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Niemeier research (no κ claim): operator factors and paid spinor circuits #606

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143 changes: 143 additions & 0 deletions research/niemeier-lattices/CATALOG.txt
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All 23 rooted Niemeier lattices: an exact low-shell census
=======================================================

The machine-readable table is catalog.json, regenerated by code/catalog.py.
Norm means squared Euclidean norm. A line is an antipodal pair. H is the
common Coxeter number of the irreducible roots, not a binary frame dimension.
This table counts geometry; it does not certify a supplier or a kappa gain.

Root system H norm 2 lines norm 4 lines in root lattice from glue
D24 46 552 85032 85032 0
D16 E8 30 360 89640 73256 16384
E8^3 30 360 89640 89640 0
A24 25 300 91080 37950 53130
D12^2 22 264 91944 42792 49152
A17 E7 18 216 93096 28836 64260
D10 E7^2 18 216 93096 33064 60032
A15 D9 16 192 93672 23757 69915
D8^3 14 168 94248 20520 73728
A12^2 13 156 94536 16458 78078
A11 D7 E6 12 144 94824 15227 79597
E6^4 12 144 94824 16092 78732
A9^2 D6 10 120 95400 10836 84564
D6^4 10 120 95400 11304 84096
A8^3 9 108 95688 8910 86778
A7^2 D5^2 8 96 95976 7358 88618
A6^4 7 84 96264 5712 90552
A5^4 D4 6 72 96552 4332 92220
D4^6 6 72 96552 4392 92160
A4^6 5 60 96840 3090 93750
A3^8 4 48 97128 2040 95088
A2^12 3 36 97416 1188 96228
A1^24 2 24 97704 552 97152

Derivation and independent checks
---------------------------------
For q^{norm/2}, the primary-source identity is

theta_L = E4^3 + (24H-720)Delta.

Using E4=1+240 q+2160 q^2+... and Delta=q-24 q^2+... gives

number of norm 2 vectors =24H,
number of norm 4 vectors =196560-576H,
number of norm 4 lines =98280-288H.

For an irreducible root lattice, the norm 4 vector counts are

A_n: 6*C(n+1,4),
D_n: 2 n+16*C(n,4),
E6:270, E7:756, E8:2160.

The A_n vectors have two +1 and two -1 entries. The D_n vectors are
+/-2 e_i or have four +/-1 entries. The exceptional counts are explicitly
enumerated inside E8 by catalog.py, also recovering their 72,126,240 roots.
In a sum R=directsum R_i, a norm 4 vector is either a norm 4 vector in one
component or a pair of norm 2 roots in two components. Hence

|R_4|=sum_i |(R_i)_4| + sum_{i<j}|(R_i)_2| |(R_j)_2|.

Subtracting half this count from the theta-derived line count gives the
last column. The script also verifies component ranks sum to 24, common
Coxeter numbers agree, root count is 24H, and determinant(R) is a square.
The glue group order is sqrt(det(R)) and is recorded in catalog.json.

What is actually different from Leech
------------------------------------
The norm 2 shell is exactly the orthogonal union of the ADE root systems.
Its ordinary Gram kernel has no cross-component entries. Merely taking
more copies of an old component repeats the old supplier; its 24-dimensional
ambient realization does not by itself produce a new interaction.

The norm 4 shell adds two genuinely relevant ingredients: pairs of roots
in different components, and glue-coset vectors. These have sharply
different tensor support. The last column shows that a root-sublattice
calculation represents the whole norm 4 shell only for D24 and E8^3.
For the other 21 systems, ignoring glue can discard most of the shell.

The same cubic parity gate is legal on every norm 4 shell:

K(v,w)=[<v,w>^3-4<v,w>]/48.

Its diagonal is 1. Every nonzero off-diagonal has odd inner product, so
B(v,w)=1+<v,w> mod 2 is a fitting binary matrix. Unlike Leech, rooted
lattices may also have off-diagonal inner products +/-3; these are allowed
by this gate and must not be accidentally omitted in finite checks.
The generic real cubic-feature bound is 2600 and the generic binary bound
is 25. These are upper bounds for a particular kernel/fit, not lower
bounds for all constructions.

