Storage and Reference as a Duality Relation: One Conservation, Three Geometries

Version 7. Storage (TD) and reference (TrD) are the two legs of a duality relation governed by TD² + TrD² = 1. Version 6 established this, correcting the linear law TD + TrD = 1 of Versions 1–5. Version 7 makes two changes, both of which narrow the paper. First, φ is removed from the probability circle. V6 derived the φ-boundary via Hurwitz's theorem — correctly — but then seated the result at the ratio TD/TrD = cot θ, an angle on the circle. That placement does not survive inspection: on the circle the elliptic norm is 1 for every angle, so it distinguishes nothing, and the hyperbolic norm evaluated at θ = arctan(1/φ) ≈ 31.7175° is 0.894427, not ±1, so the point is not a unit either. Hurwitz extremality is a property of the real quadratic field Q(√5), whose geometry is a hyperbola. V7 relocates φ to that sector, where its powers φⁿ = F₋ₙ₋₁₌ + Fₙφ sit on the unit-norm hyperbola exactly with N = (−1)ⁿ, and keeps the circle for probability alone. A consequent negative prediction: no two-path interferometer should find φ at a preferred angle. Second, the linear and quadratic laws are reconciled rather than one discarded. TD + TrD = 1 and TD² + TrD² = 1 are the same Parseval identity read at two levels — TD/TrD as energies (which sum linearly) versus as amplitudes (which sum in quadrature). V1–V5 were not writing a false equation; they were naming energies and reading them as observables. EGY's measured observables are amplitudes, which is why the quadratic form is the physical one. The error was a labelling error, not an arithmetic one. The organising frame. Every conservation statement in the programme is N(x) = ±1 for the norm of a quadratic a + bω, and the sign of the discriminant Δ = s² − 4p selects the geometry: Δ < 0 gives the circle (probability, bounded), Δ = 0 the parabolic seam (one invariant speed), Δ > 0 the hyperbola (growth, φ, the leak). These are the three Galois behaviours of one object, and probability and φ live on different ones. What survives. TD² + TrD² = 1 is the Englert–Greenberger–Yasin duality relation for pure states (verified over 20,000 random pure qubit states). The radial deficit is exactly twice the linear entropy, 1 − (TD² + TrD²) = 2(1 − Tr ρ²), verified to 5.6×10⁻¹⁶ over 20,000 random mixed states — the framework's one genuine result. φ is Hurwitz-extremal. Pisot is the class for which self-similar subdivision terminates (verified at 400 digits). What V7 resolves that V6 left open. V6 recorded that every quantity is a ratio and nothing internal supplies a scale. V7 identifies that gap as the seam: the elliptic side fixes shape, the hyperbolic side supplies scale, they meet only at Δ = 0 where the multiplier is exactly 1, and so no rate exists there to carry across. The missing unit is a structural consequence of sector separation, not an omission — a negative result. Scope. V7 makes no prediction that departs from standard quantum mechanics. The sector identifications (elliptic = quantum, hyperbolic = time/growth, parabolic = lightlike) are asserted mappings, not derivations, and are labelled as such. Withdrawn and still withdrawn from V5: the φ×√n simulation evidence (uniform weighting with no φ scores 0.9950 against φ's 0.2419), the measurement-problem and information-paradox claims, and Prediction 1. Every correction from V5 onward originated in a test the previous version failed; the tests are published alongside the paper at github.com/0x-auth/darmiyan-fs under review/ and substrate/, with companion notes at github.com/0x-auth/darmiyan-recognition under docs/.