Open the walk-count sub-problem (route-b combinatorial core) + W2 by DrMurphyIsIn · Pull Request #154 · DrMurphyIsIn/Arda

DrMurphyIsIn · 2026-08-29T10:33:53Z

What

Opens the walk-count sub-problem — the reduced, combinatorial core of route (b) from BG_CAPACITY_ATTACK_SPEC.md. Replaces the transcendental free-energy by local, polynomial spectral moments m_k=(1/n)Tr N^{2k} (weighted closed-walk densities), and reduces the upper bound to a moment-body optimization: max_{m∈𝓜_K} Σc_k m_k = log ρ* over achievable tree spectral-moment vectors.

Key structural findings

Different trees maximize different m_k (paths → high moments, hubs → low moments); the extremal caterpillar maximizes none individually.
The envelope coeffs alternate sign (c≈[+.50,−.24,+.12,−.035]), so the free-energy is the balanced alternating combination c_1m_1−|c_2|m_2+… — the caterpillar balances it. So "bound each m_k" is the wrong sub-problem; it's the joint moment SDP (finite shadow of the free-energy concavity).
Two nested relaxations: (S1) Hankel-PSD moment SDP, (S2) tree-specific walk-count cuts. Milestones W1 (SOS envelope) → W5 (poly(n)).

W2 first result (essentially closed)

max_T m_1 = max (2/n)Σ_e 1/(deg_i deg_j) = the path (n+1)/(2n)→½ for small n, a double-broom for n≥10, monotone decreasing/bounded (≤5/8, sharp at n=4). A clean rational bound via degree-sequence optimization under the handshake — the entry moment is controlled.

Why this level

The moments are local/polynomial, so bounding them is combinatorial — sidestepping (a)'s collective wall (in the per-vertex log a_v) and Koiran's SOS no-go (which blocks certifying the permanent, not moment inequalities). Open risk: whether (S1)+(S2) close tightly to ρ*^n·poly(n) — real research, but now finite-dimensional. conjecture1_proved = False.

🤖 Generated with Claude Code

https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…W2 first result Deliberate effort: BG_WALK_COUNT_SUBPROBLEM.md. Reduces route-(b) free-energy to a MOMENT-BODY optimization max_{m in M_K} sum c_k m_k = log rho* over achievable tree spectral-moment vectors (m_k=(1/n)Tr N^{2k}=weighted closed-walk density, LOCAL/polynomial). KEY STRUCTURAL FINDINGS: different trees maximize different m_k (path high-k, hubs low-k); caterpillar maximizes NONE individually; the free-energy is the balanced ALTERNATING combination (c~[+.50,-.24,+.12,-.035]) which the caterpillar maximizes = finite shadow of free-energy concavity => 'bound each m_k' is WRONG sub-problem, need the joint moment SDP. Milestones W1(SOS envelope)-W5(poly(n)). W2 FIRST RESULT: max_T m_1=(2/n)sum 1/(deg deg) is path (n+1)/(2n) small-n, double-broom n>=10, monotone DECREASING/bounded (<=5/8 ->~1/2); entry moment closed by degree-seq optimization. Next: W3 joint moment SDP (S1 Hankel-PSD + S2 tree walk-count cuts). conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…P + walk-count cut W3 (done): per-moment bounds overshoot (+0.025); Hankel-PSD SDP + m_1 overshoots at caterpillar m_1=0.52 (+0.004), closes only at m_1<=0.505 (excludes caterpillar). Cut LOCATED: envelope c_2<0 pushes m_2 to Hankel floor m_1^2=0.270, below caterpillar m_2=0.308 -- trees don't reach it. Missing = even-moment LOWER bound m_2>=phi(m_1)>m_1^2. W4 (progress): tree (m_1,m_2) lower boundary traced by caterpillar family (a=7->0.308, a=13->0.287), strictly above Hankel floor. m_2=(1/n)Tr N^4 is an EXPLICIT LOCAL degree formula (cherry-returns + neighbor-of-neighbor 4-walks) => the S2 cut is a per-neighborhood inequality, PROVABLE (vs transcendental per-vertex log a_v). Remaining: derive tight phi, re-solve SDP with cut, W5 poly(n). Route (b) is now a FINITE local moment-SDP+walk-count problem -- no transcendental obstruction left. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…validated) (1) m_2 local degree formula VERIFIED == (1/n)Tr N^4 all trees n=4..9 (cherry-return + neighbor-of- neighbor 4-walks): the cut is a concrete provable per-neighborhood degree inequality. (2) adding tree m_2 lower-bound cut collapses the SDP gap: none +0.0043 -> m_2>=0.287 +0.0028 -> m_2>=0.308(caterpillar) +0.0008 (essentially closed). residual = K=4 envelope order + missing m_4 cut. => moment-SDP + PROVABLE LOCAL walk-count cut drives bound to log rho*; route (b) genuinely evades the collective wall (which lived in transcendental per-vertex log a_v). Remaining: derive tight phi/m_4 cuts closed-form, higher-K envelope, W5 poly(n) -- all FINITE/combinatorial, no transcendental obstruction. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

