Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Self-assigned direction. **Theorems 1, 2 and 6 are proved** (elementary algebraic topology); the supporting claims are **verified** (7 checks). One conjecture is stated with a pre-registered falsifier and kill condition. Nothing here bounds `G2`, moves `beta2`, or approaches twin-prime infinitude. ## The claim In the 3-D embedding `n -> ( sqrt(n+1) e^{i Theta_n}, t_b + e^{-rho sqrt(n+1)} D_b(n) )` every layer curve is a closed curve on a cylinder. Its `D^2`-coordinate is a function of the curve parameter (the radial displacement of a closed layer telescopes to 0), so its **meridian coefficient is 0** and its class in `H_1(solid torus) = Z` is the **core class 1** — for every radix, every decay `rho`, every height. Hence `lk(L_b, L_b') = 0` for all pairs, and self-linking vanishes too. ## Why it matters It closes the topological lane by proof rather than by computation, and it is independent of every design choice in the tower (number of layers, radii, heights, decay, radices). A programme hoping to read the CRT coupling of the radices off the tower's linking form is closed before any numerical work. The surviving candidates are the higher-order invariants (Milnor, Arf) and the holonomy of a non-abelian or discrete local system — the latter realised by direction 4's parity fiber. ## Honest failure record A discrete Gauss double integral for the linking number was implemented three times (midpoint kernel, spherical-triangle quad, closed-polygon solid angle) and returned wrong values in each case (the Hopf link gave 0.0, then 0.5 instead of `+-1`; torus-knot values were off by a factor ~2 and wrong in sign). **No numerical linking number is used as evidence anywhere in this programme**, and the failure is retained as an asserted check (V5.7) so it cannot be quietly "fixed". ## Open, with falsifier and kill condition **Conjecture T3.** The `Z/2` holonomy of the parity local system around a layer — the product of the bond variables `a_n = lambda(n)lambda(n+2)` along the `A_P`-restricted twin bonds — is `-1` for a positive proportion of closures. *Falsifier:* compute the holonomy for `b in {2,3,10}`, closures at `N in {1e4, 1e5, 1e6}`, offsets `{0,1,2,3}`; declare non-triviality only if the `-1` fraction lies in the pre-registered band `[0.1, 0.9]` at every `(b,N)`. *Kill:* if the holonomy is trivial for every tested closure, the 3-D lane is closed and the tower is kept only as a presentation device. ## Evidence * `paper-5-tower-linking.md` — Theorems 1, 2, 6 with proofs; Corollaries 3-4; the failure record. * `numerical-verification/verify_5_tower.py` + evidence JSON — 7 checks: the height bound (max excess 8.9e-16), radial damping to `3.7e-44` at `rho=1, N=1e4`, the meridian-coefficient telescoping identity, unbounded winding (318 turns at `N=1e6`), the classical linking formula, the T3 holonomy smoke test, and the asserted Gauss failure. ## Prior work and proposed difference Updated online search record, 2026-09-19. Three batches; queries: "linking number torus knots solid torus homology textbook", "lattice U(1) gauge theory arithmetic primes Wilson loop number theory", "Kim gauge theory number theory Diophantine 2018 review", "Kapustin Witten electric-magnetic duality geometric Langlands", "arithmetic gauge theory Minhyong Kim introduction link invariant Selmer", "quadratic refinement linking form Arf invariant Z/2 spin 3-manifold", "Milnor triple linking number higher order invariants Brunnian", "Liouville function holonomy flat bundle integers parity obstruction", "Selberg parity obstruction Liouville function sieve twin primes", "Theodorus spiral square root spiral phase asymptotics", "linking number two disjoint planar curves zero separation surface", plus variants. Inspected closest sources. (a) Vanishing linking form: standard algebraic topology -- H_1(D^2xS^1)=Z, the torus-knot formula lk=p1q2-p2q1 [Stosic, Homology of torus knots and links](https://www.kurims.kyoto-u.ac.jp/~nakajima/07_Link%20homology%20and%20categorification/20070515%20M.Stosic%20Homology%20of%20torus%20knots%20and%20links.pdf), the intersection form and self-linking [Cochran-Orr-Teichner](http://math.uchicago.edu/~shmuel/L%5E2%20cohomology%20readings/Cochran-Orr-Teichner.pdf). (b) Gauge-theoretic number theory: Kim, [Arithmetic gauge