W(3,3): the executable exceptional-geometry atlas

One finite geometry. Thousands of exact artifacts. Named maps. Reproducible certificates. Public corrections.

Start with the symplectic space F_3^4. Its totally isotropic points and lines form W(3,3): 40 points, 40 lines, 240 incident point-pairs, and the collinearity graph SRG(40,12,2,4). This repository is the executable atlas grown from that object: exact homology, integral lattices, modular representations, error-correcting codes, Schläfli/E₆ carriers, Hecke algebras, cycle and selector geometry, and finite transport systems.

This is not a pile of numerology organized by pass number. Its strongest line is an object-level bridge: three 432-state carriers are explicitly identified with directed Schläfli edges, mapped equivariantly into an 81-dimensional constituent, resolved integrally by exact Smith forms, and followed through their bad-characteristic extensions and Hecke corners. The corpus also keeps the failed versions, so a correction has an executable owner instead of being silently overwritten.

What this is, stated positively

One finite object, pushed as far as exact computation goes. The mathematics below is not conditional on anything:

• Named theorems with machine-checkable witnesses — the two-branch and k-branch gluing laws, the coalescence theorem, pencil rigidity, the all-m trace-valuation theorem at q=3, and one Smith-form theorem that unified two agents' independently built towers.
• A complete modular picture of the literal 26-dimensional Hecke algebra — decomposition and Cartan matrices at p = 2,3,5, projective indecomposable dimensions, Loewy and radical series, and primitive idempotent systems lifted through p⁶. The ambient group block has the cyclic-defect Brauer tree 1−24−81−64−6; it is related to, but not identical with, the Hecke radical.
• A separate, explicit selector orbital algebra — the 120-selector action has 83 orbitals and rational Wedderburn algebra Q⁷ ⊕ M₂(Q)² ⊕ M₃(Q)³ ⊕ M₄(Q) ⊕ M₅(Q), realized by 83 exact matrix units.
• Exact integral arithmetic of every eigenlattice — Smith forms, discriminant identities, prime-by-prime gluing, and a rigidity theorem showing the gluing support is an invariant of the ring Z[S], not of the matrix.
• Canonical named maps, not matching integers — every one of the 540 frames carries a unique A₄-equivariant cross-matching, and the 540 of them cover the 240 edges exactly 9-to-1.
• An all-odd-q strongly regular family — regular symplectic spreads form an exact two-intersection scheme with closed parameters and eigenvalues; the q=27 Ree–Tits spread supplies a complete seven-weight, exactly 9-divisible [730,5]₂₇ code, and four named q=27 families have distinct complete spectra.
• Three controller objects, finally separated — the abstract controller has order 48 and minimal faithful rational degree 4, its canonical single-J image has order 24, and the overlapping rank-three carrier is the infinite arithmetic group SL₃(Z) with no rational common inverter.
• Every quadratic intertwiner, then its symmetry type — all 50 quadratic Hom maps from the signed-edge 90 are explicit and surjective. Their phase/outer action is exactly 16·1 ⊕ 16·sgn ⊕ 9·std for S₃, which explains both the balanced 25+25 outer split and the 32+18 phase split.
• A correction ledger with executable owners. Refuted claims keep their failure certificates instead of being silently overwritten, and several were found by the authors auditing themselves.

Where the boundary falls. The finite mathematics is exact. The physical readings — which combinatorial object is a generation, a coupling, an optical mode — are CONDITIONAL, because identifying a combinatorial object with a physical one is a map that must be built, not inferred from a matching integer. Two of fourteen published constant formulas survive σ-testing, and the table showing which twelve fail is in this README rather than in a drawer. That is the standard the whole corpus is held to, and it is the reason to trust the rest.

What is already in hand

Result family	Best current result	Canonical owner
Geometry, topology, and code	The canonical W(3,3) model, H₁ ≅ Z^81, and the ternary [[240,81,3]]₃ sector	master paper · Passes 373–374
Integral spectral arithmetic	Exact adjacency and signed-turn Smith forms, prime-by-prime gluing, ramified kernel growth, and the coalescence theorem	integral frontier
Exceptional carrier bridge	432 → 81 → 216, one-colour Smith profile 1^15,2^6,4^8,8^29,40^23, colour index 3^81	Pass 1147
Modular representation closure	The nonsplit 58|23 frame extension, one-dimensional directed Ext¹ spaces, exact H₂₆ radicals, Cartan matrices, PIM dimensions, and lifts through p^6	Pass 1335 · Passes 1340–1344
Global selector geometry	The length-4 simple-cycle orbit of size 120 is globally minimal over all lengths 3…40; adding a primitive copy idempotent gives the global orbit minimum 360	GAP witness
Selector orbital algebra	The 120-selector action has 83 orbitals, a 79-dimensional Terwilliger algebra, and 83 explicit rational Wedderburn matrix units	Passes 1355–1384
Steinberg carrier, named	The three 432-orbits carrying the 3×81 are conjugate, stabiliser S₅; the later refinement gives S₅ ∩ PSp(4,3) = A₅	Pass 1134 owner · Pass 1375 refinement
Frame cross-matching	Every frame has a unique collinearity transversal; independently it is the unique A₄-equivariant matching, and all 540 cover the 240 edges 9-to-1	Pass 1355 owner · Pass 1390 refinement
Exact cover frontier	Two disjoint 100,000-cover searches hit the same 327 complete PSp(4,3) orbits, containing 3,547,800 covers in total; this is a certified lower bound, not a global completeness claim	Pass 1510 audit
q=5/Reye action closure	The selected q=5 cover’s moving twelve is explicitly T12_165; the 312-row 2-(13,6,60) multidesign reconstructs Reye as its 16 zero-containment triples, with line action T16_1034; exact character pairings separate carrier parity, the outer point/heavy involution, and the point-side sign twist	Passes 5667–5674 · GAP witness
Regular-spread family	For every odd prime power, the q+1 intersection relation is an explicit SRG with eigenvalues q(q−2),−q; a q=27 Ree–Tits slice already has five nonregular intersection sizes	Passes 2200–2206
Complete q=27 spread spectra and codes	Ree–Tits has complete spectrum 1⁷³⁰,10⁴⁵⁶³,19⁹⁶¹⁷⁴,28⁴⁰⁸²⁹⁴,37³⁶⁵⁰⁴,46⁴⁹¹⁴,55⁷⁰² and an exactly 9-divisible [730,5]₂₇ code; regular/Kantor/Thas–Payne/Ree–Tits spectra are pairwise distinct	Pass 2300 · Pass 2304
Controller representations	Abstract (C₄×C₆):C₂ has order 48 and minimal faithful rational degree 4; the single-J image has order 24; the overlapping 3D carrier is SL₃(Z) and has no common inverter	Pass 2306
Complete quadratic map module	Full PSp(4,3) Hom dimensions are Sym=(3,6,5,12), Λ=(3,4,5,12) on targets (15,24,30,81); the combined S₃ module is 16·1⊕16·sgn⊕9·std	Pass 2301 bases · Pass 2307 character theorem
Canonical Weil outer action	At q=7,11, complex conjugation realizes the nonsquare outer similitude on both parity constituents and reverses the realified complex structure, giving exact D₄ relations	Pass 2302
Chamber Hecke and chiral carrier	The two 160-chamber panels generate the 8D type-C₂ Hecke image; Ω has a literal, uniformly isoclinic point/line 24+24 carrier with squared coupling 3/8	Passes 4324–4334
Chamber logic and finite control plane	The rank-48 packet is literally M₂(Q) on a two-state multiplicity coordinate repeated over 24 lanes; the separate ten-dimensional ternary residual gives F₃¹⁰ ⋊ PGSp(4,3) with 17 exact offset classes	Passes 4936–4937
Steiner three-cover and group firewall	The 120 Steiner triangles form a canonical 4-class, 40×3 refinement of the Q(4,3) line-side carrier; the marked-residue S₆×C₂ and duad–syntheme Aut(S₆) both have order 1440 but are nonisomorphic	Passes 4870, 4873–4874
Executable recursive runtime	HoloBox gives addressed mailbox/run, immutable path-copy checkpoints, one leaf/network loader, 4,201,025,641 level-six stateful VMs represented by seven uniform node blobs, and independent Python/GAP certificates	runtime guide · evidence card

Those are the compact front doors. The larger certified backbone below gives exact statements, tiers, and owning artifacts without forcing a reader to guess which of several historical versions is strongest.

