Original ledger status
HOR.CLOCK.10
THE ZETA BUDGET, BOTH ROUTES COMPUTED EXACTLY (the reduced seam route carries pure anchor-ratio anomalies; the naive 4d route does not - the budget computation structurally selects the reduced seam reading): Euclidean Nariai = S^2 x S^2, so the temporal dS2 sector continues to the SAME unit sphere and both candidate routes are exactly computable. (1) THE REDUCED ROUTE [I]: each 2d sector carries the HOR.CLOCK.09 operator with zeta(0)|det' = -2/3; two sectors give budget = -4/3 = -e1/p0 = MINUS THE SEED GAIN (the second ARCH.TRIAD.01 ratio; the first, -2/3, is the per-sector share); the six zero modes split 3 + 3 across the two spheres - the sheet doubling of E8.GRADE.01/HOR.CLOCK.07 IS the two-sphere structure of Euclidean Nariai (= the horizon pair, HOR.NARIAI.01). (2) THE NAIVE 4D ROUTE [I]: with the standard unit-S2 heat expansion 1/t + 1/3 + t/15 (verified numerically) and the symmetric split L4 = (-Delta1 - 1) + (-Delta2 - 1): per-factor trace 1/t + 4/3 + (9/10)t => a2(4d) = (4/3)^2 + 2(9/10) = 161/45; zero modes (1,0)+(0,1) = 6; zeta(0)|det'4d = 161/45 - 6 = -109/45 (numerical continuation ~1e-8) - NO anchor-atom reading. (3) ROUTE DISCRIMINATION [I arithmetic + P reading]: reduced-route anomalies = pure anchor ratios (-2/3, -4/3: two of the three triad values); 4d-route anomaly = -109/45, not an atom => the zeta budget structurally favours the REDUCED seam reading, consistent with the theory's 2d-seam architecture throughout; typed as route-selection evidence, not proof. (4) REMAINING [P] (recorded): the graviton/ghost shares on S^2 x S^2 (Volkov-Wipf-class computation, external standard) complete the budget; both scalar-sector shares pinned exactly - the remaining work is a finite list of known heat coefficients
Recorded status: Identity [I] (exact heat-kernel arithmetic both routes + numerics) + Physical [P] (route reading + remaining shares)
Dependencies, status detail and source
Canonical status: Identity: reduced budget = -4/3 = -seed gain (per sector -2/3); 4d route = -109/45 no atom; zero modes 3+3 = horizon pair; route selection -> reduced seam reading
Dependencies: HOR.CLOCK.09; ARCH.TRIAD.01 (triad); E8.GRADE.01; HOR.NARIAI.01
Scripts: v133_zeta_budget.py
Location: tfpt_horizon_readouts
Original CSV, line 218Original ledger status
FLAV.DUAL.01
THE DUAL ANCHOR LEMMA (the inverse flavor response of the anchor IS the Nariai root; the third purely algebraic leg of the flavor-horizon bridge; external review 2026-06-12, validated): (1) THE DUAL ANCHOR [I]: d := a^T R^-1 = a^T L^-1 = (-1/2, -1/2, 1) with d.1 = 0, d.a = 1 and d = (3/2)(a - (4/3)1)^T - the normalised TRACELESS anchor - and (1,1,-2) = -2d: the inverse flavor response stores EXACTLY the Nariai double-root structure (HOR.NARIAI.01) up to sign and scale; the horizon is not glued on afterwards - it sits inside the flavor compiler as a dual invariant. (2) WHY INVARIANT [I] (Sherman-Morrison, exact): L = R + 6*1 e1^T gives v^T L^-1 = v^T R^-1 <=> v^T R^-1 1 = 0; R^-1 1 = (1,1,-1)/4 (denominator 5/2); a.