Add multilinear tensor framework formalizing 'least adequate power' axioms#25
Add multilinear tensor framework formalizing 'least adequate power' axioms#25ib-bsb-br wants to merge 1 commit into
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Reviewer's GuideAdds a new markdown document defining a multilinear tensor-based decision framework for selecting the least powerful adequate representational scaffold, including tensors, scoring functional, escalation penalty, and an operational algorithm, plus an Einstein-notation compact form and axiom mapping. Sequence diagram for evaluating and selecting a scaffoldsequenceDiagram
actor Designer
participant Framework
participant Tensors
participant CandidateScaffolds
Designer->>Framework: Provide candidate scaffolds i = 0..m
Framework->>CandidateScaffolds: Enumerate scaffolds by expressive power
Framework->>Tensors: Estimate C, A, S, I, U, W for each scaffold
Tensors-->>Framework: Return estimated tensor values
Framework->>Framework: Compute alpha_i from A
Framework->>Framework: Discard scaffolds with alpha_i = 0
Framework->>Framework: Compute J and Xi_i for remaining scaffolds
Framework->>Framework: Compute Phi(i) using tensors, weights, lambdas, Xi_i
Framework->>Framework: Select î = argmax Phi(i) with minimum-index tie-break
Framework-->>Designer: Return selected least adequate power scaffold î
Class diagram for tensor objects and scoring functionalclassDiagram
class IndexSets {
int m
int n
int p
int r
int u
int g
set T
}
class ExpressivePowerVector_e {
float e_i[m+1]
}
class TensorC {
float C_ijt[(m+1)*n*|T|]
}
class TensorA {
int A_iqt[(m+1)*p*|T|]
int alpha_i[m+1]
compute_alpha_i()
}
class TensorS {
float S_irt[(m+1)*r*|T|]
}
class TensorI {
float I_iut[(m+1)*u*|T|]
}
class TensorU {
float U_ijt[(m+1)*n*|T|]
}
class TensorW {
float W_igt[(m+1)*g*|T|]
}
class TensorJ {
float J_i_to_i1_qt[m*p*|T|]
compute_J(int i, int q, int t)
}
class EscalationPenalty {
float Xi_i[m+1]
compute_Xi_i()
}
class Weights {
float wC_jt[n*|T|]
float wS_rt[r*|T|]
float wI_ut[u*|T|]
float wU_jt[n*|T|]
float wW_gt[g*|T|]
}
class Lambdas {
float lambda_C
float lambda_S
float lambda_I
float lambda_U
float lambda_W
float lambda_E
}
class ScoringFunctionalPhi {
float Phi_i[m+1]
compute_Phi(int i, int alpha_i, float Xi_i, Lambdas lambdas, Weights weights, TensorC C, TensorS S, TensorI I, TensorU U, TensorW W)
}
IndexSets --> ExpressivePowerVector_e
IndexSets --> TensorC
IndexSets --> TensorA
IndexSets --> TensorS
IndexSets --> TensorI
IndexSets --> TensorU
IndexSets --> TensorW
IndexSets --> TensorJ
TensorA --> EscalationPenalty : provides_alpha_i
TensorJ --> EscalationPenalty : provides_J
Weights --> ScoringFunctionalPhi
Lambdas --> ScoringFunctionalPhi
TensorC --> ScoringFunctionalPhi
TensorS --> ScoringFunctionalPhi
TensorI --> ScoringFunctionalPhi
TensorU --> ScoringFunctionalPhi
TensorW --> ScoringFunctionalPhi
EscalationPenalty --> ScoringFunctionalPhi
Flow diagram for the operational scaffold selection algorithmgraph TD
Start([Start])
Enumerate["1. Enumerate scaffolds i = 0..m ordered by expressive power"]
Estimate["2. Estimate tensors C, A, S, I, U, W for all candidates"]
ComputeAlpha["3. Compute adequacy indicators alpha_i from A"]
FilterAdequate{"alpha_i = 1 ?"}
Discard["Discard scaffold i (inadequate)"]
Keep["Keep scaffold i (adequate)"]
ComputeXi["5. Compute escalation penalties Xi_i using J"]
ComputePhi["6. Compute multilinear scores Phi(i)"]
SelectHatI["7. Select î: maximize Phi(i) subject to alpha_i = 1"]
TieBreak["Tie-break: choose minimum index among maximizers"]
Output([Output selected scaffold î])
End([End])
Start --> Enumerate --> Estimate --> ComputeAlpha --> FilterAdequate
FilterAdequate -- No --> Discard --> FilterAdequate
FilterAdequate -- Yes --> Keep --> ComputeXi --> ComputePhi --> SelectHatI --> TieBreak --> Output --> End
File-Level Changes
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Hey - I've found 2 issues, and left some high level feedback:
- In the multilinear scoring functional, the contractions ⟨𝒞, w^C⟩, ⟨𝒮, w^S⟩, etc. currently suppress the scaffold index i in the notation; consider making the i-dependence explicit (e.g., writing ⟨𝒞_i, w^C⟩ or specifying that the contraction is taken over j,t with fixed i) to avoid ambiguity about what is being summed over.
