The Non-Injective Foundations Sub-Programme

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:

genuine emergence implies non-injectivity with $S_\Pi > 0$ under ENI's surjectivity, informational-completeness and non-isomorphism hypotheses;
immediate information loss along non-injective transitions; temporal order and the arrow of time require the unestablished acyclicity hypothesis [H-acyc];
a finite $S_3$ countermodel showing that the published axioms do not select the discrete Heisenberg group;
the supplied Heisenberg group and non-trivial central character, the resulting Heisenberg/Schrödinger representation, and the distinct associated Weil action, with downstream results conditional on those inputs;
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.

Current published status. Working-paper version 1.5 separates proved, conditional and supplied ingredients. Immediate information loss and projective incompleteness under A2 are proved; temporal order requires [H-acyc], and irreducibility requires a representation-theoretic dictionary. HeisenbergStructure 2.1 supplies the finite obstruction. The parity involution is defined on central-character labels; its Born–Infeld interpretation is open.

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, Stone–von Neumann identifies the irreducible Heisenberg/Schrödinger representation $\pi_c$ on $\mathcal H_c=L^2(\mathbb Z/q\mathbb Z)$. The associated Weil action is distinct further symplectic data. What is open is the edge selecting the group 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 and Strict Information Loss in Projective Effective Descriptions. 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 order conditional on [H-acyc]. Theorem 5.7 proves the carrier-selection obstruction; irreducibility requires a representation-theoretic dictionary. Non-commutation follows for an irreducible action in dimension greater than one when the pair generates the acting group. Projective incompleteness follows from A2 alone.
HeisenbergStructure — A Finite Obstruction to Heisenberg Carrier Selection from Admissibility Constraints. Version 2.1 incorporates the six-element $S_3$ countermodel: the preceding algebraic properties do not force a central commutator, so the admissibility contract does not select a finite Heisenberg group. Stone–von Neumann identifies $\pi_c$ on $\mathcal H_c$ only when the group and non-trivial central character are supplied; it does not supply the separate Weil action. $S_3$ is not proposed 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 $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, its Schrödinger representation $\pi_c$ on $\mathcal H_c$, the separate associated Weil action, and the Born–Infeld bound $A_n \leq c_\chi/\sqrt{\lambda_n}$, and the parity involution $c \leftrightarrow q-c$ (a structural hypothesis: by O18's no-go, Born–Infeld evenness does not force it).
Note 2 (Emergent Geometry) — takes the supplied spectral carrier/action data and four-dimensional Carnot convergence as geometric inputs. Q5a 3.2 proves the zero-form Mosco limit on its canonical Fourier filtration; the non-trivial spatial-limit operator is the unestablished hypothesis [H-L] of Q5b.
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. The Born rule is not established by Q1–Q3. Q3's singlet and correlator require a supplied bipartite carrier and an unestablished diagonal-$2I$ invariance hypothesis; deriving the rule in any sector remains open.
Continuous Hilbert limit. Q5a 3.2 proves the zero-form Mosco limit on its canonical Fourier filtration. Its normalisation no-go assumes [H-depth] and [H-w′]. The spatial-limit hypothesis [H-L] is defined in Q5b and remains unestablished.
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, version 1.5, 2026. https://doi.org/10.5281/zenodo.20548383