(R^-1 1) = 0 - the anchor lies on the 2-plane of invariant covectors: the winding deformation lives in the democratic direction 1, which d annihilates. (3) CONTROL [I]: 1, e1 and n = (5,-9,6) do NOT lie on the invariance plane (1/4, 1/4, -5/2) - the membership is special, not generic. (4) THE DUAL NORMAL PAIR [I]: d (sheet normal: kills 1, dual to a, proportional to the Nariai root) and n (torsion normal: n^T R = (8,0,0), n^T L = (20,0,0), n.a = 8, n.1 = 2 = |Z2| cf. FLAV.DIAMOND.02) form the dual basis of the flavor boundary. (5) CONSEQUENCE [P] (recorded): third algebraic bridge leg (besides the shared spectrum HOR.CLOCK.04 and the entropy power law HOR.CLOCK.06); supports the reduced seam reading (HOR.CLOCK.10); the R4' selector programme (FLAV.H2.05) acquires a geometric target: derive the PAIR (d, n), not n alone
Recorded status: Identity [I] (exact dual-vector + Sherman-Morrison + controls) + Physical [P] (bridge reading)
Dependencies, status detail and source
Canonical status: Identity: a^T R^-1 = a^T L^-1 = Nariai root/(-2); invariance <=> orthogonality to R^-1 1; dual normal pair (d n); R4' target = the pair
Dependencies: HOR.NARIAI.01; FLAV.DIAMOND.02; FLAV.H2.05 (selectors); HOR.CLOCK.04; HOR.CLOCK.06
Scripts: v134_dual_anchor.py
Location: tfpt_2; tfpt_horizon_readouts
Original CSV, line 219Original ledger status
HOR.CLOCK.11
THE VOLKOV-WIPF FIREWALL (R1 typed closure: the external full-4d graviton/ghost value is pinned and the edge part is exactly two copies of the reduced seam budget; external values verified against arXiv:2506.02142 Table 4.1 which states agreement with Volkov-Wipf Nucl. Phys. B582 (2000) 313 = hep-th/0003081): (1) EXTERNAL DATUM [I typed external]: alpha_bulk = 22/45, alpha_edge = -8/3, alpha_PI = -98/45 for d = 3 (S^2 x S^2, 4d); arithmetic 22/45 - 8/3 = -98/45 exact; the review's arXiv citation 0204021 corrected to 0003081. (2) THE EDGE MATCH [I]: alpha_edge = -8/3 = 2 x (-4/3) - the external edge factor consists of TWO IDENTICAL COPIES (one per horizon) and the per-copy magnitude 4/3 = e_1/p_0 is EXACTLY the reduced seam budget of HOR.CLOCK.10 (= the seed gain; -2/3 per sector); the TFPT reduced route lands on the EDGE sector of the external bulk/edge split; the bulk 22/45 is the ideal graviton gas which the seam clock does not contain; two copies = the horizon pair = the sheet doubling (HOR.NARIAI.01; HOR.CLOCK.10's 3+3 zero-mode split). (3) FIREWALL ARITHMETIC [I]: -98/45 + 60/45 = -38/45 with 38 = 2*19; neither -98/45 nor -38/45 nor 22/45 is an anchor atom (census) - consistent with HOR.CLOCK.10 where the naive scalar 4d route (-109/45) was already non-atomic (different operator content: scalar split vs full graviton + ghosts + isometries, recorded). (4) TYPING DECISION [P] (recorded): R1 = the reduced zero-mode/RPA seam clock = an EDGE-sector object; the full 4d Volkov-Wipf determinant is the GRAVITY FIREWALL (its -98/45 absorbed in the mu_0/(Lambda G)^3 normalisation), not a TFPT atom; per-copy sign/orientation (the edge factor enters inverted) and the Lorentzian edge-mode reading stay [P]
Recorded status: Identity [I] (exact arithmetic vs external values) + Physical [P] (typing decision) + External (VW hep-th/0003081; arXiv:2506.02142)
Dependencies, status detail and source
Canonical status: Identity: -98/45 = 22/45 - 8/3 external; edge = 2 x reduced budget (-4/3); diff -38/45 no atom; typing: seam clock = edge object full 4d determinant = firewall
Dependencies: HOR.CLOCK.10; HOR.NARIAI.01; HOR.CLOCK.03-09 (the clock chain)
Scripts: v138_vw_firewall.py
Location: tfpt_horizon_readouts
Original CSV, line 223