel and has five residue classes. For $k=0$, congruence modulo zero means equality; the normal form instead forces $x=y=0$, and all shifts $s(0,t)$ coincide. There is no five-element kernel in that case. Over the integers themselves, $C$ is injective; the five-element ambiguity concerns positive finite moduli divisible by five.

The corresponding image condition is

$$\exists x,y\in \mathbb{Z}:C(x,y)\equiv (r,s)(\mathrm{mod}5k)⟺5\mid 2r+s.$$

Indeed, $2C(x,y{)}_1+C(x,y{)}_2=5x$, proving necessity even after reduction modulo $5k$. Conversely, if $2r+s=5q$, then $(x,y)=(q,2q-r)$ has output exactly $(r,s)$. This equivalence also holds at $k=0$ as the image condition for the integer map.

At modulus $5040=5\cdot 1008$, the five shifts are $(1008t,2016t)$ for $t=0,1,2,3,4$. Their output differences are $(0,5040t)$. This is the same factor-five obstruction; the factor seven contributes no additional kernel state to this readout. It supplies no implication for Robin’s criterion or RH.

The obstruction concerns framed return-parameter observations. Distinct residue parameters do not by themselves prove distinct unmarked manifolds, a general hyperbolic realization, or a counterexample to the minimum-six CFMP conjecture. Sections 133–139 separately give the fuller dynamics and geometric families; their finite auxiliary checks are in cfmp_fibonacci_ramification_check.py.