@A

…boundary) Closed-form m_1(a), m_2(a) for the infinite periodic a-arm caterpillar via local degree formulas (verified vs empirical <2e-3): m_1(a)=(2/(1+2a))[1/(a+2)^2+a/(2(a+2))+a/2]; m_2(a)=[N4_spine+a N4_mid+ a N4_leaf]/(1+2a) with explicit N4_* degree terms. Boundary curve (0.520,0.372)@A=1 -> (1/2,1/4)@A->inf. S2 cut: m_2(T)>=phi(m_1(T)), phi=this curve (parametric, monotone/invertible). Remaining W4d: PROVE m_2>=phi(m_1) all trees (per-neighborhood inequality on explicit local formulas; caterpillar tight => Lagrange/rearrangement over degree seqs) + m_4 cut for residual +0.0008 + re-solve SDP higher-K envelope => exact closure; then W5 poly(n). All finite/combinatorial. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…nequality m_2>=phi(m_1) SDP closure: m_1<=.52 +0.0043 -> +m_2>=.308 +0.00078 -> +m_2,m_4>=.140 +0.00078 (UNCHANGED). m_4 cut NOT needed; residual +0.0008 is K=4 envelope order (reducible higher-K), not a missing moment. So route (b) upper bound reduces to a SINGLE per-neighborhood degree inequality m_2(T)>=phi(m_1(T)) all trees (phi = closed-form caterpillar boundary W4c), + higher-K SOS envelope + W5 poly(n). Explicit local formulas in hand; caterpillar tight => provable by local-move/rearrangement/Lagrange. Sole remaining theorem of route (b). conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…uality claim) RETRACT: 'route (b) reduces to ONE inequality m_2(T)>=phi(m_1(T)) for every tree' is OVERSTATED. Exhaustive n<=14: m_2>=phi(m_1) FAILS low-m_1 (58 violations, min slack -0.11 at m_1~0.14). Global (m_1,min-m_2) boundary = STARS at low m_1 (m_2=m_1), caterpillar family only near m_1~0.52 (extremum band). TWO independent implementations converged on this scoping. CORRECT remaining theorem: caterpillar maximizes LOCAL POLYNOMIAL functional G(T)=sum c_k m_k over trees (scoped to extremum m_1 band, NOT universal cut) -- still local/polynomial (provable by rearrangement, real gain over transcendental F) but the FULL extremal optimization. Independently de-risked: exact m_1 formula, trees >Hankel floor by >=+0.08, SDP-overshoot measure tree-unreachable, caterpillar m_2 values -- all reproduced from scratch by a second implementation. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…_k = log rho* G over large families: cat a=7=0.205140 (MAX), a6/a8=0.20511, a5/a10=0.20497, path 0.18882, star 0.00231, broom 0.06558; log rho*=0.205098 (cat a=7 +0.00004 = K=4 envelope order). Corrected scoped W4d' theorem (caterpillar maximizes local polynomial G) confirmed numerically. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…edge-discharging potential Attacking the W4d' proof surfaced a scoping refinement (exact-rational, reproducible): - max_|T|=n F(T) decreases to log rho* FROM ABOVE (single edge = (1/2)log2 > log rho*); per/prod_deg <= rho*^n is FALSE (ratio 1.327 at n=2). rho* is a thermodynamic-limit growth rate, not a finite max. Finite bound: per/prod_deg <= C rho*^n, C>=1.327. - W4d' 'caterpillar maximizes G over trees' holds only asymptotically/among bulk families. - Finite-n maximizers are explicit length-2-arm caterpillars (single hub odd n, two hubs even n -- the cherry-parity oscillation). - Verified exact per-vertex LOCAL formulas for m_1,m_2 (0 mismatches, 47 trees): both are averages of a local 1-neighbourhood functional => certificate is an antisymmetric edge discharging potential w(x,y)=-w(y,x) with a telescoping per-vertex bound, tight at the 3 caterpillar vertex types -- structurally the folded cavity-potential that closed Laplacian Phi<=1. Adds runnable reproduction: telperion/docs/bg_walk_counts_reproduce.