From 90612e712cd6f8b8817623a741daa14b28171fa8 Mon Sep 17 00:00:00 2001 From: "github.com/ib-bsb-br/ib-bsb-br.github.io" Date: Tue, 28 Apr 2026 17:55:52 -0300 Subject: [PATCH] Add multilinear tensor framework for adequacy-power axioms --- tensor_framework.md | 181 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 181 insertions(+) create mode 100644 tensor_framework.md diff --git a/tensor_framework.md b/tensor_framework.md new file mode 100644 index 0000000..78d3460 --- /dev/null +++ b/tensor_framework.md @@ -0,0 +1,181 @@ +# Multilinear (Tensor) Formulaic Framework for the Principle of Least Adequate Power + +## 1) Entities and Index Sets + +Let: + +- $i \in \{0,1,\dots,m\}$ index representational scaffolds ordered by expressive power. +- $j \in \{1,\dots,n\}$ index downstream dimensions of impact. +- $k \in \{1,\dots,7\}$ index the seven axioms. +- $t \in \mathbb{T}$ index lifecycle phase (authoring, transmission, interpretation, reuse, preservation). + +Define scaffold $i=0$ as the weakest adequate candidate and larger $i$ as more expressive forms. + +## 2) Core Tensor Objects + +### 2.1 Expressive-Power Vector + +\[ +\mathbf{e} = (e_i) \in \mathbb{R}^{m+1}, \quad e_{i+1} \ge e_i. +\] + +### 2.2 Cost Tensor (Axiom 1) + +\[ +\mathcal{C} \in \mathbb{R}_{\ge 0}^{(m+1) \times n \times |\mathbb{T}|}, +\] + +with component + +\[ +\mathcal{C}_{i j t} = \text{cost imposed by scaffold } i \text{ on dimension } j \text{ at phase } t. +\] + +### 2.3 Adequacy Tensor (Axiom 2) + +\[ +\mathcal{A} \in \{0,1\}^{(m+1) \times p \times |\mathbb{T}|}, +\] + +where $p$ is the number of required purpose-constraints. $\mathcal{A}_{i q t}=1$ iff scaffold $i$ satisfies requirement $q$ at phase $t$. + +Define adequacy indicator: + +\[ +\alpha_i = \prod_{q=1}^{p}\prod_{t\in\mathbb{T}} \mathcal{A}_{i q t} \in \{0,1\}. +\] + +### 2.4 Structure-Explicitness Tensor (Axiom 3) + +\[ +\mathcal{S} \in [0,1]^{(m+1) \times r \times |\mathbb{T}|}, +\] + +where $r$ indexes relevant structural features; higher values mean structure is explicit rather than hidden in behavior. + +### 2.5 Constraint-Intelligibility Tensor (Axiom 4) + +\[ +\mathcal{I} \in \mathbb{R}_{\ge 0}^{(m+1) \times u \times |\mathbb{T}|}, +\] + +where $u$ indexes independent-agent tasks (parse, validate, reason, transform). + +### 2.6 Ecosystem Utility Tensor (Axiom 5) + +\[ +\mathcal{U} \in \mathbb{R}^{(m+1) \times n \times |\mathbb{T}|}, +\] + +capturing future-life value, not just authoring convenience. + +### 2.7 Standardization Reach Tensor (Axiom 6) + +\[ +\mathcal{W} \in [0,1]^{(m+1) \times g \times |\mathbb{T}|}, +\] + +where $g$ indexes communities/ecosystems; values represent shared comprehensibility/adoption. + +### 2.8 Escalation Justification Tensor (Axiom 7) + +For candidate escalation from $i$ to $i+1$: + +\[ +\mathcal{J}_{i\rightarrow i+1,\,q,t} = \max\big(0,\,R_{q t} - \mathcal{A}_{i q t}\big), +\] + +where $R_{q t}=1$ is required adequacy. Escalation is justified only if any component is strictly positive. + +## 3) Multilinear Scoring Functional + +Define weighted contraction operators: + +- $\langle \mathcal{C},\,\mathbf{w}^C\rangle = \sum_{j,t} w^C_{j t}\,\mathcal{C}_{i