Choose your route

Reader	Start here	Then go deeper
General reader	Live atlas · W33 for Everyone	Practical implications
Mathematician / researcher	Master paper · source	Result index · canonical vocabulary
Reproducer / reviewer	Reproduction commands	certificates · tests · correction ledger
Lattice / deformation researcher	Determinant-law paper	eigenlattice table
Photonic / systems reader	Photonic Holonet · source	HOLONET.md; treat implementation claims as conditional
Runtime / distributed-systems builder	HoloBox evidence card · CLI	finite controller card · runtime guide · focused regression
Assessing whether to fund this	Machine blueprint, Part I	then What is not built and the errata index, both at the end of that document

The corpus is too large to navigate by filenames. Search the result itself in RESULTS_INDEX.md before re-deriving it.

The three manuscripts

All three share one reader convention: cream boxes are plain language, blue boxes carry exact statements with scope, and rose boxes retain claims this project published and then withdrew together with the measurement that overturned them. The machine blueprint has full plain-language coverage; the two research atlases currently provide a reader guide and selected plain-language entries, not a cream-box paraphrase of every section.

Document	Pages	What it is
holonet_machine_blueprint.tex	205	A computer specified by the geometry — instruction set, gate counts, thermodynamics. Six parts, each opening in plain language.
w33_paper.tex	477	The research atlas: everything established about W(3,3), evidence-tiered.
photonic_holonet.tex	347	One self-entangled photon as computer, network and clock.

All three build with zero errors and zero undefined references, enforced in CI (.github/workflows/manuscripts-compile.yml) — an undefined reference is only a warning in LaTeX, so three of them once shipped through hundreds of clean builds.

The finite ISA design space, in one table

Two independent asymmetries in the instruction set mean four possible machines, not one. Reported together because no two are the same design and the prices do not substitute:

machine	opcodes	p/f swap	mixing	ρ(B)	localisation peak	entropy production
A — biased, irreversible (shipped)	4	no	15	5.7469	0.6129	infinite
B — symmetric, irreversible	6	yes	12	8.7621	0.4604	infinite
C — biased, reversible	8	no	16	5.7469	0.6129	0
D — symmetric, reversible	12	yes	13	8.7621	0.4604	0

C shares A's spectrum exactly: closing an instruction set under inverses adds no new undirected edges, so it changes directed thermodynamic bookkeeping without changing that simple graph. Exact conjugation by the p/f pair swap proves B and D are symmetric and A and C are not. Machine D therefore removes both named asymmetries in the finite model. Its 0.4604 peak is symmetric within-pair localisation, not residual p/f bias. These rows are analytic opcode/graph measurements; only A/C have earlier generic-cell synthesis, so B/D hardware pricing remains open.

Latest exact closure: the chamber Hecke machine

The 160 W(3,3) chambers carry two native three-way switches: change the line at a fixed point (P) or the point on a fixed line (L). GAP proves

[ P^2=2P+3I,\qquad L^2=2L+3I,\qquad PLPL=LPLP, ]

and the generated algebra has dimension 8: the full q=3 type-C2 Iwahori–Hecke image. In the 320-state oriented Levi basis,

[ B_{\rm Levi}=\begin{pmatrix}0&L\P&0\end{pmatrix}, \qquad B_{\rm Levi}^2=\operatorname{diag}(LP,PL). ]

The chirality Ω=LP−PL has rank 48 and exact projector Π₄₈=−Ω²/60; on that packet, Ω/√60 is a complex structure. The old folded cubic now has the exact normal form

[ F=-68\Pi_{48}-31X-\frac{21}{2}\Omega+\frac23X\Omega, \qquad (F+68\Pi_{48})^2=-689\Pi_{48}. ]

Pass 4334 makes the 24+24 count literal. Lift the eigenvalue-2 projectors of the W33 point graph and dual line graph to chamber projectors Qₚ,Qℓ. Their rank-24 images meet only in zero and span im Π₄₈; moreover

[ Q_pQ_\ell Q_p=\frac38Q_p,\qquad Q_\ell Q_pQ_\ell=\frac38Q_\ell. ]

All 24 principal angles therefore have cosine √6/4, and for Q=Qₚ+Qℓ the orthogonal span projector is Π₄₈=(8/5)(2Q−Q²). The conjugate packet is exactly the joined point and line eigencarriers, not merely a matching dimension count.

Pass 4936 now splits that same four-dimensional packet algebra into literal rational matrix units eᵢⱼeₖℓ=δⱼₖeᵢℓ. Consequently it is exactly M₂(Q) on a two-state multiplicity coordinate repeated over 24 representation lanes. With Z=e₁₁−e₂₂ and S=e₁₂+e₂₁,

[ Z^2=S^2=\Pi_{48},\qquad ZS=-SZ,\qquad (SZ)^2=-\Pi_{48}. ]

This is an exact algebraic logic switch, not 24 physical qubits or a synthesized unitary gate. The three chart-dependent HoloBox HP selectors sum to the intrinsic point panel, and the three HL selectors sum to the line panel. Only those complete family aggregates compress to the two packet reflections. No individual-selector packet intertwiner is asserted.

Reproduce it with the GAP witness and focused regression, plus the Pass-4334 carrier witness and regression, then the Pass-4936 matrix-unit witness and regression. The operators and family checksum are exact finite relations; a deterministic three-way selector and synthesized chamber datapath are not yet built.

Exact finite control-plane candidate: the missing ten

Pass 4864 identifies the ten-dimensional ternary quotient outside the oriented K₃,₃ span with sp₄(F₃). Pass 4861 independently proves that full three-port matching removes the local S₃⁴⁵ sheet gauge and leaves one global PGSp(4,3) frame. Pass 4937 composes those certified objects into the affine update

[ v\longmapsto vA_g+w,\qquad \mathbb F_3^{10}\rtimes PGSp(4,3),\qquad |\mathbb F_3^{10}\rtimes PGSp(4,3)|=3{,}061{,}100{,}160. ]

The 59,049 offsets fall into exactly 17 PGSp(4,3) orbits, so the result already supplies a complete finite state taxonomy. It does not supply a canonical-representative compiler or a HoloBox opcode. The equal-order group Sp₄(F₃[ε]/ε²) is not this controller: their center orders are 2 and 1 respectively, and the square-zero tangent kernel is abelian even though its transported Lie algebra is not. See the public controller card, exact GAP witness, and byte-exact regression.

Cross-q action closure: Reye is intrinsic to the q=5 design

The previously pending q=5 action gate has now executed. The selected 13-cover has one fixed vertex and a moving twelve on which the stabilizer is exactly T12_165, with an explicit conjugator to the independently built Klein-Latin action:

[1,9,4,8,12,7,10,2,5,3,6,11]

That closes the object-level chain from the q=5 cover to the twelve antipodal short-root pairs of F₄. More importantly, the q=5 object explains the Reye configuration internally. The 312 outside vertices give a 2-(13,6,60) multidesign whose triple-containment spectrum is exactly 0^16,16^30,24^240; the sixteen zero triples avoid the fixed point and form 12₄16₃. Their Levi automorphism group has order 576, acting as T12_165 on points and T16_1034 on lines.

The same abstract group is odd on the twelve-carrier but wholly even on the sixteen-carrier. Its point-side sign kernel is SmallGroup(288,1025) ≅ 2⁴:(C₃×S₃), literally transported to the even-Latin group already linked by Pass 5300 to the Hoffman stabilizer quotient. And on the natural twelve-symbol carrier, every one of the 23,760 placements of the natural degree-seven PSL(2,7) (plus five fixed symbols) generates full S₁₂ together with T12_165—not a hypothetical order-4032 intermediate.

These are finite permutation-group and multidesign theorems. The PSL(2,7) statement is scoped to the natural 7+1⁵ action, and the result does not imply continuum dynamics or a physical unification. The point and heavy-support twelves are exchanged by an explicit outer involution of the source group: a displayed order-eight lift squares to an inner automorphism, and exactly 48 inner re-gaugings give order-two outer representatives. That involution is not the point-side sign twist. The point, heavy, and line permutation characters have self inner products 3 and pairwise inner products 2, whereas the sign-twisted point character has inner product 0 with both heavy and line. Thus sign tensoring is neither the source outer involution nor the even line carrier. See the synthesis, 56-check certificate, and byte-exact regression.

HoloBox: a recursive network that executes as one VM

analysis/holobox.py now turns the 40-ary Holonet law into an executable, immutable runtime object. A leaf VM and a network of 40 child VMs use the same state media type and loader. More importantly, the identity is operational: address a nested guest, route it a mailbox value, execute it, and checkpoint the result by replacing only the digests on that one path.