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…close The discharging LP (min B over antisymmetric w s.t. per-vertex g - sum w <= B for all bulk-realizable profiles) converges to min B = 0.23099, a hard floor +0.026 above log rho* = 0.20510. Pinned exactly (hand-verified, cap-independent) by the profile pair {leaf (1,{2}) g=0.21164, path-interior (2,{1,2}) g=0.25034}: B >= (g_leaf+g_path)/2 = 0.23099. So the LINEAR discharging relaxation alone has an integrality-gap plateau at the small-path free energy -- it cannot tell 'these profiles coexist only in a low-density path' from 'they tile a high-density tree'. The missing constraint is measure-realizability = the Hankel-PSD moment body (ingredient 2). Concretely validates the plan's moment-SDP emphasis: the linear cuts (ingredient 3) need Hankel PSD (ingredient 2) to close. Reproduction: bg_discharge_lp.py. conjecture1_proved = False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…t is measure-realizability (a) Moment-SDP proper (cvxpy): Hankel-PSD alone stays at (1/2)log2=0.34657 (single-edge delta_1 admissible at every Lasserre order); +m1 cut vacuous; +m2>=phi(m1) convex caterpillar-envelope cut CLOSES to gap +0.0010 (K=4/6), argmax pinned at m1~0.523 = the caterpillar (matches prior W4 +0.0008). The m2 cut is the load-bearing constraint. (b) K=4 discharging: tighter envelope makes the 1-hop antisymmetric potential TIGHT at the caterpillar (all 3 vertex types -> log rho* exactly, residual 3e-5; K=2 residual 8e-3), but the global plateau barely moves (P4 floor 0.22913 vs 0.23099). Higher order makes the extremizer locally certifiable but does NOT remove the path-profile floor -- residual gap is measure-realizability, not locality/order. (c) The m2>=phi(m1) cut is NOT elementary convexity: Cauchy-Schwarz+Jensen give only m2>=2m1^2-m1 (valid, 0/2287 violations) but useless at the band (0.021 vs true 0.308), because caterpillar x_v variance is negligible (0.0015) so m2~2m1^2-avg(Q/d^2) and the cut is an upper bound on avg(Q_v/d_v^2) at fixed m1 -- needs the joint degree distribution (local flag-algebra / moment SDP), dual-certifiable with Telperion Hankel/SOS. Consolidated: route-(b) certificate = Hankel-PSD + the m2 cut (closes to +0.001); sole open theorem = the cut (caterpillar minimizes m2 at fixed m1, band), provable by a degree-distribution moment/flag SDP with SOS dual -- NOT rearrangement (interior, W5), NOT elementary convexity (W7c). Reproductions: bg_moment_sdp.py, bg_k4_discharge.py, bg_c_convexity.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…icit verified dual certificate Both moments are LINEAR in the vertex-type distribution pi(t), t=(d;{e_1..e_d}), so the m2 cut is an LP: min m2 s.t. m1=M over pi>=0 with the realizability constraints a real tree satisfies (normalization, tree handshake mean-degree-2, and MASS TRANSPORT / unimodularity: (d,e)-edge count equal from each side). The independent-profile discharging floored at 0.231 (W6); adding mass transport lifts min m2 onto the caterpillar boundary across the band -- at the caterpillar's own m1, min m2 = caterpillar m2 to ~1e-4 (DMAX=7 +1.4e-4, 8 +1.1e-4, 9 +0.9e-4, shrinking). The LP DUAL is the certificate (verified): the equality multipliers give a per-type inequality 2x^2-q >= b0 + b1*d + b2*x + sum_a w(d,e_a), valid over ALL 3431 types (worst slack -5.6e-17), with w(d,e)=-w(e,d) the antisymmetric discharging potential read off from the mass-transport duals. Summed over a tree the w-terms telescope -> m2 >= b0+2b1+b2*m1, tight at the caterpillar. This is the folded discharging potential, now VALID because mass transport supplies the missing coupling. Honest caveat: the 1-hop LP is a relaxation (mass-transport+mean-degree necessary not sufficient), so it certifies m2 >= phi(m1) - O(1e-3); the gap grows mildly with the degree cap (DMAX=10 +3e-3). Exact tightness needs higher-order flag constraints = Hankel-PSD on degree-type moments (flag-SDP), same HankelJensenCertificate machinery as the spectral side. Route-(b) architecture now concrete: (1) spectral moment-SDP on m_k (Hankel + m2 cut) closes G to log rho*+O(1e-3); (2) degree-distribution flag-SDP (mass-transport LP + degree Hankel) certifies the m2 cut with explicit antisymmetric-potential dual. Both kernel-gateable via hankel_jensen.py+cone.py. Reproductions: bg_flag_lp.py, bg_flag_robust.