# Appendix artifact — Structural cancellation of the non-zeta directions in the X(3,6) alpha'-expansion (weights 3 and 4) Prepared 2026-07-09 as the machine-checked record behind Appendix A of the note. File names below are relative to the proof record's own directory; the `core/` files named here are the record's working files and are not part of this bundle. The same holds for the `div1/`, `div2/`, `xver/` and `x0/` files and the companion logs named below. **STATUS: VERIFIED-CLOSED (step-4 cross-verification complete, 2026-07-09; slots in Section 7 filled 2026-07-10 from CROSSVERIFY.md).** All three proof routes were reproduced by an independent second verification, which added a fourth (two-prime modular) route of its own; plant audit caught both planted errors; all nonzero controls fired. This artifact is citable. --- ## 1. Setting and theorem **Kinematics.** For each 3-element subset t of {1,...,6} there is a rational exponent s_t, subject to the six relations sum_{t containing a} s_t = 0 (a = 1,...,6). Six of the twenty exponents are determined by these relations and enter no formula below; the remaining fourteen, indexed by t in {145, 156, 345, 456, 135, 136, 146, 235, 245, 346, 236, 246, 256, 356}, are free coordinates, written s. "Identically in s" means: as an identity of rational functions in these fourteen variables over Q. **Expansion.** In its convergence chamber the X(3,6) string integral has an alpha'-expansion whose weight-w coefficient c_w(s) is a finite sum c_w(s) = sum_k W_k(s) * C_k over a fixed list of universal constants C_k (one-, two- and three-dimensional logarithmic moment integrals attached to a complete simplicial fan in R^4 with 52 unimodular cones; the fan is independent of s). Each weight W_k(s) is an explicit rational function of s, computable in exact rational arithmetic, homogeneous of degree w - 4. At weight 4 there are 54 constants. **Declared table.** Each weight-4 constant carries a 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, tau2, tau3 are three irreducible three-dimensional integrals; and there is a declared relation tau1 + tau2 = sum_{g in B16} r_g g with fixed rationals r_g of height at most 91/24. Some coordinates are derived in closed form; the rest were determined by high-precision integer-relation fits at one benchmark kinematic point and then frozen. 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). **Theorem 1 (weight 4).** Identically in s, as rational functions over Q: (i) F_tau3 == 0; (ii) F_tau1 == F_tau2; (iii) F_g + r_g * F_tau1 == 0 for every g in B16 except zeta4 — seventeen identities in all. Consequently, in the declared expansion, c_4(s) = [ F_zeta4(s) + r_zeta4 * F_tau1(s) ] * zeta4 for every s in the convergence chamber: the vanishing of every non-zeta4 direction at the benchmark point is structural, not a property of that point. **Theorem 2 (weight 3).** The zeta2 ln2- and ln2^3-coefficient functions of c_3(s) vanish identically in s. En route, the pair-shape constants are settled exactly: E5 = 0, E6 = E1 = (3/4) zeta3, and E3 = E2 (Section 4 defines these constants). **Conditionality.** Both theorems are statements about the declared rational table (the frozen coordinate vectors gamma and the relation r). As table statements they are decided in exact arithmetic and hold unconditionally. Their reading as statements about the real numbers c_3(s), c_4(s) is conditional on exactly one thing: the fitted table entries being true identities of real numbers — the same per-constant fits already underlying the benchmark-point result. No new fitted input, and no assumption of Q-linear independence of the spanning constants, enters anywhere. The theorems therefore upgrade the benchmark-point cancellation to the whole kinematic space at zero additional transcendence exposure. --- ## 2. Proof summary: three independent routes All three routes decide the same seventeen identities; each is exact-rational throughout (no floating point, no sampling in the final step), and each carries a nonzero control showing the pipeline cannot manufacture zeros. **Route A — monomialization and empty numerator.** Every per-cone exponent kappa is a homogeneous integer linear form in s depending only on the ray, not the cone; the fan has exactly 16 rays, giving 16 distinct linear forms whose span is the full 14-dimensional dual space (2080 per-cone/per-ray integer identities checked). Fourteen of the forms are a coordinate system; in these coordinates every weight is a sum of degree-4 polynomials over products of four of the 16 forms, and each identity, cleared against the common degree-16 denominator, becomes a polynomial in fourteen variables. For all seventeen identities the exact sparse expansion of the 52-cone sum cancels coefficient by coefficient to the empty polynomial; the zeta4 direction, run through the same machinery, yields a nonzero polynomial with 14116 monomials. Artifacts: `core/step1_rays.json`, `core/step1b_formcheck.json`, `core/step2_basis.json`, `core/step3_gate.json`, `core/symcensus.json`, `core/skeptic3_hand.json`, `core/step4_summary.json`, `core/identity_out/` (17 zero certificates + nonzero control), `core/step5_sd_gamma.json`. Full statement and proofs: `PROOF_W4W3.md`. **Route B — residues on ray walls and Liouville.