tier n
0
Two closed forms — a triangular seat layer doubled by the bipartite exclusion, F(n) = (n+1)(n+2), and an antipodally-paired belt admissible only on the equator of the collective roll, B(n) = 2(n+1) — reproduce 2, 8, 20, 28, 50, 82, 126 sequence-exact with no fitted coefficient and no assigned spin–orbit coupling, and continue to closure(7) = 184. Everything on this page is either computed live from those two forms or a measured number with its source named. One quantity is read from the data and is flagged where it occurs: the descent onset n = 3.
Two-dimensional close packing is triangular; the bipartite exclusion doubles each seat; a radial rod rolls slip-free at one latitude only. Those three facts give two capacities.
| n | T(n+1) | F(n) = (n+1)(n+2) | B(n) = 2(n+1) | R(n) = n(n+1) |
|---|
The exclusion that forces the doubling is like never gears like: a proton gears only with neutrons, a neutron only with protons. A seat is therefore a rod presenting two faces, one p and one n, so the contact graph is bipartite — no odd cycles, no frustration, and traction guaranteed rather than checked. The rolling constraint that forces the belt onto one latitude is the zero-slip condition ωA = −ωB at contact; off the equator of the collective roll a radial rod's contact slips, which is forbidden.
Layer n holds T(n+1) rod seats; each seat carries a p-face and an n-face; F(n) = 2·T(n+1) = (n+1)(n+2).
Accumulating this form alone gives the first three closures, with nothing assigned: F(0) = 2, then 2 + F(1) = 2 + 6 = 8, then 8 + F(2) = 8 + 12 = 20. Three of the seven. F(0) = 2 is the alpha core's own pair — the core obeys the same formula as every tier above it.
Excess neutrons arrive as rigid n–p–n triton rods, seated radially. A radial rod is slip-free on one latitude only, so the belt is a ring, not a surface — linear capacity, antipodally paired.
The dimensional difference between the forms — quadratic surface against linear ring — is what sets the observed closure spacing: wide at low count (2→8→20, gaps of 6 and 12), compressed where belts dominate (20→28, gap 8), then widening again as remainders re-enter (28→50→82→126, gaps 22, 32, 44).
| closure step | gap | composition |
|---|---|---|
| 2 → 8 | 6 | F(1) — surface only |
| 8 → 20 | 12 | F(2) — surface only |
| 20 → 28 | 8 | B(3) — belt only (the descent) |
| 28 → 50 | 22 | R(3) + B(4) = 12 + 10 |
| 50 → 82 | 32 | R(4) + B(5) = 20 + 12 |
| 82 → 126 | 44 | R(5) + B(6) = 30 + 14 |
| 126 → 184 | 58 | R(6) + B(7) = 42 + 16 |
From the fourth tier the belt separates and descends inward to seal the closure beneath it. This produces 28 — and its onset is the one quantity here read from the measured sequence.
Through tiers 0–2 the surface fills and closes. From n = 3 the belt drops to the prior shell and seals it; the tier above then completes on a sealed floor. The three ages of a triton follow: docked → paired → sealed. Positions are fixed by tier geometry, filling order by antipodal pairing with the lone odd rod inward, and occupancy by the neutron ledger at closure time — frozen once the next tier rests on it.
The descent onset n = 3 is READ from the measured sequence, not derived. The construction fixes what a descending belt does once it descends; it does not yet fix from mesh and void geometry why separation begins at the fourth tier. Candidate mechanism: a field-cost comparison — the tier at which sealing below first becomes cheaper than spreading above — and that computation is owed (residual NP33 debt). The provenance block in sdt::laws::nuclear carries this disclosure at the definition site.
All seven closures from the two forms and one alternation rule, then one step further.
| k | added | arithmetic | closure | measured | Δ |
|---|
The forms carry no termination condition and were not conditioned on the extent of the measured set, so closure(7) = 184 costs no additional assumption. It is the same recursion advanced one step, and it is the page's falsification exposure: an eighth closure located at any other count refutes the schedule that carries the other seven.
If a closure is a sealed floor rather than a filled level, the compaction should not track how many tritons occupy the belt. Measured, single-pass, on an instrument validated on known answers: it does not.
| isotone | members | Z (n_t) → kink, milli-fm | mean | RMS const | RMS prop | outcome |
|---|
Two isotones (N = 20 and N = 126) carry a single computable member each and are therefore not adjudicable — the pre-registration required at least two, and they are reported rather than counted. N = 8 has no computable members at all: the neighbour radii are unmeasured. Of the three adjudicable lines, all three select constant over occupancy-proportional, and the |kink| span ratios cluster tightly at 0.76, 0.78, 0.76.
