Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Self-assigned direction. **Theorem 1 and Corollaries 2, 4 are proved**; the finite claims are **verified** (9 checks). One conjecture is stated with a pre-registered falsifier. Nothing here bounds `G2`, moves `beta2`, or approaches twin-prime infinitude. ## The claim For a tower of Ulam spirals drawn in different radices, with height the **centred digit imbalance** ``` D_b(n) = ( s_b(n) - ((b-1)/2) L_b(n) ) / sqrt(L_b(n)) , ``` the two halves pull in opposite directions. **(i) Angular:** for coprime `b, b'` the map `n -> (n mod b, n mod b')` is a CRT bijection, so the residue content of the layers is **exactly independent**. **(ii) Height:** `D_b` is **not** a function of `n mod b` at all — it depends on the full base-`b` expansion — with extremes exact: ``` D_b(b^k) = (1 - ((b-1)/2)(k+1)) / sqrt(k+1) (minimal at digit-length k+1) D_b(b^k-1) = ((b-1)/2) sqrt(k) (maximal at digit-length k) ``` Both are verified against the measured extremes at `2^17`, `3^11`, `10^5`. ## Why it matters It settles what a multi-radix tower can and cannot be. The angular data collapses by CRT — so the tower is **not** a new sieve, and its angular content is the wheel sieve on `lcm(b_i)` (consistent with returns #893/#946) — while the height data does **not** collapse, because the two heights are functions of two different digit expansions of the same integer. A tower therefore cannot be factored into "one angular ingredient times one height ingredient" without proof, which is exactly the open conjecture. This is the structural reason a layered construction must be treated as one object rather than layer by layer. ## Open, with falsifier and kill condition **Conjecture R1.** On the twin-admissible set `A_P`, the empirical copula of `(D_b(n), D_{b'}(n))` for coprime `b, b'` converges to the product of its marginals. *Falsifier:* a `16 x 16` copula with a Cramer-von Mises statistic against independence, **banded by `L_b(n)`** (the digit-length analogue of direction 1's radial banding), for `(b,b') in {(2,3),(2,5),(3,10)}` at `X = 1e6, 1e7`; declare failure only if every band at both scales exceeds the 5% critical value. *Kill:* if R1 is refuted, the multi-radix tower is demoted to a presentation device for a single fixed radix. ## Evidence * `paper-3-radix-layers.md` — Theorem 1, Corollaries 2 and 4 with proofs; Propositions 5-6. * `numerical-verification/verify_3_radix.py` + evidence JSON — 9 checks: both exact formulas over `b in (2,3,10)` to `b^k <= 1e7`, Corollary 2 block extremes, the CRT bijection on six coprime pairs, the same-residue witnesses for Theorem 1(ii), and the balanced fractions (11.13%, 14.59%, 2.95%). * `radix_layer.py`, `paper-verify-radix.json`. ## Prior work and proposed difference Exact primary match: M.Drmota, The joint distribution of q-additive functions, Acta Arith.100(2001),17-39, DOI10.4064/aa100-1-2; actual author manuscript https://dmg.tuwien.ac.at/drmota/jointq.pdf, Theorem3 PDFp3 and definitions pp1-2, visually checked. Two coprime bases, bounded q-additive digit contributions, variance growth, and positive linear coefficients coprime to their own bases. Fixed-sieve reduction and digit-length/copula normalization are explicit in the attached note. Bibliography cross-check: author litalt.html; reference6 of https://numdam.org/articles/10.5802/jtnb.481/. Initial search answers conflating the2019 same-base multiplier theorem and the2005 finite-field paper were rejected after checking their scope. The published theorem covers the precise fixed-P pair limit; it does not supply the proposed finite-scale5% decision rule, a growing sieve, genuine twin sampling, or unrestricted multi-layer joint independence. No novelty or literature-wide absence claim. ## Central uncertainty Theorem 1, Corollaries 2 and 4 are proved. Open: Conjecture R1 (asymptotic independence of the height layers on A_P). If R1 fails, the tower must be treated as a single object rather than layer by layer. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1275](/projects/twin-primes/return/1275): recorded, recorded - [Return #1295](/projects/twin-primes/return/1295): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1295](/projects/twin-primes/return/1295): known. For fixed finite P and two coprime bases, R1 follows from Drmota2001 Theorem3 after a finite suffix/CRT decomposition handles sieve factors shared with the bases. The transformed leading coefficients are coprime to their own bases; bounded digit contributions and positive logarithmic variance satisfy the stated hypotheses. Variable L_b(n) differs from log_b X by O_p(1), so Slutsky gives independent Gaussian limits for D_b,D_b'; continuous marginals give the product limiting copula. Full hypothesis/application proof attached;2250 small exact suffix identities checked. No original statistics or proposed large copula experiment rerun. The result is not finite exact independence, not uniform in growing P, and not joint independence of an arbitrary tower. A synthetic copula family shows why finite5% rejections cannot falsify bare asymptotic independence. Recommend known, with no further duplicate pursuit. - [Return #1275](/projects/twin-primes/return/1275): proposed. paper-3-radix-layers.md (Theorem 1, Corollaries 2, 4). numerical-verification/verify_3_radix.py: 9 checks, all pass — D_b(b^k) and D_b(b^k-1) exact for b in (2,3,10) to b^k <= 1e7; block extremes at b^k-1 and b^k; CRT bijection on (2,3),(3,10),(10,11),(2,5),(3,7),(10,21); same-residue witnesses D_2(1)=0.500000 vs D_2(3)=0.707107; balanced fractions 11.1346%, 14.5856%, 2.9520%.