Concrete structural cases
-------------------------
* D24+: the added half-integral coset has minimum norm 24/4=6. Thus the
norm 4 shell is exactly 24 axis lines and 8*C(24,4)=85008 four-support
lines. Sparse coordinate support makes this an explicit baseline.

* E8^3: norm 4 vectors occupy one block with norm 4, or two blocks with
norms 2+2. There are 3*1080+3*(240*240/2)=89640 lines. Three-block cubic
features vanish, removing 8^3=512 dimensions from the 2600-dimensional
homogeneous cubic space. The surviving 360 one-block and 1728 two-block
features explain the 2088 real-rank construction. Two-block support also
gives a binary frame-width bound 17 for those factors. The exact cost
and dyadic factorization are treated separately in this package.

* D16 E8: this is D16+ orthogonally summed with E8. The additional 16384
norm 4 lines come from the D16 spinor coset, whose 32768 minimal vectors
have sixteen coordinates +/-1/2. There are only two coordinate blocks;
the E8^3 three-block-vanishing shortcut therefore does not apply.

* D12^2: the two glue classes (s,v),(v,s) have norms 3+1, with 2048 spinor
vectors and 24 vector-coset vectors per component. Together they add
2048*24=49152 lines. Their sum class (c,c) first appears at norm 6.
These low-rank block products deserve separate kernel/label treatment.

* D8^3: a glue code generated by (s,v,v),(v,s,v),(v,v,s) has three
norm profiles 2+1+1 and three profiles 2+2+0 at norm 4. Here a spinor
coset has 128 minimal vectors, and a vector coset has 16. Consequently
the glue contributes [3*128*16^2+3*128^2]/2=73728 lines. The all-spinor
sum class first appears at norm 6. Both kinds of profile must be counted.

* The remaining entries are candidates for a glue-aware shell treatment,
not rejected lattices. D4^6 offers a small 64-word hexacode glue group;
E6^4 has a 9-word group and triple-component norm 4 glue; A2^12 and A1^24
have large coding-theoretic glue contributions. The census identifies
their available geometry but does not claim a competitive factorization.

Natural root-only binary ranks
------------------------------
For the norm 2 roots, B=1+<r,s> mod 2 is legal. After quotienting the
radical, its rank is 1+rank_F2(Cartan), namely

A_n: n+1 for even n, n for odd n;
D_n: n-1 for even n, n for odd n;
E6: 7, E7: 7, E8: 9.

These are ranks of this specified fitting matrix, not proved minimum
frame dimensions. They are recorded per component in catalog.json.
For D_n, the coordinate factor has n centers of width n-1, so its center
bill equals its n(n-1) root-line count; this particular unpunctured factor
has no rank deficit. Other factors, puncturing, and shared helper
structures require their own accounting.

See SOURCES.txt for exact primary-source locations. No conclusion here
rules out all suppliers, all kernels, or all schedules of any lattice.
172 changes: 172 additions & 0 deletions research/niemeier-lattices/CONTINUATION.txt
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Further construction search, 2026-10-12
=====================================

No new kappa is certified. This pass attempts actual routing constructions
and alternative fitting kernels. This package is submitted as separate
research following PR #598; it does not change the selected exponent.

Pinned comparison context
----------------
Fetched origin/main at 5996f9043c27bafa90b23e1c937de92a738f1a76.
The initial open numerical comparison inspected was PR #600, Jeremy Kothe's
dfb7b827990d7b586b3cd778e3d750c050dd4778, reporting
kappa = 0.001612722852614221. Its helper profiles are pinned in
workbench/continuation/endpoint-costs/sources/. These are inherited inputs,
not a fresh independent audit of that PR's multiplication theorem.