j t}$, +- $\langle \mathcal{S},\,\mathbf{w}^S\rangle = \sum_{r,t} w^S_{r t}\,\mathcal{S}_{i r t}$, +- $\langle \mathcal{I},\,\mathbf{w}^I\rangle = \sum_{u,t} w^I_{u t}\,\mathcal{I}_{i u t}$, +- $\langle \mathcal{U},\,\mathbf{w}^U\rangle = \sum_{j,t} w^U_{j t}\,\mathcal{U}_{i j t}$, +- $\langle \mathcal{W},\,\mathbf{w}^W\rangle = \sum_{g,t} w^W_{g t}\,\mathcal{W}_{i g t}$. + +Composite objective: + +\[ +\Phi(i)= +\alpha_i\Big[ +-\lambda_C\langle \mathcal{C},\mathbf{w}^C\rangle ++\lambda_S\langle \mathcal{S},\mathbf{w}^S\rangle ++\lambda_I\langle \mathcal{I},\mathbf{w}^I\rangle ++\lambda_U\langle \mathcal{U},\mathbf{w}^U\rangle ++\lambda_W\langle \mathcal{W},\mathbf{w}^W\rangle +\Big]-\lambda_E\,\Xi_i, +\] + +where $\Xi_i$ is an escalation penalty (defined below), and all $\lambda_\bullet>0$. + +## 4) Escalation Penalty and Feasibility + +Let weakest adequate index: + +\[ +i^* = \min\{i\,|\,\alpha_i=1\}. +\] + +Define unjustified-escalation measure: + +\[ +\Xi_i = \sum_{h=0}^{i-1}\mathbf{1}\!\left[\sum_{q,t}\mathcal{J}_{h\rightarrow h+1,\,q,t}=0\right]. +\] + +Interpretation: each unnecessary step to stronger form incurs penalty. + +## 5) Selection Rule (Least Adequate Power Principle) + +Primary rule: + +\[ +\hat{i}=\arg\max_{i}\Phi(i) +\quad\text{subject to}\quad \alpha_i=1. +\] + +Normative tie-break: + +\[ +\hat{i}=\min\left\{i:\,i\in\arg\max_{\alpha_i=1}\Phi(i)\right\}. +\] + +This yields the least powerful scaffold among equally adequate high-scoring options. + +## 6) Axiom-to-Tensor Mapping (Explicit) + +1. **Expressive power has cost**: monotone risk captured via $\partial \mathcal{C}_{i j t}/\partial e_i \ge 0$ (expected trend). +2. **Adequacy precedes minimization**: hard gate $\alpha_i=1$ before optimization. +3. **Explicit structure over hidden behavior**: maximize $\mathcal{S}$ contribution. +4. **Constraint creates intelligibility**: maximize $\mathcal{I}$ under bounded expressive freedom. +5. **Downstream use matters**: include lifecycle utility tensor $\mathcal{U}$ across $t$. +6. **Standard weak forms create public power**: include adoption/comprehension tensor $\mathcal{W}$. +7. **Escalation must be justified**: penalize unjustified transitions via $\Xi_i$ and $\mathcal{J}$. + +## 7) Operational Algorithm (Finite Candidate Set) + +1. Enumerate candidate scaffolds $i=0..m$ from weakest to strongest. +2. Estimate tensors $\mathcal{C},\mathcal{A},\mathcal{S},\mathcal{I},\mathcal{U},\mathcal{W}$. +3. Compute adequacy gate $\alpha_i$. +4. Discard all $i$ with $\alpha_i=0$. +5. Compute escalation penalty $\Xi_i$ from pairwise justifications. +6. Compute multilinear score $\Phi(i)$. +7. Select $\hat{i}$ by constrained maximization + minimum-index tie-break. + +## 8) Compact Einstein-Notation Form + +Using implied summation: + +\[ +\Phi(i)=\alpha_i\left(-\lambda_C w^C_{jt} \mathcal{C}_{ijt} ++\lambda_S w^S_{rt} \mathcal{S}_{irt} ++\lambda_I w^I_{ut} \mathcal{I}_{iut} ++\lambda_U w^U_{jt} \mathcal{U}_{ijt} ++\lambda_W w^W_{gt} \mathcal{W}_{igt} +\right)-\lambda_E\Xi_i. +\] + +This is the requested multilinear tensor framework: a constrained, lifecycle-aware, escalation-sensitive formalization of the stated phenomena.