At six levels, a uniform HoloBox denotes 105,025,641 internal network VMs plus 4,096,000,000 addressable leaf VMs: 4,201,025,641 stateful VMs represented by only seven unique node blobs. In the frozen fresh transition, a depth-six delivery creates seven path-state blobs plus one receipt; recipient execution creates seven path-state blobs. These are upper bounds for arbitrary writes, because a content-identical replay may allocate no new CAS key. Untouched sibling digests remain byte-identical. Recursive routes use no stored next-hop table and take at most two W33 line transactions per radix-40 address digit. This 2n logical metric is distinct from BT827's 8n chart-aware lowering (3 cube + 5 chart-web moves per digit). The frozen Python reference witness is 19/19 PASS, with an independent 7/7 GAP route certificate.

python3 analysis/holobox.py build --output /tmp/holobox --levels 6 \
  --program RECV,HALT
python3 analysis/holobox.py send /tmp/holobox --source 0/0/0/0/0/0 \
  --target 3/10/17/24/31/38 --message 13 --output /tmp/holobox-message
python3 analysis/holobox.py run /tmp/holobox-message --address 3/10/17/24/31/38 \
  --commit /tmp/holobox-run
python3 analysis/holobox.py verify /tmp/holobox-run

BT339 owns the earlier 2n hierarchy assertion, BT350 the nested-VM framing, Passes 2642--2644 the same-port recursive hardware module, and the older Witting architecture the CID-container, WASM/OCI, policy, receipt, and Projection Engine design. HoloBox implements the previously missing nested lifecycle and content graph; it does not claim those earlier ideas. Its bundle is OCI-shaped, not yet OCI-conformant, and the Python model is not Linux/KVM isolation, a guest kernel, confidential-computing attestation, or a performance result. See the runtime guide, 19-check certificate, and GAP route witness.

Current correction ledger

• Pass 4253's Z₍₂₇₃₁₎ lift has an explicit zero-voltage eight-cycle, hence girth exactly 8, not at least 16. The valid current high-girth construction is the much larger Pass 4261 Z₍₇₅₀₀₁₉₎ lift.
• The irregular Kotani–Sunada non-real-pole annulus has square roots. The former no-square-root “band filling”/“closest irregular Ramanujan” score is withdrawn.
• Affine translations act on 81 frames but descend to neither projective 40-carrier; they do not force a point-side projective load port.
• Pass 4331's linear-opcode mismatch incidence census detects 1656/1920 = 69/80 differential rail substitutions and 0/960 shared-control substitutions. It is narrower than the Pass 4367/4374 intrinsic flag-register comparator, which detects 36/39 = 12/13 arbitrary one-register substitutions at q=3. The former 95.71% result used a golden run.
• The complete universal-set census is 360, and the shipped ISA is tied 7th–12th by the reported ρ value, not strictly 12th.

The exact correction witness is Passes 4328–4333; the readable evidence map is here.

Evidence tiers

Tier	What it means
PROVED	A mathematical proof or named formal theorem. If Lean-owned, build the specific module; do not infer a green library from a file's existence.
CERTIFIED	Exact computation with a deterministic witness, certificate, and focused test.
CONDITIONAL	The finite mathematics is sound; an interpretation or implementation map is still missing.
OPEN	A precise question with no completed witness.
RETRACTED	Previously promoted, then refuted; retained with the failure certificate.

Canonical objects: names that must not be conflated

Name	Canonical meaning
Γ = W33	The graph obtained from symplectic orthogonality on PG(3,3), not an arbitrary SRG(40,12,2,4); 28 graphs share the parameters.
G₀	PSp(4,3), order 25,920, the inner projective symmetry.
G = Aut(Γ)	PGSp(4,3) ≅ W(E6), order 51,840. The same-order group Sp(4,3) is a central double cover, not this faithful projective action.
H₁(Γ)	Z^81, the first homology of the clique complex.
Y₄₈₀	The 480 directed edges of W33, carrying the signed-turn operator K.
X₄₃₂	W(E6)/S5, equivalently the directed edges of the Schläfli graph.
81₋	The Pass-1147 constituent in Λ²(Aug(Q^27)); it is not silently identified with H₁(Γ).
H₂₆	End_G(X₄₃₂), the literal 26-dimensional coset Hecke algebra.
Selector orbital algebra	End_H(Q^120), dimension 83; its 83 rational matrix units do not belong to H₂₆.
Γctrl	The abstract independent-clock group (C₄×C₆):C₂, order 48, requiring two complex phase registers for faithfulness over Q.
ΓJ	The canonical single-J quotient C₁₂:C₂, order 24; kernel ⟨(2,3,0)⟩.
Arithmetic phase carrier	The overlapping three-coordinate action ⟨R₄,U₆⟩=SL₃(Z); infinite and not a smaller representation of Γctrl.

For aliases, superseded names, and pass ownership, use RESULTS_VOCABULARY.md, data/ALIAS_REGISTRY.json, and data/w33_pass_namespace_registry_v2.json.

Certified finite backbone

flowchart TD
    V["(F₃⁴, alternating form)"] --> W["W(3,3): 40 points, 40 lines, 240 edges"]
    W --> C["clique complex: H₁ ≅ Z⁸¹"]
    W --> SP["spec(A)=12¹,2²⁴,(−4)¹⁵<br/>spec(A−I)=11¹,1²⁴,(−5)¹⁵"]
    W --> A["adjacency and signed-turn lattices"]
    A --> L["Smith forms and prime-by-prime gluing"]
    L --> CO["coalescence: the p-part is carried<br/>by eigenvalues colliding mod p"]
    W --> Y["Y₄₈₀ directed-edge carrier"]
    W --> F["540 frames = disjoint line pairs<br/>stab in PSp: C₂×S₄ (order 48)<br/>stab in PGSp: C₂²×S₄ (order 96)"]
    F --> FA["derived subgroup A₄ acts faithfully<br/>on each line's 4 points"]
    FA --> FM["canonical 4-edge cross-matching<br/>540 frames → 240 edges, 9-to-1"]
    S["Schläfli graph on 27 lines"] --> X["X₄₃₂ = W(E₆)/S₅"]
    X --> T["rank-81 odd transform; 216 tight-frame lines"]
    T --> I["integral bad primes {2,5}"]
    X --> ST["3×81 Steinberg carrier:<br/>three conjugate 432-orbits, stabiliser S₅"]
    ST --> STP["S₅ ⊄ PSp(4,3); S₅ ∩ PSp(4,3) = A₅"]
    X --> H["H₂₆ = End_G(X₄₃₂), three-carrier triality"]
    H --> J["234 → 78 → 52; Hecke bad primes {2,3,5}"]
    J --> R["modular radicals: 21→17→13→7→2→0 at p=2"]
    X --> B["ambient p=5 group block:<br/>Brauer tree 1−24−81−64−6<br/>Ext¹(23,58)=Ext¹(58,23)=1"]
    R --> P["Cartan/PIM at p=2,3,5; idempotents through p⁶"]
    H --> Q["global cycle orbit 120 = 40 lines × 3 matchings<br/>cycle + copy orbit 360"]
    Q --> SA["120-selector orbital algebra, dimension 83"]
    SA --> MU["83 rational matrix units:<br/>Q⁷⊕M₂(Q)²⊕M₃(Q)³⊕M₄(Q)⊕M₅(Q)"]
    Q --> QO["no maximal subgroup holding a 432-selector<br/>stabiliser contains the S₅"]
    W --> RS["regular spreads for every odd q:<br/>closed SRG parameters"]
    RS --> RT["q=27 Ree–Tits control:<br/>five nonregular intersection sizes"]
    CTRL["abstract controller, order 48<br/>minimal faithful Q-degree 4"] --> CJ["single-J image, order 24"]
    CTRL --> AR["overlap phase planes in rank 3"]
    AR --> SL["SL₃(Z); no rational common inverter<br/>R₄²U₆ has spectral radius φ"]

