# Identical vanishing of the non-zeta coefficient functions of the X(3,6) moment expansion (weights 3 and 4) Draft appendix material. All claims are exact-arithmetic statements; every machine-checked step lists its artifact under `core/` (paths relative to the proof record's own directory; the `core/` files are the record's working files and are not part of this bundle). ## 1. Setting and objects **Kinematics.** For each 3-element subset t of {1,...,6} there is a rational exponent s_t, subject to the six momentum-conservation relations sum_{t contains a} s_t = 0 (a = 1..6). Six exponents (the constant minors 123, 124, 125, 126, 134, 234) are determined by the relations and do not enter any formula below. The remaining fourteen, s = (s_145, s_156, s_345, s_456, s_135, s_136, s_146, s_235, s_245, s_346, s_236, s_246, s_256, s_356), are free coordinates; "identically in s" means as an identity of rational functions in these fourteen variables over Q. Ten of the minors (135, 136, 146, 235, 245, 346, 236, 246, 256, 356) are polynomials in the four integration variables; call these the POLY set. Each POLY minor t has a fixed Newton polytope N_t in Z^4 (its exponent vectors after extracting monomial content). **Fan and per-cone data.** The common refinement of the ten normal fans of the N_t is a complete fan in R^4 with 48 maximal cones, triangulated into 52 simplicial subcones, each spanned by 4 primitive integer rays with |det| = 1. The fan is independent of s. For a subcone c with rays r_1,...,r_4, the local integral representation carries per-ray exponents kappa_i(s) (linear forms in s) and, for each POLY minor t, a lattice polynomial Q_t with constant term 1 whose exponent tables depend only on the fan data. Write P_c(s) = kappa_1 kappa_2 kappa_3 kappa_4. **Census weights.** The weight-4 moment coefficient c_4(s) of the X(3,6) string integral expands as a finite sum c_4(s) = sum_k W_k(s) * C_k over a list of 54 universal constants C_k (one-, two- and three-dimensional log-moment integrals: K1, U2, U3, H22, five pair shapes in four orientation families Fu/Fv/Gu/Gv, eight oriented pair-pair shapes Gd, and 22 triple shapes T). Each weight W_k(s) is an explicit rational function: a sum over the 52 subcones of (a polynomial of total degree 4 in the kappa_i and the linear forms s_t, sigma_i) divided by P_c, where sigma_i(s) is the sum of s_t over the factors with a nontrivial single-axis restriction on axis i. The weights are homogeneous of degree 0. The generating program for these weights (the "census") is the object pinned by the target; all statements below are about its output as exact rational functions. **Declared table.** Each constant C_k carries a declared coordinate vector gamma_k in Q^19 over the spanning list B19 = B16 union {tau1, tau2, tau3}, where B16 = {zeta4, zeta3*ln2, zeta2*ln2^2, ln2^4, Li4(1/2), zeta3*ln3, zeta2*ln2*ln3, zeta2*ln3^2, ln2^3*ln3, ln2^2*ln3^2, ln2*ln3^3, ln3^4, Li4(1/3), Li4(-1/3), Li4(2/3), Li4(-1/2)} and tau1..tau3 are the three irreducible triple integrals; and there is a declared relation tau1 + tau2 = sum_{g in B16} r_g g with fixed rationals r_g. For each direction g in B19 define the coefficient function F_g(s) := sum_k W_k(s) * gamma_k(g), a rational function in Q(s). ## 2. Statements **Theorem 1 (weight 4).** Identically in s (as rational functions over Q): (i) F_tau3(s) == 0; (ii) F_tau1(s) == F_tau2(s); (iii) F_g(s) + F_tau1(s) * r_g == 0 for every g in B16 except zeta4. Consequently, in the declared expansion, c_4(s) = [F_zeta4(s) + F_tau1(s) r_zeta4] * zeta4 for every s in the convergence chamber, and the vanishing of every non-zeta4 direction at the benchmark point is structural, not point luck. **Theorem 2 (weight 3).** The zeta2*ln2- and ln2^3-coefficient functions of the weight-3 moment coefficient c_3(s) vanish identically in s. In particular the pair-shape constants satisfy, exactly: E5 = 0, E6 = E1 = (3/4) zeta3, and E3 = E2 (transposition), closing the previously open bookkeeping between the six assembled weights and the five canonical pair shapes. ## 3. Proof of Theorem 1 **Lemma 1 (ray-form lemma).