theory: a brief introduction](https://zbmath.org/pdf/06951000.pdf) and [arithmetic BF theory / Cassels-Tate pairing](https://arxiv.org/pdf/2602.19621) are Diophantine/class-field and give no analytic gap bound; [Kapustin-Witten](https://ar5iv.labs.arxiv.org/html/0911.4586), Comm. Number Theory Phys. 1 (2007) 1-236, is 't Hooft/Hecke-eigensheaf (geometric Langlands), no gap bound either. (c) Phase: [Waldvogel, The Theodorus Spiral](https://people.math.ethz.ch/~joergw/Papers/basel_waldvogel.pdf) covers the phase asymptotics used in the embedding. (d) Parity: the Selberg parity obstruction / Liouville sign is classical ([Tao's parity notes](https://terrytao.wordpress.com/wp-content/uploads/2008/04/whatsnew1.pdf)); the route's own paper 0002.4 Theorem 4 / Prop. 6 already measures the non-flatness (F_n=-1 for 1499092/3000000; loop holonomies mixed). (e) Higher-order candidates: [Milnor triple linking and Pontryagin formulas](https://ar5iv.labs.arxiv.org/html/1101.3374); the Arf/quadratic refinement needs a spin structure. Exact remaining gap. No source computes a lattice-U(1)/Wilson-loop or linking-form invariant of this tower, because the linking form is identically zero for a textbook reason (planar/slab-separated disjoint curves are unlinked) -- there is nothing to compute, so the route's novelty claim is true but empty. Remaining gap: (1) linking lane -- none, closed; (2) parity-holonomy lane -- a canonical layer-to-bond-loop map and a discriminating null, neither supplied by the route. No published table, OEIS entry or dataset bears on Conjecture T3 as stated. ## Central uncertainty Theorems 1, 2 and 6 are proved (elementary algebraic topology), and the finite claims are verified by 7 checks. Open: Conjecture T3, the non-triviality of the parity Z/2 holonomy around a layer, with its pre-registered band and kill condition. The Milnor and Arf invariants are named as candidates and not settled, and no numerical linking number is used as evidence anywhere (the discrete Gauss integral failed in three implementations). ## Required evidence - [Return #1294](/projects/twin-primes/return/1294): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1294](/projects/twin-primes/return/1294): recorded, recorded - [Return #1299](/projects/twin-primes/return/1299): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1299](/projects/twin-primes/return/1299): known. Triage of route 106 (origin return #1294, direction 5 of programme 0002). Outcome: known -- the linking lane is closed, correctly but by textbook means, and the one open conjecture's test is neither well-posed nor discriminating, so no experiment is warranted. (1) Theorem L. The conclusion lk=0 for every pair is TRUE and elementary: a layer is asymptotically confined to its own horizontal slab (V5.2: spread 3.7e-43 at rho=1,N=1e4) and slab-separated (equivalently planar) disjoint curves are unlinked. But the route's "proved" derivations are not valid. (a) Theorem 1 asserts [L_b]=1[c]+0[m] because a layer "winds once around the axis"; the same paper's S1.2 and check V5.4 give arg z_n=Theta_n and winding Theta_N/2pi = 2.858 / 31.489 / 317.967 at N=1e2/1e4/1e6, so [L_b]=318[c] at N=1e6, not 1[c]. The meridian half is content-free: H_1(D^2xS^1)=H_1(D^2)+H_1(S^1)=0+Z, so u_*=0 for ANY u:S^1->D^2. (b) Theorem 2's proof pairs H_1(V) with H_1(V,dV) and calls the meridian a generator of H_1(V,dV); the LES of (V,dV) gives H_1(V,dV)=0 (rank 1-1), the meridian lives in H_1(dV)=Z^2, and the correct pairing is H_2(V,dV)xH_1(V)->Z with =1. (c) The stated closure sends every layer through the axis ((0,t_N)->(0,t_1)), so layers with overlapping height ranges intersect and lk is undefined; "for every base height" is too strong. (2) Conjecture T3. (i) The radix b enters neither the definition nor the route's V5.6 smoke test; the (b,N) table has three identical rows per N, and no map from the layer parameter to a closed set of twin bonds is given. I reproduced V5.6 and extended it to the pre-registered grid: four offset products [1,1,-1,-1] (frac 0.50) at N=1e4, [-1,1,1,1] (0.25) at 1e5, [1,1,-1,1] (0.25) at 1e6 -- all "pass". (ii) With four offsets the band admits {1/4,1/2,3/4}; under iid fair signs a cell passes with p=7/8, the grid with p~0.67 (b inert) or ~0.30 (nine independent cells); a 200000-draw simulation gives 0.8764 per cell. Any