OMEGA PAPER IX — EXACT NEIGHBORHOOD-COVER CONTINUATION
Research checkpoint, 1 November 2026. This is a companion calculation, not
a manuscript redraft. The original construction and rational constants are
unchanged.

COMPLETED RESULT

For the original output-conditioned witness problem on the unbounded integer
domain, the certified interval improves from 307..353 to

    307 <= ell* <= 352 bits.

The same maximum occurs on the unbounded nonnegative domain by the preserved
phase-realizability/translation proof. The lower endpoint does not automatically
apply to a bounded input cube. The upper bound applies to every subset of the
input domain, including any such cube. Output and shared metadata are excluded
from this witness-width accounting.

The key new integer is an UPPER BOUND on one 13-dimensional projected volume:

    U = 59,529,412,393,701,779,352,833,006.

This package proves

    F_max^13 * Delta^13 <= U^120 < (2^352 * Delta)^13.

Thus F_max < 2^352. Averaging and the preserved full-width box realization give
F_max > 2^306, so ceil(log2(F_max)) lies between 307 and 352.

No exact maximum fiber, exact optimal witness width, exact collision moment
C_10, exact 13-dimensional projected volume, or efficient codec is computed.
The result does not establish physical or perceptual claims.

FIXED DEFINITIONS AND PROVENANCE

    Q = 339949771344778
    S = 10^15
    D = 25*S
    M = 13*I + A, where A is the canonical 600-cell edge adjacency matrix
    original output T(d) = floor(Q*M*d / D), componentwise mathematical floor
    B = {0,...,73}^120
    G = Z^120 / M*Z^120
    F(c) = #{u in B : [u]=c}
    F_max = max_c F(c)
    ell* = ceil(log2(F_max))
    N = 74^120
    Delta = det(M) = 2^80 * 5^27 * 11^36 * 13^25 * 41^9 * 211^4.

The source baseline/verify_structure.py is preserved byte for byte:
    SHA-256 57100e240ba7d1a8c10c133e51937461cf6b9b0cb1ef1602671095bd4375917b
The graph's pinned canonical JSON digest remains:
    40bc6a1d52efab3d75cf343e65ac06b314f9c9c375fc02bf6d45bf6bb21f9529
The integer inverse denominator remains h = 4,948,372,000, and the infinity
norm numerator remains 801,320,608. The baseline proof of the exact reduction
is preserved in baseline/STRUCTURE_REDUCTION_AND_PROOFS.md. Its old bit-bound
labels are historical and are not edited to make it appear to prove this
later result.

The previous 307..353 block package and subsequent lower-bound/Fourier
research remain prior work. This new package rechecks the original 307-bit
lower inequality; it does not independently recertify every later lower-bound
refinement or moment obstruction, and does not supersede their proofs.

1. UNIFORM PACKING ARGUMENT

Let Z = [0,73]^120 + M[0,1]^120. Fix any coset c and representative u_0.
Every u in B in that coset has a unique form u=u_0+M*z with z integral.
The cells z+[0,1)^120 are disjoint. Their images

    u_0 + M*(z+[0,1)^120) = u + M[0,1)^120

remain disjoint, each has volume Delta, and all lie in Z. Therefore

    F(c)*Delta <= vol_120(Z), for every c.

F(c)*Delta is the volume of this union of cells. It is not being treated as
the cardinality of a discrete set when applying a continuous volume theorem.

2. EXACT UNIFORM COVER BY CLOSED NEIGHBORHOODS

Take W = {0} union N(0) in the fixed graph. In its pinned vertex order this is

    [0,33,34,35,36,37,38,39,40,41,42,43,44].

It contains 13 vertices and 42 internal edges. The projected generator list
of Z has 46 distinct directions after identical vectors are combined.

The verifiers construct 120 explicit coordinate permutations by exact left
quaternion multiplication on the original vertex coordinates. A pair (a,b)
represents (a+b*phi)/2 with phi^2=phi+1. Every multiplication and divisibility
check uses integer pairs, not approximated quaternion coordinates.

For each permutation p the checker verifies all of the following:
  * p is a permutation of all 120 coordinates;
  * p carries every vertex's neighbor set exactly to the neighbor set of p(v);
  * the projected weighted generators on p(W), in transported coordinate
    order, are identical to those on W.

The 120 translated sets are explicitly checked to cover every coordinate
exactly 13 times. No claim about having enumerated the entire automorphism
group is needed. Each of their projected volumes equals V_W=vol_13(pi_W Z).

