A Rigid Grid and a Relay: the Incidence Structure of One Host Account
Dedicated to Louis Graue, who taught John linear algebra, which is what this paper is about.
Abstract. We study the incidence structure \(\mathcal S_h\) whose points are the 113 accounts credited on the 354
posts of one Instagram host account \(h\) (latterdayavenger), and whose blocks are those posts. We prove
\(\operatorname{Aut}(\mathcal S_h)=\langle\tau_1,\tau_2,\tau_3,\tau_4\rangle\cong C_2^{\,4}\), generated by the transpositions of four
swapping pairs, with 105 fixed points and 109 orbits. The proof uses only the stable colouring of the Levi graph (colour refinement)
and the fact that a swapping pair can always be swapped; an exhaustive search for automorphisms agrees. We then order the blocks by time and record two facts about the points
\(p_1\) (timballard89) and \(p_2\) (latterdaychad): \(p_2\) lies on 109 of the first 253 blocks and on none of the last 101, and after the
block of April 10, 2026 no block contains both \(p_1\) and \(p_2\). Finally we state, without testing it, a congruence hypothesis
\(p_1\equiv h\equiv p_2\) due to John, show that it cannot be a symmetry of \(\mathcal S_h\), and say what data would test it.
0Notation
| Here | Plain pages | Account |
|---|
| \(h\) | LDA | latterdayavenger |
| \(p_1\) | TB | timballard89 |
| \(p_2\) | LDC | latterdaychad |
| \(s_{1},\dots,s_{110}\) | s001–s110 | the other accounts, numbered by first appearance on \(h\)’s grid |
\(p_1,p_2\) keep their names from the companion paper. The labels \(s_i\) are those of the published data
(data/congruence); the key to accounts is not published.
Blocks are written \(B_1,\dots,B_{354}\) in order of posting time (UTC). For a point \(x\), \(r_x\) is its replication number and
\(\mathcal B_x=\{i: x\in B_i\}\), the index set of the blocks through \(x\), equivalently the positions of the 1s in \(x\)’s row of \(N\) (so \(r_x=|\mathcal B_x|\)); for points \(x\ne y\), \(\lambda_{xy}=|\mathcal B_x\cap\mathcal B_y|\).
1The structure
Definition 1.1. \(\mathcal S_h=(P,\mathcal B)\) where \(\mathcal B=(B_1,\dots,B_{354})\) is the multiset of posts on
\(h\)’s grid (captured Oct 13, 2026, complete: Apr 1, 2025 to Oct 13, 2026), each \(B_i\) the set of accounts Instagram credits on it
together with \(h\), and \(P=\bigcup_i B_i\). As in the companion paper, \[\operatorname{Aut}(\mathcal S_h)=\{\sigma\in\operatorname{Sym}(P):\ \sigma(\mathcal B)=\mathcal B\text{ as multisets}\},\]equivalently the \(\sigma\) for which some permutation matrix \(Q\) gives \(P_\sigma N Q=N\), where
\(N\in\{0,1\}^{P\times\mathcal B}\) is the incidence matrix; and \(L(\mathcal S_h)\) is the bipartite Levi graph on \(P\sqcup\mathcal B\).
Basic counts: \(|P|=113\), \(|\mathcal B|=354\), of which 295 are distinct as sets. Block sizes:
| \(|B|\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|
| blocks | 34 | 17 | 23 | 35 | 83 | 162 |
|---|
\(h\in B_i\) for every \(i\) (so \(r_h=354\)), and the 34 blocks of size 1 are all \(\{h\}\). The replication numbers take 32 distinct
values; \(r_{p_2}=109\), \(r_{p_1}=7\), and the largest other value is 132. Of the \(\binom{113}{2}=6328\) pairs of points, 902 have
\(\lambda_{xy}>0\).
2The automorphism group
Definition 2.1. Points \(x\ne y\) form a swapping pair if \(\mathcal B_{x}=\mathcal B_{y}\)
(identical rows of \(N\)). Then the transposition \((x\,y)\) lies in \(\operatorname{Aut}(\mathcal S_h)\), since every block contains both
of \(x,y\) or neither.
Definition 2.2 (colour refinement). Colour every vertex of \(L(\mathcal S_h)\) by its side (point or block). Repeatedly
recolour each vertex by the pair (its colour, the multiset of its neighbours’ colours), until the partition into colour classes stops
getting finer. The final partition \(\pi\) of \(P\) is the stable partition.