- The escalation justification tensor 𝒥 and penalty Ξ_i are defined conceptually but their shapes/domains are not stated as explicitly as for 𝒞, 𝒜, 𝒮, etc.; you could add a brief statement of 𝒥 ∈ ℝ^{m × p × |𝕋|} (or similar) and a range for i in Ξ_i to keep the tensor definitions uniform and easier to reason about.
Prompt for AI Agents
Please address the comments from this code review:
## Overall Comments
- In the multilinear scoring functional, the contractions ⟨𝒞, w^C⟩, ⟨𝒮, w^S⟩, etc. currently suppress the scaffold index i in the notation; consider making the i-dependence explicit (e.g., writing ⟨𝒞_i, w^C⟩ or specifying that the contraction is taken over j,t with fixed i) to avoid ambiguity about what is being summed over.
- The escalation justification tensor 𝒥 and penalty Ξ_i are defined conceptually but their shapes/domains are not stated as explicitly as for 𝒞, 𝒜, 𝒮, etc.; you could add a brief statement of 𝒥 ∈ ℝ^{m × p × |𝕋|} (or similar) and a range for i in Ξ_i to keep the tensor definitions uniform and easier to reason about.
## Individual Comments
### Comment 1
<location path="tensor_framework.md" line_range="117" />
<code_context>
+
+## 4) Escalation Penalty and Feasibility
+
+Let weakest adequate index:
+
+\[
</code_context>
<issue_to_address>
**issue (typo):** Add missing article and verb in this phrase for grammatical correctness.
You could rewrite this as "Let the weakest adequate index be:" to make the sentence grammatically complete while preserving the meaning.
```suggestion
Let the weakest adequate index be:
```
</issue_to_address>
### Comment 2
<location path="tensor_framework.md" line_range="129" />
<code_context>
+\Xi_i = \sum_{h=0}^{i-1}\mathbf{1}\!\left[\sum_{q,t}\mathcal{J}_{h\rightarrow h+1,\,q,t}=0\right].
+\]
+
+Interpretation: each unnecessary step to stronger form incurs penalty.
+
+## 5) Selection Rule (Least Adequate Power Principle)
</code_context>
<issue_to_address>
**issue (typo):** Add an article in "incurs penalty" to improve grammatical correctness.
Consider rephrasing to: "each unnecessary step to stronger form incurs a penalty" so the sentence reads more naturally.
```suggestion
Interpretation: each unnecessary step to stronger form incurs a penalty.
```
</issue_to_address>Help me be more useful! Please click 👍 or 👎 on each comment and I'll use the feedback to improve your reviews.
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| ## 4) Escalation Penalty and Feasibility | ||
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| Let weakest adequate index: |
There was a problem hiding this comment.
issue (typo): Add missing article and verb in this phrase for grammatical correctness.
You could rewrite this as "Let the weakest adequate index be:" to make the sentence grammatically complete while preserving the meaning.
| Let weakest adequate index: | |
| Let the weakest adequate index be: |
| \Xi_i = \sum_{h=0}^{i-1}\mathbf{1}\!\left[\sum_{q,t}\mathcal{J}_{h\rightarrow h+1,\,q,t}=0\right]. | ||
| \] | ||
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| Interpretation: each unnecessary step to stronger form incurs penalty. |
There was a problem hiding this comment.
issue (typo): Add an article in "incurs penalty" to improve grammatical correctness.