** For every subcone c and ray r of c, the per-ray exponent is kappa_r(s) = - sum_tau s_tau (mindeg(tau) . r) - sum_{t in POLY} s_t max_{m in N_t} (m . r), a homogeneous integer linear form in s that depends only on the ray vector r, not on the cone. Across the whole fan there are exactly 16 distinct rays, giving 16 distinct linear forms L_1,...,L_16, pairwise distinct within every cone; their span is the full 14-dim dual space. *Proof.* A maximal cone of a normal-fan refinement selects, for each t, the vertex v_t of N_t maximizing the linear functional; by definition of the cone, v_t . r = max_m (m . r) for every ray r of that cone. Substituting into the generator's kappa formula gives the display. The count, distinctness, and rank are finite integer computations. *Machine:* per-(cone, ray) coefficient identity between the generator's kappa and the display: 2080 integer equalities, all pass; 16 rays, 16 forms, bijective, per-cone multiplicity 1; exact rank 14; ray determinants all +-1 (`core/step1_rays.py` -> `core/step1_rays.json`; `core/step1b_formcheck.py` -> `core/step1b_formcheck.json`). The factor exponent tables are independent of s (same artifact). **Lemma 2 (ray-form coordinates).** Fourteen of the sixteen forms, suitably chosen, are a basis of the dual space: setting x_j := L_{S(j)}(s) for that 14-subset S defines a linear isomorphism s -> x of Q^14 with exact rational inverse, under which the two remaining forms become L15 = x_5 + x_6 + x_7 - x_9 and L16 = x_3 + x_4 - x_8 + x_13. Hence every per-cone denominator P_c is a product of four distinct elements of {x_1, ..., x_14, L15, L16}, and D(x) := x_1 ... x_14 * L15 * L16 (degree 16) is a common denominator for all weights. *Machine:* basis selection, exact Gauss-Jordan inverse, 196 entrywise identity checks of M * M^{-1}, and exact reconstruction of both leftover forms (`core/step2_basis.py` -> `core/step2_basis.json`). **Lemma 3 (symbolic census).** For every subcone c and constant C_k there is an explicit polynomial n_{c,k}(x) of total degree 4 such that the census weight satisfies, for all rational s off the 16 hyperplanes L_j = 0, W_k(s) = sum_c n_{c,k}(x(s)) / P_c(x(s)). *Proof.* The census weight is by construction a finite sum, over cones and slots, of det * (monomial in the kappa_i) * (monomial in the s_t and sigma_i) / P_c with fixed shape bookkeeping; substituting the exact linear forms of Lemmas 1-2 for each factor (dets are all 1; the shape data are s-independent; a factor with s_t = 0 contributes zero to every slot, so the generator's skip of such factors is immaterial) yields the display term by term. *Machine:* the symbolic census (1414 per-cone numerator polynomials, 40714 terms) agrees with the live generator EXACTLY — all 62 weight values (54 constants plus the weight-2/3 auxiliaries) at three independent rational points (`core/step3_symcensus.py` -> `core/step3_gate.json`, census stored in `core/symcensus.json`); and an independent from-scratch re-derivation of one cone's full 35-weight contribution at a fourth random point, sharing no code with the generator, agrees exactly (`core/skeptic3_hand.py` -> `core/skeptic3_hand.json`). **Lemma 4 (cleared-numerator identities).** For each of the 17 target combinations I in { gamma(tau3); gamma(tau1) - gamma(tau2); gamma(g) + r_g * gamma(tau1), g in B16 \ {zeta4} }, the polynomial M_I(x) := sum_c [ sum_k I_k n_{c,k}(x) ] * (D/P_c)(x) in Q[x_1..x_14] is IDENTICALLY ZERO: the exact sparse expansion of the 52-cone sum cancels coefficient by coefficient to the empty polynomial. The same machinery applied to the zeta4 direction gives a NONZERO polynomial with 14116 monomials (all of the homogeneous degree 16, as required), so the pipeline does not manufacture zeros. *Machine:* `core/step4_clear.py` -> `core/step4_summary.json` and one artifact per identity in `core/identity_out/` (17 files with n_terms = 0, plus the nonzero control). This is plain exact-Fraction polynomial arithmetic: no sampling, no probabilistic identity testing. **Conclusion.