The Bollobas-Thomason uniform-cover inequality consequently gives

    vol_120(Z)^13 <= product_p vol_13(pi_p(W) Z) = V_W^120.

The exact-cover hypothesis matters. Merely having coordinate weights at least
one is insufficient for an unrestricted continuous-volume assertion: on a
one-dimensional interval of length 1/2, using its sole coordinate twice with
weight one would incorrectly assert 1/2 <= 1/4.

For completeness, the required volume theorem follows from the discrete
entropy cover inequality and a grid limit. For a finite set X and a uniform
random point x in X, expand H(x_W) in increasing coordinate order. Conditioning
on fewer earlier coordinates can only increase conditional entropy. Summing
over an exact k-cover therefore gives sum_W H(x_W) >= k*H(x). Also H(x_W) <=
log|pi_W X| and H(x)=log|X|. Hence |X|^k <= product_W |pi_W X|. Apply this to
the grid points in a full-dimensional compact convex body, multiply by the
corresponding powers of the grid spacing, and let the spacing tend to zero.
The exact-cover equality makes the dimensions cancel, and convex-body
boundaries have measure zero. This proves the volume inequality used here.

3. VOLUME POLYNOMIAL AND ITS COEFFICIENT UPPER BOUND

For a row set W of size m let

    V_W(t) = vol_m(t*[0,1]^m + pi_W(M[0,1]^120)).

The standard zonotope determinant formula, expanded over the coordinate-axis
generators, yields

    V_W(t) = sum_{r=0}^m a_r*t^(m-r),
    a_r = sum_{I subset W, |I|=r} sum_{J subset {0,...,119}, |J|=r}
              |det(M_{I,J})|,
    a_0 = 1.

Indeed, selecting m-r axis generators forces the complementary r rows I.
Laplace expansion leaves t^(m-r) times the displayed minor. Summing over all
generator choices gives the formula. The general determinant-volume formula
is also proved by segment addition in the preserved baseline proof.

For a nonnegative square matrix K, the triangle inequality applied to its
Leibniz determinant expansion proves

    |det(K)| <= perm(K).

M has only the nonnegative entries 0,1,13, so define

    p_r = sum_{|I|=|J|=r} perm(M_{I,J}).

Then a_r <= p_r coefficient by coefficient. Since t=73 is nonnegative, replacing
any uncomputed a_r by p_r gives a rigorous upper bound on the projected volume.

4. EXACT PERMANENT COEFFICIENTS WITHOUT ENUMERATING ALL MINORS

The p_r are computed by a small matching recurrence on the m rows of W.
After processing columns 0,...,j-1, let D_j(T) be the total weight of injective
matchings covering exactly the row subset T using those columns. Initially
D_0(empty)=1, with all other states zero. For the next column j,

    D_{j+1}(T) = D_j(T) + sum_{i in T} M_{i,j} * D_j(T\{i}).

The first case does not use column j; the other cases match it to exactly one
row i. These cases are disjoint and exhaustive. This proves the recurrence.
An injective matching between fixed r-row and r-column sets contributes
exactly one term of the corresponding permanent, so

    p_r = sum_{|T|=r} D_120(T).

The implementation processes masks in descending order, ensuring that every
predecessor is still its previous-column value. Its memory is 2^13 = 8192
integer states, and an elementary operation bound is O(120*13*2^13).

For W above the complete coefficient list is:

 r   p_r (exact permanent sums)
 0   1
 1   325
 2   47352
 3   4091656
 4   233723200
 5   9310255536
 6   265780471680
 7   5498521219872
 8   82326531807696
 9   879734614473680
10   6510322192690880
11   31533042290406720
12   89385221732384160
13   111664804452677280

These values are independently controlled on ten small rectangular matrices
by exhaustive row/column subset enumeration and direct permutation sums.
The general justification is the recurrence proof, not those finite controls.

5. EXACT LOW-ORDER COEFFICIENTS AND THE NEW BOUND

For each I subset W of size r<=5, the checker forms the weighted nonzero
projected columns of M, with no added box-axis segments. Their r-dimensional
zonotope volume is sum_J |det(M_{I,J})|. Summing these exact small volumes over
I computes a_r. Combining identical projected columns is valid: any determinant
selecting two identical columns is zero; for distinct directions the product
of their multiplicities counts every column choice.

The executed values and numbers of enumerated generator subsets are:

 r   a_r (exact)    enumerated generator subsets
 0   1             coefficient fixed by definition
 1   325           26
 2   45408         780
 3   3647440       28536
 4   188983720     1010815
 5   6697090056    21108096

There are 2379 low-dimensional projection cases and 22,148,253 total minor
subsets. Every determinant has dimension at most five. This replaces a direct
13-dimensional enumeration of C(46,13)=101,766,230,790 minors.