Consider rephrasing to: "each unnecessary step to stronger form incurs a penalty" so the sentence reads more naturally.
| Interpretation: each unnecessary step to stronger form incurs penalty. | |
| Interpretation: each unnecessary step to stronger form incurs a penalty. |
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| - $k \in \{1,\dots,7\}$ index the seven axioms. | ||
| - $t \in \mathbb{T}$ index lifecycle phase (authoring, transmission, interpretation, reuse, preservation). | ||
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| Define scaffold $i=0$ as the weakest adequate candidate and larger $i$ as more expressive forms. |
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Stating that i=0 is already the “weakest adequate candidate” conflicts with the rest of the framework, which later computes adequacy via \alpha_i, discards inadequate candidates, and defines i^* = \min\{i\mid\alpha_i=1\}. If implementers follow this line literally, i=0 is forced to be adequate by definition and the adequacy gate becomes inconsistent/redundant, which changes the selection procedure semantics.
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| \mathcal{J}_{i\rightarrow i+1,\,q,t} = \max\big(0,\,R_{q t} - \mathcal{A}_{i q t}\big), | ||
| \] |
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The escalation-justification term for i→i+1 is computed only from \mathcal{A}_{i q t} (current scaffold adequacy), so any inadequate i automatically marks the next step as justified even when i+1 provides no improvement. This causes \Xi_i to miss unnecessary intermediate escalations through equally inadequate scaffolds, weakening the intended penalty and biasing selection toward stronger representations without evidence of added adequacy.
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Code Review
This pull request introduces a formal multilinear tensor framework for the Principle of Least Adequate Power, defining various tensors for cost, adequacy, and utility across lifecycle phases. The review feedback highlights several areas for improvement: removing the unused index
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| - $i \in \{0,1,\dots,m\}$ 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. |
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The index
| - $k \in \{1,\dots,7\}$ index the seven axioms. | ||
| - $t \in \mathbb{T}$ index lifecycle phase (authoring, transmission, interpretation, reuse, preservation). | ||
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| Define scaffold $i=0$ as the weakest adequate candidate and larger $i$ as more expressive forms. |
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Defining
| Define scaffold $i=0$ as the weakest adequate candidate and larger $i$ as more expressive forms. | |
| Define scaffold $i=0$ as the weakest candidate and larger $i$ as more expressive forms. |
| where $p$ is the number of required purpose-constraints. $\mathcal{A}_{i q t}=1$ iff scaffold $i$ satisfies requirement $q$ at phase $t$. | ||
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| Define adequacy indicator: | ||
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| \[ | ||
| \alpha_i = \prod_{q=1}^{p}\prod_{t\in\mathbb{T}} \mathcal{A}_{i q t} \in \{0,1\}. |
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The adequacy indicator
| 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\}. | |
| where $p$ is the number of purpose-constraints. Let $R \in \{0,1\}^{p \times |\mathbb{T}|}$ be the requirement tensor where $R_{qt}=1$ if constraint $q$ is mandatory at phase $t$. $\mathcal{A}_{i q t}=1$ iff scaffold $i$ satisfies requirement $q$ at phase $t$.\n\nDefine adequacy indicator:\n\n\[\n\alpha_i = \prod_{q=1}^{p}\prod_{t\in\mathbb{T}} (1 - R_{qt} + R_{qt}\mathcal{A}_{i q t}) \in \{0,1\}.\n\] |
| \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, | ||
| \] |
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The composite objective
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| - $i \in \{0,1,\dots,m\}$ 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. |
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📝 Info: Index variable
At tensor_framework.md:9, the index
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| - $\langle \mathcal{C},\,\mathbf{w}^C\rangle = \sum_{j,t} w^C_{j t}\,\mathcal{C}_{i j t}$, | ||
| - $\langle \mathcal{S},\,\mathbf{w}^S\rangle = \sum_{r,t} w^S_{r t}\,\mathcal{S}_{i r t}$, | ||
| - $\langle \mathcal{I},\,\mathbf{w}^I\rangle = \sum_{u,t} w^I_{u t}\,\mathcal{I}_{i u t}$, | ||
| - $\langle \mathcal{U},\,\mathbf{w}^U\rangle = \sum_{j,t} w^U_{j t}\,\mathcal{U}_{i j t}$, | ||
| - $\langle \mathcal{W},\,\mathbf{w}^W\rangle = \sum_{g,t} w^W_{g t}\,\mathcal{W}_{i g t}$. |
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📝 Info: Weighted contraction notation hides dependence on scaffold index
At tensor_framework.md:94-98, the weighted contraction operators are written as
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| \[ | ||
| \Xi_i = \sum_{h=0}^{i-1}\mathbf{1}\!\left[\sum_{q,t}\mathcal{J}_{h\rightarrow h+1,\,q,t}=0\right]. | ||
| \] | ||
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| Interpretation: each unnecessary step to stronger form incurs penalty. |
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📝 Info: Escalation penalty
The escalation penalty tensor_framework.md:126 counts unjustified steps from
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2 issues found across 1 file
Prompt for AI agents (unresolved issues)
Check if these issues are valid — if so, understand the root cause of each and fix them. If appropriate, use sub-agents to investigate and fix each issue separately.
