Records the degree-5 double-contraction proof skeleton: Kempe chains as
Heawood face-chains (flip-set + same-side sign rule), the chain-transport
hypothesis L1, the inner/outer nested reductio, H'' as per-interface
contraction along the chain, and the single emptiness Claim it reduces to,
with the Errera oracle as the stress test.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Section 4 no longer states the floor as a proven Proposition. Now: prove
interior-free disks attain 2^(n-2) (ear-peeling) and the un-stacking
lemma, state |Phi(D)| >= 2^(n-2) as a Conjecture, and give an honest
status remark -- holds for the Apollonian class, reduces to the
irreducible case, empirically strict (5/4), but |Phi| is NOT monotone
(the earlier freedom-positive monotonicity claim was wrong) and both
natural elementary proofs provably fail. Soften the note's observation to
match.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
transversal.py: a STRONG transversal (n-2 faces whose Boolean assignment
single-valuedly and injectively determines the boundary) would give a
constructive proof. It exists in 0/2948 disks with k>=1 -- once there is
any interior vertex, fixing n-2 faces leaves boundary-visible completion
freedom, so the boundary is never single-valued in them. Works only at
k=0 (base case). Both elementary routes (reduction localization, direct
transversal) now closed.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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>
Section 4: un-stacking lemma (degree-3 removal preserves Phi, proved),
ear-peeling base case (k=0 => 2^(n-2)), reduction to the irreducible
case, and the irreducible lemma as the sole open conjecture (|Phi| >=
5/4 * 2^(n-2), tight at the degree-4/5 patch; wheel = floor(2^n/3) is not
extremal). Records the two dead ends (monotonicity false, universal
toggles insufficient) and ties each claim to its experiment.
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>
Remark: a disk with k interior vertices has 2k+n-2 faces (Euler) but only
k interior constraints, so each interior vertex adds two degrees of
freedom against one constraint -- depth is freedom-positive and Phi can
only retain or enlarge below the interior-free floor 2^(n-2). Motivates
the lower bound and replaces the prior TODO sketch.
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>
Add section 4: define the achievable boundary set Phi(D) of a triangulated
disk and state the constraint-floor proposition |Phi(D)| >= 2^(n-2), with
the attainment direction proved (fan injectivity) and the lower bound left
as a marked gap with strategy. Remark records the zonotope structure and
the short-interface concentration of difficulty.
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>
The bounded-face sum omits the outer face at outer-boundary vertices, so
restrict the gluing identity to interior vertices (where all cluster
interfaces live) and recover a colouring by carrying a single +/-1 label
on the unbounded face f_inf, giving Heawood's identity on the full cubic
dual for the Tait step.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add the two-sided cluster decomposition proposition: a vertex's full
Heawood face-sum splits as exactly one child-cluster contribution plus
one parent-cluster contribution (the at-most-two-clusters bound makes the
pairing binary and complete). Explain why this fails per-tire -- a vertex
on many same-depth tires has only a fragment of its face-star in any one
tire -- and recast the chain-pigeonhole and 4CT conjectures to nested
clusters with a cluster restriction relation.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Define a connected tire cluster (union of same-depth tires joined by
shared vertices, transitive closure), prove same-depth tires meet only
in vertices, and prove every vertex lies in at most two clusters (one at
each of two consecutive depths) -- the bounded coarsening of the
unbounded per-vertex tire count.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Define a +/-1 Heawood face-labelling of a tire, its induced boundary
Heawood sequences and restriction relation, and interface compatibility
(0<->0, +1<->-1 = vertex face-sum vanishes mod 3). State the Heawood
chain-pigeonhole conjecture and a tire route to the Four Colour Theorem,
parallel to the medial programme.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add a new paper stub referencing the nested tire decompositions paper,
with intro, Heawood bibliography entry, and an empty restrictions section.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis