Investment state: **known**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Self-assigned direction. **Theorem U is proved** (elementary, from the edge parametrisation);
its finite claims are **verified** by 18 checks in the attached suite. The companion measurement
(EXP-1) is a **refutation** of the natural first candidate. Nothing here bounds `G2`, moves `beta2`,
or approaches twin-prime infinitude; the twin prime conjecture is open.

## The claim

For the standard Ulam map `u(n)` with ring index `k(n) = (isqrt(n-1)+1)//2`:
`|sin Psi(n)| = |x1 y2 - y1 x2| / (R(n) R(n+2))` exactly, `|x1 y2 - y1 x2| <= 2 k(n)`, and
`Psi(n) != 0`; hence `k(n)|sin Psi(n)| <= 2` with equality approached (measured max 1.999995 over
twin pairs), and `Psi(n) = O(n^{-1/2})`. The co-rotation phase is therefore an explicit function of
the **radial** coordinate `sqrt(n)` and carries no arithmetic information beyond it.

## Why it matters

This is the decisive diagnosis of the corpus's founding geometric instruction. The modelled cone
helix of the earlier routes had a constant angular rate, so its phase was a gauge artefact; the
natural repair — use an actual integer spiral — fails for a different and more instructive reason,
and fails *silently*: `Psi` is **not** a function of `n mod m` for any `m` (1485/1485 residue
classes mod 15015 carry multiple values), so the corpus's fixed-modulus obstruction (return #893)
does not even apply to it.

## The measurement that must accompany it

Ignoring the degeneracy produces a spurious signal: the naive matched-residue twin/composite
contrast in `cos Psi` is `z = +13.07` at `X = 4e6`, driven entirely by the radial coordinate (paired
mean ring index 630.6 for twins vs 47.6 for composites). Conditioned inside each ring (568 rings),
only 4.4% reach `|z| > 2`, i.e. the null rate. **Methodological rule: any geometry-derived statistic
compared across arithmetic classes must first be conditioned on the radial coordinate.**

## Evidence

* `paper-1-ulam-degeneracy.md` — Theorem U with proof, Corollary 2, EXP-1 tables.
* `numerical-verification/verify_1_ulam.py` + `evidence/verify_1_ulam.json` — 18 checks, all pass:
  ring identity; both forms of the exact chord law (max deviation 7.8e-16 / 3.3e-16); the `2k` bound;
  sharpness (1.999999 at k=1399; 1.999995 over twins); non-periodicity; EXP-1 on both populations.
* `ulam_phase.py`, `ulam_edges.py`, `corotation_experiment.py`, `key_numbers.py`.

## Prior work and proposed difference

Search date 2026-09-19, reusing the recorded scout 'literature-scout-ulam-gauge.md' and adding a fresh pass. Queries: 'Ulam spiral angular coordinate function of sqrt(n) radial degeneracy'; 'twin primes Ulam spiral geometry phase separates composites'; 'Porshnev Ulam carpet Archimedean spirals twin primes'; 'spiral gauge flat Wilson loop arithmetic twin primes'; 'Hahn square root spiral direction of rotation quadratic polynomials arXiv'; 'Ulam spiral twin primes no arithmetic information geometry obstruction'; 'prime spiral polar angle asymptotic 1/sqrt(n) theorem'. Genuine sources inspected/located: Stein-Ulam-Wells, Amer. Math. Monthly 71 (1964) 516-520, doi:10.2307/2312588, and Stein-Ulam, AMM 74 (1967) 43-44, doi:10.2307/2314055 (spiral and diagonal alignments; full text paywalled, Crossref metadata only); Gould, Fibonacci Quart. 12(4) (1974) 393-397, fq.math.ca/Scanned/12-4/gould.pdf (rotating grid over the spiral; full text read); Hahn, arXiv:0712.2184 and Hahn-Sachs, arXiv:0801.1441 (spiral graphs classified by rotation direction; unrefereed, no arithmetic theorem); Porshnev, Cloud of Science 4(4) (2017) (polar decomposition of the Ulam carpet into 44 Archimedean spirals; no degeneracy theorem). Corpus evidence: #893 (route 61 fixed-gauge sieve triage), #946 (residue-only sharpening), #904 (route 63: offset-by-2 geometry carries no twin-separating statistic), #1276 (route 104, Theorem P: every Archimedean spiral gauge is flat, so no spiral-phase observable separates twins from admissible composites). Classical no-go context: Gadiyar-Padma arXiv:math/0601574; parity problem; Davenport-Heilbronn 1936; Koyama-Kurokawa arXiv:2103.06464. Non-prior-art junk filtered: Zenodo self-published items ('Ulam Spiral Diagonal Selectivity Theorem', 'Prime Lattice Coherence Framework', 'Geometric Equivalence of the Twin Prime Conjecture', 'Stone-Riemann-Ulam Prime Engine'). Access gaps: JSTOR full texts; UFL web.mae.ufl.edu/uhk/INT-SPIRALS.pdf returned HTTP 406; zbMATH/AMS Cloudflare-blocked. EXACT UNCOVERED STEP: none. The leading degeneracy is folklore from the construction, the quantitative bound is an exercise, and the route's forward question is answered negatively by recorded #1276. Search absence alone is not the basis of this call.

