Original ledger status
HOR.CLOCK.03
THE RESUMMED CLOCK (R1's bend acquires a closed form; the HOR.CLOCK.02 job description transforms): (1) THE SPECTRUM READS (1 - n/N)^{p2} [I]: the frozen transfer eigenvalues {1, (2/3)^6, (1/3)^6} (v54/v69/v95) are EXACTLY lambda_n = (1 - n/N_fam)^{p2} for n = 0, 1, 2, with p2 = 6 = |R+(A3)| the hexagon size. (2) CLOSED-FORM CLOCK [I]: rate(n) = -p2 ln(1 - n/N_fam) reproduces {0, Delta, 6 ln 3} exactly; the HOR.CLOCK.02 bend rate(2)/rate(1) = log_{3/2}3 becomes a one-line consequence. (3) THE LINEARISATION CARRIES THE SHEET [I]: rate(n) = |Z2| n + n^2/3 + 2n^3/27 + ... - the linear (one-loop) term has slope exactly |Z2| = 2 relative to the integer classical Ginsparg-Perry clock {0,-1,-2} (HOR.CLOCK.01): the quantum clock is the SHEET-DOUBLED classical clock plus the geometric tail sum_k (n/N)^k/k. (4) THE WALL AT N_fam [I]: rate -> infinity as n -> N_fam (pole of the resummed logarithm at the family count) => the cusp ladder ends at n = N_fam - 1 = 2: the three-level transfer spectrum is forced by the clock itself. (5) R1 RE-SHARPENED [P] (recorded, not claimed): closed-form semiclassical target - derive rate(n) = -p2 ln(1 - n/N_fam) from the near-Nariai one-loop at kappa = 1/(3pi); the linear response must give slope |Z2| x classical and the resummation must exponentiate the geometric series; all spectra frozen in v54/v69/v95/v104/v107 - nothing fitted
Recorded status: Identity [I] (exact spectral reading + series) + Physical [P] (semiclassical derivation)
Dependencies, status detail and source
Canonical status: Identity: transfer spectrum = (1 - n/N)^6; rate = -6 ln(1 - n/3); linearisation slope |Z2|; pole at N_fam forces three levels; R1 target now closed-form
Dependencies: HOR.CLOCK.02; HOR.CLOCK.01; FR.KOIDE.06 (transfer spectrum); SYNTH.INVENTORY.02 (R1)
Scripts: v124_resummed_clock.py
Location: tfpt_horizon_readouts
Original CSV, line 209Original ledger status
GATE.METRIC.08
THE GLUE Q-SYSTEM (the index-4 mu4 extension exists as an explicit Longo Q-system; the R2 gate sharpens from 'find a Q-system' to 'match the constructed one'): (1) Q-SYSTEM AXIOMS [I] on A = C[Z4] (the glue group algebra; multiplication delta_a x delta_b -> delta_{a+b}, unit delta_0): associativity, unit law, FROBENIUS (m x 1)(1 x m*) = m* m = (1 x m)(m* x 1), and SPECIALNESS m m* = |Z4| id all hold exactly => a special C* Frobenius algebra = a Longo Q-system with dim theta = 4: JONES INDEX [B:A] = 4 = |mu_4| - exactly the GATE.METRIC.06 Carrier Index Lemma value; KLM bookkeeping mu_carrier = 16 = 4^2 x mu_E8 re-verified. (2) LOCALITY = ISOTROPY [I] (GATE.METRIC.07 re-derived in Q-system language): on the Lagrangian glue <(1,1)> the discriminant form q = (5x^2+3y^2)/8 gives q(k(1,1)) = k^2 = (0,1,4,9), all integers - the Q-system braiding obstruction vanishes (local/bosonic extension); the two Lagrangian glues = the sheet pair are the only maximal isotropic choices. (3) THE HALFWAY SUB-Q-SYSTEM [I]: {0,2} < Z4 closes multiplicatively => the unique intermediate Q-system C[Z2] = the SO(16)_1 step of the tower (GATE.METRIC.07/CAR.QFREE.01), index factorisation 4 = 2 x 2. (4) R2 SHARPENED [P] (recorded, not claimed): the gate's algebraic object now EXISTS explicitly; what remains is the operator-algebraic IDENTIFICATION 'the seam-Calderon inclusion of the 16-Majorana net realises THIS Q-system' - matching two constructed objects, not a search; the three loads (metric + carrier + QBL, SYNTH.INVENTORY.02) hang on this one identification; gate typing [P]/[A] unchanged [2026-08-03 LIII: the index side is now MEASURED on the CAR ladder (v726, 57/57, GNET.PPINDEX.01): the mu4-average expectation has optimal Pimsner-Popa constant EXACTLY 1/4 and Watatani index EXACTLY 4 via an explicit quasi-basis with FORCED weights, with mu4 DERIVED from the clock (not inserted) and the Ramond sector healed state-preservingly by the v679 Klein dressing; the remaining half of the gate is the IDENTIFICATION of this ladder Q-system with the constructed GATE.METRIC.08 object -- registered as the open contract half of GNET.RAMIFIED.01 (kills: Frobenius mismatch, index != 4 on the state-preserving ladder, dressing incompatible with the sigma descent). No status move.] [2026-08-05 LXII (v779): the finite precursor of the R2 identification is measured -- the coherent COMMUTING system of index-4 expectations along the Haar tower (restriction coherence, [E_I, E_J] = 0, joint idempotency, all exact) is the finite half of the Longo Q-system identification, and the exact dim law d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) makes the 4-element C[Z4] group-layer quasi-basis obstruction exactly 2^-l at every finite level (d_0 - d_2 = 2^l > 0), restored only asymptotically -- the crossed-product structure is exactly asymptotic; d_2(1) = 0 IS the l = 1 index-3 anomaly. No status move.]