Loading

Mathematical object	Strongest current result	Tier	Canonical entry
Symplectic quadrangle	SRG(40,12,2,4), spectrum 12^1,2^24,(−4)^15, Aut ≅ W(E6)	PROVED	master paper
Clique complex	H₁ ≅ Z^81; qutrit CSS sector [[240,81,3]]₃ with (d_X,d_Z)=(3,4)	CERTIFIED	Passes 373–374
Integral adjacency	SNF(A)=diag(1^16,2^8,8^15,24); saturated gluing (Z/2)^6⊕(Z/6)^9⊕Z/120	CERTIFIED	pass827
Ramified gluing	Kernel growth 40,80,119,158,182 reconstructs Z/8⊕(Z/2)^15 at p=2	CERTIFIED	Pass 1002 release
Signed directed edges	spec(K)=(−6)^81,2^120,4^24,10^15; exact four-branch gluing	CERTIFIED	pass826
Schläfli/E₆ carrier	X₄₃₂ maps with rank 81 to 216 antipodal tight-frame lines; three colours give rank 243, and adjoining the disjoint rank-45 cubic block gives rank 288 with residual 1952	CERTIFIED	Pass 1147
Integral Schläfli frame	Smith profile 1^15,2^6,4^8,8^29,40^23; internal bad primes {2,5}; colour split index 3^81	CERTIFIED	Pass 1147
Saturated frame mod 5	Nonsplit 0→I₅₈→S₅→K(W33)₍₅₎⊗sgn→0; Ext¹(23,58)=Ext¹(58,23)=1 over both groups, so this is the unique nonzero extension type up to endpoint rescaling	CERTIFIED	Pass 1147 · Pass 1335
Three-carrier Hecke/triality	Commutants 234 → 78 → 52; six-channel SNF 1,1,1,12,12,24; Hecke bad primes {2,3,5}; invariant cycles do not select a copy	CERTIFIED	Passes 1325–1329
Modular H₂₆	Radical powers at p=2,3,5 are 21,17,13,7,2,0; 22,16,10,4,0; 6,2,0; the exceptional p=5 scalar Ext quiver is doubled A₃, a condensation shadow of the same cyclic-defect block	CERTIFIED	Passes 1330–1334 · Pass 1335
Rational degree-20 model	Exact 20×20 rational standard generators satisfy C²=D⁹=(CD)¹⁰=I; GAP affords faithful images of order 51,840 and uniquely matches CTblLib row 11. The reported literal-480 derivation remains provenance, not rebuilt here.	CERTIFIED	Pass 1341 analysis
Binary quadratic-residue code	Corrected code [[137,1,21]]; exact affine/real-Clifford towers and explicit parity boundaries	CERTIFIED	Passes 358–367
Section trace tower	For every m≥2, min_c v_λ(tr(D_c^m)) = 2(m+[m odd]) at q=3	CERTIFIED	Pass 541
H₂₆ Cartan/PIM and p-adic refinement	C₂=diag(1,22), C₃, C₅=I₆⊕[[2,1,1],[1,1,0],[1,0,2]]; PIM dimensions (2,22), (9,6,10,1), (3,2,1,1,1,1,4,2,3); primitive systems verified through p⁶; Smith and Loewy filtrations differ	CERTIFIED	Passes 1340–1344
Global cycle/copy selector bound	Exact GAP path-stabilizer proof: global simple-cycle orbit minimum 120 at length 4; a primitive copy idempotent gives 360; cycles alone act as C⊗I₃	CERTIFIED	GAP witness
Shifted adjacency	spec(A−I) = 11¹,1²⁴,(−5)¹⁵, m_D(t)=(t−11)(t−1)(t+5); the historical cubic (t+1)[(t+1)²−36] annihilates no eigenspace (rank p_old(D)=40)	CERTIFIED	erratum
Steinberg carrier stabiliser	Three conjugate 432-orbits, stabiliser S₅ = SmallGroup[120,34]; the later refinement gives S₅ ∩ PSp(4,3) = A₅ and the maximal-subgroup obstruction	CERTIFIED	Pass 1134 owner · Pass 1375 refinement
Tomotope, from its own paper	Γ(T)=[96,227]=2⁴:S₃, Γ(T)′=[48,50]=2⁴:C₃, built from the published generators. Aut(T) satisfies the intersection condition — Mon(T) is what fails	CERTIFIED	Pass 1376
Frame cross-matching	Pass 1355 owns the unique collinearity transversal; Pass 1390 independently characterizes it as the unique A₄-equivariant bijection and proves uniform 9-to-1 coverage	CERTIFIED	Pass 1355 owner · Pass 1390 refinement
Frames are not polytope facets	O_h is a string C-group {4,3}, but no rank-4 string C-group extends it in PSp(4,3)	CERTIFIED	Pass 1377
Exact-cover orbit frontier	Two disjoint deterministic prefixes independently hit the same 327 complete PSp(4,3) orbits, whose sizes sum to 3,547,800; global completeness remains open	CERTIFIED	Pass 1510 audit
All-odd-q regular-spread graph	v=q²(q²−1)/2, k=q(q−2)(q²+1)/2, λ=q(q³−4q²+7q−8)/2, μ=q(q−2)(q−1)²/2; nontrivial eigenvalues q(q−2),−q	PROVED / CERTIFIED	Passes 2200–2206
Controller representation trichotomy	Finite abstract order 48 / canonical order 24 / infinite SL₃(Z) are distinct; minimal faithful rational degree 4 and common-inverter nullity 0	CERTIFIED	Pass 2306
Complete q=27 named-family codes	All four standard coordinate families have hyperplane sections 1 mod 9; the regular code is 27-divisible [730,4]₂₇, while three nonregular codes are exactly 9-divisible [730,5]₂₇	CERTIFIED	Pass 2304
Complete quadratic Hom bases	Every nonzero basis map is target-surjective; full dimensions total 26 symmetric and 24 alternating, with outer-even and outer-odd halves both dimension 25	CERTIFIED	Pass 2301
Quadratic Hom S₃ character	Sym=13·1⊕3·sgn⊕5·std, Λ=3·1⊕13·sgn⊕4·std, combined 16·1⊕16·sgn⊕9·std; explains 25+25 and 32+18	CERTIFIED	Pass 2307

The flagship bridge, in one paragraph

Each of the three 432-state A₂ colours is the directed-edge set of SRG(27,16,10,8). GAP constructs the explicit odd transform into 81₋; its 432 vectors form 216 antipodal lines with G²=3200G and angles 0,1/15,1/5. One colour has Smith profile 1^15,2^6,4^8,8^29,40^23; the three-colour Fourier split adds index 3^81. Modulo 5, the saturated 81-space is not 58⊕23: it is the nonsplit length-two module 0→I₅₈→S₅→K(W33)₍₅₎⊗sgn→0, with a unique proper nonzero submodule. Pass 1335 identifies the cyclic-defect Brauer tree and proves both directed cross-Ext¹ spaces have dimension one, so this nonsplit module exhausts the previously open extension class. The Hecke radical records a condensed doubled-A₃ shadow; it is not the module itself. The next algebra layer is explicit as well: Pass 1340 computes the Cartan and projective-indecomposable data at 2,3,5, while Pass 1343 lifts complete primitive systems through p^6 and proves that the Smith and Loewy filtrations are genuinely different. Separately, GAP proves that the 120-element length-4 cycle orbit is globally minimal, so selecting one of the three species-20 copies costs a minimum orbit 360; this quantifies a gauge choice without pretending the choice is canonical. This is an exact theorem about named W(E6) modules and integral lattices. It does not identify generations, Yukawa couplings, particles, or optical modes.

Find the canonical result, not the newest filename

1. Search a formula, integer sequence, or code parameter in RESULTS_INDEX.md.
2. Resolve aliases and retractions in RESULTS_VOCABULARY.md and data/ALIAS_REGISTRY.json.
3. Open the owning synthesis, then the executable witness and JSON certificate.
4. Run the focused test. A later pass that repeats the number is not a new owner.

Every promoted bridge should name its source object, target object, and map. Matching dimensions or group orders are evidence to investigate, not maps.

---

The derivations, symbolically

The finite-geometry tables immediately below are derived from (q, k, λ, μ) = (3, 12, 2, 4) and the graph itself. Later sections explicitly name any additional representation-theory, coding-theory, experimental, or interpretive input. Status is honest: PROVED = machine-checked or proved in the paper; CERTIFIED = exact computation with an idempotent JSON certificate; OPEN = stated, not settled; RETRACTED = we published it, then killed it.