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…s a generalized caterpillar Degree-cap convergence at fixed m1=0.520: min m2 = 0.31246/0.30784/0.30485/0.30251 for DMAX 8/9/10/11 -- monotone decreasing with SHRINKING increments (-4.6,-3.0,-2.3e-3), converging from below to the true phi(0.520)~0.302-0.305. Refines the W8 caveat: - It is a valid CONVERGENT lower bound (relaxation of the degree-capped tree min), not a blow-up. - True phi(0.520) is BELOW the uniform caterpillar 0.30841: a direct family search finds mixed-arm/ multi-hub caterpillars beat the uniform a=7 cat at m1=0.520 (a1=8/a2=7 -> m2~0.305), LP<=real confirmed. So the exact extremal boundary is traced by GENERALIZED (period>1, mixed-arm) caterpillars = the cherry-parity/multi-hub oscillation of the finite-n maximizers (W5iii). - W8's 'tight to 1e-4 at DMAX=9' was the converging sequence crossing the uniform-cat level, not exact tightness vs the uniform family. Proof consequence: the target is a HIERARCHY LIMIT (as W5 established -- log rho* is a thermodynamic- limit growth rate). Route (b) = a convergent family of finite kernel-gateable certificates indexed by envelope order K and flag cap DMAX, each proving density <= log rho* + eps(K,DMAX), eps->0. Complete proof STRUCTURE for sup-density = log rho*, modulo standard hierarchy convergence + Hankel/2-ball acceleration for a compact emitter. Uniform caterpillar = near-optimal reference, not exact extremizer. Reproductions: bg_converge.py, bg_lp_vs_real.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…gated (W10) The W8 mass-transport flag-LP dual becomes a first-class Telperion certificate. FlagDischargeCertificate.from_flag_lp(dmax, m1_target) solves the flag-LP, reads the unimodularity duals as the antisymmetric edge potential w(d,e)=-w(e,d), rationalizes to denom 720, and sets b0 to the EXACT rational infimum of the per-type residual so the per-vertex inequality 2x^2 - q >= b0 + b1*d + b2*x + sum_a w(d,d_a) holds by construction. check() re-verifies it EXACTLY over all 3431 degree-<=7 types (worst slack 0, tight at the extremal caterpillar profile) + antisymmetry. Telescoping (sum w=0) + handshake (sum d=2n-2) assemble the certified cut m2(T) >= -1937/3600 + (13/360)(2-2/n) + (1081/720) m1(T) for max-degree <= 7, with -2b1/n the W5 surface term; at the a=5 caterpillar m2 >= 0.32026 vs 0.32164 (gap +0.0014 = rationalization order). lean_module() emits frozen examples/bg_flag_discharge/frozen/BGFlagDischarge.lean of norm_num-checked rational atoms (leaf/arm/hub/tight), wired into the rh_lean FROZEN library as RH.BGFlagDischarge (build.py --check: OK) so the kernel gate re-checks every atom. Generator generate.py; tests test_bg_flag_discharge.py (6 green: exact check, antisymmetry, independent per-type re-derivation, valid-lower-bound-at-caterpillar, atom shape, frozen==generated). Existing BG/hankel tests still green. One finite level of the W9 convergent hierarchy, now kernel-gateable end-to-end. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…d, correct object identified Pushing on tightening the 1-ball flag relaxation (W9) maps the hierarchy: - LINEAR pair/2-ball lift is VACUOUS: joint edge var E(a,b)>=0 with marginals + symmetry adds NOTHING beyond 1-hop mass transport (min m2 unchanged to 1e-5 at DMAX=4,5) -- given mass transport the joint is always fillable. Residual gap NOT closable by linear flag constraints. - NAIVE edge-matrix PSD lift is INVALID: constraining E>>0 OVERSHOOTS (DMAX=4/5 min m2~0.400 > real tree 0.368), excluding real trees. Direct check: a=2 caterpillar's E has eigenvalues incl -68.7 (strongly indefinite -- bipartite hub<->arm<->leaf adjacency, +-symmetric spectrum, never PSD). - CORRECT object: reflection positivity (Lovasz) is PSD on the moment matrix of ROOTED partial- subtree homomorphism densities M[F,F']=t(F u F' at root, T), NOT the raw 2-point adjacency. That matrix IS PSD for every tree -- the valid flag-SDP tightening; building it (rooted-star features + reliable SDP solver) is the genuine remaining construction. Until then the sound finite certificate is the degree-capped 1-ball emitter (W10). Reproductions: bg_level2.py, bg_psd_lift.py, bg_edge_matrix_indefinite.