local_star_fan.py: the size step |Phi(D-v)| <= |Phi(D)| localizes to a
star-vs-fan contribution comparison at v. But |Star(t)| >= min_root
|Fan(t)| is FALSE (6586 violations) -- the star's extra v-constraint
(sum mu ≡ 0) can make it realize fewer boundary vectors than the fan when
the link has interior vertices. So Strategy A is globally true (100%) but
NOT via per-vertex local domination; the size inequality needs a global
union/choice-of-v argument. Useful byproduct: the Boolean-bit / mod-3
incidence reformulation of Phi.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
reduction_exists.py: every disk with k>=1 (and every irreducible disk)
has a Phi-NON-INCREASING vertex removal (|Phi(D-v)| <= |Phi(D)|), 100%
over thousands of disks -- so induction to the k=0 base case is viable.
BUT the clean set-inclusion Phi(D-v) subset Phi(D) holds for only ~8%, so
the size step cannot be proved by "re-inserting v loses no sequence"; it
needs a genuine cardinality injection between non-nested sets.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
wheel_extremal.py: |Phi(wheel_n)| = floor(2^n/3) exactly (ratio ->4/3),
but the wheel is NOT the irreducible minimiser for n>=6. The extremal disk
is a single MINIMAL-degree interior vertex (degree 4 or 5, both tie),
giving |Phi| = (5/4)*2^(n-2) = 5*2^(n-4). The ratio rises monotonically
with center degree, 5/4 -> 4/3, so minimal degree is extremal. Sharpens
the irreducible lemma to |Phi| >= (5/4)*2^(n-2), tight at the degree-4/5
patch.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
irreducible_floor.py: over 10k+ irreducible disks (k>=1, min interior
degree >=4), |Phi| never violates 2^(n-2) and never sits on it -- min is
5*2^(n-4) = (5/4)2^(n-2), the wheel being the minimizer. Universal toggles
are dead (99.9% have zero boundary-only faces). Since un-stacking degree-3
vertices preserves Phi and terminates at a k=0 or irreducible residue, the
whole lower bound reduces to: every irreducible disk has |Phi| >= 2^(n-2).
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
monotonicity_test.py inserts interior vertices and checks |Phi|. Degree-3
stacks preserve Phi exactly (confirms un-stacking, 100%), but degree-4
insertions can SHRINK Phi (6->5, 30->28) and Phi(D') subset Phi(D) fails
~13% -- so the reduce-to-base-case proof of the 2^(n-2) floor via
monotonicity does not work. Violations stay above the floor, so the floor
is protected by something stronger; redirect to a direct n-2 toggle
construction.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
The stacked-only search missed non-stacked disks, and cocircular boundary
points gave degenerate Delaunay (invalid disks, spurious sub-floor |Phi|).
Add floor_diverse_disks.py: 1700+ validated disks per n (convex non-
cocircular boundary, face-count and boundary-edge checks) confirm min|Phi|
= 2^(n-2). Note records that interior structure tends to ENLARGE Phi
(wheel 5 vs fan 4) and that depth adds two faces per one constraint.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Experiments probing the cluster restriction set R_K / Phi: R_K is a Z/3
zonotope (not a GF(3) subspace), the "richness" invariant is an artifact
of non-shrinking annuli, the interface gluing always works on interior
cycles (forced by 4CT), and the maximal constraint achievable on an
n-cycle is a floor of 2^(n-2) -- already reached by the trivial tire.
Note boundary_restriction_structure.tex writes these up.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis