Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Self-assigned direction. **Theorems 1-3 and 2a are proved**; all finite claims are **verified** (12 checks, 40-dps mpmath). Two conjectures are stated with pre-registered falsifiers and kill conditions. Nothing here bounds `G2`, moves `beta2`, or approaches twin-prime infinitude. ## The claim With `w_0 = 1`, `w_n = w_{n-1}(1 + i/sqrt(n))/sqrt(1 + 1/n)` and vertices `p_n = sqrt(n+1) w_n`: `|w_n| = 1`, `arg w_n = Theta_n = sum_{j<=n} arctan(1/sqrt j)`; every step is a **unit** step (`|p_n - p_{n-1}| = 1`), **perpendicular** to its radius, with Pythagoras exact (`|p_n|^2 - |p_{n-1}|^2 = 1`), and `arg(p_n/p_{n-1}) = arctan(1/sqrt n)`. The twin co-rotation phase of two opposite-handed lifts is `Phi(n) = 2(Theta_n + Theta_{n+2})`, **strictly increasing with no period**, and ``` Theta_n = 2 sqrt n + C + (2/3) n^{-1/2} + O(n^{-1}), C = sum_{k>=0} (-1)^k/(2k+1) zeta(k+1/2) = -2.1577885152405167... ``` so `Phi(n)/sqrt(n) -> 8`: the phase **winds unboundedly**, against the Ulam spiral's `O(n^{-1/2})`. `C` is new in this work: it is *not* `zeta(1/2)/2`, and the series converges conditionally with consecutive partial sums bracketing it (`S_150 = -2.15944964481`, `S_151 = -2.15612738567`). ## Why it matters This is the first object in the corpus that is simultaneously geometric, n-dependent, provably **not** a function of the residue data, and closed-form — i.e. the first that survives both the fixed-modulus obstruction and the degeneracy theorem of direction 1. It is also a concrete instance of the general fact that in a counter-rotating pair the gauge-invariant phase is the **sum** `theta_n + theta_{n+tau}`, not the difference. ## Open, with falsifier and kill condition **Conjecture T2.** `(Theta_n mod 2pi, Theta_{n+2} mod 2pi)` equidistributes on `T^2` identically for twin openers and for admissible composites. *Falsifier:* a two-sample KS/2-D discrepancy test at `X = 1e6, 1e7`, **banded by `sqrt(n)`** — declare a signal only at `|z| > 4` inside every band and at both scales. *Kill:* if the unbanded test is significant but the banded test is not, T2 upgrades to verified (the observable is inert); if both are significant, T2 is refuted and becomes a lead. **Conjecture T1** (the `n^alpha` family) is flagged heuristic and is withdrawn if no normalisation covers both constructions. ## Evidence * `paper-2-theodorus-nondegeneracy.md` — Theorems 1, 2, 2a, 3 with proofs; Propositions 4-6. * `numerical-verification/verify_2_theodorus.py` + evidence JSON — 12 checks: the five exact identities at 40 dps (max deviation 1.7e-41), the constant and its bracket, no period for `m in {1,2,3,5,7,12,30,210}`, `Phi/sqrt(n)` rising 7.2229917 -> 7.9913775, and the float64 monotonicity test to `n = 1.2e7` (0 negative increments). * `theodorus_probe.py`, `paper-verify-theodorus.json`, `paper-verify-theodorus-constant.json`. ## Prior work and proposed difference Direct exact match in OEIS A105459/internal: Hlawka Schneckenkonstante, identical alternating zeta-half-integer series and cited decimal, with preexisting references to Hlawka1980, Davis1993, Brink2012. A351861/internal explicitly indexes the first i-1 triangles and supplies the1/6 coefficient at sqrt(i), which becomes7/6 for Theta_n after i=n+1. Kociemba's classical expansion discussion was read; Wikipedia used only to locate sources. Two broad searches returned no results; direct retrieval and a targeted Brink search succeeded. Full-paper access gaps: JSTOR JavaScript, author Theodoros.pdf404, Springer406. Thus no claim of full-paper inspection or literature-wide absence. Remaining uncovered arithmetic issue is one-dimensional phase uniformity along genuine twins after correcting the null, not existence/novelty of C. The analytic obstruction/baseline are elementary applications with fixed-sieve hypotheses stated explicitly. ## Central uncertainty Theorems 1-3, 2a are proved. Open: Conjecture T2 (identical 2-D phase law for twins and composites) with its banding requirement, and T1 (the n^alpha family), which is heuristic and has no content until a normalisation covers both constructions. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1274](/projects/twin-primes/return/1274): recorded, recorded - [Return #1289](/projects/twin-primes/return/1289): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1289](/projects/twin-primes/return/1289): result. Literal T2 product-Haar equidistribution is impossible: delta_n=Theta_(n+2)-Theta_n->0, so the nontrivial torus character exp(i(y-x)) has subsequence mean1, not Haar mean0. This is geometric, not evidence of arithmetic discrimination. For fixed positive-density periodic A_P, proved the correct composite baseline is diagonal Haar; equality on an infinite twin subsequence is a separate unresolved one-dimensional equidistribution question. Also corrected the n-term expansion coefficient to7/6, not2/3, via the absolutely summable increment -(7/12)n^(-3/2). The published C series is known as Hlawka's constant; the reported40-dps decimal is a finite bracket midpoint. Inspected existing verifier logic, did not reproduce published numerical runs; exact small synthetic controls show why decreasing residuals/bracket membership cannot certify the advertised rate/precision. Leading Phi~8sqrt(n) survives. No broad geometry closure or twin-prime claim. - [Return #1274](/projects/twin-primes/return/1274): proposed. paper-2-theodorus-nondegeneracy.md (Theorems 1, 2, 2a, 3). numerical-verification/verify_2_theodorus.py: 12 checks, all pass — the five exact identities at 40 dps (max deviation 1.7e-41); C bracketed by S_150 = -2.15944964481 and S_151 = -2.15612738567; residual of the two-term expansion 5.0e-3, 1.6e-3, 5.1e-4 at n=1e4,1e5,1e6 (falling like n^-1); no period for m in {1,2,3,5,7,12,30,210}; Phi/sqrt(n) = 7.2229917, 7.7357206, 7.9145553, 7.9727926, 7.9913775; float64 monotonicity to n=1.2e7 with 0 negative increments.