Paper 6: Conservation Laws, Records, and Time-like Structure — The Σλ² Non-conservation Theorem, the Freezing Theorem, the B₄ Gauge, the 1+3 Polar Decomposition, the Record Theorem, and the Null Structure

This paper establishes the kinematic core of the reciprocal dual model (Papers 1–5). The assumptions are only three — (1) the reciprocal duality νλ = 1, (2) the zero point 1/2, and (3) one asymmetric bit, namely that the additive conservation law rides on only one of the two ledgers — and this paper organizes, as theorems, how much structure follows from this inventory. First, the freezing theorem: under conservation of Σν², every nontrivial split strictly increases Σλ² (a one-line proof by Cauchy–Schwarz). As a corollary, if both sums of squares were conserved simultaneously, no nontrivial split would exist, and no cascade or time-like structure would arise. The existence of the asymmetric one bit is itself the existence condition for time-like structure. Second, the gauge structure: the exact symmetry group of the counting condition is the hyperoctahedral group B4 (order 384), and within each energy shell the distribution over axes, the signs, and the orientations are pure gauge. The additive conservation laws close the ledger with just two entries, Σν² and the Z2 parity. As a by-product, the fine structure of shells that are degenerate in energy and split by shape (137 states at R = 3 → 7 gauge-invariant classes) is laid bare. Third, the 1+3 polar decomposition: it is not that one of the four axes becomes time. In the polar decomposition ℝ⁴ ≅ ℝ₊ × S³ of the 4-dimensional frequency space, the radial direction, on which the manifestation of the asymmetric bit concentrates, appears as time-like, while the angular directions untouched by the duality (the B4 gauge) appear as space-like. The 1+3 role differentiation is obtained without introducing the imaginary unit or the Minkowski negative sign. We also show that there is no way to decide from the inside whether one is at the apex of the hierarchy (hierarchy relativity). Fourth, the record theorem and the null structure: measurement is recording, and recording is accumulation. Since only a non-conserved monotone ledger can accumulate, every record remains as a configuration on the λ side, and ν can never be a direct read-out target. The integer lattice is reinterpreted not as an assumption but as the resolution structure of unit records (a downgrade in the assumption inventory). Furthermore, in the log-conjugate plane (q, p) = (log λ, log ν) the constraint νλ = 1 is a null line, the freedom of distributing the conjugate widths is the SO(1,1) boost (squeeze), and uncertainty is built in as the boost-invariant minimal area. The negative sign is not an axiom but a hyperbola — though the aggregation of the four pairs of (1,1) structures into a single (1,3) signature remains as an explicit construction task. This paper does not claim to have derived physical time, space, or measurement. What it does claim is that, from the inventory of three assumptions, the role differentiation into time-like, space-like, record, uncertainty, and null structure follows as theorems. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus three figures (PNG).