Recorded status: Identity [I] (exact finite Frobenius/Q-system axioms + isotropy) + Physical [P] (identification)
Dependencies, status detail and source
Canonical status: Identity: C[Z4] is a special Frobenius algebra (Longo Q-system) of index 4 = |mu4|; locality = isotropy; halfway C[Z2] = SO(16)_1; R2 = identification of constructed objects
Dependencies: GATE.METRIC.06; GATE.METRIC.07; CAR.QFREE.01; SYNTH.INVENTORY.02 (R2)
Scripts: v125_glue_qsystem.py
Location: tfpt_research_contracts
Original CSV, line 210Original ledger status
HOR.CLOCK.04
THE CLOCK AND THE WALL SHARE A SPECTRUM (the v124 resummed-clock weights ARE the Mehta-Seshadri parabolic weights; R1's first checkpoint passes classically): (1) THE WEIGHTS ARE THE PARABOLIC WEIGHTS [I]: the arguments n/N_fam = {0, 1/3, 2/3} of the resummed logarithm are EXACTLY spec A0* (the exact U_wall residue, GATE.UWALL.07); the spectral bridge lambda_n = (1 - alpha_n)^{p2} over alpha in spec A0* reproduces the frozen transfer spectrum {1, (2/3)^6, (1/3)^6} - the horizon clock and the flavor wall share their spectrum through the complement map alpha -> 1 - alpha: the transfer eigenvalues are the hexagon-power of the COMPLEMENTARY parabolic weight. (2) CHECKPOINT 1 PASSES CLASSICALLY [I]: the resummed clock's linear coefficient p2/N_fam = 2 = |Z2| equals the established classical entropy-deviation rate 2H = |Z2| x Hubble (HOR.CLOCK.01) - the slope-2 linear response demanded by HOR.CLOCK.03 is already the classical entropy rate; the genuinely QUANTUM content of R1 isolates to the geometric tail sum_{k>=2} alpha^k/k. (3) THE DETERMINANT IS THE ENTROPY CURVATURE [I]: det(1 - A0*) = (1)(2/3)(1/3) = 2/9 = |Z2|/N_fam^2 - exactly the established gravity entropy curvature at the anchor (HOR.ORIENT.01), a fourth independent appearance; tr(1 - A0*) = 2 = |Z2|; det A0* = 0 (the parabolic degree-0 marker). (4) R1 RE-TARGETED [P] (recorded, not claimed): derive 'rate = -p2 ln(1 - alpha)' with alpha the parabolic weight - first order is classical (HOR.CLOCK.01's entropy rate), the tail is the quantum part; ONE SPECTRUM, TWO GATES, ONE OBJECT A0*
Recorded status: Identity [I] (exact spectral bridge + coefficient identities) + Physical [P] (re-target)
Dependencies, status detail and source
Canonical status: Identity: clock weights = parabolic weights (spec A0*); lambda = (1-alpha)^6; linear term = classical entropy rate |Z2|H; det(1-A0*) = entropy curvature 2/9; quantum content = geometric tail
Dependencies: HOR.CLOCK.03; HOR.CLOCK.01; HOR.ORIENT.01; GATE.UWALL.07 (A0*); SYNTH.INVENTORY.02 (R1)
Scripts: v126_clock_wall_bridge.py
Location: tfpt_horizon_readouts
Original CSV, line 211Original ledger status
HOR.CLOCK.05
THE CLOCK IS A LOG-DETERMINANT (the geometric tail of HOR.CLOCK.03/04 is the standard ring/RPA resummation; the 'O(1) resummation' demanded by HOR.CLOCK.02 is now TYPED): (1) LOG-DET IDENTITY [I]: for a rank-1 insertion P (tr P^k = 1), -ln det(1 - alpha P) = -ln(1 - alpha) = sum_k alpha^k/k - so the resummed clock is EXACTLY rate = -p2 Tr ln(1 - alpha P): p2 = 6 identical one-loop ring towers, ONE PER HEXAGON SITE (= positive root of A3), with coupling alpha = the Mehta-Seshadri parabolic weight. (2) RING LADDER [I]: the k = 1 ring is the CLASSICAL term (6 alpha = |Z2| n = the HOR.CLOCK.01/04 entropy rate); the k = 2 ring is the first quantum correction (1/3 at weight 1); the partial sums converge to the exact rates {Delta, 6 ln 3} monotonically from below. (3) COUPLING CROSS-LINK [I, audit]: the established seam coupling kappa = (c/24pi)(Lambda/Mbar^2) at Lambda = Mbar^2 with c = 8 (HOR.CLOCK.02) is 8/(24pi) = 1/(3pi) = alpha_1/pi - the seam coupling is the first parabolic weight over pi. (4) THE WALL = RING DIVERGENCE [I]: radius of convergence |alpha| < 1; at alpha -> 1 (n -> N_fam) the sum diverges - the HOR.CLOCK.03 wall at the family count is the standard RPA instability. (5) R1 FINAL FORM [P] (recorded, not claimed): show that the near-Nariai one-loop effective action of the seam mode, with insertion strength = parabolic weight and multiplicity p2, is the ring sum - 'the seam clock is the RPA determinant of one mode on six sites'; the resummation type is identified, the derivation itself remains open
Recorded status: Identity [I] (log-det + ring ladder + coupling arithmetic) + Physical [P] (derivation)
Dependencies, status detail and source
Canonical status: Identity: clock = -p2 Tr ln(1 - alpha P) (ring/RPA determinant; one tower per hexagon site); k=1 ring = classical entropy rate; wall = RPA instability; kappa = alpha_1/pi
Dependencies: HOR.CLOCK.04; HOR.CLOCK.03; HOR.CLOCK.02; FLAV.H2.02 (hexagon); SYNTH.INVENTORY.02 (R1)
Scripts: v127_ring_resummation.py
Location: tfpt_horizon_readouts
Original CSV, line 212Original ledger status
E8.GRADE.01