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…sure-extremality Building the reflection-positive moment matrix carefully closes the question of whether any finite flag-SDP level exactly closes the m2 cut. It does not, structurally: - k=1 (single-root) moment matrix M[j,k]=E_v[cnt_v(j)cnt_v(k)] is VACUOUS -- a covariance of 1-ball features, auto-PSD for any pi; adding M>>0 changes min m2 by 0 (DMAX=5). - DICHOTOMY: single-vertex features (k=1, linear moments) -> covariance/transportation, auto-satisfied -> vacuous; edge/two-vertex features (raw E, cavity-pair) -> bipartite adjacency, indefinite for real trees -> invalid. No finite matrix over local types is both valid (PSD on all trees) and biting. - REASON: mass transport + local moments enforce exactly belief-propagation/cavity consistency, and the BP fixed-point set is strictly larger than genuine tree limits. The flag gap IS the gap between locally-consistent (BP) and extremal tree-limit measures -- a measure-extremality/Gibbs-uniqueness phenomenon invisible to any finite convex relaxation. Why route (b) is a convergent hierarchy (W9) with no finite exact level, and why log rho* is a thermodynamic limit (W5) -- same fact three ways. - CONSTRUCTIVE EXACT PATH: trees are loopless => cavity/BP is EXACT per tree (no RSB, no gap); the exact optimum is a variational problem over BP fixed points maximized by the caterpillar's fixed point = the folded cavity-potential that closed Laplacian Phi<=1. Exact route = cavity/interpolation, not a finite moment-SDP. Route (b) delivered what a moment hierarchy can: convergent bound + kernel-gated finite cert (W10); exact closure is the cavity variational proof. Reproduction: bg_k1_moment.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…d; naive local potential hits same wall Built the exact cavity/Bethe machinery (route-a foundation for the matching object): - Exact cavity free energy: messages x_{u->v}=sum_{c!=v} w_uc/(1+x_{c->u}), w=1/(d_u d_c); logZ = sum_v log(1+sum_a w_va q_av) - sum_e log(1+w_e q q). Reproduces log(per/prod deg) EXACTLY (max err 9e-16 over 47 trees n<=8; exact on trees per Heilmann-Lieb). bg_cavity.py. - rho* LOCALIZED as a cavity variational optimum: infinite length-2-arm caterpillar explicit fixed point (x_leaf->AM=0, x_AM->H=1/2, hub quadratic); per-cell density F(a) maxed at a*=7.016 with F(a*)=log rho*=0.205098 (to 2e-8), integer a=7 hits it. rho*=max_a[caterpillar cavity density]. bg_cavity_caterpillar.py. - BUT naive local cavity potential hits the SAME wall (W12 confirmed in cavity space): per-vertex pv=log A_v - (1/2)sum log B_va + message-discharge sum[P(x_av)-P(x_va)] (telescoping) + handshake beta*d, min over configs -> plateaus at 0.331 (gap +0.126, DMAX=4), not log rho*. Same cause: per- config relaxation admits non-realizable message configs + single-edge boundary. Exact-cavity vars do NOT dodge the realizability wall. bg_cavity_potential.py. RIGHT GLOBAL TOOL: Heilmann-Lieb (monomer-dimer has NO phase transition) => cavity recursion is a CONTRACTION with UNIQUE fixed point; the gap is realizability not Gibbs multiplicity. Exact bound should come from a global monotone/contraction argument (Guerra interpolation T<->caterpillar, or a Lyapunov of the contraction), NOT a local potential (provably plateaus). Cavity foundation now in place to support it. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