Geometry and spectrum

Quantity	Symbolic derivation	Value	Status	Witness
Eigenvalues r, s	r,s = ½[(λ−μ) ± √((λ−μ)² + 4(k−μ))]	2, −4	PROVED	SRG theory
Spectral gap r−s	√((λ−μ)² + 4(k−μ)) = √36	6	PROVED	”
Multiplicities f, g	f,g = ½[(n−1) ∓ (2k+(n−1)(λ−μ))/(r−s)]	24, 15	PROVED	”
Edge count	nk/2 = 40·12/2	240	PROVED	”
Ramanujan bound	|λ_nontrivial| ≤ 2√(k−1) = 2√11 ≈ 6.633	4 ≤ 6.633 ✓	PROVED	w33_paper.tex
H_1 of clique complex	dim = |E| − rank d₁ − rank d₂ = 240−39−120	Z^81	CERTIFIED	w33_pass682_*

The Ihara zeta and its zeros

Quantity	Symbolic derivation	Value	Status
Zero locus	roots of 1 − λu + (k−1)u² = 0, per eigenvalue λ	—	PROVED
Zero radius	|u| = 1/√(k−1)	1/√11 ≈ 0.3015	PROVED
Zero phase	φ = arccos( λ / (2√(k−1)) )	—	PROVED
Gauge phase	φ_g = arccos(2/2√11) = arctan√Φ₄(3), Φ₄(3)=3²+1	72.45°	PROVED
Chiral phase	φ_c = arccos(−4/2√11), involves Φ₆(3)=3²−3+1	127.09°	PROVED
Graph RH ⟺ Ramanujan	standard equivalence (Terras) — not new	—	PROVED

The 72.45° and 127.09° phases are real, exact, and were in w33_paper.tex before three separate "discoveries" of them. See the retractions.

Eigenlattices, gluing, and the E₈ boundary — the 2026-07 arc

This is the newest frontier and the one with live theorems. L_c = ker(A − cI) denotes a saturated eigenlattice.

Quantity	Symbolic derivation	Value	Status	Witness
Two-branch gluing	S(S−cI)=0, S=[[cI,Y],[0,0]] ⟹ Z^n/(L_c⊕L_0) ≅ ⊕ᵢ Z/(c/gcd(dᵢ,c)), dᵢ = Smith(Y)	—	PROVED	pass806 + Lean
k-branch gluing	Nᵢ=∏_{j≠i}(S−c_j), Dᵢ=∏_{j≠i}(cᵢ−c_j) ⟹ Z^n/⊕Lᵢ = Z^n/⋂ᵢ ker(Nᵢ mod Dᵢ)	—	CERTIFIED	pass809
Coalescence theorem	for v_p(M)=1, M=lcm(Dᵢ): p-part = (Z/p)^{r_p}, r_p = rank_{F_p} of the Nᵢ with p∣Dᵢ; and p∣Dᵢ ⟺ cᵢ≡c_j (mod p)	—	CERTIFIED	pass828
⤷ in words	the p-part is carried entirely by eigenvalues that collide mod p	—	”	”
Adjacency 3-branch gluing	Z^40/(L₁₂⊕L₂⊕L₋₄)	(Z/2)⁶⊕(Z/6)⁹⊕Z/120	CERTIFIED	pass827
⤷ primary form	—	(Z/2)¹⁵⊕Z/8⊕(Z/3)¹⁰⊕Z/5	”	”
Collision structure	{12,2,−4} ≡ {12},{2,−4} mod 3; {12,2},{−4} mod 5	ranks 10, 1	CERTIFIED	pass828
K four-branch gluing	Z^240/⊕Lᵢ, spectrum {−6,2,4,10}	(Z/32)¹⁴⊕(Z/8)⊕(Z/4)⁶⁶⊕(Z/2)²³⊕(Z/3)¹⁰⊕(Z/5)²³	CERTIFIED	pass826
Discriminant identity	∏ᵢ det(Lᵢ) = [Z^n : ⊕Lᵢ]² = |gluing|²	2³⁶·3²⁰·5² ✓	CERTIFIED	pass829
det(L₂)	Gram determinant of the +2-eigenlattice	2¹⁶·3¹⁰·5	CERTIFIED	”
det(L₋₄)	forced by the identity; not in the paper	2¹⁷·3¹⁰	CERTIFIED	”
L₂ discriminant group	L₂^#/L₂	(Z/2)¹⁶⊕(Z/3)¹⁰⊕Z/5	PROVED	w33_paper.tex
Rigidity	(a−b) ∣ f(a)−f(b) ∀f∈Z[x] ⟹ collisions are functorial ⟹ gluing support is an invariant of Z[S], not S	—	PROVED	pass876
⤷ consequence	eigenlattices split ⟺ gap is a unit; adjacency gaps are 10,16,6 ⟹ no f(A) splits Z^40	—	PROVED	”
Coalescence = p-rank	for an {r,s} collision, r_p = rank_{F_p}((A−kI)(A−rI)) — a classical SRG p-rank	—	CERTIFIED	pass983
Gluing ≻ spectrum	on cospectral T(8) / Chang: (Z/2)⁶⊕Z/4 vs (Z/2)⁷⊕Z/4 — separates them	—	CERTIFIED	pass984
Signed edge action	Aut acts on oriented edges by signed permutations; commutes with K (8/8)	—	CERTIFIED	”

The flat block (Heisenberg–Weyl track)

Quantity	Symbolic derivation	Value	Status
Flat-block quadratic	F² + 2F − (q²−1)I = 0	eigenvalues −1±q, gap 2q	PROVED
Order bridge	S = F + (q+1)I ⟹ S² − 2qS = 0	node, branches {0,2q}	PROVED
Abstract Ext quiver	over Z_p: (Ext¹_self, Ext¹_cross, Ext²_self, Ext²_cross)	(0, Z/p^{v_p(2q)}, Z/p^{v_p(2q)}, 0)	PROVED
q=2 fibre = S8 commutant	Z₂[S]/(S²−4S), Ext = Z/4, Kuranishi cone xy=0	—	PROVED
Key congruence	F ≡ −I (mod q)	verified q ≤ 13	CERTIFIED
Real gluing	Z^n/(L_{q−1}⊕L_{−(q+1)}) = im((F+(q+1)I) mod 2q)	(Z/2)^{(q−1)²/2}	CERTIFIED
⤷ at q=3,5,7	—	(Z/2)², (Z/2)⁸, (Z/2)¹⁸	CERTIFIED
Burnside orbit count	|Fix_all(g)| = (pⁿ)^{c⁺(g)}, |SL(2,Z/pⁿ)| = p^{3n−2}(p²−1) on (p^{2n}−1)/2 pairs	all odd Z/pⁿ	PROVED
⤷ exact values	F_3 → 7; F_5 → 2,034,735; Z/9 → 228100045392509153077600971330057241	—	CERTIFIED

The physics chain — seven steps from a finite geometry

This is the program's most ambitious arc and its most contested. Every row's arithmetic is exact and verified; the physical interpretation of each is CONDITIONAL — the identification of a combinatorial object with a physical one is a map that must be built, not inferred from a matching integer. Read the retractions alongside this table.

Step	Symbolic identity	Reading	Status
1. Geometry	W(3,3) = isotropic points/lines of (F_3^4, ω)	the substrate; no free parameters	PROVED
2. Homology	H_1(clique complex) = Z^81, 81 = 3^4	"homology reveals matter"	PROVED / CONDITIONAL
3. Vertex split	40 = 1 + 24 + 15 (eigenvalue multiplicities)	1 vacuum, 24 = dim adj SU(5), 15 Weyl spinors/generation	CERTIFIED / CONDITIONAL
4. Generations	240 = 40 × 3 × 2: each K_4 line has 3 perfect matchings (labelled by GF(3)), each 2 edges	three generations from `	GF(3)
⤷ refined	240 = 72 + 6 + 81 + 81 = 3 × (24 + 2 + 27 + 27), per-generation 80 = 4+4+36+36	Sp(4,3) is edge-transitive — a single orbit	CERTIFIED
5. Gauge group	k = (k−μ) + q + 1 = 8 + 3 + 1 = 12	dim SU(3)=8, dim SU(2)=3, dim U(1)=1	CERTIFIED / CONDITIONAL
⤷ forced identity	2q = λ + μ (6 = 2+4) holds automatically for W(q,q)	the split is not chosen	PROVED
6. Matter sector	v − 1 − k = 40 − 1 − 12 = 27	fix a vacuum vertex: 27 non-neighbours carry E_6 fundamental, since |Aut| = 51,840 = |W(E_6)|	CERTIFIED / CONDITIONAL
⤷ branching	27 = 16 + 10 + 1 under E_6 ⊃ SO(10) ⊃ SU(5)	one generation + Higgs + singlet	PROVED (rep theory)
6b. E₈ boundary	240 = |Φ(E_8)|	The global W33-edge map is obstructed; the distinct 40×3×2 local-axis endpoint carrier has an explicit integral lift onto all 240 signed roots; a different transitive subgroup embedding remains open	CERTIFIED / OPEN (local-axis lift)
7. Curved 4D	KO-dim = 6 = 2q (Connes–Barrett)	4D spacetime as a derived quantity	CONDITIONAL
α (fine structure)	Hashimoto operator B on 480 = 2×240 directed edges	a spectral identity on the non-backtracking carrier, not a fit	CONDITIONAL
Koide / flavour	residual packet 98 · 17 · 208, 208 = 4·dim(F_4) = 4·52	factor arithmetic closed; physical identification open	OPEN
CKM from Ihara phases	δ_CP ≟ φ_gauge = 72.45°	REFUTED — see below	RETRACTED

The honest summary of this arc: the decompositions are exact and the group theory is real. Whether 24 = dim adj SU(5) is physics or coincidence is exactly the kind of claim this repository has learned to tier rather than assert.