THE GRADED HULL + THE ZERO-MODE MULTIPLICITY (both named computations gain their structural anchors): R2 HALF - (1) EXPLICIT COSET CONSTRUCTION [I]: the 240 E8 roots, built as explicit norm-2 vectors in D5 (+) A3 + glue coordinates, decompose over the glue group Z4 as 240 = 52 + 64 + 60 + 64 (C0 = carrier roots 40+12; C1 = (16, 4bar) from norm 5/4 + 3/4; C2 = (10, 6) from 1 + 1; C3 = (16bar, 4)) - exactly the GATE.METRIC.06 branching, now as vectors. (2) THE GRADING IS EXACT [I]: for ALL 6720 root pairs whose sum is again a root, coset(r1+r2) = coset(r1) + coset(r2) mod 4 - E8 IS A Z4-GRADED LIE ALGEBRA over the carrier subalgebra, and the grading group IS the glue = the GATE.METRIC.08 Q-system: the checksum hull is the Q-system's graded module realised in 248 dimensions (not merely 'compatible' with it). R1 HALF - (3) THE MULTIPLICITY IS THE ZERO-MODE COUNT [I]: the Ginsparg-Perry l = 1 sector (the zero modes, lambda in {0,-1}, HOR.CLOCK.01/02) has 2l+1 = 3 modes; sheet doubling gives 2 x 3 = 6 = p2 = |R+(A3)| - the ring multiplicity of the HOR.CLOCK.05 log-det clock equals the sheet-doubled zero-mode count; [P] reading (recorded, not claimed): zero modes are exactly the modes that force nonperturbative resummation in de Sitter one-loop physics - the natural home of the six RPA towers. (4) RESIDUE [P]: R1 = compute the RPA determinant on the six sheet-doubled l=1 zero modes; R2 = identify the seam-Calderon inclusion with the Z4-graded structure now exhibited inside E8 itself
Recorded status: Identity [I] (explicit construction + exact grading + arithmetic) + Physical [P] (anchors)
Dependencies, status detail and source
Canonical status: Identity: E8 = Z4-graded Lie algebra over the carrier (grading = glue Q-system; 6720 pair checks); ring multiplicity 6 = sheet-doubled GP zero modes
Dependencies: GATE.METRIC.08; GATE.METRIC.06; HOR.CLOCK.05; HOR.CLOCK.01; E8.GLU.01
Scripts: v128_graded_hull.py
Location: tfpt_research_contracts; tfpt_horizon_readouts
Original CSV, line 213Original ledger status
HOR.CLOCK.06
THE CLOCK IS AN ENTROPY POWER LAW (the transfer eigenvalues are the Nariai entropy fractions to the hexagon power; the ring coupling is the fractional entropy deficit): (1) THE THREE ENTROPY LEVELS [I]: the established Nariai bookkeeping (HOR.NARIAI.01) has S/S_dS in {1, 2/3, 1/3} (pure dS; two-horizon total; single horizon - deficit S_dS/3 per step) - EXACTLY the complementary cusp weights 1 - n/N_fam. (2) THE CLOCK IS AN ENTROPY POWER LAW [I]: lambda_n = (S_n/S_dS)^{p2} = the frozen transfer spectrum {1, (2/3)^6, (1/3)^6}; rate(n) = p2 ln(S_dS/S_n) = {0, Delta, 6 ln 3} - hexagon-weighted log-entropy deficits; audit: exponent p2 = 6 = 2h reads as the two-point form of a weight h = 3 = N_fam operator (recorded, not claimed). (3) THE COUPLING IS THE ENTROPY DEFICIT [I] (triple identity): alpha = n/N_fam = the Mehta-Seshadri parabolic weight (GATE.UWALL.07/HOR.CLOCK.04) = 1 - S_n/S_dS = DeltaS_n/S_dS - the RPA insertion strength is the fractional entropy deficit of the SdS configuration: the coupling is not a free parameter in ANY of its three readings. (4) ORIENTATION CONSISTENCY [I]: weight 0 <-> pure dS (max entropy, rate 0: the stationary attractor); weight 2 <-> single horizon (lowest level, fastest decay) - monotone, the established repeller -> attractor orientation (FR.KOIDE.05/HOR.NARIAI.01/HOR.ORIENT.01). (5) R1 THERMODYNAMIC FORM [P] (recorded, not claimed): derive Gamma_n ~ (S_n/S_dS)^{p2} from the near-Nariai one-loop - a Gibbons-Hawking-type entropy power law with exponent = the hexagon size; the exponent (weight h = N_fam) is the remaining quantum content
Recorded status: Identity [I] (entropy-level + power-law + triple-coupling identities) + Physical [P] (derivation)
Dependencies, status detail and source
Canonical status: Identity: lambda = (S/S_dS)^p2; rate = p2 ln(S_dS/S); alpha = entropy-deficit fraction = parabolic weight; orientation consistent; R1 = Gibbons-Hawking power law
Dependencies: HOR.CLOCK.05; HOR.CLOCK.04; HOR.NARIAI.01; HOR.ORIENT.01; GATE.UWALL.07
Scripts: v129_entropy_power_law.py
Location: tfpt_horizon_readouts
Original CSV, line 214Original ledger status
HOR.CLOCK.07
THE BORN SQUARE (the amplitude weight is h = N_fam = half the zero-mode count; the clock exponent p2 = 2h is its Born square - the HOR.CLOCK.05 multiplicity and the HOR.CLOCK.06 weight are ONE object): (1) EXPONENT BOOKKEEPING [I]: in the standard collective-coordinate treatment each zero mode trades a Gaussian integral for a moduli factor ~ S^{1/2}; with n_zero = 6 the transition AMPLITUDE scales as A ~ (S/S_dS)^{n_zero/2} = (S/S_dS)^3 and the PROBABILITY is its Born square Gamma = |A|^2 = (S/S_dS)^6 = the HOR.CLOCK.06 entropy power law => h = n_zero/2 = 3 = N_fam and p2 = 2h: the weight audit is DERIVED from mode counting; the factor 2 between weight and exponent is the Born rule, nothing else. (2) THE THREE ARE THE SPHERE'S CONFORMAL DIRECTIONS [I]: the l = 1 scalar eigenspace on S^2 has dimension 2l+1 = 3 = the number of proper conformal Killing vectors of the sphere (gradients of the l = 1 harmonics; the classic zero modes of near-Nariai physics, Ginsparg-Perry/Bousso-Hawking class, external standard); the doubling x2 is the HORIZON PAIR = the SdS mass-line deck (HOR.NARIAI.01): n_zero = |Z2| x 3 = 6 = p2 - every factor an established object. (3) CONSISTENCY [I]: multiplicity (E8.GRADE.01) = exponent (HOR.CLOCK.06) = Born square of the amplitude weight; h = N_fam ties the clock's conformal weight to the family count - the same 3 counts generations and the sphere's conformal directions. (4) RESIDUE [P] (recorded, not claimed): justify the per-zero-mode S^{1/2} collective-coordinate normalisation on the Nariai instanton for exactly these six modes - a textbook-class computation; no exotic object remains in R1 (log = ring sum, power law = entropy fractions, exponent = Born-squared mode count)
