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. An audit now separates results with different epistemic status:
- 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);
- a finite $S_3$ countermodel showing that the published axioms do not select the discrete Heisenberg group or its Weil representation;
- the Heisenberg/Weil carrier as a supplied realization, with downstream results conditional on that input;
- 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.
Status correction. Version 1.3 of the presentation note records the
correction. The earlier PDFs remain available as part of the record, but their
carrier-selection theorem must not be read as established.
Featured video
One Reality, Multiple Descriptions introduces the structural necessity of non-injectivity, the first result in this sub-programme's logical chain.
Notation bridge. The video's $\Omega$ is the fine-grained configuration space of ENI's
framework-independent theorem. It is not the preliminary white paper's substrate symbol $\chi$, and the
sub-programme does not identify them.
Read the associated paper overview or watch on YouTube or Instagram.
Logical chain
$$ [X,Y]\neq e \;\not\Longrightarrow\; [X,Y]\in Z(G) \;\not\Longrightarrow\; G\simeq\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}). $$
ENI proves non-injectivity under its own explicit hypotheses. If a finite Heisenberg group and a non-trivial central character are supplied, the Stone–von Neumann endpoint remains valid. What is open is the edge selecting that carrier from A1–A3 and BI parity.
- 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. Axiomatic architecture and temporal ordering results. Its published Heisenberg/Weil selection theorem is not established.
- HeisenbergStructure — Heisenberg Structure from Admissibility. Published carrier-selection proof attempt. Its centrality and prime-order steps fail; its final representation-theoretic step is valid only when the carrier data are supplied.
- HeisenbergCarrierObstruction — finite countermodel to carrier selection. The six-element $S_3$ construction proves that the preceding algebraic properties do not force a central commutator. It does not propose $S_3$ as the physical carrier.
- 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.
- ProjCE — Projection as Conditional Expectation and the Capacity Ceiling. Under four graded hypotheses (fibre-invariance forced by ENI; fixity on resolved content; linearity; non-amplification), the amplitude-level action of $\Pi$ is the conditional expectation onto the distinguishable-record sigma-algebra, by two classical characterisations. Consequences: strict ensemble-power suppression (law of total variance) and the parameter-free capacity ceiling $(N-1)/N$ — identically the Bessel envelope of the low-$\ell$ capacity note.
- EffComp — Support and Preparation Criteria for Independent Effective Subsystems. Names the transversal contract K4 (admissibility $\to$ composable effective subsystems). The support layer is completely characterised under the proposed contract: independence blocks are exactly the bicliques of the projected support relation. The preparation layer is resolved for two reference classes — the full simplex, and strictly positive cylindrical conditionings, which are product-closed iff $\operatorname{rank} W = 1$ — and remains open in general. Support rectangularity does not imply preparation independence. The contract itself is proposed, not derived from A1–A4; its states-and-effects layer's finite, mono-system, and bipartite-composition content is now resolved by STEFT below, which leaves the general preparation layer, the tensor product, and any phase/modulus law open in turn.
- STEFT — Finite Criteria for Simplicial States and Local Tomography. Continues EffComp's effective-composition track: the raw cylindrical-conditioning family is never itself convex (unconditional, any local outcome set of size at least two); an explicit classical-randomisation postulate closes it onto the full simplex, rank-independently. Affine effects on a simplex correspond to a cube of vectors (representation, not physical availability); a coarse-graining axiom yields the full Boolean effect algebra for free, a dual randomisation axiom the full cube. The two randomisation postulates are logically independent, unified only under an explicit control-independence clause. Given physical existence of a composed bipartite effect, the pointwise product is forced, not postulated; local composition reaches only rectangle effects, a CHSH-shaped witness excluding the diagonal. Local tomography holds exactly when local effect families linearly span their ambient spaces. No Born rule or quantum state geometry is derived.
- HMPL — Refinement Consistency Obstructs Nonlinear Power Lifts of Finite History Measures. Takes up the phase/modulus question STEFT's own conclusion names as its natural next step, without deriving it: for any finite branching system, the power-lift family $\mu^\alpha$ of its induced valuation is Kolmogorov-consistent, at a branching state reached with positive $\mu$-valuation, iff $\alpha=1$ — a self-contained theorem, independent of this sub-programme's corpus. A row-renormalised escort lift restores consistency trivially but generally departs from the pointwise lift. Specialised, conditional on a named, non-derived maximal-entropy/Parry postulate, to the Heisenberg trajectory-branching automaton. No Born rule, complex carrier, or coherent sum is established; this paper's directory sits at the programme root, alongside its trajectory-branching source, not in this sub-programme's tree.
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
Other sub-programmes consume either proved outputs or explicitly supplied realizations.
- Note 1 (Spectral Admissibility) — assumes $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, $V_\rho$, 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
- Carrier selection. Derive an independently motivated condition that excludes the $S_3$ countermodel and selects a central extension, or retain the Heisenberg/Weil carrier as a declared model choice.
- 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