** Each F_g has at most simple poles, and only on the 16 hyperplanes where a ray form vanishes. The residue on each wall was certified zero by an exact, wall-local computation: the deepest (codimension-2) content localizes to 312 flags, each fed by exactly two cones sharing a facet, where the two contributions cancel by the gluing mechanism of Section 3 together with two exact properties of the declared table (the derived constants' columns are pure zeta4, and 3 gamma_U2 + 2 gamma_U3 = 0 on every non-zeta4 direction); the remaining wall content cancels by exact polynomial divisibility. A rational function with no poles that is homogeneous of degree 0 is a constant, and the exact zero at the benchmark point forces the constant to be 0. The zeta4 direction fails the wall certificate on every wall — its poles are real, and the failing flag polynomials carry the derived coordinate 7/4. Artifacts: `div1/rays.json`, `div1/wall_00.json` ... `div1/wall_15.json`, `div1/residue_summary.json`, `div1/mechanism.json`, `div1/flags.json`. Full log: `DIVERSITY_RESIDUE.md`. **Route C — independent symbolic clearing.** A from-source reimplementation of the weight computation (sharing no code with Route A's generator and never evaluating at a kinematic point after its one validation) carried the fourteen exponents as symbols and cleared each identity against the true least common denominator — the product of the 16 ray forms, a polynomial with 5,472,948 terms. All seventeen cleared numerators are the zero polynomial. Controls: the zeta4 direction gives a nonzero numerator with 35,103,860 terms, and a deliberate rational poisoning of one table entry (by 3/1000003) gives a nonzero numerator with 1,637,378 terms. Artifacts: `div2/stage0_result.json`, `div2/stage1_forms.json`, `div2/identity_*.json` (17 zero certificates + 2 controls), `div2/forms_census.json`. Full log: `DIVERSITY_SYMBOLIC.md`. Audit of the acceptance procedure itself by planted errors (corruptions planted in scratch copies — an orientation flip and a dropped cone — both caught multiple independent ways): `core/plant_copy.md`, `core/plant_audit.json`. --- ## 3. The mechanism: why every direction but zeta4 dies The wall residues localize to flags where exactly two cones meet along a facet. Across each facet the geometry obeys an exact wall-crossing law: the axis datum sigma is the same on both sides, and the two opposite ray exponents satisfy kappa + kappa' = -sigma. This is the substitution y -> 1/y on the underlying one-dimensional integrals: the two half-line pieces int_0^1 y^{kappa-1}(1+y)^{sigma} dy on either side of the facet glue into the full-line integral, which is the Beta value Gamma(kappa) Gamma(-kappa-sigma) / Gamma(-sigma), whose logarithmic expansion is pure zeta at every weight. Individually, each cone's piece is impure; only the glued pair is pure. At the level of the declared table this becomes a parity statement. Under the involution (kappa, sigma) -> (-kappa - sigma, sigma), the slot monomials split into odd and even. Every non-zeta4 column is supported on the odd combinations, which cancel in pairs across each facet — for the one-dimensional slots this is exactly the exact table relation 3 gamma_U2 + 2 gamma_U3 = 0, the table shadow of the analytic identity "3 U2 + 2 U3 = (3/4) zeta4" forced by the Beta expansion. The zeta4 column alone collects the even invariants: the two derived constants (K1 = (7/4) zeta4 and H22 = (5/8) zeta4) sit precisely in the even slots, and the odd-slot remainder on the zeta4 direction is 3 gamma_U2 + 2 gamma_U3 = 3/4, not 0. So the survivor is selected by parity under the facet involution — and the fitted table satisfies the parity relations exactly, a consistency check the fits were never told about. --- ## 4. The weight-3 leg At weight 3 the two-dimensional pair constants are E(Q) = int_0^1 int_0^1 log[ Q(u,v) / (Q(u,0) Q(0,v)) ] du dv / (uv) for the five pair polynomials Q that occur. Exact factorizations settle three of them: for Q = (1+u)(1+v) the integrand vanishes, so E5 = 0; Q = 1 + v + uv + uv^2 factors as (1+v)(1+uv), so E6 = E1; and E1 (for Q = 1 + uv) is the alternating sum sum_{k>=1} (-1)^{k+1}/k^3 = (3/4) zeta3. Transposed shapes have equal constants, so E3 = E2. With the classical values Li3(-1) = -(3/4) zeta3 and int_0^1 ln^2(1+y) dy/y = zeta3/4, and 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 — so both weight-3 coefficient functions are the zero combination of the weights, identically in s, with no summation over the fan required. Remaining fitted inputs at weight 3: E2 = (13/24) zeta3 (blind fit, with an independent derivation whose one intermediate step is itself pinned by an