The r47-regen-diff / casestudy CI job runs 'telperion.cli verify' which requires every examples/*/generate.py to be listed in telperion.toml (a family cannot silently exist). The W10 bg_flag_discharge generator was unlisted -> MANIFEST INCOMPLETE failure. Add it (group=quick). Regenerates without drift; all generators now listed. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…ib-only (sympy-only CI) telperion CI installs only 'sympy pytest' (numpy/scipy/networkx are optional deps). The W10 generator called from_flag_lp (needs scipy) at generation time, so 'verify --group quick' and the unit job would break importing scipy. Refactor: embed the exact rational LP dual (b0,b1,b2,w, denom=720) as literals; certificate() reconstructs via the plain constructor (fractions/itertools only) -- from_flag_lp remains the offline derivation tool. Verified: frozen Lean BYTE-IDENTICAL (no drift), generate+certificate+build+check() run with numpy/scipy/networkx import-blocked, tests 6/6 green. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…Yang = W13 contraction) Reviewed RH Telperion skillset (turan/jensen/hankel_jensen/interlacing/toeplitz/weil_positivity/ trig_nonneg) for BG leads. Connecting fact (verified 1e-9): per/prod deg = |char_N(i)| = prod sqrt(1+lam^2), char_N real-rooted (Heilmann-Lieb = Lee-Yang, an already-proven RH-analog for the matching polynomial). Three leads: (1 deep) Heilmann-Lieb -> Stieltjes cavity continued fraction = the W13 global contraction made rigorous, AND the realizability constraint the local potential lacked (messages are Stieltjes values, not free); interlacing.py = the certificate vocabulary. (2 buildable) kernel-gate the moment-SDP Hankel-PSD via HankelJensenCertificate/WorstCorner (Hermite criterion, in-kernel over brackets) -- with W10 flag cut = fully kernel-gated route-b level. (3 angle) Weil-positivity template for the caterpillar second-variation PSD. Extra: matching numbers c_k=e_k(lam^2) nonneg + Newton log-concave = constraint beyond generic Hankel. Reproduction: bg_rh_toolkit_lead.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…nv); Stieltjes messages tighten ~44% (A) Cavity map is a strong contraction: leaf-perturbation decays per-hop ratio ~0.008-0.045; Bethe free energy converges in ~2-3 sweeps (4e-6->2e-10->machine zero). F(T) determined by depth-2/3 local structure (deeper <1e-4). Heilmann-Lieb 'no phase transition => contraction' made quantitative -- a GEOMETRICALLY convergent hierarchy vs the slow moment/flag one (W9). bg_cavity_contraction.py. (B) Stieltjes-realizable messages tighten the cavity bound ~44%: DMAX=5 free-message density bound 0.2286 (gap +0.0235) -> realizable-message 0.2182 (gap +0.0131). The realizability the W13 local potential lacked genuinely helps, as the RH/Stieltjes lead predicted. bg_stieltjes_potential.py. Honest: tightens but doesn't alone close (+0.013 DMAX=5 coarse) -- per-config relaxation still lacks joint mass-transport (W8); per W12 no finite local relaxation closes exactly. BUT strong contraction => geometric convergence, so the certificate = a finite cavity level (realizable msgs + mass transport + fine res) with a certified geometric error F <= log rho* + C rho^d (rho~0.03). Synthesis = Lead1 (Stieltjes/exact cavity) + W8 (mass transport) + Lead2 (kernel-gate via RH WorstCorner). conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…s to BG (kernel-gating foundation) RH hankel_minors + WorstCornerCertificate (Hermite: PSD iff leading minors > 0 over rational brackets) certifies BG moment-body facts in EXACT rationals. Caterpillar exact moments m1..m4 -> Hankel leading minors D1=1, D2=610291/8820000>0, D3=116095997089/185220000000000>0 -- exactly what WorstCorner gates. Buildable kernel-gated route-b cert = W10 FlagDischargeCertificate (m2 cut atoms) + RH Hankel-minor cert (moment-body PSD) + W15 geometric error F<=log rho*+C rho^d, at one finite level. Note: primal moment-PSD trivial for a real tree; kernel-gating value is the moment-SDP DUAL bound combining these certs -- the formalization the RH toolkit now makes reachable. Leads status: L1 (Stieltjes, real progress) + L2 (kernel- gating foundation) + L3 (Weil template, untried). conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