Physics constants — every derivation, verified or flagged

The repository contains 50+ constant tables of varying quality. This one is built by evaluating every closed form and comparing against PDG-2025 in experimental σ, not percent. Two things follow, and both matter more than any individual row.

First: of the 14 closed forms in the most-cited ledger, only 5 evaluate to their own stated value. The numbers may well be right; the formulas as written are not. A reader who checks will find this in minutes, so it is recorded here rather than reproduced.

Observable	Closed form as written	Evaluates to	Claimed	PDG-2025	σ	Verdict
N_ν	q	3	3	3 (exact)	—	✅ exact
sin²θ₂₃ (PMNS)	7/13	0.53846	0.5385	0.546 ± 0.021	0.4	✅ agrees
m_t (pole)	v_EW/√2	173.948 GeV	173.95	172.57 ± 0.29	4.8	⚠️ formula OK, value excluded
sin²θ_W (dressed)	q/(q²+q+1) = 3/13	0.230769	0.23077	0.23122 ± 0.00003	15.0	⚠️ formula OK, value excluded
α⁻¹ (integer skeleton)	k² − (|r|+|s|+1) = 144−7	137	137	137.035 999 178(8)	—	✅ integer only; the .036 is not derived
|V_us|	√(3/v)·k	3.286	0.2253	0.2245 ± 0.0008	—	❌ formula ≠ claim
m_H	1/(q⁻⁵) = q⁵	243	125.0	125.25 ± 0.17	—	❌ formula ≠ claim
m_W	v_EW√((1−3/13)/2)	152.56	80.44	80.369 ± 0.013	—	❌ formula ≠ claim
H₀	12/q!	2.0	67.0	67.4 ± 0.5	—	❌ formula ≠ claim
n_s	1 − 2/(q·q)	0.7778	0.9667	0.965 ± 0.004	—	❌ formula ≠ claim
Ω_Λ	1 − 1/(k·Φ₄/10)	0.9167	0.6833	0.685 ± 0.007	—	❌ formula ≠ claim
sin²θ₁₂ (PMNS)	3/(4·13) = 3/52	0.05769	0.3077	0.307 ± 0.013	—	❌ formula ≠ claim
sin²θ₁₃ (PMNS)	3/(6·29)	0.01724	0.02198	0.0220 ± 0.0007	—	❌ formula ≠ claim
α⁻¹ (ledger form)	k² + (k−1)² + λ	267	137.036	137.036	—	❌ formula ≠ claim

And most of the broken rows cannot be repaired. Searching 7,128 expressions built from eighteen W(3,3) atoms, four targets — m_H, Ω_Λ, sin²θ₁₃ and m_W/v_EW — are reached by nothing at all, so they should be withdrawn, not rewritten. The rest do have hits, but 36 hits for sin²θ₁₂ is what chance gives in a space that size: a hit found by search is a candidate for a derivation, not a derivation. (pass1010)

Second: even the formulas that evaluate correctly are mostly excluded by experiment. sin²θ_W is 15σ from the measured value and m_t is 4.8σ. Exactly two rows survive both tests — N_ν = q = 3, and sin²θ₂₃ = 7/13 at 0.4σ. That is the honest state of the constant program: one exact integer count, one genuine agreement, and a great deal that needs its formulas re-derived before it can be called a derivation.

The combinatorial identities in the physics chain above are a different matter — those are exact and verified. The gap is between counting the geometry, which works, and predicting a dimensionful constant, which so far does not.

A verified structural result: E₈ inside W(3,3)

Not a constant, but the strongest physics-adjacent claim that survives checking. The eight vertices [7, 1, 0, 13, 24, 28, 37, 16] induce a subgraph of W(3,3) that is the E₈ Dynkin diagram:

Check	Result
induced degree sequence	[1,1,1,2,2,2,2,3] — E₈ Dynkin exactly
Gram 2I − A_sub	positive definite
det(Gram)	1 — the E₈ Cartan determinant

So E₈'s Cartan matrix is realised on eight points of the geometry.

But the 240 = 240 edge–root correspondence is now known to be obstructed. The repository's own solvers recorded that the edge graph is 22-regular and the root graph 56-regular, so no graph isomorphism exists, and spent many passes seeking an equivariant bijection instead. That map does not exist either, for the embedding they assumed:

	orbits under the 51,840-element group
240 W(3,3) edges	one orbit (transitive, stabiliser 216)
240 E₈ roots, under E₆ × A₂	four orbits: 72 + 6 + 81 + 81

An equivariant bijection carries orbits to orbits of equal size, so one orbit cannot map onto four. The failed searches were not failing for want of effort. (pass1012)

The obstruction is embedding-specific, not group-theoretic: Aut(W(3,3)) ≅ PSp(4,3):2 ≅ W(E₆) does act transitively on 240 things — it does so on the edges. What remains open is whether some other conjugacy class of order-51,840 subgroups of W(E₈) acts transitively on the roots. That is a GAP question, and it is the live form of the E₈ problem.

Codes, groups, lattices

Quantity	Symbolic derivation	Value	Status
|Sp(4,3)|	q⁴(q²−1)(q⁴−1), q=3	51,840	PROVED
|W(E₆)|	—	51,840	PROVED
The real coincidence	|W(E₆)| = |Sp(4,3)| — E₆, not E₈	—	PROVED
[W(E₈) : Sp(4,3)]	696,729,600 / 51,840	13,440	PROVED
dim e₈	|roots| + rank = 240 + 8	248	PROVED (textbook)
QR-CSS code	exact length-137 construction	[[137,1,21]]	CERTIFIED
2-rank of A	#{invariant factors = 1} in SNF(A)	16	CERTIFIED
E₈ shadow rank	#{invariant factors = 2}	8	PROVED

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How this repository grew

Era	Passes	What happened
Genesis (2026-01)	Parts I–LXIV	Initial archive. Roman numerals. Ambition unbounded.
Physics sprint	PART_*, BT*	Yukawas, CKM, neutrinos, RG running, E₆/E₇/E₈ bridges
The audit	322–346	Discovery that the rank law was already published (Sastry–Sin; Chandler–Sin–Xiang) and already in this corpus. ~19 passes wasted. Produced RESULTS_INDEX.md and the rediscovery guard.
Selection layer	346	Closed: chirality is hostable but not selectable from inside. Don't reopen.
Exact frontier	479–541	Flat block, trace valuations, all-exponent q=3 theorem, chain rings
Deformation arc	641–830	2-adic tower, Ext quivers, the two-branch and k-branch gluing theorems, coalescence
Cross-track	806–828	Two agents' independent constructions unified by one Smith-form theorem
Audit again	856–984	Three external batches audited at intake; several headline claims refuted
Exceptional/modular closure	1002–1391	Ramified reconstruction; 432→81→216; Brauer/Cartan/PIM closure; selector orbital and frame-matching algebras
Cover-resolution atlas	1408–1975	Certified 327-orbit cover frontier, signature compression, decoders, arithmetic multiplicity order, and SL₃(Z) phase carrier
Spread and controller frontier	1976–2206	Regular-spread classification for every odd prime power, Ree–Tits control, exact outer-even Hom multiplicities, and the canonical order-24 controller
Complete spectra and representations	2300–2307	Complete q=27 named-family spectra/codes, all quadratic Hom bases, q=7/11 Weil inversion, controller trichotomy, and the induced quadratic-map S₃ character

Two agents work this repository in parallel. Neither reads the other's filenames. That is a structural cause of rediscovery, not a discipline problem — hence RESULTS_INDEX.md, the guards, and the pass-number reservation protocol.