Recorded status: Identity [I] (exponent bookkeeping + mode-count identities) + Physical [P] (measure normalisation)
Dependencies, status detail and source
Canonical status: Identity: h = n_zero/2 = N_fam; p2 = 2h (Born rule); 3 = proper CKVs of S^2; x2 = horizon pair; R1 residue = standard zero-mode measure
Dependencies: HOR.CLOCK.06; HOR.CLOCK.05; E8.GRADE.01; HOR.NARIAI.01
Scripts: v130_born_square.py
Location: tfpt_horizon_readouts
Original CSV, line 215Original ledger status
HOR.CLOCK.08
THE MEASURE IS THE AREA (the per-mode S^{1/2} of the Born square is the zero-mode NORM, computed exactly; the R1 residue narrows to one determinant-ratio cancellation): (1) ZERO-MODE NORM = AREA [I]: on the sphere S^2(r) the L^2 norm^2 of each of the three normalised l = 1 harmonics is EXACTLY r^2 = A/(4pi) (all three modes integrated exactly) - the norm of every zero mode IS the horizon area, hence the entropy, up to the fixed 4pi. (2) JACOBIAN BOOKKEEPING [I]: the collective-coordinate Jacobian per zero mode is the mode norm ~ S^{1/2} (standard change of variables, no model input); with the six sheet-doubled modes (HOR.CLOCK.07) the amplitude ratio is (S_n/S_dS)^3 and the Born square gives Gamma_n/Gamma_0 = (S_n/S_dS)^6 = the HOR.CLOCK.06 entropy power law - every exponent factor now has a derivation-level origin: norm = area -> Jacobian = S^{1/2} -> six modes -> amplitude S^3 -> Born -> S^6. (3) UNIT CONSISTENCY [I]: the 4pi in norm^2 = A/(4pi) is the same Gauss-Bonnet 4pi that builds c3 = 1/(2 x 4pi) (ARCH.QUAD.01/P1 hardening); it CANCELS in all ratios - exactly why the clock sees only entropy FRACTIONS S_n/S_dS. (4) THE LAST STEP [P] (recorded, not claimed): show the non-zero-mode fluctuation determinants cancel in the ratio, det'_n/det'_dS = 1 + O(deficit) - plausible (same local near-Nariai geometry) but the one remaining computation; R1 = norm (exact, done) + Jacobian (standard) + Born (rule) + det-ratio (open)
Recorded status: Identity [I] (exact sphere integrals + bookkeeping) + Physical [P] (det-ratio)
Dependencies, status detail and source
Canonical status: Identity: ||Y_1m||^2 = A/(4pi) exactly (all three modes); Jacobian = S^(1/2) per mode; 4pi cancels in ratios; R1 residue = det-ratio cancellation only
Dependencies: HOR.CLOCK.07; HOR.CLOCK.06; HOR.NARIAI.01; ARCH.QUAD.01 (c3 primitive)
Scripts: v131_measure_is_area.py
Location: tfpt_horizon_readouts
Original CSV, line 216Original ledger status
HOR.CLOCK.09
THE DET-RATIO ANOMALY IS THE KOIDE CONSTANT (the HOR.CLOCK.08 'last step' acquires its first exact invariant): (1) SPECTRAL SETUP (re-pinned): the Ginsparg-Perry sphere operator on the unit S^2 is L = -Delta - 2 (the -2 = -|Z2| = the HOR.CLOCK.01 modulus mass); eigenvalues mu_l = l(l+1) - 2 = (l+2)(l-1), degeneracy 2l+1; kernel = the 3 zero modes (l = 1); one negative mode (l = 0). (2) THE HEAT COEFFICIENT IS MADE OF ATOMS [I]: a_1 = (1/4pi) Int(R/6 - E) dA = 1/3 + 2 = 1/N_fam + |Z2| = 7/3 - the curvature third and the sheet mass, both compiler atoms. (3) THE ANOMALY IS THE KOIDE CONSTANT [I]: zeta(0)|det' = a_1 - dim ker = 7/3 - 3 = -2/3 = -|Z2|/N_fam, independently confirmed by numerical continuation of the heat trace (constant term of Tr' e^{-tL} - 1/t extrapolates to -2/3 at ~1e-7) - the EIGHTH appearance of 2/3 in the inventory (SYNTH.INVENTORY.01 counted seven) and the first as a SPECTRAL anomaly. (4) CONSEQUENCE [I]: det'(cL) = c^{zeta(0)} det'(L): the determinant ratio between configurations is NOT automatically 1 - the S^2-sector share of the scaling budget is exactly -2/3 per Hubble-unit conversion; the dimensionless tower itself is configuration-independent (pure-number GP spectrum, HOR.CLOCK.01): all scale dependence = this one anomaly number + the zero-mode Jacobians (HOR.CLOCK.08). (5) THE BUDGET STATEMENT [P] (recorded, not claimed): show the FULL zeta(0) budget of the 4d Nariai fluctuation problem (dS2 sector + gauge/ghost sectors + this S^2 sector) vanishes or is absorbed into the established power law; the S^2 share is pinned exactly at -|Z2|/N_fam - nothing claimed about the total
Recorded status: Identity [I] (heat-kernel arithmetic + numerical continuation) + Physical [P] (full budget)
Dependencies, status detail and source
Canonical status: Identity: zeta(0) of the det'-operator = -2/3 = -|Z2|/N_fam (a1 = 1/N_fam + |Z2| = 7/3); det-ratio question = one zeta-budget statement; S^2 share pinned
Dependencies: HOR.CLOCK.08; HOR.CLOCK.01; SYNTH.INVENTORY.01 (2/3 inventory); HOR.NARIAI.01
Scripts: v132_detratio_anomaly.py
Location: tfpt_horizon_readouts
Original CSV, line 217Original 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 223Original ledger status
HOR.CLOCK.12
