Minimal Axioms of Causal Inheritance: An Axiomatic Construction from Update Identity to Structural Generation
Year: 2026
Type: preprint
Abstract: Minimal Axioms of Causal Inheritance presents a parameter-free axiomatic construction in which two structural channels preserve an identical parent-to-child update law. The minimal causal conditions select $$q=2,\qquadq_\sharp=q+1=3.$$ From the derived pair $(q,q_\sharp)$, the structural sequences $$X(n,s)=q_\sharp q^n+s,\qquadX(n):=X(n,0),$$ and $$Y(n)=q^{n+1}-1.$$ are generated. Their normalized product map is $$P(n)=\frac{X(n)Y(n)}{q_\sharp}.$$ At the distinguished arguments $1$, $q$, and $q^2$, $$\bigl(P(1),P(q),P(q^2)\bigr)=(6,28,496).$$ which coincide with the first three even perfect numbers. Version 1.0 reorganizes the downstream construction so that the structural ratios $R$ and $S$ are defined directly from the generated $X/Y$ hierarchy: $$R=\frac{X(q)_\sharp}{X(1)}=\frac{13}{6},$$ $$S=\frac{Y(q^2)}{X(q_\sharp)}=\frac{31}{24},$$ The second relation gives the compact terminal ratio $$\frac{q}{S}=\frac{X(q^2)}{Y(q^2)}=\frac{48}{31}.$$ The product-map expressions remain as exact equivalent closure representations, $$R=q+\frac{1}{P(1)},$$ $$S=\frac{P(q^2)}{P(1)q^{P(1)}},$$ and $$R-S=\frac{P(q)}{q^{P(1)-1}}=\frac{7}{8}.$$ rather than as the primary definitions of $R$ and $S$. The resulting logical structure is $$\text{causal inheritance}\longrightarrow(q,q_\sharp)\longrightarrow(X,Y).$$ with the two downstream constructions $$(X,Y)\longrightarrow P,\qquad(X,Y)\longrightarrow(R,S).$$ The framework is formulated at a pre-observational structural layer (L1). No observational constant is used to determine $q$, $q_\sharp$, $X$, $Y$, $P$, $R$, or $S$.
Author: Tanaka, Yasuo
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7125627954
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