<file name="tensor_framework.md">
<violation number="1" location="tensor_framework.md:12">
P2: The text incorrectly assumes `i=0` is already adequate, which contradicts the later adequacy-gating logic. Define `i=0` as the weakest candidate (not weakest adequate).</violation>
<violation number="2" location="tensor_framework.md:163">
P2: The algorithm does not handle the empty-feasible-set case after adequacy filtering, so selection becomes undefined when no candidate satisfies requirements.</violation>
</file>
Reply with feedback, questions, or to request a fix. Tag @cubic-dev-ai to re-run a review.
| - $k \in \{1,\dots,7\}$ index the seven axioms. | ||
| - $t \in \mathbb{T}$ index lifecycle phase (authoring, transmission, interpretation, reuse, preservation). | ||
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| Define scaffold $i=0$ as the weakest adequate candidate and larger $i$ as more expressive forms. |
There was a problem hiding this comment.
P2: The text incorrectly assumes i=0 is already adequate, which contradicts the later adequacy-gating logic. Define i=0 as the weakest candidate (not weakest adequate).
Prompt for AI agents
Check if this issue is valid — if so, understand the root cause and fix it. At tensor_framework.md, line 12:
<comment>The text incorrectly assumes `i=0` is already adequate, which contradicts the later adequacy-gating logic. Define `i=0` as the weakest candidate (not weakest adequate).</comment>
<file context>
@@ -0,0 +1,181 @@
+- $k \in \{1,\dots,7\}$ 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
</file context>
| Define scaffold $i=0$ as the weakest adequate candidate and larger $i$ as more expressive forms. | |
| Define scaffold $i=0$ as the weakest candidate and larger $i$ as more expressive forms. |
| 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 $\alpha_i=0$. |
There was a problem hiding this comment.
P2: The algorithm does not handle the empty-feasible-set case after adequacy filtering, so selection becomes undefined when no candidate satisfies requirements.
Prompt for AI agents
Check if this issue is valid — if so, understand the root cause and fix it. At tensor_framework.md, line 163:
<comment>The algorithm does not handle the empty-feasible-set case after adequacy filtering, so selection becomes undefined when no candidate satisfies requirements.</comment>
<file context>
@@ -0,0 +1,181 @@
+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 $\Phi(i)$.
</file context>
| 4. Discard all $i$ with $\alpha_i=0$. | |
| 4. Discard all $i$ with $\alpha_i=0$; if no candidates remain, terminate and report that no scaffold satisfies required adequacy. |
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Motivation
Description
tensor_framework.md, which defines index sets and core tensor objects for expressive power, cost (\mathcal{C}), adequacy (\mathcal{A}), structure-explicitness (\mathcal{S}), intelligibility (\mathcal{I}), ecosystem utility (\mathcal{U}), standardization reach (\mathcal{W}), and escalation justification (\mathcal{J}).\Phi(i)that combines weighted contractions of the tensors with an adequacy gate\alpha_iand an escalation penalty\Xi_ito favor the least powerful adequate scaffold.Testing
nl -ba tensor_framework.md 2>&1 | fold -w 1500 | sed -n '1,260p', which returned the expected sections and formulas.nl -ba tensor_framework.md 2>&1 | fold -w 1500 | sed -n '172,180p'to confirm the final Einstein-notation block is present and syntactically corrected.python - <<'PY' ... PYto fix escaped-token issues in the compact formula, and revalidated the affected lines, which succeeded.rg --files 2>&1 | fold -w 1500, which includedtensor_framework.mdin the results.Codex Task
Summary by Sourcery
Documentation:
Summary by cubic
Adds
tensor_framework.md, a multilinear tensor model that formalizes the “least adequate power” axioms and selects the weakest adequate scaffold. It introduces a lifecycle-aware score with adequacy gating and escalation penalties to guide choice.e, costC, adequacyA(with gatealpha_i), structure explicitnessS, intelligibilityI, ecosystem utilityU, standardization reachW, escalation justificationJ.Phi(i)with weighted contractions across lifecycle phases and an escalation penaltyXi_i.Phi(i)over adequate candidates, tie-break by smallest index.Written for commit 90612e7. Summary will update on new commits. Review in cubic