DET-RATIO CANCELLATION EXACT WITHIN THE SdS FAMILY (the v131 step closed in the within-family sense): e2-RIGIDITY - the traceless SdS cubic (t^3-3t+2c) forces r_b r_c = 1 - Delta^2/3 EXACTLY (Delta = horizon split; no higher corrections); with the v132 anomaly zeta(0)|det' = -2/3 per sphere, det'(L_r) = r^{4/3} det'(L_1), so the two-sphere non-zero-mode determinant ratio against Nariai is det'_n/det'_N = (1 - Delta^2/3)^{4/3} in CLOSED FORM: NO first-order term in the split (d/dDelta|_0 = 0 exactly) - v131's '1+O(deficit)' DERIVED from e2-rigidity + the established anomaly, not assumed. DEFICIT FORM (audit): eps = 1-c = (3/8)Delta^2 + O(Delta^4) => ratio = 1 - (32/27) eps + O(eps^2), coefficient 32/27 = |mu4|(|Z2|/N_fam)^3 = 4(2/3)^3 (the v103 canonical curvature x glue order; audit-typed). HONEST SCOPE [P]: the finite-weight comparison (S_n/S_dS in {1,2/3,1/3}) still runs through the v132/v133 anomaly budget - a naive global-rescaling reading would shift the Born exponent 6 -> 14/3 (not an atom), so the absorption must proceed via the Hubble-unit conversion (v132's reading); the frozen transfer spectrum is untouched
Recorded status: Identity [I] (exact rigidity + closed-form ratio + audit coefficient) + [P] (finite-weight absorption)
Dependencies, status detail and source
Canonical status: Identity [I]: within-family det-ratio cancellation derived; absorption obstruction stated sharply
Dependencies: HOR.CLOCK.07; HOR.CLOCK.08; HOR.CLOCK.09; HOR.CLOCK.10; v101; v103
Scripts: v144_detratio_family.py
Location: tfpt_horizon_readouts
Original CSV, line 229Original ledger status
HOR.CLOCK.13
THE CLOCK IS A GAUSSIAN ZERO-MODE INTEGRAL AND THE BEND IS DET'-CLEAN (R1's derivation structured to its floor): (1) GAUSSIAN MODEL THEOREM - for p2 modes with per-mode variance ratio (1-alpha) (FORCED by norm = area, v131), the Born-squared partition ratio is (1-alpha)^{p2}, i.e. rate = -p2 ln(1-alpha) EXACTLY = the v124 closed form, and the ln series IS the v127 ring sum term by term: the 'one-loop = ring sum' job discharged at the Gaussian-zero-mode-model level; (2) THE BEND IS DET'-CLEAN - weights n = 1,2 are bookkeeping levels of ONE Euclidean geometry (Nariai S2 x S2, v101), so det'_1 = det'_2 IDENTICALLY and the bend log_{3/2}3 (v107 target) carries NO determinant correction; (3) the single across-topology step is the alpha = 0 reference (round dS vs Nariai), where the v132/v133 budgets and the v144 6->14/3 obstruction live - the quantum content is clean, the classical side of the reference is matched (v126 slope 2 = |Z2|); (4) kappa_seam = 1/(3pi) = alpha_1/pi re-pinned. RESIDUE [P]: the measure identification (gravitational zero-mode measure = this Gaussian model) + the reference normalisation only
Recorded status: Identity [I] (model theorem + det'-clean bend + localisation) + [P] (measure identification)
Dependencies, status detail and source
Canonical status: Identity [I]: ring sum = Gaussian zero-mode integral; bend det'-clean; residue = measure + reference
Dependencies: HOR.CLOCK.12; HOR.CLOCK.07; HOR.CLOCK.08; v101; v126; v127; v131
Scripts: v147_clock_gaussian.py
Location: tfpt_horizon_readouts
Original CSV, line 232Original ledger status
GATE.METRIC.18
THE FINAL IDENTIFICATION, FORCED AS FAR AS IT GOES (the v161/GATE.METRIC.17 one-particle statement carries NO new open content; premise (A) factors into the two already-open items A2 + R1): the residual was 'the seam transport one-particle symbol is the gapped Calderon Bogoliubov map - (alpha) fixed = rank-8 K_Sigma, (beta) gap = (2/3)^6'. (1) THE GAP VALUE IS FORCED [I] - the cusp weights = spec(A0*) = {0,1/3,2/3} (the parabolic residue, forced by anchor+torsion, v114-v117), the transfer eigenvalue is (1-alpha)^{p2} with p2 = 6 = |R+(A3)| (v124/v126), giving {1,(2/3)^6,(1/3)^6} and gap 6 log(3/2) - every entry carrier data, no free knob; (2) MU4-EQUIVARIANCE FORCES DIAGONAL-IN-CUSP [I] - sum_{k=0}^{3} U^k X U^{-k} = |mu4| diag(X) for U = diag(1,i,-i) and the commutant of U is exactly the diagonal algebra, so a mu4-equivariant gapped generator is diagonal with spectrum = the cusp weights (the gapped equivariant generator is unique, v116/v140); with (1) its gap is (2/3)^6; (3) CUSP GRADING = A3 GRADING [I] - degrees (1,2,3) = Spec(Q+) = A3 exponents, h(A3)=4=|mu4|, three levels = N_fam (v137), so the lift from the 3-dim generation space to the 16 Majoranas is the established A3 subnet grading; (4) FIXED = rank-8 CALDERON POLARIZATION [I] - DtN symbol |k|>=0 (v156/v157), polarization P=(1+iA)/2 rank 8 = rank E8 = c (v113), the IR fixed point of the gapped free-boundary RG is the conformal free-fermion vacuum with this covariance. CLOSURE [P]/[A]: with v161 (cone gap = one-particle gap) and v160 (unique quasi-free fixed point => kappa_{2n}=0), (A) CLOSES given only mu4-equivariance + admissibility of the seam RG generator, whose two non-derived inputs are (i) 'the seam RG generator is mu4-equivariant + admissible' = the R1 seam-clock identification (HOR.CLOCK.*; v105 relocation) and (ii) the operator-algebraic net existence = the A2 residual (Kawahigashi-Longo). BOTH already open => premise (A) carries NO independent open content; it factors exactly into A2 + R1 with all numbers forced, the count of independent open items does NOT increase, and (A) is closed CONDITIONALLY on A2 + R1 (a unification, NOT an unconditional proof)
Recorded status: Identity [I] (gap forced by mu4-equivariance + spec(A0*); fixed = rank-8 Calderon) + [P]/[A] (A) factors into the already-open A2 (net existence) + R1 (seam clock)
Dependencies, status detail and source
Canonical status: Identity [I]: gap (2/3)^6 forced + fixed = rank-8 K_Sigma; (A) factors into the already-open A2 + R1 (no new open content)
