Generative Dependency Mathematics: Minimal Bases, Necessity, Synergy, and Irreducibility in Forward-Generating Systems

This paper introduces a mathematical framework for analyzing systems in which a finite set of primitive inputs generates a substantially larger set of downstream structures through explicit forward dependency rules. The motivating application is the Emergence Canvas Model, but the framework is formulated independently of any particular physical theory. The Central Structure A finite generative system is represented by a primitive set P, an output set O, and a monotone generation map \Gamma: 2^P \rightarrow 2^O, where \Gamma(A) is the set of outputs generable from primitive subset A. For each output o \in O, we define its family of minimal primitive supports \mathcal{H}_o = \{ S \subseteq P : o \in \Gamma(S), \ o \notin \Gamma(S') \ \forall S' \subsetneq S \}. Key Theorems · Minimal-Support Representation Theorem: For every output o and every A \subseteq P, o \in \Gamma(A) \iff \exists S \in \mathcal{H}_o \text{ such that } S \subseteq A. Once \mathcal{H}_o is known, the primitive-level generative logic of o is completely determined.· Uniqueness of the Support Representation: The antichain \mathcal{H}_o is unique. No two different antichains can produce the same generation behavior.· Primitive Necessity Criterion: A primitive P_i is necessary for output o iff P_i \in \bigcap_{S \in \mathcal{H}_o} S.· Robustness–Transversal Theorem: The minimum number of primitive deletions required to destroy output o equals the transversal number of the support hypergraph: \rho(o) = \tau(\mathcal{H}_o). Necessity Is Not Robustness Consider two outputs: · \mathcal{H}_A = \{ \{P_1, P_2, P_3, P_4\} \}: Every primitive is individually necessary, yet \rho(A) = 1.· \mathcal{H}_B = \{ \{P_1, P_2\}, \{P_3, P_4\} \}: No primitive is individually necessary, yet \rho(B) = 2. Thus, a primitive may be necessary while the output is fragile, or unnecessary while the output is robust. These properties are mathematically distinct. Generative Functions Need Not Be Submodular A single output with \mathcal{H}_o = \{ \{P_1, P_2\} \} demonstrates: f_o(\{P_1\}) = 0, \quad f_o(\{P_2\}) = 0, \quad f_o(\{P_1, P_2\}) = 1. Submodularity would require 0 \ge 1. Therefore, monotone generative output functions are not generally submodular. The reason is primitive complementarity: two primitives that generate nothing individually may produce a new capability jointly—increasing rather than diminishing returns. Complexity of the Minimum Generative Basis Problem The minimum-cardinality generative basis problem is NP-hard. The proof reduces Set Cover to the problem: create a primitive for each set and an output for each universe element, with singleton minimal supports corresponding to set membership. A minimum generating basis is exactly a minimum Set Cover. Boolean–LCM Primitive Lattices For pairwise coprime primitive periods 2, 3, 5, 7, the Boolean lattice of primitive subsets is isomorphic to the square-free divisors of 210: 2^{\{2,3,5,7\}} \cong D_{\mathrm{sf}}(210), with union and intersection represented by least common multiple and greatest common divisor. This provides a lossless multiplicative encoding of the primitive composition lattice. Application to the Canvas Model Applying the framework to a restricted set of explicitly documented Canvas structural outputs yields a unique sufficient basis containing all eight primitives. The paper therefore establishes structural irredundancy relative to the present documented derivations. A proposed reduction of the eight primitives to four dynamic/property dual generators is tested and fails. The four obstructions are: · Order does not uniquely generate chirality: P_1 \not\Rightarrow P_5· Amplitude does not uniquely generate dimension: P_2 \not\Rightarrow P_6· Acceleration does not uniquely generate angle: P_3 \not\Rightarrow P_7· Polarity does not uniquely generate charge multiplicity: P_4 \not\Rightarrow P_8 The pairings remain structurally meaningful, but they are complementary rather than derivational. The resulting Primitive Projection No-Go result distinguishes genuine reduction from mere informational repackaging. Fewer symbols do not constitute a true reduction unless total independent specification information decreases. A General Research Programme The paper proposes the following programme for any candidate foundational system: 1. Determine the complete primitive set2. Construct the explicit forward dependency graph3. Identify minimal supports for all important outputs4. Perform primitive ablation5. Identify alternative derivations6. Calculate robustness and synergy7. Solve the minimum-basis problem8. Test whether alternative bases are merely coordinate-equivalent9. Measure total informational cost rather than symbol count10. Subject the primitive basis itself to attempted reduction This converts the statement "these assumptions are fundamental" into a sequence of mathematical questions. Main Results 1. Every finite monotone output possesses a unique antichain of minimal primitive supports.2. Output generation is exactly characterized by containment of one of these minimal supports.3. Individual primitive necessity is exactly the intersection of all minimal supports.4. Minimum destructive ablation equals the transversal number of the minimal-support hypergraph.5. Necessity and robustness are mathematically distinct.6. Primitive synergy is represented directly by multi-element minimal supports.7. Generative output functions need not be submodular.8. Minimal generating bases form an explicit combinatorial optimization problem.9. The general minimum-cardinality basis problem is NP-hard.10. Pairwise-coprime primitive labels provide a lossless Boolean–LCM composition encoding.11. The Canvas dynamic periods 2,3,5,7 instantiate this encoding.12. A restricted audited Canvas support system requires all eight primitives simultaneously.13. A naive reduction from eight primitives to four dynamic/property generators fails under present Canvas mathematics. The Ultimate Question The framework provides a formal way of asking one of the deepest questions available to any foundational theory: Which parts of the foundation are truly doing indispensable explanatory work? The ultimate optimization problem is: \min I(P, \Gamma) subject to recovery of the required consequence space. In fundamental physics this becomes: How little independent structure must be specified before the rest of physics becomes forced? That is the mathematical problem of generative minimality. Keywords: generative dependency, minimal support, primitive necessity, ablation robustness, synergistic order, minimal generating basis, Set Cover, hypergraph transversal, Boolean–LCM correspondence, Canvas Model, primitive projection no-go, fundamental physics