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.
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.
This paper continues a constructive “forced-sequence” program that introduces only representational structure required by earlier residues and refinement-stability constraints. Building on (i) a minimal phase-bearing carrier with a quadratic yield map, (ii) closed evolution as yield-invariant internal update with unitary/antiunitary symmetry pressure, and (iii) refinement corridors that force compressed transport and germ connection fields, the paper isolates the next forced step: refinement-stable closedness. If a transport family represents internal (closed) evolution under arbitrary refinement, then yield preservation must persist at the germ level (the “yield corridor”). Using a tier-based (non-limit) notion of first-order content and an explicit accumulation principle for repeated near-identity composition, the paper derives infinitesimal symmetry pressure: admissible germ generators of closed transport must lie in the infinitesimal symmetry class of the quadratic yield structure. Specializing to the minimal complex-carrier realization, this yields a Hamiltonian corridor: germ generators are constrained to the antiunitary-compatible infinitesimal class underlying norm-preserving phase evolution, without assuming Hilbert space axioms, Schrödinger dynamics, derivatives, or a continuum limit as primitives. Finally, the paper imposes a residue-closure constraint: interaction residues (graded bracket / curvature-tier contributions) must remain admissible under the yield corridor at their proper tier.
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.
Complex Hilbert space is usually assumed at the start of quantum mechanics. This paper formulates a reverse construction for the one-particle and operator-algebraic setting. Starting from a real Hilbert space carrying an orthogonal representation, the admissible imaginary units are identified with the skew-adjoint square roots of − Id in the Real commutant of the complexified representation. The novelty is not the isolated use of complexification, spectral orientation, or polar decomposition, but their order of dependence: first the symmetry representation defines an intrinsic operator-valued phase space; then an oriented internal time flow selects a distinguished point of that phase space; only afterward are the usual complex one-particle, observable, probability, and Weyl–CCR structures invoked under their standard hypotheses. In the terminology used here, this oriented internal flow is the reversible or inner-generative side of a dual-time distinction. It is not a second observer clock and not an additional Lorentzian coordinate: mathematically it is the strongly continuous orthogonal group U ( t ) = exp ( t A ) on the real Hilbert space. Its Stone-dual variable is energy/frequency λ ; canonical conjugation pairs λ with − λ ; positive energy orients this pair and selects the canonical phase on the dynamic sector through A = J | A | = | A | J . Thus inner-generative time is the source of phase orientation, while irreversible record time, projection, memory, and decoherence enter only as later reduced-dynamics layers. No spectral gap at zero is required, and zero modes remain visible as residual phase data rather than being hidden by the construction. The selected phase is invariant under time-orientation-preserving symmetries and changes sign under time reversal, giving the usual antiunitary action after complex reconstruction. The paper also makes explicit the monoidal limitation: two selected local phases do not make the raw real tensor product the standard composite system; the standard composite is recovered only after imposing the J -balanced, equivalently complex, tensor product rule. Thus the result is a reconstruction of compatible complex kinematics on the selected one-particle/operator-algebraic sector, not a derivation of all quantum foundations from purely real data.
A forced-structure route from quantization constraints to a Schrödinger-class update law. This preprint continues the Forced Sequence / Corridor program. It derives an operational “local update” corridor that avoids global recomputation, then shows how refinement stability constrains admissible updates into a linear, norm-preserving class with a phase-bearing generator. The result is presented as a structural slot (what must hold under the corridor constraints), not as an assumed physical postulate. Prior papers in the sequence provide the prerequisite corridor objects and invariants.
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.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.