… generalized-caterpillar family peaks at caterpillar = log rho* The Lead1+W8 synthesis must certify no degree structure beats the ~7-arm length-2 caterpillar. Confirmed over a rich explicit family (arm-counts 1-19, arm-lengths 1-3, hub-periods 1-3): exact cavity F maximized at (arms=7,arm_len=2,hub_period=1)=length-2 7-arm caterpillar, F=0.205160 (excess +6e-5 finite-spine boundary); top six all arm_len=2/hub_period=1/arms 5-10, no mixed structure near. Ingredients validated: W15 realizable msgs tighten ~44%, W8/W9 mass-transport moment relaxation -> caterpillar, W15 contraction -> geometric conv. Remaining build = combined relaxation (exact cavity F over reversible/mass-transport degree-message dist w/ Stieltjes messages) run to convergence -> geometric to log rho* + certifiable error F<=log rho*+C rho^d. Family confirmation + ingredient validation show it's the right object w/ caterpillar as unique max. Reproduction: bg_synthesis.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…-config bounds sup_T F (finite) not the density Built combined relaxation: exact cavity + Stieltjes-realizable messages + FULL mass-transport discharge P(d,x) on half-edge states (sender degree+message), telescoping. (A) P(d,x) beats P(x): DMAX=5 bound 0.218->0.209224 (gap +0.0131->+0.0041, 69% cut). Progression: free P(x) 0.331 -> realizable P(x) 0.218 -> realizable P(d,x) 0.209. (B) BUT bound INCREASES with degree cap: DMAX 4/5/6/7 = 0.2047/0.2092/0.2124/0.2146 monotone up -- it bounds sup over degree-<=DMAX bulk trees of F, and every finite tree has F>log rho* (W5), so the sup exceeds log rho* and grows with degree. log rho* is the n->inf THERMODYNAMIC DENSITY, NOT a per- config quantity -- no local relaxation (moment/cavity/combined) reaches it. Definitive W5/W12 wall. Density IS obtained directly on infinite trees (W13 exact, caterpillar=log rho* to 2e-8) + maxed by caterpillar over rich family (W17); RH-lead ingredients tighten the finite local bound dramatically but can't cross to the density. Exact route = infinite-tree variational argument; contraction (W15) gives analytic control not a finite convex certificate. Reproduction: bg_combined_relaxation.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

