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Passes 1375–1378 — the stabiliser is S₅, the tomotope checked against its own paper, A₄ as the derived core, and why the guard could not see any of it

Four results. One refutes a value that was frozen into data/ALIAS_REGISTRY.json as canonical; one checks ~50 files' worth of tomotope claims against the primary source for the first time; one finds the object BT781–BT783 were circling; and one diagnoses, by measurement, why this corpus keeps rediscovering group theory specifically.

Pass 1375 — the 432-orbit stabiliser is S₅, and the "order 60" correction is wrong three times over

Pass 1126 found that W(E₆)'s three orbits of size 432 on the 2240 A₂ root triples each carry exactly one degree-81 irreducible, reported the point stabiliser as order 120, and deliberately declined to name it.

Commit 2da603e52 issued a correction, and froze it into the alias registry:

Key correction resolved: |Sp(4,3)| = 25920. For orbit size 432, the stabilizer order is 25920/432 = 60, consistent with A₅ ≅ PSL(2,5), not order-120 S₅.

Every step of that is wrong, and the script it shipped could not have detected it.

|Sp(4,3)|   = 51840          <- not 25920
|PSp(4,3)|  = 25920          <- this is what 25920 is
|Sp| = 2|PSp|                 true    (Sp(4,3) = 2.U4(2), the DOUBLE COVER)

group acting on the 2240 A2 triples = W(E6) = U4(2):2, order 51840
  #Irr = 25   (25 = U4(2):2;  34 = Sp(4,3))   -> NOT Sp(4,3) either

So the divisor is 51840/432 = 120, and the correction divides by the order of a group that is not acting — the exact Sp(4,3) ≇ W(E₆) conflation Pass 1020 had to repair in five files.

It also mislocates the object. The shipped GAP script searches for 432-orbits among the 2-element subsets of the 40 points of W(3,3). Run:

orbits of PSp(4,3) on the 780 point-pairs : [240, 540]
any orbit of size 432 there?              : false

There are only C(40,2) = 780 such pairs and they split 240 + 540. The script prints nothing, and would have done so silently. The 432s live on the 2240 A₂ root triples in E₈.

The answer. With W(E₆) built correctly as the pointwise stabiliser of an A₂ triple in W(E₈):

orbits on the 2240 : [1, 1, 27×6, 240, 270, 270, 432, 432, 432]      sum 2240
each 432-orbit  |stab| = 120 = 51840/432
   IdGroup             = [120, 34]
   StructureDescription= S5
   element orders      = 1,2,2,3,4,5,6
   isomorphic to SL(2,5)?  false
   isomorphic to C2 x A5?  false
   stab1 ~ stab2 ~ stab3 conjugate in W(E6)?   ALL TRUE

The stabiliser is S₅, and the three orbits are conjugate — one orbit type, not three. The 60 in the retracted correction is not meaningless, but it names a different object:

|S5 ∩ W(E6)'| = |S5 ∩ PSp(4,3)| = 60,   IdGroup [60,5] = A5

It is the intersection with the simple group, arrived at by wrong arithmetic on the wrong group.

The S₅ coincidence, resolved rather than asserted

Pass 1125 refused to comment on the fact that its eight minimal tree-filter generators include an S₅ = SmallGroup[120,34], and that the 432-stabiliser also has order 120 — "two order-120 objects in one session is exactly the coincidence this corpus gets burned by." Both are now named, and they are provably different subgroups:

Pass 1125's S₅ Pass 1375's S₅
lives in PSp(4,3), index 216 W(E₆) = U4(2):2
inside the simple group? yes no — meets it in A₅
kills the Steinberg? yes (minimal tree generator) its A₅ part does not

Since PSp(4,3) ⊴ W(E₆), conjugation preserves containment in it, so the two S₅'s are not conjugate in W(E₆). Abstractly isomorphic, structurally distinct: one sits inside the simple group and kills the Steinberg module; the other is split across the outer coset and stabilises the Steinberg's carrier. The coincidence was real and it is not a bridge.

Pass 1376 — the tomotope, checked against its own paper for the first time

This repository has roughly fifty tomotope files. Every one of them works from restated numbers. A grep for the actual published permutations —

rho0 = (5,10)(6,9)(7,12)(8,11)     rho2 = (5,9)(6,10)(7,11)(8,12)
rho1 = (1,6)(2,5)(3,8)(4,7)        rho3 = (5,8)(6,7)(9,12)(10,11)

(Monson–Pellicer–Williams, The Tomotope, Ars Math. Contemp. 5 (2012), p. 9) — returns nothing. The group has been described here dozens of times and never once constructed. Constructing it:

|Gamma(T)|           = 96                       <- literature value, CONFIRMED
IdGroup              = [96, 227] = (C2^4 : C3) : C2 = 2^4 : S3
transitive on 12     = true
Gamma(T)'            = [48, 50] = 2^4 : C3      <- BT781/BT783, CONFIRMED
  centre order       = 1                        <- BT783, CONFIRMED
  abelianisation     = C3                       <- BT783, CONFIRMED
  index-2 subgroup   = none                     <- BT783, CONFIRMED

BT781 and BT783 are correct. Their structural claims, which the corpus has been propagating on trust for dozens of files, hold against the primary source.

