Phase is forced by cancellation, before probability or physics enters the story. This paper continues a constructive sequence deriving only what is logically forced once incompatible closure criteria induce non-commuting stabilization operations. When distinct stabilization routes matter, a state-only description becomes lossy, but explicit route-tracking produces unbounded descriptive growth (“route explosion”). To summarize non-equivalent routes without erasing forced distinctions, the framework is compelled to introduce a contribution carrier and a combination rule. Regrouping invariance forces associativity; null contributions force an identity element; cancellation adequacy forces inverses, pushing the contribution space toward an abelian group (on the generated representational classes). Purely nonnegative scalar accumulation cannot support cancellation, so a phase-like orientation invariant appears prior to probability assignments, amplitudes, Hilbert space axioms, or physical interpretation. Finally, requiring descriptive boundedness under iterative route concatenation pressures the minimal orientation structure toward a compact (periodic) group. The forward pressure is then explicit: extract a nonnegative invariant magnitude from phase-bearing summaries in a way that is phase-invariant, additive over exclusive outcomes, and stable under composition—setting up the next paper’s derivation of a quadratic (Born-like) invariant.
Phase is forced by cancellation, before probability or physics enters the story. This paper continues a constructive sequence deriving only what is logically forced once incompatible closure criteria induce non-commuting stabilization operations. When distinct stabilization routes matter, a state-only description becomes lossy, but explicit route-tracking produces unbounded descriptive growth (“route explosion”). To summarize non-equivalent routes without erasing forced distinctions, the framework is compelled to introduce a contribution carrier and a combination rule. Regrouping invariance forces associativity; null contributions force an identity element; cancellation adequacy forces inverses, pushing the contribution space toward an abelian group (on the generated representational classes). Purely nonnegative scalar accumulation cannot support cancellation, so a phase-like orientation invariant appears prior to probability assignments, amplitudes, Hilbert space axioms, or physical interpretation. Finally, requiring descriptive boundedness under iterative route concatenation pressures the minimal orientation structure toward a compact (periodic) group. The forward pressure is then explicit: extract a nonnegative invariant magnitude from phase-bearing summaries in a way that is phase-invariant, additive over exclusive outcomes, and stable under composition—setting up the next paper’s derivation of a quadratic (Born-like) invariant.
Born-like squared magnitude is forced by stable composition, not assumed from physics. This paper derives a Born-like squared-magnitude rule as a representational necessity in a pre-physical program. Building on prior results that force (1) plural incompatible stabilization criteria, (2) route dependence with cancellation-capable summaries (phase-like structure), and (3) the need for a stable nonnegative "yield" map, we formalize a yield function rho from summaries to nonnegative real numbers under minimal requirements: phase invariance, additivity over exclusive outcomes, multiplicativity under independent composition, and symmetry under balanced regrouping (no preferred exclusive coordinates). Under mild regularity, multiplicativity restricts rho to a power-law family, and regrouping invariance uniquely selects the quadratic case. The result is a squared-magnitude extraction rule obtained without assuming Hilbert spaces, wavefunctions, inner products, dynamics, hbar, or measurement postulates. The paper concludes by identifying the next forced pressures toward complex realization and norm-preserving (unitary-like) evolution.
Incompatible projections force context, history, and interference-like combination. This paper derives the next forced structures that follow once plural, incompatible closure criteria induce projection-like stabilization maps. Building on prior results establishing ordered asymmetry, thresholded measure, and non-commuting stabilizations, we show that any nontrivial use of multiple projections requires (i) an explicit selector indicating which stabilization regime is enforced and (ii) a representation of route-dependence under sequences of stabilizations. Non-commutation implies the existence of distinct admissible routes yielding distinct stabilized outcomes, so a state-only description becomes lossy. To preserve forced distinctions without enumerating all histories, the framework is compelled to introduce an interaction residue and a route-combination operator (oplus). The result is an "interference-like" structural phenomenon obtained prior to probability assignments, amplitudes, Hilbert space axioms, or physical interpretation, and it identifies the next forced step: determining what algebraic properties of (oplus) are required to prevent unbounded route explosion while preserving descriptive adequacy.