The full program, by domain

Everything below descends from the same 40 points. Tiers are the domain's overall standing, not any single claim's.

Domain	What it contains	Tier
Finite geometry & groups	W(3,3), Sp(4,3), PSp(4,3), W(E_6), ovoids, spreads, generalized quadrangle combinatorics	PROVED
Spectral & zeta	adjacency/Hashimoto/Ihara–Bass, Ramanujan property, closed-form zeta, non-backtracking dynamics	PROVED
Lattices & gluing	eigenlattices, E_8 shadow, Smith forms, critical groups, the k-branch/coalescence theorems	PROVED / CERTIFIED
Deformation theory	flat block, 2-adic tower, Ext quivers, Kuranishi cones, conductors, Burnside orbit counts	PROVED / CERTIFIED
Codes & QEC	[[137,1,21]] QR-CSS, stabilizer cascades, syndrome structure, Clifford recovery protocol	CERTIFIED
Representation theory	E_6/E_7/E_8 chains, 27/78/248, H_27 middle layers, Loewy structure, ATLAS matrices	PROVED / CERTIFIED
Topology & homology	clique complex, H_1 = Z^81, Hodge-style force classification, cohomology of the selector	PROVED / CONDITIONAL
Moonshine & modular	Niemeier/Leech material, McKay–Thompson series, Hecke operators, j-function arithmetic	CONDITIONAL / much RETRACTED
Holonet (the machine)	GKP tower A_2 < D_4 < E_8, degree-2 symplectic + degree-3 E_6 cubic gates, routing, schedulers, contextuality tax	CONDITIONAL
Photonics	dual-rail single-photon runtime, interference-phase predictions at 72.45°/127.09°, lab packets	CONDITIONAL
Selector / tomotope	selector frames, braid registers, Reye/Q4 configurations, orientation quotients	CERTIFIED / CONDITIONAL
Physics program	masses, Yukawas, CKM/PMNS, α, neutrinos, cosmology, RG running	CONDITIONAL / several RETRACTED
Tooling & audit	Thousands of witnesses and certificates, executable guards, RESULTS_INDEX.md, pass-reservation protocol, and batch-intake harness	—

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Things we got wrong, on purpose and in public

This is the section that makes the rest trustworthy.

Claim	What killed it	Pass
Flat-block gluing = (Z/q)^{(q²−1)/2}	Glued eigenlattice images (unsaturated) with a buggy hand-rolled Smith routine. Truth: (Z/2)^{(q−1)²/2}, pure 2-torsion.	808
"Deformation–Burnside bridge"	(q−1)²/2 ≠ (q²−1)/2 always. The rank match was the bug.	808
Tower theorem for all n	The modulus-qⁿ flat block fails its quadratic in every entry at (3,2) and (5,2).	807
Factorial trace law	Deviates below the law — opposite sign to the proposed mechanism.	508
CKM from Ihara phases	In experimental σ: θ₁₂ 28.8σ, θ₁₃ 62.9σ, λ_W 35.7σ. Reported as "11% agreement." Source file tried four θ₁₂ formulas and kept the closest.	981
[W(E₈):Sp(4,3)] = 480	It's 13,440.	981
5 orthogonal E₈ in Leech	5×8 = 40 > 24 = rank(Leech). Dimensionally impossible.	981
A₅ splits 240 edges into 4×60	17 verified A₅ subgroups, all with profile (60,60,30,30,20,20,10,10). 240=4·60 satisfies orbit counting, but divisibility ≠ freeness.	982
Ihara Φ₄(3)=10 = coalescence rank	Held for W(3,3) and T(8) with the values correctly swapped — then died on T(12) (predicts 3, actual 11).	983

The five failure modes this repo has actually produced, in increasing order of how hard they are to catch: coordinate artefacts · over-reads · unbuilt objects · unbuilt halves · rediscovery. The last one cannot be self-checked, because novelty is a property of the corpus, not of the claim. It can only be searched for.

The 2026-08 instruction-layer arc

Claim	Why it was withdrawn	Pass
"The frame Cayley graph"	It is a Schreier graph on a coset space. Schreier graphs collide by construction; the misnomer made regularity feel obligatory and caused the next three entries.	4203
"The instruction layer can be Ramanujan"	The five-generator graph measured was the discrete torus C₃⁴; its only Clifford generator draws no edges at all, every one duplicating a translation edge.	4201 → 4204
"The instruction graph misses Ramanujan by 3.23%"	2√(k−1)/k is a k-regular bound. This graph has degrees 2–8, so it has no claim on 0.866. The measurement (|λ₂| = 0.893992320) is exact and stands; only the grade was withdrawn.	3042 → 4213
"78 = dim E₆ identifies W(3,3)"	All 28 Spence graphs give the same 78 poles: it is 2(v−1), a property of the parameter set (40,12,2,4), not of this graph.	4281
"|Aut| singles out W(3,3)"	51,840 is attained by two of the 28 — the point graph and the line graph of one GQ(3,3).	4287 → 4296
"Add S_f to unfreeze the register"	S_f moves x₃, not x₂ — the reasoning went from a coordinate's name to an opcode whose subscript matched. No pool opcode can unfreeze it, and a control improved mixing as much as either candidate.	4244 → 4245
"Every defect traces to the load port"	Localisation does; the arrow of time does not. One-way transitions rise as load ports are added, and the machine with none still has 216. Two independent asymmetries.	4314

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How this repository checks itself

Each of these exists because the same mistake was made more than once, and each has been verified against a planted fault — a checker that has only ever reported clean has unknown recall.

Check	Catches	Verified by
check_tex_insert_pitfalls.py	six LaTeX fault families across 287 inserts	test_checker_recall.py — 6/6 planted faults, silent on a clean file
check_labels.py	duplicate labels, dangling references	planted duplicate + planted dangling ref, both caught
find_orphaned_inserts.py	finished write-ups no manuscript includes	census went 114 → 0; CI baseline now 0
route_orphaned_inserts.py	inserts routed away from the section they cite	planted cross-referencing pair, co-location confirmed
check_site_is_current.py	the CDN serving a stale page behind a green deploy	caught a 116 KB-behind artifact reporting status: built
check_certificates.py	certificates that cannot reproduce their own digest	a certificate unverifiable from birth

Two lessons worth stating in the open, because both cost real time:

• Zero LaTeX errors is not zero undefined references. Both are now CI failures.
• Planted-fault recall measures the families you have, never the ones you lack. The pitfall checker reported a clean scan across all 287 inserts while two of them failed to compile, because the fault family did not yet exist.

---

Certificate contract

The evidence tiers apply repository-wide. A certificate is idempotent: rerun its producer with --check and it must reproduce byte-identically, or it fails. CERTIFIED describes the named finite computation only; it never upgrades an attached physical interpretation.

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Lean build status (read this before trusting any PROVED tier)

A whole-repository lake build in formal/ does not currently complete on the machine it was measured on. There is no Lean badge in this README because nothing green has been demonstrated.

Verify a Lean-owned claim by building its named module alone:

cd formal && lake build W33.<TheModule>

Open the historical build autopsy and repaired-module ledger

An earlier version of this section said "20 modules with real compile errors", then "19". Both were wrong by roughly a factor of three. The correction is recorded here rather than quietly edited away.

What happened: a whole-library build reported ~20 failures and they were taken at face value. Nearly all were failed to read file …/Mathlib/….olean at line 1, column 0 — the import line — naming a different mathlib file on each run. A genuinely corrupt artifact fails identically every time; varying targets mean transient I/O, and the builds had been running concurrently. lake exe cache get reports the cache complete and the named files are present on disk.

Settled 2026-07-25 by building every suspect module one at a time, with nothing else running (leanprover/lean4:v4.32.0-rc1, prebuilt mathlib):

.lean files under formal/W33/	40
imported by formal/W33.lean (so reachable by lake build)	39
all seven originally-broken modules	FIXED — Pass447, Pass491, Pass450, Pass565, Pass502, Pass488, Pass570
newly revealed once they built	1 — Pass575CyclotomicDVRKernel, which had never been compiled because it imports Pass570
falsely accused by the contended build, and fine	12
never imported at all, so never type-checked by anything	4 (now 3 imported, 1 left out — see below)

Every one of the seven was mathlib drift, not bad mathematics. A renamed constant, a tactic that moved, a missing noncomputable, or a lemma absorbed upstream. Two were instructive: Pass491 was re-proving Matrix.det_conjTranspose, a @[simp] lemma mathlib already had; and Pass488 resisted three tactic swaps because its ring A is only [Ring A] — possibly noncommutative — so ring, ring_nf and linear_combination were never applicable. What makes that theorem true is that algebraMap lands in the centre, which is now what the proof uses.