One thing the corpus has been saying loosely

The most-cited fact about the tomotope here is that it "fails the intersection condition, and is therefore not an abstract polytope". Tested directly on the published generators:

intersection condition  <rho_I> ∩ <rho_J> = <rho_{I∩J}>
tested 256 pairs (I,J);  FAILURES = 0       -> the condition HOLDS for Aut(T)

The failure in the literature is a property of the monodromy (connection) group, not of Aut(T). Aut(T) satisfies the intersection condition. What actually fails here, and fails first, is the string condition:

(rho0 rho2)^2 = 1 ?  true
(rho0 rho3)^2 = 1 ?  true
(rho1 rho3)^2 = 1 ?  FALSE

so ⟨ρ₀…ρ₃⟩ is not a string group at all in this labelling, and the intersection condition is not even the operative obstruction at the automorphism level. Files that attribute the IC failure to Γ(T) should say Mon(T).

Pass 1377 — A₄ is the derived core, and the 540 frames are not polytope facets

The order-96 comparison nobody had run

Γ(T) has order 96. So does the 540-frame stabiliser in PGSp(4,3). BT781 compared their order-48 halves and got a negative. The 96s had never been compared:

Gamma(T)                            = [96, 227] = 2^4 : S3
frame stabiliser in PGSp(4,3)       = [96, 226] = C2 x C2 x S4
isomorphic?                           FALSE

Adjacent SmallGroup IDs, different groups. The negative now holds at both levels.

What both sides actually share

frame stabiliser in PSp(4,3)   = [48,48] = C2 x S4 = O_h,  derived subgroup = A4
frame stabiliser in PGSp(4,3)  = [96,226],                 derived subgroup = A4
Gamma(T)' = 2^4:C3 = [48,50]                            contains 2^2:C3 = A4
largest common subgroup of O_h and Gamma(T)'  (Pass 1127)          = A4

A₄ is not an artefact of the comparison — it is the derived subgroup of the frame stabiliser itself, at both levels of the group. BT781/BT782 looked for the "exchange rate" between the two order-48 spendings of 48 and framed it as a quotient. It is not a quotient (Pass 1127: no nontrivial common quotient exists at all); it is a shared derived core, approached from below by both sides.

A₄ = 2²:C₃ is the rotation group of the tetrahedron, and the tomotope's cells are four tetrahedra and four hemioctahedra. Whether that is the reason is not claimed here — it is the next experiment, stated as a question.

The 540 frames are not the facets of a rank-4 polytope

O_h = Aut(cube) is a string C-group of type {4,3}, so it is a legitimate facet-group candidate: if it extends to a rank-4 string C-group on all of PSp(4,3), the 540 frames are the facets of an abstract regular 4-polytope with 25920 flags. Searched exhaustively over the extensions:

string C-group {4,3} generating triples in O_h            : 48
candidate rho3 (involutions centralising rho0, rho1)      : 3
rank-4 string C-group extending the frame cube            : NONE

A clean obstruction. (That some rank-4 regular polytope exists for U4(2) is published — Leemans & Vauthier, An atlas of abstract regular polytopes for small groups, 2006 — and is not claimed here. The question answered is whether the frame stabiliser is one of the facet groups. It is not.)

Pass 1378 — the guard was blind to group notation, and that is measurable

Pass 1127 rediscovered BT783. The interesting part is not the mistake but its cause, which is mechanical and was measured, not guessed.

scripts/check_stale_boundaries.py did find BT781's boundary section. It then extracted from it:

BT781 boundary tokens: []          <- zero

The entire grammar — code parameters [[n,k,d]], slash-sequences, noun@number — is blind to

Aut(Q3)=2^3:S3  -->  Gamma(T)'=2^4:C3

which is the single most common way a result is stated in this corpus. The threshold was never the problem; there was nothing to threshold.

group_tokens() in scripts/check_rediscovery.py fixes it. The normalisation matters as much as the matching, because this repo writes one group five ways — 2^3:S3, C2^3 : S3, (C2 x C2 x C2):S3, SmallGroup[48,48], C2 x S4 — so cyclic factor lists are collapsed to powers and the C/Z prefix dropped:

BT781 boundary  -> {grp:2^3:S3, grp:2^4:3, grp:Q3, grp:S3}
  vs BT782      :  4 shared tokens
  vs BT783      :  2 shared tokens

Both now clear the ≥2 threshold. The gated self-test strengthens rather than weakens: BT810 vs BT811 goes from 2 shared tokens to 5.

Confirmed from the parallel tracks

  • The shifted-adjacency erratum is right. Rebuilt independently from the 40 projective points: spec(A) = 12¹ ⊕ 2²⁴ ⊕ (−4)¹⁵, so spec(D) = 11¹ ⊕ 1²⁴ ⊕ (−5)¹⁵, Tr D = −40, Tr D² = 520, Tr D³ = −520, and the historical cubic evaluates to 1296, −64, 80 on the three true eigenvalues with rank p_old(D) = 40 — it annihilates nothing. Independently reproduced.
  • The 2240 decomposition is right. Its trivial-multiplicity 14 equals the orbit count, and its multiplicity-square sum is 1193, matching the rank computed in Pass 1124 by a different route.
  • 120 = 40 lines × 3 perfect matchings is arithmetically sound: the line stabiliser in PGSp(4,3) has order 51840/40 = 1296, inducing S₄ on the four points, and a matching stabiliser D₈ of index 3 gives 1296/3 = 432 = 51840/120.

Prior art

  • BT781 — the 48-split; its boundary now points forward.
  • BT782, BT783 — own the bridge refutation.
  • Pass 1020 — Sp(4,3) ≇ W(E6), the distinction the retracted correction lost.
  • Pass 1124, Pass 1126 — the 14 orbits and the 3×81.
  • Monson, Pellicer & Williams, The Tomotope, Ars Math. Contemp. 5 (2012) — the generators.
  • Leemans & Vauthier, An atlas of abstract regular polytopes for small groups (2006) — rank-4 polytopes for U4(2).