Closed evolution forces norm-preserving symmetry on the minimal complex carrier. This paper continues a forced-sequence program deriving only what is structurally required by prior steps. Given a phase-bearing summary with phase-invariant quadratic yield, we show that faithful representation of continuous compact phase requires a two-dimensional real realization (equivalently, a complex carrier). Defining closed evolution as internal yield-preserving update, we show that yield-preserving additive maps are pressured into the norm-preserving symmetry class (unitary-like, with antiunitary-like orientation reversal remaining logically admissible). The forward pressure is toward interaction via non-commutation when multiple contexts/operators coexist, and toward generator structure from refinement-stable incremental composition. V2- Minor clarifications only (no change in claims): minimality of U(1), antiunitary permissibility, and tighter non-commutation forward linkage.
Finite determinacy yields an allocation geometry and a causal cone but not quantummechanics: its axioms admit a classical model, so superposition and Born’s rule cannot followfrom them. This paper asks how far one may nonetheless go, and locates the remaining step.The minimal carrier of a continuous recombination is shown to be C, by classification of theconnected abelian Lie groups admitting the loop action by automorphisms, so the complexunit is not imported. Transport equivariance then forces contributions to combine by additionuniquely, while convex recombination is excluded for failing to close under composition. Thedependence of a record’s probability on the carrier is fixed at the square by an involutivesplitter, its normalisation following from involutivity rather than from any norm, with no otherexponent surviving repeated splitting. Phase-invariance of records and the density ceiling arederived. The chain then closes to an inner-product space, the amplitude being a unit vectorin L2 of the phase space; what is not obtained is the reduction to configuration space, and ofthe two polarisations available one rests on a primitive and the other on a theorem, neitherbeing forced. Several negative results are established unconditionally, among them that nopositive transport of a non-negative content can produce cancellation, and that no phase canbe built from a scalar or from any translation-invariant function of a cell.
Quantum structure follows from refinement-stable bookkeeping, not postulates. Soft selection is forced when corridor-level action ties make hard selection require forbidden tie-break (micro-history) data. The corridor therefore retains all admissible configuration classes and aggregates them using action-only weights. Interference stability pressures a phase-bearing carrier, concatenation stability pressures exponential phase weights, and gauge-meaningful scalar prediction forces quadratic magnitude extraction (the Born slot). No Hilbert axioms, path-integral primitives, or calculus limits are assumed.
This paper is Paper 003 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the Born Ledger: the probability scalarization of Binary Mechanics. Paper 002 derived finite polarized graph histories and the primitive finite amplitude law. In that construction, amplitudes are not assigned to instantaneous graph states, vertices, or hidden local phase registers. Instead, they arise as finite sums of Z4-weighted admissible histories grouped into coarse record classes. The present paper begins from that result and asks what probability ledger is forced once record-disjoint coarse alternatives are required to receive stable nonnegative weights. The central result is a no-fence scalarization theorem: the only admissible nonnegative scalarization of a finite complex record amplitude is its squared modulus. When normalized over an exhaustive record-disjoint partition, this yields the Born-type probability rule. The derivation does not assume Hilbert space, a wavefunction, a measurement postulate, a continuum path integral, primitive probability, or primitive branch selection. Interference appears before probability as cancellation inside finite history amplitudes, and probability appears only as the record-level scalarization of those amplitudes. No fine history is primitively deleted or selected; the ledger assigns weights to record-disjoint coarse alternatives generated from the full admissible finite history structure. The paper therefore supplies the Born-weighting step in the Binary Mechanics construction: finite graph histories first, finite amplitudes second, record ledgers third, normalized probabilities last.