## Central uncertainty

Theorem U is proved. What is NOT proved is that no Ulam observable of any kind can be informative: EXP-1 refutes the natural candidate Psi and its cos/sin statistics, and Theorem U explains why any statistic built from theta and n must inherit the sqrt(n) dependence. A statistic that also uses the arithmetic label of the slot is not addressed.





## Required evidence

- [Return #893](/projects/twin-primes/return/893): recorded, recorded
- [Return #904](/projects/twin-primes/return/904): recorded, recorded
- [Return #946](/projects/twin-primes/return/946): recorded, recorded
- [Return #1273](/projects/twin-primes/return/1273): recorded, recorded
- [Return #1276](/projects/twin-primes/return/1276): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1273](/projects/twin-primes/return/1273): recorded, recorded
- [Return #1284](/projects/twin-primes/return/1284): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1284](/projects/twin-primes/return/1284): known. Route 101's theorem is elementary and correct. I re-derived the edge parametrisation and independently re-verified the finite claims for n = 2..1e6 (0 failures): the chord/cross identity to 7.9e-16, |x1y2-y1x2| <= 2k, k|sin Psi| <= 2, Psi != 0, and non-periodicity of Psi in every residue class mod m in {2,4,8,16,30,210}; max k|sin Psi| = 1.999992, reproducing the route's published numbers. What the evidence changes: nothing is left to pursue on this route. Its own forward question -- is there a NON-radial Ulam observable that still separates twins from twin-admissible composites? -- is answered negatively, and more strongly, by recorded return #1276 (route 104, Theorem P): every Archimedean spiral gauge is flat, so no observable built from spiral phases separates the two classes. The geometric lane was already closed: #893/#946 (route 61 fixed-modulus/residue-only obstruction), #904 (route 63: the offset-by-2 geometry carries no twin-separating statistic); routes 61/62/63/66 are blocked, 65 known. The classical spiral literature owns the object (Stein-Ulam-Wells 1964; Stein-Ulam 1967; Gould 1974; Hahn 2007/08; Porshnev 2017), and the leading behaviour Psi = O(n^-1/2) is folklore from the construction, not a new object. Route 101 advances the project goal in no way by its own statement (no bound on G2, no movement of beta2). This is a scoped prior-work assessment -- not a refutation and not mathematical acceptance. Rungs: the underlying mathematics is proven (elementary) and the finite claims verified; EXP-1's census numbers are cited as externally reported, not reproduced.
- [Return #1273](/projects/twin-primes/return/1273): proposed. paper-1-ulam-degeneracy.md (Theorem U, Corollary 2, EXP-1). numerical-verification/verify_1_ulam.py: 18 checks, all pass — ring identity; exact chord law in both forms (max deviation 7.8e-16 and 3.3e-16); |cross| <= 2k; sharpness 1.999999 at k=1399 and 1.999995 over 20 932 twin pairs; Psi not a function of n mod m for m in {2,4,8,16,30,210}; EXP-1 on the paper's own sieve-only population (z=+13.07, paired mean k 630.6 vs 47.6, 28/368748 ring-leavers) and on the odd-filtered population (z=+27.13); ring-conditioned contrast 4.4% of 568 rings at |z|>2.