Dependencies: GATE.METRIC.16; GATE.METRIC.17; v56; v113; v115; v137; v156; v157
Scripts: v162_seam_transport_identification.py
Location: tfpt_research_contracts
Original CSV, line 246Original ledger status
GATE.METRIC.19
PREMISE (A) MAXIMAL HONEST CLOSURE (the two residual gates A2 + R1 pinned to their floor; (A) carries ZERO independent open content): (1) A2 (NET EXISTENCE) CLOSES BY ASSEMBLY [C] - every ingredient is a constructed object: 16 free Majoranas => free-fermion conformal net (Buchholz-Mack-Todorov / Boeckenhauer-Evans), holomorphic c=8 lattice net = (E8)1 (Frenkel-Kac-Segal VOA / Dong-Xu net / Kawahigashi-Longo machinery), mu4 simple-current extension = Longo Q-system index 4 (v125), KLM mu(B)=16/16=1 => holomorphic, modular data checked 248=120(NS)+128(R), character E4/eta^8=j^{1/3} (v156); the ONLY TFPT-specific identification (seam Calderon kernel = free-fermion 2-point function) is DONE since the DtN/Calderon symbol is universally |k| (Lee-Uhlmann, v156/v157) => A2 is citation/assembly of published theorems, not a finite computation we owe (re-typed [A]->[C]); (2) R1 (SEAM CLOCK) REDUCES TO GATE.QGEO [C] - mu4-equivariance forces the generator diagonal in the cusp basis (sum_k U^k X U^{-k}=|mu4| diag X), and the flat mu4-equivariant admissible generator on the carrier is UNIQUE = A0* (v115/v116), spectrum {0,1/3,2/3} => transfer eigenvalues (1-alpha)^{p2}={1,(2/3)^6,(1/3)^6}; so IF the seam boundary generator is in that class its gap and grading are FORCED (seam clock = flavour wall = horizon clock, v54), and the residual is ONE geometric identification 'seam collar boundary = the flat mu4-parabolic connection on P^1 minus mu4' = GATE.QGEO (the parabolic realisation of Q; cohomological half established v137: b1=3=N_fam, cusp grading = A3 cohomology) - NOT a new gate; (3) IRREDUCIBLE CORE IS A THEOREM [O] - the No-Unit Theorem (v153) makes {pi, v_geo} the provably minimal irreducibles, and closing (A) introduces NO new irreducible; the seam=flavour=horizon geometric postulate behind GATE.QGEO is in that same irreducible class. FINAL ACCOUNTING: premise (A) = (A2 assembly [C]) + (R1 = GATE.QGEO, already open) + (the {pi,v_geo} core, a theorem) => ZERO independent open gates; (A) ceases to be its own gate. Honest scope: a unification + re-typing, NOT an unconditional proof
Recorded status: Identity/assembly [C] (A2 net existence by established AQFT; R1 reduced to GATE.QGEO via A0* uniqueness) + [O] (the irreducible {pi,v_geo} geometric core, a theorem by v153)
Dependencies, status detail and source
Canonical status: Closure [C]: A2 = net-construction assembly, R1 = GATE.QGEO; (A) has zero independent open gates, irreducible core {pi,v_geo} a theorem
Dependencies: GATE.METRIC.16; GATE.METRIC.17; GATE.METRIC.18; v115; v116; v125; v137; v153; v156
Scripts: v165_premise_a_closure.py
Location: tfpt_research_contracts
Original CSV, line 247Original ledger status
QGEO.REALIZE.01
THE SEAM-COLLAR REALISATION (the physical half of GATE.QGEO; the one Tier-1 realisation premise, stays [C]/[O]): the QGEO Rigidity Theorem (QGEO.RIGID.01/v168) closes the PHYSICS only if its hypotheses -- genus-0 seam collar, faithful D4 action, one orbit of four parabolic marks -- follow from the seam construction (P1) itself. The cohomological half is established (b1=N_fam=3, cusp grading = A3 exponents), but the identification 'seam-collar boundary = P^1 minus mu4' is the realisation premise. HONEST TYPING: RIGID is [E] (the finite math), REALIZE stays [C]/[O] (the seam-side identification) -- not silently promoted; this is the single remaining structural realisation premise (the Tier-1 hinge of v167) PRECISION (2026-07-21, v503/QGEO.EMERGE.LIGHT.01 -- marker UNCHANGED [C]/[O]): the residual is now LOCATED, not moved -- the mark count 4 is derived (v216) and the square modulus is derived GIVEN the order-4 clock (v493; the DtN mod-4 indicator has an isolated zero exactly at tau = i, and free dynamics selects the HEXAGONAL point, which carries no order-4 element), so the irreducible rest of the realisation premise is precisely 'why does the raw seam carry the order-4 clock' (clock = Coxeter element of W(A3) = the carrier datum) -- IDENTICAL with the P2-typing rest R1 of v499 (module identification with the cusp connection): QGEO-R1 == P2-R1, one common residual. The square and the mark count are derived; the clock stays the one [C] carrier input, its dynamical realisation stays [O]. PRECISION (2026-07-21, v506/SEAM.CLOCK.RIGIDITY.01 -- marker UNCHANGED [C]/[O]): the rest is now ONE bit: the central position of the deck in the mark-D4; order/uniqueness/orbit/tau = i FOLLOW (v506) -- the deck has exactly two Moebius square roots, both automatically order 4, and a marks-preserving root exists iff the deck is the central involution of the mark-stabiliser D4 (harmonicity alone is insufficient, exact counterexample); the fermionic side is FORCED (U^2 = (-1)^F exact on the NS seam, nonsplit Z4, Z8 tower): the fermionic Z4 forces the ORDER of the clock, the alignment bit remains the one [C] carrier input, its dynamical realisation stays [O]. PRECISION (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED [C]/[O]): the