… = unique max (variational proof structure) Since local relaxations bound sup_T F not the density (W18), the exact route is the infinite-tree variational argument (max over unimodular tree measures of exact cavity density = log rho*), whose crux is CONCAVITY. (1) F(a) strictly concave in arm-count: F(a+1)-2F(a)+F(a-1)<0 for all a=4..10, max at a~7 (=a*=7.016). (2) No mixed structure beats the caterpillar: spatial mixes (p hubs a=9, rest a=5, flanking a*=7) all F_mix<=F(a*). Consistent with concavity + unique maximizer. PROOF STRUCTURE (Lead 3 = Weil-positivity): caterpillar is stationary (F'(a*)=0), second variation negative (F''<0, no mix exceeds), => if full Hessian neg-def over unimodular perturbations, strict local max; with global concavity, GLOBAL max = log rho*. The neg-def second variation is a PSD quadratic-form fact = natural home for RH WeilPositivityCertificate/ WorstCorner (certify -Hessian PSD over rational brackets). Remaining: prove full-Hessian concavity over all unimodular directions (numerically supported), then kernel-gate. Reproduction: bg_concavity.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

…l direction (local-max half via contraction) Exact route (W19) reduces to concavity; its local half = caterpillar is a strict local max. Six independent single-site perturbations of uniform a=7 length-2 caterpillar ALL strictly decrease F: +arm -1.0e-6, -arm -7.9e-7, arm 2->3 -7.1e-5, arm 2->1 -2.4e-4, spine-branch -1.2e-4, cherry-end -7.1e-5. Strict local max in arm-count/length/spine-branching/arm-end-degree -- every direction. ANALYTIC REASON: strict local max of Bethe density = BP fixed-point stability = Bethe-Hessian negative-definite = the strong contraction (W15, rate ~0.03). So local-max half REDUCED to the established contraction. Variational proof assembled: (i) caterpillar unique stationary pt a*=7.016 (W17), (ii) strict local max every direction (W20) via contraction (W15), (iii) concave along tested families (W19). ONE remaining piece = GLOBAL concavity/no-other-local-max over all unimodular measures (hard analysis, numerically supported W17). With it, caterpillar = global max = log rho* = BG density bound; neg-def Hessian kernel-gateable via RH WeilPositivity/WorstCorner. Route (b) fully mapped W5-W20. Reproduction: bg_localmax.py. conjecture1_proved=False. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01U5GjxUBoxVBMPwNwUH9RKy

Dr. Murphy and others added 25 commits August 29, 2026 06:33

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Open the walk-count sub-problem (route-b combinatorial core) + W2 - #154

Open the walk-count sub-problem (route-b combinatorial core) + W2#154
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DrMurphyIsIn commented Aug 29, 2026

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W2 first result (essentially closed)

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