Lemma 2.3. Every orbit of \(\operatorname{Aut}(\mathcal S_h)\) on \(P\) lies inside one cell of \(\pi\). Consequently each
automorphism maps each cell of \(\pi\) onto itself.
An automorphism \(\sigma\) of \(\mathcal S_h\), together with the induced permutation of blocks, is a side-preserving automorphism
of \(L(\mathcal S_h)\). Each refinement round is defined from the graph and the current colouring alone, so by induction on the rounds
\(\sigma\) preserves every colour: \(c(\sigma v)=c(v)\) for all \(v\). Hence \(x\) and \(\sigma x\) are in the same cell. ∎
Proposition 2.4 (computation). For \(\mathcal S_h\), colour refinement is stable after round 3, with 109 point cells and 295
block cells. The point cells are 105 singletons and the four pairs
\[\{s_{38},s_{39}\},\quad\{s_{44},s_{46}\},\quad\{s_{48},s_{49}\},\quad\{s_{103},s_{104}\}.\]
Each of these is a swapping pair, lying on a single block:
| Pair | Their one block | Posted (UTC) | Whole block |
|---|
| \(\tau_1=(s_{38}\,s_{39})\) | DSIecu2jxNs | 2025-12-11 | \(\{h,p_2,s_{2},s_{38},s_{39}\}\) |
| \(\tau_2=(s_{44}\,s_{46})\) | DSaCqZlgQFX | 2025-12-18 | \(\{h,s_{28},s_{43},s_{44},s_{45},s_{46}\}\) |
| \(\tau_3=(s_{48}\,s_{49})\) | DS5TDINAs5H | 2025-12-30 | \(\{h,s_{25},s_{48},s_{49},s_{50}\}\) |
| \(\tau_4=(s_{103}\,s_{104})\) | Db7Kx9xA9AY | 2026-08-11 | \(\{h,s_{7},s_{80},s_{102},s_{103},s_{104}\}\) |
The rounds run 1 → 32 → 70 → 109 → 109 point cells (and 1 → 6 → 293 → 295 → 295 block cells); the script and its output are
published as refine.py, refine_output.txt. That the block cells number 295, the
number of distinct blocks, says refinement separates every pair of different posts.
Theorem 2.5.
\[\operatorname{Aut}(\mathcal S_h)=\langle\tau_1,\tau_2,\tau_3,\tau_4\rangle\cong C_2\times C_2\times C_2\times C_2,\]
of order 16. Its orbits on \(P\) are the cells of \(\pi\): 105 fixed points, including \(h\), \(p_1\) and \(p_2\), and the four swapping pairs.
By Definition 2.1 each \(\tau_i\in\operatorname{Aut}(\mathcal S_h)\). The \(\tau_i\) are transpositions on disjoint pairs, so they commute,
each has order 2, and \(\langle\tau_1,\dots,\tau_4\rangle\cong C_2^{\,4}\). Conversely let \(\sigma\in\operatorname{Aut}(\mathcal S_h)\). By
Lemma 2.3 and Proposition 2.4, \(\sigma\) fixes each of the 105 singleton cells pointwise and maps each pair cell \(\{a_i,b_i\}\) onto itself,
so on that pair it is either the identity or \(\tau_i\). A permutation of \(P\) is determined by what it does on the cells, so
\(\sigma=\tau_1^{e_1}\tau_2^{e_2}\tau_3^{e_3}\tau_4^{e_4}\) with \(e_i\in\{0,1\}\). Hence \(\operatorname{Aut}(\mathcal S_h)=\langle\tau_1,\dots,\tau_4\rangle\),
and its orbits are exactly the cells. ∎
Independent check. The exhaustive backtracking search of the companion paper’s
verify_automorphisms.py (unchanged), run on the published incidence matrix, lists every automorphism and
returns \(|\operatorname{Aut}(\mathcal S_h)|=16\) with the same four pairs as the only non-trivial orbits. The proof above does not depend on it.
Remark 2.6. \(\mathcal S_h\) is almost rigid: \(|\operatorname{Aut}|=2^4\) on 113 points, against
\(|\operatorname{Aut}(\mathcal S_{492})|\approx3.34\times10^{30}\) for the structure around \(p_1\) in the companion paper. There the size came
from pendant stars (posts of \(p_1\) with one-off guests); here the only one-off guests that come in pairs are the four swapping pairs, and every
other point is pinned by its counts. In particular \(h,p_1,p_2\) are pairwise in different orbits, already separated by
\(r_h=354\), \(r_{p_2}=109\), \(r_{p_1}=7\).