integer-relation fit) and E4 = -(5/12) zeta3 (fit only). Everything else in the weight-3 table is derived. Artifact: `core/step6_w3.json`. --- ## 5. Scope for weight 5 What transfers to the weight-5 expansion verbatim, because it is a property of the fan and the fixed {0,1}-polynomials alone (all machine-checked here, independent of the weight): the 16 rays and the bijection between rays and pole hyperplanes; simple poles only; the flag structure (each codimension-2 residue fed by exactly two facet-adjacent cones); and the wall-crossing law kappa + kappa' = -sigma with sigma stable across the facet — hence the entire Beta-gluing/parity template of Section 3. What weight 5 must establish for itself: its own weight table (new slot families, including genuinely four-dimensional integrals); its own constants and their coordinate vectors, with their own reduction relations; the weight-5 analogs of the parity relations (the finitely many weight-5 Beta-expansion combinations of the new one-dimensional constants must have pure-zeta5 coordinates — expected to be derivable in closed form from the log-Gamma expansion, but not yet derived); and its own wall certificates at the new degrees. One template step changes: the weight-5 coefficient functions are homogeneous of degree +1, not 0, so pole-freeness leaves a linear form rather than a constant — the final pinning step needs either fourteen independent rational points per direction or a symbolic identity, not the single benchmark evaluation that sufficed at weight 4. No global miracle beyond these ingredients is required by the template; but the statements in this paragraph beyond the verbatim-transfer list are expectations, not theorems. --- ## 6. Corollary: the expansion at any kinematic point, with no new fits Because the constants and their coordinate vectors are independent of s, Theorem 1 makes the non-zeta ledger of ANY rational kinematic point a finite exact-rational computation: evaluate the weight table at the new point (exact fractions, about a third of a second), contract with the frozen coordinate table and the declared relation r, and read off c_4 = Q_4(s) zeta4; the seventeen non-zeta4 identities are theorems and must return exactly zero (a nonzero flags an input or implementation error, not new mathematics). Likewise c_2(s) and c_3(s) come out as explicit rationals times zeta2, zeta3. Zero new fits and zero new integer-relation searches are needed at the new point. This was exercised at the independently recorded second kinematic point: the harness reproduced the separately computed ledger at that point exactly — all seventeen identities zero, the common tau coefficient equal to -12470507884/2384452215, and Q4(s2) = 5195290564957251462484448119654052781955295437809234600769228667 / 26939566599931690440035677084523431778704442682163408434038400, in exact agreement with the second-point computation performed before this proof existed. Recipe and reusable-constants interface: `INTERFACE_TRACE.md`, Section U3; harness: `x0/assemble.py`. --- ## 7. Verification sentences - Table-level proof, Route A: seventeen cleared numerators identically zero by exact polynomial arithmetic; nonzero zeta4 control. Step-4 cross-verifier verdict: PASS — all seventeen zero-certificates reproduced in the verifier's independent run (identity artifacts in xver/, e.g. identity_tau3.json status PASS), with the zeta4 control nonzero (35,103,860 numerator terms) and the eps-poison control nonzero. - Route B: zero residue certified on all 16 walls for all seventeen identities; zeta4 control fails every wall. Step-4 verdict: PASS — 16 walls recounted with the CTRL_z4 control firing (nonzero residue) on every wall; the plant audit fired and caught both planted errors (step4_fired true, caught true, 11 identities driven nonzero, audit_pass true). - Route C: independent implementation, all seventeen cleared numerators zero over the 16-form denominator; two nonzero controls. Step-4 verdict: PASS — plus a FOURTH route no prior step used: the verifier's own census reimplementation (checked, 54/54 keys at two kinematic points) cleared all seventeen numerators to zero modulo two independent 61-bit primes (2259404181720454079 and 1748385269457203879), control nonzero at exactly 35,103,860 terms at both primes. - Point checks (already run, independent of step 4): the identity set holds exactly, in rational arithmetic, at 56 scattered rational points and on three full lines, and the assembled table reproduces the benchmark-point ledger exactly (`core/step5_sd_gamma.json`); at the second recorded kinematic point the harness reproduces the independently computed ledger exactly (Section 6). - All seventeen identities were verified by three independent exact-arithmetic routes and reproduced by an independent second verification that added a fourth, two-prime modular route of its own; every planted error was caught and every nonzero control fired. Verdict: VERIFIED-CLOSED (CROSSVERIFY.md).