bit is characterised as a tetrad (clock exists <=> tau = i AND deck = the CM-fixed half-period translation <=> Stab = D4 AND deck central <=> deck pairs harmonically separated <=> NS Fock lift nonsplit) and the class of the deck IS derived (half-period translation of the seam double: every mark-free collar involution descends from a half-period translation, v507); only its position (which half-period) stays the input -- the common torus origin does NOT force the alignment (in-family silver witness), so the bit is no tautology; its fermionic face is the non-split NS Fock extension class (Fidkowski-Kitaev-type). PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED [C]/[O]): the alignment bit is HALVED -- its position half (deck central vs edge in the mark-D4) is TOPOLOGY: the collar deck is a covering deck, hence free on the seam circle; the unique free involution is the antipode (complete dihedral census), NONSPLIT <=> FREE over all 17 involutions, and the edge class is excluded (it is the restriction of the BRANCHED pillowcase involution -- v510); the remaining carrier input is the square-modulus datum tau = i alone (equivalently: the clock / the CM-fixed half-period / the non-split extension class on the free seam circle); its dynamical realisation stays [O]. PRECISION (2026-07-22, v512/SEAM.TAU.FLAG.01 -- marker UNCHANGED [C]/[O]): the remaining input can equivalently be stated as the flag-transitivity postulate -- v512 (the four marks are also SIDE-indistinguishable: V4 -> D4 symmetry lift <=> tau = i, 12-fold exact equivalence web; bare mark-transitivity is delta-blind and no local jet sees the side bit); the search space for a future [E] closure is narrowed to the RP positivity of the odd sector (the negation is concretely measurable: odd-split (2/pi) ln cot(delta/2) != 0).
Recorded status: Physical [C] (seam-side identification) + [O] (open realisation)
Dependencies, status detail and source
Canonical status: Physical [C]/[O]: seam-collar = P1-mu4 stays the one realisation premise (RIGID [E], REALIZE [C]); rest PRECISED by v503 and REDUCED by v506: marks + square derived, the order-4 clock reduced to ONE alignment bit ('the collar deck is the central involution of the mark-D4' -- order/uniqueness/orbit/tau = i follow, the fermionic Z4 order is forced, U^2 = (-1)^F); QGEO-R1 == P2-R1 = this single bit -- marker unchanged; v507 derives the CLASS of the deck (half-period translation) and characterises the bit as a tetrad with the non-split NS Fock extension class as its physical face -- only the position stays the input; v510 halves the bit: the position half is topology (covering-deck freeness excludes the edge class), the rest is the square-modulus datum tau = i alone; v512 restates the rest as the flag-transitivity postulate (V4 -> D4 symmetry-lift bit) and narrows the [E]-closure search space to the RP positivity of the odd sector
Dependencies: QGEO.RIGID.01; GATE.QGEO.01; GATE.METRIC.19; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; v503; SEAM.CLOCK.RIGIDITY.01; v506; SEAM.BIT.ORIGIN.01; v507; SEAM.BIT.FREEDOM.01; v510; SEAM.TAU.FLAG.01; v512
Scripts: v168_qgeo_rigidity.py
Location: tfpt_research_contracts;origin_theory
Original CSV, line 253Original ledger status
QGEO.MARKS.01
THE SEAM MARKING OBLIGATION (genuine open obligation 1 of 2; the structural residual of the theory, NOT a finite computation): the RAW RP seam collar must canonically produce a genus-0, four-parabolic-marked boundary with a faithful D4 action -- the four marks coming from the anchor atom e1(a)=4=|mu4| (NOT an arbitrary marking), with mark collisions excluded (else b1=N_fam=3, the D4 faithfulness or the H^1 character split collapse), and the sheet reflection tau and clock rotation rho satisfying tau.rho.tau=rho^{-1} (D4). Proof route: RP one-sidedness => sheet involution tau; normal-double compactification + P1 hardening => genus-0/P^1 normalisation; e1(a)=4 => the four-point divisor; non-collision + the D4 relation. Given QGEO.MARKS.01, the geometry side closes via QGEO.UNIFORM.01/COHOM.01/MODULE.01 (all [E], v177). A constructive-geometry statement, left honestly OPEN (needs a human argument or proof assistant, not arithmetic)
Recorded status: Physical/geometric [O] (open constructive obligation)
Dependencies, status detail and source
Canonical status: Open [O]: raw RP seam collar -> genus-0 four-marked D4 boundary (one of the two genuine obligations)
Dependencies: QGEO.TREE.01; QGEO.UNIFORM.01; ARCH.SPINE.01; v177_seam_marking_kernel.py
Scripts: v177_seam_marking_kernel.py
Location: tfpt_research_contracts
Original CSV, line 256Original ledger status
QGEO.REDUCE.01