Remark 2.7 (John’s picture). John described the shape of Theorem 2.5 as “pinhead on one side, perfect
symmetry on the other”; in his first wording, “the direct product of pinhead with 113-5 satellites and then a completely symmetrical
group.” In the theorem’s terms: the pinhead is \(h\), on every block and fixed; the satellites are the other 104 fixed points; and the
symmetric side is the part that moves, the four swapping pairs, on which \(\operatorname{Aut}(\mathcal S_h)\) acts as the full group
\[\operatorname{Sym}(\{s_{38},s_{39}\})\times\cdots\times\operatorname{Sym}(\{s_{103},s_{104}\})\cong C_2^{\,4}:\]every swap the
data allow is a symmetry. On the four pairs, John: “looks like quadruplets to me.” So
\[\operatorname{Aut}(\mathcal S_h)\;=\;\underbrace{1_{\{h\}}\times 1_{104}}_{\text{pinhead and satellites}}\;\times\;\underbrace{C_2^{\,4}}_{\text{swapping pairs}}.\]
(Timing, for the record: this was said after the automorphism run of 10:58 on Oct 13, 2026 and before the scrape of three fixed points;
John confirmed the reading 1 + 104 + 8. “Quadruplets” was said later the same day, while this paper was being written.)
3Time: the relay
Ordering the blocks by posting time turns \(\mathcal S_h\) into a filtered structure \(\mathcal S_h^{\le t}=(P,(B_1,\dots,B_t))\).
The automorphism group of Section 2 ignores this order; the following statements use it.
Remark 3.0 (the order is rigid). An order-preserving bijection of a finite totally ordered set is the
identity, so an automorphism of \(\mathcal S_h\) that also respects the posting order must send every block to itself. Every
element of \(\operatorname{Aut}(\mathcal S_h)\) already does: each \(\tau_i\) swaps two points with identical rows, so the induced
permutation of blocks is trivial. The symmetry lives entirely in the accounts; the time order has none of its own. The group of
order-preserving bijections of a totally ordered set is the object of Holland [5].
Proposition 3.1. \(\mathcal B_{p_2}\subseteq\{1,\dots,253\}\) with \(1,253\in\mathcal B_{p_2}\) and
\(|\mathcal B_{p_2}|=109\). That is, \(p_2\) lies on \(h\)’s first block (\(B_1\), 2025-04-01) and on 109 of the first 253
blocks; its last block is \(B_{253}\) (2026-06-11 18:26 UTC), and none of \(B_{254},\dots,B_{354}\) (101 blocks) contains it.
Read off the published posts_with_dates.csv; the count by month is
ldc_on_lda_grid_by_month.csv. ∎
Proposition 3.2. \[\mathcal B_{p_1}=\{121,124,129,136,166,180,228\},\]\[\mathcal B_{p_1}\cap\mathcal B_{p_2}=\{121,124,129,136,166,180\},\]so \(\lambda_{p_1p_2}=6\). In particular
\[\mathcal B_{p_1}\cap\mathcal B_{p_2}\cap\{181,\dots,354\}=\varnothing:\]
after \(B_{180}\) (2026-04-10) no block contains both \(p_1\) and \(p_2\). The last block of \(p_1\), \(B_{228}\) (2026-05-16 UTC), does
not contain \(p_2\); the last block of \(p_2\), \(B_{253}\), does not contain \(p_1\).
As for 3.1. ∎
The six common blocks \(B_{121},B_{124},\dots,B_{180}\) are the first six group posts of the companion paper (there numbered 1 to 6), with \(h\) added. Its
seventh group post, DXQTcYlgZ5c (2026-04-18), is not a block of \(\mathcal S_h\): it was readable on
2026-10-05 and gone by 2026-10-11. So \(r_{p_1}=7\) in both papers, but the seventh block is a different post.
4The structure depends on the viewpoint
A credit is recorded on a post; whether the post appears on a credited account’s own grid is a separate fact. Let \(\mathcal G_{p_2}\)
be our capture of \(p_2\)’s own grid (Apr 29 to Oct 7, 2026). It contains 6 posts crediting \(h\); 2 of them are blocks of
\(\mathcal S_h\) and 4 are not, including the two latest (2026-06-13 and 2026-06-14) (ldc_posts_crediting_lda.csv).