A caution the count itself teaches. Fixing the seven did not make lake build green: it exposed Pass575CyclotomicDVRKernel, which imports Pass570 and had therefore never been compiled at all. A failing module masks everything downstream of it, so any count taken from a failing build is a lower bound. The honest statement is that seven are fixed and one is newly visible.

Both fixed modules were mathlib drift, not bad mathematics, and that is the likely character of the rest. Pass447 assumed a subst direction: in rintro v (rfl | rfl) the disjunct v = p eliminates p, so later haves mentioning p fail with Unknown identifier p — establishing them before the rintro fixes it. Pass491 was reinventing an upstream lemma: it hand-proved (Mᴴ).det = star M.det via Matrix.det_transpose_eq_det_map, a constant that no longer exists, while mathlib has had Matrix.det_conjTranspose as a @[simp] lemma with exactly that statement. Deleting the proof in favour of the upstream name fixed it in 20 seconds.

To settle a module, build it alone — a whole-library build on this machine is not a reliable measurement:

cd formal && lake build W33.<TheModule>   # exit 0, run with nothing else building

Pass828CoalescenceArithmetic is deliberately not imported: it cannot compile, because line 91 asks Lean to synthesise Decidable (¬∃ k, gluing_order = k^2), an unbounded existential over ℕ. It is left out with a comment rather than patched over or sorry-ed.

Why this was not visible. Not because CI lied — because two thirds of the Lean CI was aimed at nothing.

• .github/workflows/lean-formal.yml targets formal/ and does enforce: its "Enforce kernel success" step fails the job unless lake build --wfail returned 0, and a second job rejects any sorry/admit. It is correct, and it must have been failing. It only triggers on formal/**, and no badge surfaced it, so its redness sat where nobody looked.
• .github/workflows/lean4.yml and lean4-weekly-verify.yml ran with working-directory: proofs/lean — a directory that does not exist in this repository. Both degraded to no-ops by design (lake build || echo "...continuing", || true, and an explicit "skipping Lean build" branch). They have been deleted; a workflow that cannot verify anything is worse than no workflow, because it looks like one.

What this does and does not invalidate. It does not touch the GAP certificates or the pytest suite, which are independent. It does mean a PROVED tier justified by "Lean" is only as good as the specific module, so check it:

cd formal && lake build W33.<TheModule>     # exit 0 means that module really is checked

Modules verified to build at that measurement: Pass806TwoBranchGluing, Pass1006RamifiedFiltration, Pass1018PencilRigidity, and the 18 other imported modules not on the broken list above.

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Reproduce the flagship results

Commands below use the Windows launcher; replace py -3 with python3 on Unix-like systems.

# Schläfli–Steinberg object map, integral frame, and focused contract
gap -q analysis/w33_pass1147_schlaefli_steinberg_fourier_bridge.g
py -3 -m pytest -q tests/test_pass1147_gap_schlaefli_steinberg_fourier_bridge.py

# three-carrier triality, transport/Hecke Smith forms, independent reconstruction
py -3 analysis/w33_pass1325_1329_triality_integral_gauge.py
py -3 analysis/w33_pass1329_independent_checker.py
py -3 -m pytest -q tests/test_w33_pass1325_1329.py

# modular H26 radicals, central blocks, selected cycles, and AtlasRep carriers
py -3 analysis/w33_pass1330_1334_modular_triality_cycle_atlas.py
gap -q analysis/w33_pass1333_atlasrep_species20.g
py -3 -m pytest -q tests/test_w33_pass1330_1334.py

# cyclic-defect Brauer tree and the complete 23↔58 extension calculation
py -3 analysis/w33_pass1335_export_hecke_gap_input.py
gap -q analysis/w33_pass1335_brauer_tree_hecke_corner.g
py -3 -m pytest -q tests/test_w33_pass1335_brauer_tree_hecke_corner.py

# ramified p=2 reconstruction and coalescence theorem
py -3 analysis/w33_pass1002_ramified_kernel_growth_gluing.py --check
py -3 analysis/w33_pass828_coalescence_theorem.py --check

# all-odd-q spread theorem, q=27 nonregular control, and corrected controller
py -3 analysis/w33_pass2201_all_q_regular_spread_scheme.py --verify-frozen
py -3 analysis/w33_pass2203_ree_tits_nonregular_control.py --verify-frozen
py -3 -m pytest -q tests/test_w33_pass2200_2206.py

# controller representation trichotomy (GAP is the owning computation)
gap -q analysis/w33_pass2306_controller_representation_trichotomy.g
py -3 -m pytest -q tests/test_w33_pass2306_controller_representation_trichotomy.py

# current complete spectra, Hom bases, Weil inversion, and S3 character layer
py -3 analysis/w33_pass2300_ree_tits_divisible_code.py --verify-frozen
py -3 analysis/w33_pass2301_complete_quadratic_hom_bases.py --verify-frozen
py -3 analysis/w33_pass2302_q7_q11_weil_outer_inversion.py --verify-frozen
py -3 analysis/w33_pass2304_known_q27_spread_spectra.py --verify-frozen
gap -q analysis/w33_pass2307_quadratic_hom_s3_decomposition.g
py -3 -m pytest -q tests/test_w33_pass2307_quadratic_hom_s3_decomposition.py

# chamber matrix units, HoloBox family checksum, and ten-trit affine controller
gap -q analysis/w33_pass4936_chamber_packet_matrix_units.g
py -3 -m pytest -q tests/test_w33_pass4936_chamber_packet_matrix_units.py
gap -q -b analysis/w33_pass4937_adjoint_dual_number_controller.g
py -3 -m pytest -q tests/test_w33_pass4937_adjoint_dual_number_controller.py

# q=5 cover -> Reye/Latin/F4, intrinsic zero shell, and the exhaustive 7-side join
gap -q analysis/w33_pass5667_5674_q5_reye_equivariant_orientation.g
py -3 -m pytest -q tests/test_w33_pass5667_5674_q5_reye_equivariant_orientation.py

# corpus and claim guards
py -3 analysis/build_results_index.py
py -3 scripts/next_free_pass.py --report      # claim a pass number safely
py -3 scripts/check_rediscovery.py <files>    # is this result already ours?
py -3 scripts/check_sigma_gate.py <files>     # percent vs experimental sigma
py -3 scripts/check_remotes_sync.py           # have the two remotes diverged?
py -3 scripts/check_mechanism_claims.py <json>

# Lean (mathlib required)
cd formal && lake env lean W33/Pass806TwoBranchGluing.lean

# the papers
tectonic -X compile w33_paper.tex --outdir <dir>

---

Recovery Packet

A self-contained, independently checkable bundle for the Clifford recovery protocol — the one artifact to reach for if you want to verify a single complete result end to end rather than navigate the atlas.

Artifact	Path
Landing page and how-to	docs/recovery_packet_landing.md
Packet index	data/bt1279_recovery_packet_index.json
Strict polar-path certificate	data/bt1275_strict_polar_path_recovery_certificate.json

py -3 tools/bt1291_verify_release_packet.py   # verifies the whole packet

Repository map

Path	Contents
analysis/	Executable Python and GAP witnesses (w33_passNNN_*)
data/	Deterministic JSON certificates; many are intentionally gitignored unless promoted
tests/	Focused pytest contracts tying prose, witnesses, and certificates together
scripts/	Corpus, rediscovery, namespace, sigma, and mechanism guards
formal/	Lean 4 + mathlib; build named modules individually
papers/	Specialist manuscripts; the master source is w33_paper.tex at the root
docs/	The live atlas, PDFs, demonstrators, and reader-facing artifacts
PASS_*, BREAKTHROUGH_*, PART_*	Synthesis and historical release documents; use the result index to find the owner

---

Citation, provenance, license

MIT. Every promoted claim must name a proof or witness path; executable packets also carry deterministic certificates, and current release notes publish their SHA-256 digests. If you find an error, the correct response is a retraction pass with a certificate — that is how the entries in Things we got wrong got there, and several of them were found by the authors auditing their own work.

"A claim you have not searched the corpus for is not new." — CLAUDE.md