Define

    U = sum_{r=0}^5 a_r*73^(13-r) + sum_{r=6}^13 p_r*73^(13-r)
      = 59529412393701779352833006.

Then V_W(73) <= U by coefficient domination. Combining packing, the exact
13-cover, and this volume upper bound proves

    F_max^13 * Delta^13 <= vol_120(Z)^13 <= V_W(73)^120 <= U^120.

The verifier checks the strict integer inequality

    U^120 < (2^352*Delta)^13.

An optional exact integer upper bound, obtained by integer-root bracketing, is

    F_max <= 6532878329630567505239614989176153262981819298946574097022310908426237298103083709098862916506832521344797.

Terms through order four alone also suffice for 352 bits, with the larger
volume upper bound 61636826066837019638356886. Order five gives the stronger
cardinality bound recorded above. Neither is an exact projected volume.

For orientation only, converting the final bound to a floating-point log
gives about 351.51017 bits. This display approximation is not used in any
decision or proof. The verifier also checks that the numerical upper bound
lies above 2^351; that is NOT a lower bound on the true F_max.

6. ADDITIONAL COMPLETED EIGHT-COORDINATE CALCULATION

The suggested connected eight-vertex test was also executed directly on

    [3,6,16,39,40,61,93,94].

It has 19 internal edges and 33 weighted projected generator directions.
The exact projected volume is

    V_8 = 7467853699089983,

from all C(33,8)=13,884,156 generator subsets. Its 120 explicitly checked
translations form an exact 8-cover. Hence F_max*Delta <= V_8^15. This is a
smaller cardinality bound than the former 7511951082105388^15/Delta, but still
lies between the numerical upper thresholds 2^352 and 2^353. The 13-coordinate
bound above is stronger and supplies the actual one-bit endpoint improvement.

The eight-vertex cluster was found by heuristic search. Neither its global
optimality among eight-vertex clusters nor a maximum internal-edge theorem is
claimed. The final 13-vertex choice is the entire closed neighborhood and does
not require the heuristic search to reproduce it.

7. AUDIT OF THE SUPPLIED EXTERNAL RESEARCH REPORT

The uniform-cover direction is valid when used with its exact-cover hypotheses.
It led to the completed 352-bit result above. Several other statements in the
report should not enter Paper IX as established facts:

* Connected overlapping clusters do not automatically yield 350 or 351 bits.
  The completed eight-coordinate volume still gives 353, while this new
  13-coordinate upper bound gives 352. No global optimality is claimed.

* The proposed higher collision-moment Fourier formula is false. With the
  unnormalized convention hat(F)(chi)=sum_c F(c)*conjugate(chi(c)), Parseval is

      C_2 = (1/Delta)*sum_chi |hat(F)(chi)|^2.

  For general p, the correct identity is

      C_p = Delta^(1-p) *
              sum_{chi_1*...*chi_p=1} product_j hat(F)(chi_j).

  It couples p characters; it is not a sum of their individual absolute pth
  powers. For example, on Z/2Z with F=(1,2), the Fourier coefficients are
  (3,-1), C_4=17, but the report's expression is (3^4+1)/2^3=41/4.

  More generally, sum_chi |hat(F)(chi)|^(2k) equals
  Delta*sum_c |F^{*k}(c)|^2 for unnormalized group convolution. It is not
  Delta^(2k-1)*C_(2k). Thus the proposed substitution into the tail bound
  cannot be used. Known lower bounds on spatial collision moments do not
  automatically become computable bounds on these spectral moments.

  A legitimate tail inequality remains available: for normalized amplitudes
  a_chi=|hat(F)(chi)|/N, a chosen character set S, and q>=1,

      sum_chi a_chi <= sum_{chi in S} a_chi +
          |S^c|^(1-1/q)*(sum_{chi in S^c} a_chi^q)^(1/q).

  It requires an actually justified spectral tail upper bound. The subtraction
  of selected-mode powers must use suitable certified lower bounds when
  subtracted from a total upper bound. The full tail is not evaluated here.
  Having certified 240 particular large modes does not prove that those are
  all large modes.

* The canonical graph has BFS distance shells [1,12,32,42,32,1]. Its squared
  graph has degree 44. The nine algebraic association classes are not nine
  ordinary graph-distance shells. The supplied degree/shell figures and
  uncertified treewidth lower bounds cannot justify impossibility statements.
  A large dense contraction table does not prove every compressed algorithm
  impossible.

