Minimal Axioms of Causal Inheritance: An Axiomatic Construction from Update Identity to Structural Generation

Minimal Axioms of Causal Inheritance presents a parameter-free axiomatic construction in which distinct structural channels preserve an identical parent-to-child update law. The minimal causal conditions select $$q=2,\qquadq_{\#}=q+1=3,$$ and therefore $$(q,q_{\#})=(2,3).$$ The derived pair generates $$X(n,s)=q_{\#}q^n+s,\qquadX(n):=X(n,0),\qquadY(n)=q^{n+1}-1.$$ From the common \(X,Y\) hierarchy, three downstream structural maps are defined: $$P(n):=q_{\#}^{-1}X(n)Y(n),$$ $$R(n):=q\,X(n)^{-1}X(n)_{\#},$$ $$S(n):=q\,X(n)^{-1}Y(n).$$ For \(q=2\), the product map becomes $$P(n)=2^n\left(2^{n+1}-1\right).$$ To retain the consecutive product sequence, define $$G_P(n):=\gcd\!\left(P(n),\prod_{k=1}^{n-1}P(k)\right).$$ For every integer \(n\ge2\), $$\frac{P(n)}{G_P(n)}=1\iffn=5.$$ The first values are $$\frac{P(n)}{G_P(n)}=6,14,5,31,1,127,\ldots\qquad(n=1,2,3,4,5,6,\ldots).$$ The four consecutive product values preceding the unique equality are $$(P(1),P(2),P(3),P(4))=(6,28,120,496).$$ Exactly three of these are even perfect numbers: $$(P(1),P(2),P(4))=(6,28,496).$$ The value \(P(3)=120\) remains an ordinary member of the generated sequence and is not discarded. The product sequence itself remains defined for all \(n\ge0\); the gcd criterion is a derived arithmetic property and does not truncate \(P(n)\). The equivalence $$\frac{P(n)}{G_P(n)}=1\iffn=5$$ is proved in the main text using the Zsigmondy theorem, and a self-contained cyclotomic and multiplicative-order proof is provided separately in the appendix. The two ratio maps are evaluated at different structural generations: $$\begin{pmatrix}R(q)\\S(q^2)\end{pmatrix}=q\begin{pmatrix}X(q)^{-1} & 0\\0 & X(q^2)^{-1}\end{pmatrix}\begin{pmatrix}X(q)_{\#}\\Y(q^2)\end{pmatrix}=\begin{pmatrix}13/6\\31/24\end{pmatrix}.$$ Thus $$R(q)=\frac{13}{6},\qquadS(q^2)=\frac{31}{24},$$ with $$\frac{q}{S(q^2)}=\frac{X(q^2)}{Y(q^2)}=\frac{48}{31}.$$ The product-map formulas for \(R(q)\), \(S(q^2)\), and \(R(q)-S(q^2)\) remain exact equivalent downstream representations rather than primary definitions. The resulting logical structure is $$\text{causal inheritance}\longrightarrow(q,q_{\#})\longrightarrow(X,Y),$$ followed by $$(X,Y)\longrightarrow P,\qquad(X,Y)\longrightarrow (R(n),S(n)).$$ After \(P\) has been generated, its divisor structure yields the separate arithmetic branch $$P\longrightarrowG_P\longrightarrow\left[\frac{P(n)}{G_P(n)}=1\iffn=5\right].$$ The framework is formulated at a pre-observational structural layer (L1). No observational constant is used to determine \(q\), \(q_{\#}\), \(X\), \(Y\), \(P\), \(R(n)\), or \(S(n)\). Finite geometry is retained as an independent mathematical consistency check. In particular, the independently selected dimensions \((2,3,5)\) reproduce the same even-perfect-number subsector \((6,28,496)\) without removing the intermediate structural level \(n=3\).