Source for The Non-Injective Foundations Sub-Programme (Presentation Note 5).
The note is aligned with Foundation version 2.6 and HeisenbergStructure version 2.1. The ENI no-go theorem establishes non-injectivity under explicit hypotheses. Within Foundation, A1 and A2 give immediate information loss along non-injective transitions; the arrow of time follows only under the acyclicity hypothesis [H-acyc], which is not established. Irreducibility of the admissible fibre is conditional on a representation-theoretic dictionary and on an additional sector reading of A3, not derived, on which a decomposition stable under the admissible generators counts as premature selection. The discreteness of quantum transitions is postulated by Foundation's A4, not derived; on the supplied Heisenberg shell realisation a realised saturation depth is an integer (O22), and the existence of such a depth is not established. A1--A3 and Born--Infeld parity do not derive the discrete Heisenberg group. Once that group and a non-trivial central character are supplied, Stone--von Neumann identifies the Heisenberg/Schrödinger representation on (L^2(\mathbb Z/q\mathbb Z)); the associated Weil action is distinct further symplectic data.
The finite countermodel of A Finite Obstruction to Heisenberg Carrier Selection from Admissibility Constraints version 2.1 uses (G=S_3), (X=(12)), and (Y=(23)). The pair is exchanged by an involutive automorphism, minimally generates (S_3), and acts faithfully and irreducibly in the standard two-dimensional representation. Nevertheless,
Thus a non-trivial commutator does not imply a central commutator or a Heisenberg presentation. The Heisenberg group, Schrödinger representation, and separate associated Weil action remain coherent supplied data, and downstream results that consume them are conditional on those choices.
| Paper | Central output | Status |
|---|---|---|
| ENI | Non-injectivity as a necessity of genuine emergence | proved |
| ENI Corollary 6 | Recursive non-injectivity; structural colour-confinement reading | proved / structural |
| Foundation M | Immediate information loss; temporal order under [H-acyc]; carrier-selection obstruction (Theorem 5.7) | proved / conditional / proved |
| HeisenbergStructure 2.1 | Finite (S_3) obstruction; Schrödinger representation once group and non-trivial central character are supplied; Weil action separate | obstruction proved / representation conditional / Weil action supplied |
| noscale | No independent dimensional parameter beyond (c_{\mathrm{BI}}) | proved under stated hypotheses |
- Carrier selection: derive an independently motivated condition that excludes the countermodel and selects a central extension, or retain the carrier as an explicit model choice.
- Born rule: derive it in any sector; Q3's singlet and correlator are conditional on a supplied bipartite carrier and an unestablished diagonal-(2I) invariance hypothesis, and do not establish the Born rule.
- Level 2 scale determination: determine whether emergent scales are fixed from (c_{\mathrm{BI}}) and the remaining structure.
- Continuous Hilbert limit: supply a spatial second-order limit operator, the unestablished hypothesis [H-L] of Q5b. On its canonical Fourier filtration, Q5a proves that the published admissibility form converges to the zero form and, under its depth and weight hypotheses, that no common scalar normalisation yields a non-trivial toric differential operator.
bash compile.shThis runs pdflatex -> bibtex -> pdflatex -> pdflatex and writes
out/NonInjectiveFoundationsNote.pdf.