The Young sector of covering growth

A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Every Young-diagram poset is a legal covering history: axis boxes are one-parent births, interior boxes are two-parent births, and every standard Young tableau is a legal birth word. Young's graph, quotiented by transposition, embeds in the covering growth graph. Transposition-invariant central measures, including Plancherel measure, descend to that Young sector. Extra covering children exist from size 3 and receive Plancherel transition probability 0; the descended law is not the path-entropy measure of Companion 37. Under Plancherel growth, typical diagrams fill the Vershik-Kerov-Logan-Shepp region. The product order on boxes is, in null coordinates, the causal order of 1+1 Minkowski space. Combined with area measure this is a Level-2 Lorentzian kernel of spacetime dimension 2 on that region. Graded chain length depends on aspect ratio, so Level 3 (longest chain as proper time) fails. Finite-N samples have L/sqrt(N) climbing toward 2 and global t(2+2) bounded away from 0. Three candidates for a sourced confluence rule fail: minimising new comparable pairs never selects a two-parent birth; labelled ports are extra input; the diamond property does not force Young (smallest counterexamples at size 5). Finite order ideals of N^q lie in Cov(q,q); their rescaled order cone is the positive orthant, polyhedral for q>=3, and is not a round Lorentzian cone in dimension >=3. The three stepwise laws of Companion 39 hand a later stage a total order or an antichain, not this sector. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. This note does not claim Lorentz invariance and does not assert that covering structures cannot produce geometry. Other laws remain open. It does not replace the parent programme manuscript, and it is not a new version of Companions 13-39.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-02
DOI
https://doi.org/10.5281/zenodo.23879512
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

The Young sector of covering growth

NANJIE MA
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Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

The Young sector of covering growth

NANJIE MA
preprint en

Abstract

A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Every Young-diagram poset is a legal covering history: axis boxes are one-parent births, interior boxes are two-parent births, and every standard Young tableau is a legal birth word. Young's graph, quotiented by transposition, embeds in the covering growth graph. Transposition-invariant central measures, including Plancherel measure, descend to that Young sector. Extra covering children exist from size 3 and receive Plancherel transition probability 0; the descended law is not the path-entropy measure of Companion 37. Under Plancherel growth, typical diagrams fill the Vershik-Kerov-Logan-Shepp region. The product order on boxes is, in null coordinates, the causal order of 1+1 Minkowski space. Combined with area measure this is a Level-2 Lorentzian kernel of spacetime dimension 2 on that region. Graded chain length depends on aspect ratio, so Level 3 (longest chain as proper time) fails. Finite-N samples have L/sqrt(N) climbing toward 2 and global t(2+2) bounded away from 0. Three candidates for a sourced confluence rule fail: minimising new comparable pairs never selects a two-parent birth; labelled ports are extra input; the diamond property does not force Young (smallest counterexamples at size 5). Finite order ideals of N^q lie in Cov(q,q); their rescaled order cone is the positive orthant, polyhedral for q>=3, and is not a round Lorentzian cone in dimension >=3. The three stepwise laws of Companion 39 hand a later stage a total order or an antichain, not this sector. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. This note does not claim Lorentz invariance and does not assert that covering structures cannot produce geometry. Other laws remain open. It does not replace the parent programme manuscript, and it is not a new version of Companions 13-39.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
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The Young sector of covering growth — NANJIE MA · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS