>[!warning] >This content has not been peer reviewed. # Integer Pivot — Definitive constraint for the Micro-Graph Generator **Status: Canonical.** This document defines the non-negotiable constraint for any graph rule that claims to *derive* macroscopic dimension from information axioms. Adopted 2025-03. --- ## 1. The tautology we avoided All previous rules (ledger, traffic, literal) used the L^1.25 norm at the microscopic edge level: $W = (S^n + N_0^n)^{1/n}, \quad n = 1.25$ If we plug $n = 1.25$ into the micro-code and the graph outputs a macroscopic dimension of $1.25$, a reviewer will reject the paper: > *"You didn't derive the dimension; you literally programmed 1.25 into the micro-code and got 1.25 out."* That is the ultimate bottom-up cheat. The dimension must **emerge** from rules that are blind to it. --- ## 2. The constraint: Planck-scale math is integer math At the absolute bottom of reality (Axiom A1), the substrate does not calculate non-linear $L^{1.25}$ norms. It calculates in **1s and 0s**. | Micro-rule requirement | Rationale | |:-----------------------|:----------| | **No $n = 1.25$** | At Planck scale there are no fractions. Only discrete bits. | | **No floating-point** | The fundamental rule must use $n = 1$ (linear addition) or pure Boolean logic. | | **No $\pi$, no $c$** | No physical constants in the micro-rule. They must emerge. | **The emergence:** The $1.25$ dimension must be the **macroscopic shadow** of an integer rule, completely un-programmed by us. --- ## 3. The resolution: n=1 Micro-Triangle (we do NOT abandon the Resource Triangle) We do not throw away the Resource Triangle. At Planck scale (Axiom A1), bits combine **linearly** — the $L^1$-norm (taxicab geometry). The pure, un-dressed micro-triangle is: $W = S^1 + N^1 = S + N, \quad \mu = \frac{S}{S + N}$ **Signal $S$:** Local structural integrity. $S = \mathrm{deg}(u) + \mathrm{deg}(v)$. **Noise $N$:** Substrate refresh cost. $N = N_0$ (constant). **Landauer tax:** P(survive) = $\mu$, P(erase) = $1 - \mu$. The $1.25$ dimension must **emerge** at the macro scale from the aggregate of $n=1$ micro-triangles. We never program $1.25$ in. --- ## 4. Implementation: `rst_automaton_triangle.py` - **Rule:** `triangle_n1` in `rst_graph_rule_candidates.py` - **Micro-triangle:** $W = S + N$, $\mu = S/(S+N)$ - **FSS:** `python rst_automaton_triangle.py --L-values 16,24,32,48,64 --N0 1.0` --- ## 5. Deprecated approaches (do not use for derivation claims) | Approach | Why deprecated | |:---------|:---------------| | Ledger (L^1.25) | Programs $n = 1.25$ into micro-code. Tautology. | | Traffic (Laplacian + N_0) | Uses L^1.25. | | Literal (1-hop + L^1.25) | Same tautology; sweep stopped. | | Markov (PageRank) | N_0 decoupled; assumes lattice (circular). | | Sandpile (integer threshold) | Generic CS model; abandons Resource Triangle. | **Current approach:** `triangle_n1` — n=1 micro-triangle. Keeps the Triangle; resolves tautology. **Note:** `rst_genesis_automaton.py` uses $\mu = S/(S^n + N_0^n)^{1/n}$ with $n=1.25$ and does **not** follow Integer Pivot — it has $n$ in micro-code. It is an alternative genesis model, not the derivation engine. --- ## Related - [[Graph rule spec]] — Rule interface; now constrained by Integer Pivot - [[Graph rule verification results]] — Record of what was tried and why we pivoted