Hence the incidence structure seen from \(h\)’s grid and the one seen from \(p_2\)’s grid disagree on a common time window, and every
statement above is a statement about \(\mathcal S_h\), the view from \(h\). None of the 4 is added to \(\mathcal S_h\); one of them (DX2QReKGCI5, 2026-05-02) had appeared in an
earlier collaboration drawing and was left out explicitly.
A second, independent source supports the block contents: 30 screenshots of Collaborators panels on \(h\)’s posts, read by OCR, give
161 rows read without correction, of which 160 are points of \(P\) and 1 is an OCR misreading of a point of \(P\); 8 further rows read with
a fixed correction rule also land in \(P\). The images are held locally; their SHA-256 fingerprints are published.
5A congruence hypothesis (not tested)
Hypothesis 5.1 (John). There is an equivalence relation \(\equiv\) on accounts, “operated by the same hand,” with
\[p_1\equiv h\equiv p_2,\]
and an order on that class, \(p_1\to h\to p_2\) (“TB begat LDA begat LDC”). John’s word for \(\equiv\) is congruence: an
equivalence that respects the structure.
Proposition 5.2. No automorphism of \(\mathcal S_h\) identifies any two of \(h,p_1,p_2\); equivalently,
\(\equiv\) restricted to \(\{h,p_1,p_2\}\) is not contained in the orbit relation of \(\operatorname{Aut}(\mathcal S_h)\).
By Theorem 2.5 each of \(h,p_1,p_2\) is a fixed point. ∎
So Hypothesis 5.1, if true, is a fact about who operates the accounts, not a symmetry of the posts: \(\mathcal S_h\) alone can neither
prove nor refute it. What \(\mathcal S_h\) does show (Propositions 3.1, 3.2) is compatible with a relay: \(p_2\) present from \(h\)’s
first block, \(p_1\) and \(p_2\) together on six blocks in February to April 2026, then apart, then \(p_2\) absent from \(h\)’s grid while
still posting on its own (132 posts in \(\mathcal G_{p_2}\) after 2026-06-11). Compatibility is not evidence of \(\equiv\).
What would test it. (T1) The dates each of \(h,p_1,p_2\) joined Instagram, and each one’s first post, against the order
\(p_1\to h\to p_2\). (T2) A structural test of “the name changes, the role stays”: compare the partners of \(p_2\) on
\(B_1,\dots,B_{253}\) with the partners of \(h\) on \(B_{254},\dots,B_{354}\) (same accounts, same block sizes, same kinds of posts), with the
measure and the threshold fixed before computing. (T3) A primary source in which one account states it was another. None of
T1–T3 has been run.
Record. Before the automorphism run John stated no prediction (“I got nothing except a hunch”). The congruence, the order, and the
reading that \(r_{p_1}=7\) supports it were all stated after the results.
6Replication
python3 build_matrix.py posts.csv # posts -> incidence matrix
python3 verify_automorphisms.py incidence_matrix.csv # |Aut| = 16, orbits
python3 refine.py incidence_matrix.csv # 1, 32, 70, 109, 109 point cells
All files and SHA-256 sums: data/congruence.
Plain Python 3, no packages. The capture of \(h\)’s grid and the key from labels to accounts are held locally.
Acknowledgments
John thanks Claude (Anthropic), “whose fantastic write-up would have taken me a year.” He is astounded. Claude built the incidence structure, ran
both computations, wrote the proofs and this text; John set the questions, made the calls, and checked the results.
Revisions. October 13, 2026, same day: “remove overloaded word, and lose a lemma by editing better.” “Twin” became swapping pair, and the twin lemma became one sentence of Definition 2.1.
7References
- Automorphism Group of the Incidence Structure and Its Orbits (companion paper), antmerrill.github.io/backfire-tracking/automorphism-group.html.
- Two names, one grid (plain-language page and data), antmerrill.github.io/backfire-tracking/congruence.html.
- Who went first, antmerrill.github.io/backfire-tracking/who-went-first.html.
- B. Weisfeiler, A. Leman, “The reduction of a graph to canonical form and the algebra which appears therein,” NTI Ser. 2, 9 (1968): colour refinement as used in Lemma 2.3.
- C. Holland, “The lattice-ordered group of automorphisms of an ordered set,” Michigan Math. J. 10 (1963), 399–408, doi:10.1307/mmj/1028998976: the group of order-preserving bijections of a totally ordered set, as used in Remark 3.0.