At the time of the research, a freshness check found draft PR #604 at
af8c23b24482386aeed314b379b92482c2fca993, reporting
kappa = 0.001620147045734211. Its then-current description stated that heavy fresh-copy
verification and CI were still running. This is DreamingOfClouds' work with
Claude Opus 5.5 via Claude Code, building on Jeremy Kothe's PR #601.
It ends fourteen cold factory helpers at four-dimensional frames. The
metadata/body is retained in workbench/continuation/PR604.json. The tensor
invoice here uses its explicitly pinned PR #600 profiles, not PR #604.
The all-stage hybrid obstruction also fails the higher reported bound.

Native tensor lift: concrete, but not competitive
------------------------------------------------
Remove an orthogonal eight-root frame from each E8 factor. The remaining
112 x 112 relative-sign pairs give 25,088 mixed ports in binary dimension
17. A same-label Walsh change separates two sectors. Each sector is a
disjoint collection of 112 native E8 blocks.

The construction has 224 native invocations, center charge 14,336 and
112,896 inherited helpers. Exact checks verify the signed transport to the
inherited Hamming-coordinate prototype, all 224 isometric frame embeddings,
the eight-dimensional spectator complements, and rank-eight center descents.
Virtual entrances and hyperplane endpoints retain their native residual
widths. The native scalar word is inherited, not re-emitted and replayed here.

Pricing the actual PR #600 profiles and paid spectator banks gives best
nominal kappa about 0.000872182222 over odd stage counts 3 through 101
(seven stages, ATM3 variant). CK is slightly worse. This is about 46% below
the comparison, not a gain. It does not exclude a different shared circuit
or prove a global bound over all stage counts for this tensor construction.

An exact edge partition provides a possible shared-circuit ingredient:
135 orthogonal eight-root cliques cover every E8 orthogonality edge once,
each root occurring nine times. If G is the root Gram matrix and H is
(G entrywise-squared minus J)/4, then

H = 3 I - (1/4) sum_C 1_C 1_C^T.

The decomposition is dyadic. Applying its clique words independently
duplicates source inputs across incomparable frames; the identity alone
does not make those repeated gathers free.

Spinor routing: actual reusable subwords
---------------------------------------
A fixed pairing of the 16 coordinates partitions the spinor ports into
128 complementary rational eight-dimensional packets, each with 128 ports.
At a common packet frame, a reversible sum of existing source registers
can feed the opposite packet's targets and then be undone. This single
matching layer needs no extra helper roles. Different pairings generally
require incompatible frame histories, so their costs cannot be combined
by simply reusing the same data registers.

A complete orthogonal 16-port clique-adjacency word uses 34 dirty helpers.
The three helper classes have paths [2,1,13] eight times, [4,3,9] twelve
times, and [8,7,1] fourteen times. Exact integer replay checks every input
column, nested/common frames, target endpoints, and restoration of all
source and helper values. Across 1,024 Hadamard cliques this is 34,816
helpers. This is a compensation subword; a compatible center phase is
still required for a full supplier.

One matching layer can be fused at the clique word's eight-dimensional
stage. Sixteen source pivots at a packet frame hold its clique-half totals.
The invertible binary mixer I+J on those 16 pivots changes each total into
the sum of the other 15. The existing fanout then supplies clique adjacency
XOR matching adjacency, with no additional paid helper paths. The mixer
must be reversed during cleanup; this fusion is over F2.

The final fused variant cleans each copied helper immediately at the
parent span, before its source pivot joins the sibling total. Its helper
paths become [2,1,1] eight times, [4,3,1] twelve times, and [8,7,1] fourteen
times per clique. The checked rank mass falls from 1,024 to 832 per clique
(18.75%). This includes source and target paths, which remain unchanged.
Across the shell, helper endpoint widths are 4 for 8,192 roles, 8 for
12,288 roles, and 16 for 14,336 roles. The final fused replay checks all
1,056 input columns of a representative 256-port block and rejects an
omitted cleanup operation. All 64 blocks have the same enumerated geometry.
Native admission still needs the exact endpoint wrappers and their paid
spectator banks; this is not an exponent certificate.

