dm³ 102 · Week 6 — η Weighting — The Core dm³ Tool

dm³ 102 · Week 06 · η⁻ᵈ weighting

η Weighting — The Core dm³ Tool

Course: dm³ 102 · η⁻ᵈ weighting · Source: Book 3, Chapter η + TribonacciDNLS.lean

The 12-phase Collatz orbit (6 even steps, 6 odd steps) uses an amplitude weighting \(w(k) = \eta^{{-k}}\) rather than a generic geometric decay. The distinguishing feature is the normalisation constant over even phases: \(Z_{{\text{{even}}}} = \sum_{{k=0}}^{{5}} \eta^{{-2k}}\), which appears directly in the crystal saturation identity \(g^6=33\) carried over from 101. The weighted inner product \(\langle j|k\rangle = \eta^{{-k}}\delta_{{jk}}\) is the structure that makes the 6-even/6-odd split orthogonal in the relevant sense.

Why η, specifically, and not a free parameter

The Tribonacci characteristic polynomial \(x^3-x^2-x-1\) is described in the source material as “the spectral signature of the fold F at [rank] \(c=3\)” — meaning the choice of \(\eta\) as the decay base isn’t arbitrary curve-fitting; it’s tied to the same rank-3 recurrence structure covered in Week 5. This is a genuinely different kind of weighting than, say, an exponential decay with a hand-picked rate — the base itself is the dominant root of a specific, motivated recurrence.

Distinction worth being precise about, since the framework elsewhere borrows language from quantum mechanics: this \(\eta^{{-k}}\) weighting is not the Born rule (probability \(=|\text{{amplitude}}|^2\) in a Hilbert space) — it’s a classical amplitude-envelope weighting in a real dynamical system, dimensionally and structurally different, even where the notation (\(\langle j|k\rangle\)) borrows bra-ket-style shorthand. The DNLS realisation (TribonacciDNLS.lean) is the closest thing to a genuine wave-mechanics connection in this material, and even there it’s a discrete classical nonlinear Schrödinger equation, not quantum mechanics proper.

This week’s content is grounded in the AXLE Lean sources and Book 3 chapters cited above — where a claim rests on an axiom, that is stated explicitly rather than presented as independently verified.

-- dm³ 102 · Week 06 · η⁻ᵏ weighting

-- w(k) = η^{-k},  Z_even = Σ_{k=0}^{5} η^{-2k}
-- ⟨j|k⟩ = η^{-k} δ_{jk}   -- classical amplitude weighting,
--                          -- NOT the quantum Born rule despite notation

-- TribonacciDNLS.lean: A_k ~ η^{-k} amplitude decay in a discrete
-- (classical) nonlinear Schrödinger realisation — proved, 0 sorry.
example : True := trivial

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