DEEPER REDUCTION OF THE TWO OBLIGATIONS (an honest attempt at QGEO.MARKS.01 + QGEO.KERNEL.01; NEITHER is closed -- they are genuine constructive-geometry/AQFT statements, not finite computations -- but each finite/algebraic CORE is now closed [E] and each obligation is reduced to ONE sharper irreducible premise [O]): (1) MARKS FINITE CORE [E] -- given a genus-0 double with an order-4 clock rho (conjugate to z->iz, fixed points {0,infinity}) and sheet reflection tau, the parabolic marks are a generic rho-orbit {1,i,-1,-i}=mu4 (|mu4|=4=e1(a), NOT the fixed points), DISTINCT iff b1=(#marks)-1=3=N_fam (any collision drops b1 below N_fam, excluded by the family count), and tau:z->1/z gives tau.rho.tau=rho^{-1} and permutes the marks => <rho,tau>=D4 of order 8, faithful; so MARKS reduces from 'produce the whole marked D4 boundary' to the SINGLE topological/clock premise 'the seam double is genus-0 and the seam clock is an order-4 conformal automorphism whose marks form one orbit'. (2) KERNEL FINITE CORE [E] (dissolves the spectrum-vs-operator worry on the finite block) -- on the 3-dim H^1 character space the mu4 characters (weights (1,2,3) -> i,-1,-i) are DISTINCT, so by Schur a mu4-equivariant operator is forced DIAGONAL (commutant of distinct-eigenvalue U = diagonal algebra) and is COMPLETELY DETERMINED by its eigenvalue per character; hence two such operators with the same forced transfer spectrum {1,(2/3)^6,(1/3)^6} (v162) and the unique residue-normalised basis Phi (v177) are EQUAL as operators, not merely isospectral -- the 'C_Sigma=U^{-1}C_mu4 U vs same spectral list' distinction VANISHES on the multiplicity-free H^1 block; so KERNEL reduces from the full operator identity to the SINGLE full-operator-reduction premise 'the full seam Calderon DtN on L^2 reduces to this 3-dim gapped mu4-moving transfer block plus the fixed rank-8 Calderon polarisation K_Sigma' (principal symbol universally |k|, Lee-Uhlmann/v156; c=8 fixed point isolated, no relevant/marginal drift, v158). NET (honest): neither obligation closed; both sharpened to milder premises, finite cores [E]; the structural residual of the whole theory is exactly {the topological clock premise, the full-operator-reduction premise}
Recorded status: Deeper reduction [E] (finite cores) + [O] (the two sharper irreducible premises)
Dependencies, status detail and source
Canonical status: Deeper reduction [E]/[O]: MARKS + KERNEL finite cores closed; each reduced to one sharper premise (topological clock / full-operator reduction); neither obligation closed
Dependencies: QGEO.MARKS.01; QGEO.KERNEL.01; QGEO.TREE.01; v156_seam_net_construction.py; v158_fixed_point_stable.py; v162_seam_transport_identification.py; v177_seam_marking_kernel.py
Scripts: v178_marks_kernel_attempt.py
Location: tfpt_research_contracts
Original CSV, line 258Original ledger status
QGEO.CONF.01
THE SEAM CONFORMAL-REALISATION PREMISE -- the SINGLE structural residual of the whole theory (the two obligations QGEO.MARKS.01 + QGEO.KERNEL.01 collapse onto it; honest unification, the premise itself stays OPEN). The two residual premises left by v178 are NOT independent: (A) MARKS's residual decomposes into three pieces, two of which are already-established TFPT facts and the third forced -- (1) GENUS-0 IS P1: the one-sided Gauss-Bonnet normalisation c3 = 1/(|Z2|*2pi*chi(S^2)) = 1/(2*2pi*2) = 1/(8pi) uses chi(S^2)=2 (TOPOLOGICAL), and chi=2-2g => g=0, so 'genus-0' IS the same chi=2 that supplies the '8' in c3 (v58/v73/SEAM.KHALF.01) -- not a new assumption but P1 itself; (2) ORDER-4 CLOCK IS THE CARRIER: the seam clock is the Coxeter element of the carrier monodromy W(A3)=S4 (v117), order h(A3)=3+1=4=|mu4| -- forced by P2, not assumed; (3) MARKS=ONE ORBIT is FORCED: the parabolic marks are not the two elliptic fixed points {0,infinity} of the order-4 rotation, which acts freely off them, so a free order-4 orbit has exactly 4=e1(a) points; so MARKS's residual is ONLY 'the carrier clock acts conformally (Moebius) on the genus-0 seam double'. (B) KERNEL's residual then FOLLOWS from that same geometry by established theorems: DtN principal symbol universally |k| (Lee-Uhlmann, v156) => the rank-8 (=c=rank E8) continuous 'fixed' polarisation K_Sigma; the c=8 holomorphic fixed point isolated, no relevant/marginal drift (v158); on a finite-volume cusped surface the spectral decomposition (Eisenstein continuous + finite residual) puts the residual/cusp-form data in H^1(P^1\\mu4)=C^3=N_fam = the 3-dim gapped mu4-moving transfer block, Schur-diagonal with the forced weights {1,(2/3)^6,(1/3)^6} (v162/v178). (C) UNIFICATION: both reduce to QGEO.CONF.01 -- 'the raw RP seam normal double is conformally (P^1,mu4) with the carrier clock as the order-4 Moebius automorphism z->iz'. Given it, genus-0 (=P1), order-4 (=h(A3)), marks=orbit (forced), the cohomology grading (v177), the DtN block + rank-8 polarisation (v178) and the (E8)1 net (v154/v175) all follow. [O] IRREDUCIBLE: QGEO.CONF.01 is a constructive-geometry statement (the conformal class of the raw seam double IS the cusped sphere with the carrier clock as a Moebius map), NOT a finite computation; it is the single structural residual of the theory, left honestly OPEN
Recorded status: Geometric realisation premise [O] (the single residual; component reductions [I] established)
Dependencies, status detail and source
Canonical status: Geometric [O]: the SINGLE residual is the seam conformal realisation (P^1,mu4) with the carrier clock as the order-4 Moebius map; MARKS+KERNEL both reduce to it
Dependencies: QGEO.MARKS.01; QGEO.KERNEL.01; QGEO.REDUCE.01; AX.P1.01; SEAM.KHALF.01; v58_seam_horizon_chain.py; v73_k_c3_half.py; v117_monodromy_weyl_a3.py; v156_seam_net_construction.py; v158_fixed_point_stable.py
Scripts: v179_conformal_realisation.py
Location: tfpt_research_contracts;origin_theory
Original CSV, line 259