The associated alternative decoder C' = J_H + M has binary rank 1,024.
Within each complementary packet pair, its quotient is

[[I, A], [A, I]], A = I+J_16, A^2 = I.

Its image consists of (z,Az). Every nonzero vector has support on both
packet halves, each of which spans its full eight-dimensional space.
Therefore every nonzero image column spans all 16 rational dimensions.
Every rank-minimal center factorization costs 1,024 x 16 = 16,384, equal
to the spinor port count. This fused route has no positive rank surplus
under that center model, despite its cheaper compensation circuit.

Changing the fitting kernel
----------------------------
The full-shell hybrid retains the original sparse/sparse and sparse/spinor
orthogonality entries, but replaces spinor/spinor entries by Hadamard-clique
blocks. Exact binary rank is 1,851, versus 815 for the original full kernel.
Its full-width center bill is 29,616 on 30,960 ports. An exact active-pressure
bound gives kappa < 0.000448646990529330 at every odd stage count, even
omitting all helper, center-pressure and idle-pressure costs.

Further exact section searches retain all quadratic Boolean functions on
four coordinates and k independent cubic terms:

k 0 1 2 3 4
spinor ports 1024 2048 4096 8192 16384
binary decoder rank 112 180 272 392 576

Pruning the full-shell spinor interactions to the corresponding disjoint
sections, while keeping all sparse and cross entries, gives full-shell
ranks 2,407, 2,033, 1,675 and 1,359 for k=0,1,2,3 respectively. These are
valid fitting matrices but no cheap complete routing words were found.

The sparse/spinor cross block has exact rank 470. A tested natural-row
factor has every decoder support spanning the full rational 16-space:
low scalar rank again fails to guarantee a cheap physical decoder.

The earlier residual D = C + J_H has a stronger obstruction than the
natural-column check: every nonzero vector in its image spans at least
nine rational address dimensions. The proof uses the degree-six Boolean
polynomial for C, invariance of J_H under H, and the minimum-weight bound
for degree-five Boolean polynomials. A support in at most eight rational
dimensions would have at most 128 spinor lines. Its H-derivatives would
have at most 256 entries, below the 512 minimum for a nonzero derivative,
forcing a whole orthogonal 16-port H-clique into that small subspace.

Consequently any rank-minimal 1,268-column factorization implemented by
full-frame aggregates and private decoder copies pays at least 11,412
descent ranks. This exceeds the spinor surplus of 7,168. The result covers
every minimal basis of that residual, but not redundant factorizations,
ordinary helper computations or fused circuits with different endpoints.

Reproduction and scope
-----------------------
workbench/continuation/ contains the scripts and exact receipts under
tensor-routes/, endpoint-costs/, spinor-routes/ and section-search/.
Portable drivers replay the published receipts with outputs outside the
checkout; use the top-level verify.py --workbench command in README.txt.
The native tensor interface is inherited rather than independently
re-emitted. Optional search scripts and partial exploratory receipts are
identified in the subtree READMEs and are not elevated into global bounds.

The experiments do not establish a new integer-multiplication exponent,
a universal obstruction to Niemeier lattices, or an O(1) indexing result.

Attribution
------------
New experiments: DominikScholz with OpenAI GPT-6 Astra in T3 Code.
The tensor control transports the existing E8/Hamming native word and
PR #600 helper profiles; it does not claim those circuits as new. Preserve
the contributor lineage in NOTICE and in the pinned upstream sources.

Related primary sources consulted for orthogonal-basis and Pauli structure:
M. Waegell and P. K. Aravind, https://arxiv.org/abs/1502.04350;
Simon Schmidt, https://arxiv.org/abs/2209.14906;
K. Abdukhalikov, E. Bannai and S. Suda, https://arxiv.org/abs/0802.1425.
The finite partitions, rank receipts and circuit deductions above are
checked locally, not attributed to those papers.
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