* The actual eigenvalues of M and their multiplicities are
      10:16, 11:36, 13:25, 16:16, 25:1,
      16+3*sqrt(5):4, 16-3*sqrt(5):4,
      15+2*sqrt(5):9, 15-2*sqrt(5):9.
  The orbit verifier recovers these multiplicities by exact traces and the
  certified annihilator. Thus the minimum adjacency eigenvalue is
  3-3*sqrt(5), and the minimum M eigenvalue is 16-3*sqrt(5).
  The annihilator constant has absolute value 24,741,860,000 = 5*h, not h/2.
  A spectral decomposition does not by itself evaluate arbitrary minors.

* The open 12-vertex neighborhood has 45 generator directions and
  C(45,12)=28,760,021,745 direct minors. The closed 13-vertex neighborhood has
  46 directions and 101,766,230,790 direct minors. This makes the coefficient
  bound useful; the report's direct-volume runtime prediction was unsupported.
  Two tetrahedra sharing a face have five vertices, not eight.

* The general all-minors volume expansion is valid. A graphical-forest shortcut
  for this particular generator matrix was not established. The cited Stanley
  PDF ending in 85.pdf is a different paper; the author's publications list
  links the graphical-degree zonotope paper as 83.pdf.

8. EXECUTION AND REPRODUCTION

Environment of the completed runs: Python 3.12.14, g++ 13.3.0, Linux x86_64.
No third-party Python packages are required. Run from any directory:

    python3 verify_neighborhood_bound.py --output my_result.json
    python3 -O verify_neighborhood_bound.py --output my_optimized_result.json

For the faster order-four bound:

    python3 verify_neighborhood_bound.py --max-order 4 --output order4.json

The two complete default neighborhood runs took approximately 15 seconds each
in this environment. Their stdout and complete result JSON were byte-for-byte
identical. Normal and optimized eight-coordinate runs likewise had identical
stdout and result JSON. Exact logs, coefficient case tables, automorphisms,
cover sets, source fingerprints, and a package manifest are included.

To reproduce the additional eight-coordinate result:

    python3 verify_orbit_cover.py --output my_orbit_result.json

The native volume kernel is unchanged from the earlier block certificate. Its
SHA-256 is c421badd2b050edc6ecddb8717074bc372eebbd79fb2d9f6ce053e34993dd556.
It uses exact Bareiss determinants, checks every division, and enforces the
original input contract: dimension <=8, entries <=13, weights <=98, and
weighted coordinate spans <=98. Here the new low-order cases have dimension
<=5 and span 25. Under the full contract every determinant is bounded by
B=8!*13^8; Bareiss product-difference intermediates are bounded by 2*B^2 and
determinant-times-weight terms by B*98^8, both below 2^127. Partial volume sums
are nonnegative and bounded by the coordinate box volume 98^8. Thus its stated
input range is safe for signed 128-bit arithmetic. All global powers, DP
weights, and integer-root comparisons use Python arbitrary-precision integers.

The finite controls check the implementations. The universal result rests on
the packing, exact uniform-cover, determinant-volume, determinant-permanent,
and matching-recurrence proofs above.

SOURCE REFERENCES USED FOR THE GEOMETRIC STEP

B. Bollobas and A. Thomason, Projections of bodies and hereditary properties
of hypergraphs, Bulletin of the London Mathematical Society 27(5), 417-424
(1995). https://doi.org/10.1112/blms/27.5.417

An accessible precise statement of the exact-cover theorem is Theorem 1.1,
equation (1.5), in S. Brazitikos, A. Giannopoulos and D-M. Liakopoulos, Uniform
cover inequalities for the volume of coordinate sections and projections of
convex bodies (2016), https://arxiv.org/pdf/1606.03779 . The displayed theorem
was checked rather than relying on the imprecisely worded abstract.

R. P. Stanley's author-maintained publications list:
https://math.mit.edu/~rstan/pubs/ ; correct linked graphical-degree zonotope
paper: https://math.mit.edu/~rstan/pubs/pubfiles/83.pdf . The determinant-volume
argument needed here is supplied in the preserved baseline and above; no
unverified theorem-number or forest specialization is required.

UNFINISHED TASK

Continue tightening a universal upper bound or producing a stronger attained
fiber lower bound until consecutive powers of two trap F_max. The present
result narrows the certified bit interval by one bit. It does not determine
how near the true optimum is to either endpoint. The Fourier tail, literal
maximum, collision moments, optimal width, and practical decoding remain
distinct unfinished tasks.
