Overview
The non-injective foundations sub-programme is the axiomatic spine of the Cosmochrony corpus. Starting from four axioms (A1–A4) governing admissible non-injective transitions between observable states, it derives as theorems, not postulates:
- the structural necessity of non-injectivity ($\Pi$ non-injective $\Leftrightarrow$ $S_\Pi > 0$ $\Leftrightarrow$ genuine emergence);
- irreversibility and the arrow of time (from A1+A2 alone);
- the discrete Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ and its Weil representation $V_\rho$ as the unique admissible fibre;
- the absence of any independent dimensional parameter beyond $c_\chi$.
The presentation note formally liquidates the white paper's preliminary vocabulary of $\chi$, relaxation, and iterated projection. The substrate is static; the admissibility constraint replaces relaxation throughout the corpus.
Logical chain
$$ \text{A1–A4} \;\Longrightarrow\; \Pi \text{ non-injective},\; S_\Pi > 0 \;\Longrightarrow\; [X,\sigma(X)] = Z \neq 0 \;\Longrightarrow\; \mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}) \;\Longrightarrow\; V_\rho. $$
- A1 (Local projective admissibility): there exists a non-empty set $F_n$ of admissible successor directions from any observable state $O_{n-1}$.
- A2 (Structural non-injectivity): distinct admissible directions may lead to the same resolved state; $\Pi_n$ is generically non-injective.
- A3 (Non-premature selection): an admissible transition preserves the full multiplicity of open directions until projection locking.
- A4 (Projection locking): resolution occurs only when continuous BI saturation meets the discrete shell support.
Constituent papers
- ENI — Non-Injectivity as a Structural Necessity of Genuine Emergence. Framework-independent no-go theorem: any surjective $\Pi$ that is informationally complete for observables and such that $\mathcal{O}$ is not structurally isomorphic to $\Omega$ must be non-injective with $S_\Pi > 0$. Recursive non-injectivity (Cor. 6) and structural colour confinement follow.
- Foundation M — Admissible Non-Injective Transitions as the Primitive. From A1–A4 alone: irreversibility, the proto-state, the non-trivial commutator $[X,\sigma(X)]=Z\neq 0$, $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ as the fibre symmetry group, $F_n \cong V_\rho$, and projective incompleteness.
- HeisenbergStructure — Heisenberg Structure from Admissibility. Detailed proof of Foundation Theorem 5.7, including admissible non-factorisability (Lemma 3.1), prime order of the centre, and the Stone–von Neumann identification of $V_\rho$. The Heisenberg uncertainty principle is derived as a structural property of the unresolved proto-state.
- noscale — No External Dimensional Scale at the Level of Admissibility. Any deformation that introduces an independent dimensional parameter $\lambda$ (with $[\lambda] \neq [c_\chi]^k$) breaks BFS-consistent spectral scaling, structural non-injectivity, or bounded admissibility closure. Charge $Q$ and a cosmological constant $\Lambda$ cannot enter the admissibility structure.
The white paper is the programme overview and not a constituent of this sub-programme: its preliminary $\chi$/relaxation/iterated-projection vocabulary is superseded by the axiomatic formulation collected here.
Outputs to downstream sub-programmes
Every other sub-programme takes one or more outputs of the foundations as inputs.
- Note 1 (Spectral Admissibility) — consumes $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, $V_\rho$, the Born–Infeld bound $A_n \leq c_\chi/\sqrt{\lambda_n}$, and the parity involution $c \leftrightarrow q-c$.
- Note 2 (Emergent Geometry) — consumes $V_\rho$ and the four-dimensional Carnot convergence as starting point of the Mosco limit.
- Note 3 (Gauge Structure) — consumes the phase fibre and projective incompleteness as the origin of gauge freedom.
- Note 4 (Spectral Gravity) — consumes $S_\Pi > 0$ as the structural condition forcing an effective gravitational sector.
Open deliverables
- Born rule for general observables. The Born rule is established within the $\mathrm{SU}(2)$ sector (Q3); extension to arbitrary observables, multipartite systems, and continuous spectra is the primary open structural problem of this sub-programme.
- Level 2 scale determination. Proving that every emergent scale (masses, couplings, cosmological constant) is uniquely determined from $c_\chi$ and the cascade (noscale Level 2) would make the framework fully predictive with a single dimensional free parameter.
References
Beau, J. The Non-Injective Foundations Sub-Programme: Presentation Note 5. Working paper, 2026. https://doi.org/10.5281/zenodo.20548383