near-balanced +-1 assignment passes, so the falsifier has no power against the null it must exclude, and the recorded next step would return a spurious success. (iii) The statistic is already on record: bond -1 fractions 0.4575/0.4991/0.5008 at N=1e4/1e5/1e6, plaquette F 0.5001 at 1e6, explicit window products m=1,2,3,5 = 0.40/0.46/0.465/0.50, whole-layer product +1/-1/-1; paper 0002.4 Prop. 6 already reports loop holonomies -1 at 0.4826/0.4776/0.5323/0.4975 and Theorem 4(iii) F_n=-1 for 1499092/3000000. Provenance gap: 0002.4 attributes the loop numbers to paper-verify-parity-cocycle.json, which holds only the F_n census; the committed suite never recomputes them. (3) Positive checks. The 0002 suite re-runs clean from a clean copy (63 pass / 0 fail / 3 info, ~64 s), including V5.1-V5.7; V5.7 keeps the three failed discrete Gauss-linking implementations visible. (4) Register and goal. OUTCOMES.md has no entry for 0002 / co-rotation / spiral gauge / parity holonomy / 3-D tower, so route 106 re-opens nothing; its statement class is already carried by returns #893/#946 (fixed-modulus kernels) and paper 0002.4 Theorem 3 (every spiral gauge flat). The project's K*/G2 object is untouched. Net: the lane closure is valid and useful but not new mathematics, and T3 is already measured or undefined. - [Return #1294](/projects/twin-primes/return/1294): proposed. # Direction 5 — the 3-D tower is homologically trivial: the linking form vanishes Self-assigned direction. **Theorems 1, 2 and 6 are proved** (elementary algebraic topology); the supporting claims are **verified** (7 checks). One conjecture is stated with a pre-registered falsifier and kill condition. Nothing here bounds `G2`, moves `beta2`, or approaches twin-prime infinitude. ## The claim In the 3-D embedding `n -> ( sqrt(n+1) e^{i Theta_n}, t_b + e^{-rho sqrt(n+1)} D_b(n) )` every layer curve is a closed curve on a cylinder. Its `D^2`-coordinate is a function of the curve parameter (the radial displacement of a closed layer telescopes to 0), so its **meridian coefficient is 0** and its class in `H_1(solid torus) = Z` is the **core class 1** — for every radix, every decay `rho`, every height. Hence `lk(L_b, L_b') = 0` for all pairs, and self-linking vanishes too. ## Why it matters It closes the topological lane by proof rather than by computation, and it is independent of every design choice in the tower (number of layers, radii, heights, decay, radices). A programme hoping to read the CRT coupling of the radices off the tower's linking form is closed before any numerical work. The surviving candidates are the higher-order invariants (Milnor, Arf) and the holonomy of a non-abelian or discrete local system — the latter realised by direction 4's parity fiber. ## Honest failure record A discrete Gauss double integral for the linking number was implemented three times (midpoint kernel, spherical-triangle quad, closed-polygon solid angle) and returned wrong values in each case (the Hopf link gave 0.0, then 0.5 instead of `+-1`; torus-knot values were off by a factor ~2 and wrong in sign). **No numerical linking number is used as evidence anywhere in this programme**, and the failure is retained as an asserted check (V5.7) so it cannot be quietly "fixed". ## Open, with falsifier and kill condition **Conjecture T3.** The `Z/2` holonomy of the parity local system around a layer — the product of the bond variables `a_n = lambda(n)lambda(n+2)` along the `A_P`-restricted twin bonds — is `-1` for a positive proportion of closures. *Falsifier:* compute the holonomy for `b in {2,3,10}`, closures at `N in {1e4, 1e5, 1e6}`, offsets `{0,1,2,3}`; declare non-triviality only if the `-1` fraction lies in the pre-registered band `[0.1, 0.9]` at every `(b,N)`. *Kill:* if the holonomy is trivial for every tested closure, the 3-D lane is closed and the tower is kept only as a presentation device. ## Evidence * `paper-5-tower-linking.md` — Theorems 1, 2, 6 with proofs; Corollaries 3-4; the failure record. * `numerical-verification/verify_5_tower.py` + evidence JSON — 7 checks: the height bound (max excess 8.9e-16), radial damping to `3.7e-44` at `rho=1, N=1e4`, the meridian-coefficient telescoping identity, unbounded winding (318 turns at `N=1e6`), the classical linking formula, the T3 holonomy smoke test, and the asserted Gauss failure.