The lone-rod-inward result (83.1%) shares its predicted sign with the pairing account: shared form, cannot discriminate between accounts. What the census tests is the direction the seat law asserts — inward — not the superiority of one account over the other.
Post-hoc, not pre-registered: closure grip peaks at N = 28 — the first descended belt — at −31.8 milli-fm, and fades with tier size (−22.2 at N = 50, −12.3 at N = 82). The pre-belt N = 20 closure is weak at −6.1 (single member). Whether closure grip is a belt property is a follow-up question, not a claim.
The mesh is built one nuclide at a time from hydrogen. Only one lock on the ladder is derived; the rest are measured numbers the picture names, and they are tagged that way.
| step | nuclide | structure | lock, MeV | tag |
|---|---|---|---|---|
| 1 | ¹H | one trefoil, no contact — the free reference rotor | — | [D] seat 1.830c |
| 2 | ²H | one p↔n gear; e⁻ rides the rolling node at v = 1.69c; gap D = (7/3)R_p = 1.963 fm | 2.200 vs 2.2246 (−1.1%) | [D] derived |
| 3 | ⁴He | p–n–p–n ring (4-cycle); tetrahedral shape, ring mesh; like-like diagonals off contact | 23.847 = 28.296 − 2(2.2246) | [measured, mesh-named] |
| 4 | ³H / ³He | 3-gear rods | 6.26 | [measured, mesh-named] |
| 5 | ⁶Li | α ring + first deuteron dock | 1.4738 (α–d separation measures 1.4743) | [measured, mesh-named] |
| 6 | ⁷Li | + lone rod | 2.467 | [measured, mesh-named] |
| 7 | ⁸Be | two α — unbound | −0.092 | [measured, mesh-named] |
| 8 | ¹²C | three α | +7.274 | [measured, mesh-named] |
| 9 | ¹⁶O | four α — tetrahedron of rings; closure 8 complete | +14.435 | [measured, mesh-named] |
The deuteron's 2.200 MeV is the ladder's only [D] — computed from the shared-electron occlusion sum E = −Σqiqj·αℏc/rij with no fitted scale. The α ring lock 23.847 MeV is convergent with the independent scission line-item (23.85 MeV, NP25) — independent origins, same number. Every other entry is a mass-table difference the mesh picture names, and the tag says so; the ledger does not inflate one label at a time.
An earlier conjecture in this line held that mesh frustration selects the closures. It is withdrawn: under like-never-gears-like the contact graph is bipartite, so it has no odd cycles and no frustration is available to select with. The successor account is shell completion — the schedule above.
The seat law says a stable odd-Z nuclide carries an odd triton count: the lone rod takes the inward seat. An odd-Z nuclide with an even triton count has no partner to seal against — a parity misfit.
| class | members | predicate | observed |
|---|---|---|---|
| mono-isotopic grips (odd n_t) | Na, Al, P, Sc, V, Mn, Co | clears — 7/7 | single stable isotope each |
| parity misfits (even n_t) | K-40, V-50, La-138, Lu-176, Ta-180m | flags — 5/5 | every one a long-lived radioactive oddity |
| suite row B38 | 12/12 combined | ||
The misfit class is exactly the five naturally occurring odd–odd quasi-stables. Counting convergent with the odd–odd rule of the prevailing account; the native content is the unpaired rod with nothing to seal against. Corroborating pair: Ca-48 completes the first triton belt (4 antipodal pairs) and is doubly magic and held; Ni-56 has an empty belt and is unstable.
F(n) is numerically identical to the oscillator-level degeneracy of the prevailing shell account, and B(n) to its intruder-orbit capacity. At sequence level the two accounts are not discriminable — they count the same objects. Shared form, claimed as consistency and nothing more. Not shared: the bipartite doubling and the rolling-equator constraint, neither of which is available to a construction resting on an assigned spin–orbit coupling. No such coupling appears at any step — not by removal, but because the symbol is absent from the permitted namespace.
The descent onset n = 3 is read from data (§4). Capacities are closed-form; the alternation's initial index is not. Two isotones are non-adjudicable at one member each; N = 8 has no computable members. The lone-rod census shares its sign with pairing and discriminates nothing. The 184 prediction is the standing exposure.