** D(x) is a product of nonzero linear forms, hence nonzero in the integral domain Q[x]; since x is a coordinate system (Lemma 2), M_I == 0 implies F-combination_I == M_I / D == 0 in Q(x) = Q(s). This proves (i), (ii), (iii). The possible poles of the F_g lie only on the 16 hyperplanes L_j = 0; in the convergence chamber all kappa are positive, so the identities hold there as honest function values. QED. **Verification sentence.** Independently of the symbolic proof, the identity set was verified exact-rationally at 56 scattered rational points and on three full lines (651 census evaluations, each identity value exactly 0 as a Fraction), and the assembled table at the benchmark point reproduces the pinned nine-line collapse ledger exactly (`core/step5_sd_gamma.py` -> `core/step5_sd_gamma.json`). ## 4. Proof of Theorem 2 At weight 3 the moment coefficient is c_3(s) = sum over cones of (1/P)[ sum_i kappa_i^2 B_i + sum_{i=1} (-1)^{k+1} w^{k-1} / k converges uniformly on [0,1], and termwise integration gives **E1 = sum_{k>=1} (-1)^{k+1}/k^3 = eta(3) = (3/4) zeta3**. - Transposed shapes have equal constants (swap u and v in the integral), so **E3 = E2**; the census merges the two orientations of the shape 1 + v + uv into one canonical key, and the two assembled orientation weights sum exactly to the census weight of that key. - Classical inputs: Li3(-1) = -(3/4) zeta3 and L2 = zeta3 / 4. With the declared values E2 = (13/24) zeta3 and E4 = -(5/12) zeta3, every constant entering c_3 has zero coordinates on zeta2*ln2 and ln2^3 in the declared weight-3 basket; hence both coefficient functions are the zero combination of the census weights — identically in s, with no census computation required. QED. *Machine:* exact shape factorizations; exact reassembly of the recorded weight-3 rational Q3 from these constants and this author's own symbolic census weights; exact reconciliation of the six assembled orientation weights with the five canonical shapes; the weight-2 rational check; and agreement of the declared E2/E4 rationals with the recorded two-hundred-digit numerics to full stored precision (`core/step6_w3.py` -> `core/step6_w3.json`). **Fitted-versus-derived ledger (weight 3).** Derived or classical: Li3(-1), L2, E1, E5, E6 (= E1), and the identity E3 = E2. Remaining fitted inputs: E2 = (13/24) zeta3 (two-hundred-digit blind fit, with an independent derivation whose one intermediate is itself PSLQ-pinned) and E4 = -(5/12) zeta3 (fit only, from the two-hundred-digit blind fits). ## 5. Conditionality Both theorems are statements about the DECLARED rational table: the frozen per-constant coordinate vectors gamma (regenerated deterministically from the recorded evidence and verified against the pinned benchmark ledger, `core/step5_sd_gamma.json`), the declared tau-sum relation r, and at weight 3 the declared E2/E4 coordinates. They are decided in exact arithmetic and hold unconditionally AS TABLE STATEMENTS. Their reading as statements about the real numbers c_4(s), c_3(s) is conditional exactly on the recorded fitted entries being true identities of real numbers (the same per-constant fits already recorded at the benchmark point — no NEW fit, and no assumption of Q-linear independence of the spanning constants, enters anywhere in the proofs). In particular the theorems upgrade the benchmark-point cancellation to the whole kinematic space at zero additional transcendence exposure, and make the non-zeta ledger of any further kinematic point a finite exact-rational computation. ## 6. Validation of the acceptance procedure by planted errors The acceptance pipeline (weight-equality check, coverage asserts, cleared-numerator zero check) was audited by planting two deliberate corruptions in scratch copies: flipping one cone's pair-pair orientation bit (caught three independent ways: 8 weight mismatches, 3 unknown-key hard asserts, 9 of 17 identities nonzero) and silently dropping one cone from the 52-cone sum (caught: 11 of 17 identities nonzero). Artifacts: `core/plant_copy.md`, `core/plant_audit.json`.