Programme note · 15 September 2026 · Nanjie Ma ·
ORCID 0009-0002-4415-1209
About or Work subpage. Not a homepage hero. Not a ranking. Not a breakthrough notice. Not cosmogenesis payment.
This page describes a line of work on discrete sequential growth under a registered covering kernel. It is written so that a reader can check the deposits. It does not ask the reader to treat the origin of the universe as solved.
The live generator is the registered kernel EK-A0007-MINIMAL-T1-T2-COVERING-V0,
read in a port-resolved convention at port budget q = 2.
Births have arity one or two.
The relative intensity of those two arities is a quantity that a selection principle would have to supply;
it is not derived from discrete general covariance alone.
On that kernel the covariant coherent dynamics, truncated at a finite horizon N, form a set affinely isomorphic to the probability simplex on the isomorphism classes of ported states. The number of those classes, by exact enumeration, is
1, 1, 2, 4, 11, 33, 118, 471, 2 126, 10 625, 58 403, 349 391
for N = 1, …, 12. At each finite-horizon extreme point the arity-two count is a class invariant, so covariance plus coherence does not pin a unique physical intensity. On the infinite growth graph the same spectrum fills the whole interval [0, 1] among constructed ergodic (tail-trivial) measures; fixing a density still leaves infinitely many pure phases. A selector, if there is one, has to import data from outside the growth graph.
The certified extremal locus used for those constructions is classified: it is exactly the admissible ladders (blockwise total orders with a singleton bottom block and no block larger than two). On that locus an independent Rideout–Sorkin p-thinning that retains the unique minimum acts as a one-parameter semigroup. Scale invariance on the locus selects the chain and nothing else. That is a statement about a flow of measures on a classified family of states. It is not a derivation of leftover r (the leftover share of arity-two births), a continuum spacetime, or a physical universe.
Proofs for this layer occupy Companion Papers 20–23. Companion Paper 24 is a related labeled-path coarse map; it is a different object, not a continuation of the covering-kernel classification. The geometry of the remaining mixed-extremality bar occupies Companion Paper 25, recorded below. The parent manuscript is the Quantum Narrative Matrix / covering-kernel text on Zenodo.
| Record | Role | Deposit |
|---|---|---|
| Parent manuscript | Programme text | 10.5281/zenodo.20630985 |
| Companion 20 | Finite-horizon covariant classification | 10.5281/zenodo.22524084 |
| Companions 21–22 | Ergodic arity spectrum; ladder locus | 10.5281/zenodo.22520894 |
| Companion 23 | Measure-level thinning semigroup on the locus | 10.5281/zenodo.22546067 |
| Companion 25 | Geometry of the covering mixed-extremality bar; not payment | 10.5281/zenodo.22769118 |
| Author | Independent researcher | ORCID 0009-0002-4415-1209 |
Parent paper · Zenodo Companion 25 · Zenodo ORCID
Please judge the deposits, the named unpaid objects, and the claim boundary on each page.
Companions 20–23 exhibit covariant coherent dynamics on the kernel. They do not exhibit a point of covering K.
A further layer, occupied in Companion Paper 25 (15 September 2026), records the already-deposited geometry of the remaining mixed-extremality bar, without claiming that bar paid. Three compacta on this axis are not the same set: the inverse-limit covering compactum of the covariant simplices; the covering-face compactum on which mixed extremality is posed; and the every-layer leaf Meyer box last ≤ 3/4. Occupancy of a Meyer-range slice is not mixed extremality of covering K.
The deposited mixed-extremality clause is a liminf: exhibit one point of the covering-face compactum whose covering birth-density sequence satisfies lim inf Fn ≤ 3/4. Fatou’s lemma reduces a Choquet clamp at threshold 1 to that inequality at threshold 3/4 without exhibiting a point. Co-density converts the box inequality into a budget that Dirac reverse-equal complete-parent edges cannot carry. An exact census through event-count 11 is on record.
A point of the covering-face compactum is exactly an all-N reverse-equal compatible family. Event-count 2 is structurally over-wall, so the last-cut question begins at event-count 3. An exact downward last-cut profile records 923 of 58 403 classes at event-count 11 remaining in the box at every event-count from 3 through 11. The associated witness polytopes are nonempty at event-counts 9, 10 and 11. Unique-minimum of covering classes is a covering-fire invariant and is not a characterisation of those polytopes. Nonemptiness of the witness polytope at every event-count remains open. That all-M statement is mixed extremality; it is not claimed here. Companion Paper 25 is a further child of the parent, not a new version of Companions 20–24. This child is 10.5281/zenodo.22769118.
Sequential growth is already a recognised way to ask how a discrete causal order can be generated rather than assumed. What this kernel adds is a small, registered generator on which several questions can be answered by exact classification instead of by slogan: which covariant laws exist at a finite horizon; that covariance and tail-triviality do not select an intensity; what the certified extremal states actually look like; and how a standard thinning flow acts on that locus.
Those no-go results are useful because they say where not to look. The remaining work is then forced to be honest: any physical selection has to come from an object that is not already implied by the growth graph, and mixed extremality of covering K has to be an inverse-limit point of the covering-face compactum, not a finite prefix and not occupancy of a Meyer-range slice. That is a narrower, checkable problem. I think the direction is worth continuing for that reason — the questions are well posed, the certificates are machine-readable, and the unpaid mouths are named rather than hidden.
If later work pays a named mouth, that should be recorded as a scoped result. Until then, the correct public sentence is the one above: the object is on record; cosmogenesis is not.