Navier-Stokes as a smooth projection of a binary radius: a discrete octave selector passes the ln2 autocorrelation test where the continuum fails
I expose two documents that belong to an ongoing program. They are not presented as closed results, but rather as a set of derivations and open points that I would like to submit to technical scrutiny. Both are written in a deliberately programmatic register: they distinguish postulates from consequences, and mark explicitly what remains open.
Document 1: geometric octave structure
It proposes an ontology in which the relational radius is quantised in binary octaves, Rϵ=2−ϵ, and space is generated as a proportion to that radius. From this one obtains, without an independent postulate, k=1/R, hence the dispersion Ω(k)=ck—identical to the linear dispersion of the continuum—but now with thresholds at kϵ=2ϵ. The period is ln2.
The result I would like to subject to examination is the following: the bare continuum, without imposed structure, shows no peak in the log-wavenumber autocorrelation at Δ=ln2; the same test, applied to the response modulated with the predicted octave periodicity, detects the peak and its harmonics. The discriminant is operative, at least in simulation on the continuum itself. The falsifiable prediction is therefore modest but sharp: log-autocorrelation in the ringing band, with a local maximum at Δ=ln2.
The technical question I would like to discuss is whether this test is truly blind to other mechanisms—for example, boundary conditions or geometric modes with built-in scale symmetry—and whether the choice of detrending (polynomial degree 3–6) and band truncation could introduce false positives. The document includes a status table (derived / postulated / open) which I consider honest, but I would welcome criticism on whether any of those labels is too optimistic.
Document 2: pending sign recursion anchor
It is an anchor note that addresses a question left open in the first document: whether the pending sign—the unresolved branch of a square root—is recursive or not. The answer I find is affirmative: n↦n2↦(n2−1,n2+1), and the difference of squares reproduces the Mersenne identity M2k=Mk(2k+1). Hence the recursion is log-periodic with period ln2ln2, intrinsically.
In addition, the factorisation of the Mersenne spectrum separates two arithmetic modes:
- Innovation: appearance of a prime not seen at any lower depth.
- Crystallisation: repetition of an already existing prime.
The first pure-crystallisation level is ϵ=6, which coincides (by two independent routes) with the first level where Ior>0, i.e. the first nested pending sign. The proton appears as the base case: 4=2 with branches 3 and 5, whose product is M4.
The limitation I declare explicitly is that this structure is arithmetic and internal; I have not demonstrated that crystals, genes or discharge structures grow by this mechanism, although the analogy is tempting.
The technical question here is: is the identification of "new information = new prime factor" a forced interpretation, or is there some deeper reason that justifies it? Is the coincidence at ϵ=6 genuinely significant, or an artefact of small-number arithmetic?
What I seek with this thread
I do not seek validation, but technical review: someone with more experience in spectral theory, signal processing, or number theory to examine whether the derivations are solid, whether the discriminant test has any hidden bias, or whether the connection between the continuum band and the octave ladder rests on some implicit normalisation I am not seeing.
I am also interested in whether the absence of a dynamical mechanism—what resolves the sign in the real world—invalidates the programme, or whether it can be treated as a geometry of possible states awaiting coupling to local boundary conditions.
The documents are written with their postulates exposed, and they do not attempt to conceal their weak flanks (especially the underived ϵ=4, and Postulate 3 as an undischarged root). Precisely for that reason, they seem to me suitable for open discussion.
I thank in advance for readings, objections, and references to analogous work I may have overlooked.
Final note: if anyone wishes to run the factorisations or the autocorrelation test, the scripts are short and described in the appendices; I can pass the code if there is interest .
