Original ledger status
QFT.FOCK.01
SEAM FOCK-SPACE READINGS (three exact audit bridges that sharpen the QFT/topological reading of already-used objects): (1) CAR CONE COMPRESSION [I] -- the even CAR algebra of the 16 seam Majoranas has dim Lambda^even(R^16)=2^15=32768 directions (cumulant blocks C(16,4)=1820, C(16,6)=8008, C(16,8)=12870), but a quasi-free Bogoliubov transport is the second quantisation Gamma(t)|_Lambda^m=Lambda^m(t), so the whole cone is generated by the 16-dim one-particle spectrum, compression 2^15/16=2048=2^11 (the explicit count behind v161; 'the QFT cone is large but not independent'); (2) WITTEN INDEX = PLUCKER NORM 11 [I] -- the 4 mu4-corner fermions span a 2^4=16=dim S+ Fock space; QBL (degree<=2) keeps sum_{k<=2}C(4,k)=1+4+6=11=||Pl(K)||_1, discards C(4,3)+C(4,4)=5=g_car, so 11=dim S+ - g_car and the quark ratio c_u/c_d=g_car*11/(N_fam^2*Delta_Q)=55/117 reads the QBL-protected boundary state-space dimension (a topological Witten index, not a free integer); (3) AFFLECK-LUDWIG g-THEOREM RIGIDITY [I]/[L] -- c_UV=c((D5)1)+c((A3)1)=5+3=8=c_IR((E8)1), so Delta c=0 along the seam boundary RG flow; a flow needs Delta c>0 to dissipate, so Delta c=0 forces an exact isometry (no relevant/marginal perturbation), the boundary-entropy form of the v157/v158 rigid fixed point. Exact identities; readings audit
Recorded status: Identity/Lattice [I]/[L] (cone count + Witten index + central-charge conservation)
Dependencies, status detail and source
Canonical status: Identity/Lattice [I]/[L]: CAR cone 2^15->2^11; Witten index 11=Plucker norm=dim S+ - g_car; g-theorem Delta c=0 rigidity
Dependencies: GATE.UWALL.06; CAR.QFREE.01; GATE.METRIC.14; v42_exterior_leg.py; v161_one_particle_reduction.py
Scripts: v174_seam_fock_readings.py
Location: tfpt_2;tfpt_5_redteam
Original CSV, line 287Original ledger status
QFT.NETRP.01
NET EXISTENCE + FULL-CONE RP DISCHARGED TO [F] (the heavy artillery; the 'three open QFT theorems' are two [F] + one irreducible [O], nothing fabricated): (1) FULL-CONE REFLECTION POSITIVITY FOR ALL m [F] -- the CAR second-quantisation functor Gamma(t)=(+)_m Lambda^m(t) of the one-particle seam contraction t acts on the COMPLETE Fock space Lambda^*(R^16) (dim 2^16=65536); by the exterior-power spectrum theorem every eigenvalue of Gamma(t) is a subset product of spec(t), verified on all 2^16 products to be in [0,1] (contraction), with eigenvalue-1 (fixed) multiplicity 2^8=256 = wedges of the rank-8 K_Sigma polarisation and sub-leading eigenvalue = the one-particle gap (2/3)^6 -- so full-cone RP/gap reduce, for ALL m (not just m<=6, v161), to the one-particle contraction (Shale-Stinespring CAR functor + quasi-free positivity); the v160 scope premise is DISCHARGED by a theorem; (2) A2 NET EXISTENCE [F] -- the (E8)1 net exists by cited theorems (16 free Majoranas -> SO(16)_1 chiral net, Buchholz-Mack-Todorov/Boeckenhauer-Evans-Kawahigashi; holomorphic c=8 lattice net = (E8)1, Frenkel-Kac-Segal/Dong-Xu/Kawahigashi-Longo; mu4 index-4 Q-system, v125), with machine-verified data: character E4/eta^8=1+248q+4124q^2+34752q^3+213126q^4+1057504q^5 (level-1=248), embedding 248=120+128, E8 Cartan even unimodular (det 1, the unique rank-8 even unimodular lattice, Minkowski-Siegel/v83); (3) BOTH COLLAPSE TO ONE [I] -- full-cone RP -> one-particle contraction, A2 -> cited net existence, both needing the SAME input: the actual seam Calderon kernel = the gapped one-particle contraction on P^1\\mu4 = QGEO.REALIZE.01 (finite half proven, v168). (4) IRREDUCIBLE PREMISE [O] -- QGEO.REALIZE.01 (seam boundary CFT = parabolic transport on P^1\\mu4) is a constructive-geometry/AQFT statement, NOT closeable by finite computation, left OPEN (needs a human constructive-QFT/geometry argument). NET: net existence + full-cone RP are [F]; the one irreducible structural premise is the seam realisation
Recorded status: Formalised/Identity [F]/[I] (all-m functoriality + verified net data) + [O] (the seam realisation)
Dependencies, status detail and source
Canonical status: Formalised [F]: full-cone RP (all m, CAR functor) + (E8)1 net existence (assembled+verified); the one [O] is the seam realisation QGEO.REALIZE.01
Dependencies: GATE.METRIC.11; GATE.METRIC.13; GATE.METRIC.16; GATE.METRIC.17; QGEO.RIGID.01; QGEO.REALIZE.01; v83_e8net_holomorphic_uniqueness.py; v125_glue_qsystem.py; v161_one_particle_reduction.py
Scripts: v175_net_existence_full_cone.py
Location: tfpt_research_contracts;tfpt_5_redteam
Original CSV, line 288Original ledger status
FR.MUONG2.01
MUON g-2 SEAM VERTEX (2026-06-15, archive integration of the old muon-g2 note, re-typed for the current kernel): the carrier fixes a second-order topological defect delta_2 = B*gamma*delta_top^2 = (5/4)delta_top^2 (delta_top=Omega_adm c3^4=48 c3^4=3/256pi^4, v2/v23; B*gamma=(3/2)(5/6)=5/4 the carrier compression quotient, v51); the seam-loop projection a_mu^seam = delta_2/(2 pi) = 45/(524288 pi^9) = 2.879e-9. (1) [E] the VALUE is an exact compiler number (trace reading delta_2 = 4! Tr_{S+}(X^2) c3^8, Tr=120=5!). (2) [C] the IDENTIFICATION of delta_2/(2 pi) as a magnetic vertex correction (the 1/2pi=4c3 seam-loop projection) is a physical bridge, NOT a theorem -- so the prediction is [C]. (3) [N] 0.81 sigma vs the dispersive Delta a_mu=(2.49+-0.48)e-9. (4) [C] HONEST: lattice/CMD-3 HVP shrinks Delta a_mu (~1.5e-9), the FIXED TFPT value then sits ~1.5 sigma high -- testable against either HVP outcome, not tuned to the dispersive one. (5) [X] dated kill test: a converged Delta a_mu outside 2.879e-9 +- 0.5e-9 excludes the seam-vertex mechanism (compiler core untouched). A downstream readout, NOT a compiler power. Wolfram-mirrored (the value is exact). (6) KILL FIRED 2026-08-04 (news round): the lattice-side HVP consensus consolidated -- WP25 (Phys. Rep. 2025) adopts the lattice HVP as the SM baseline and the BMW hybrid evaluation (Nature 2026) confirms it, Experiment - SM = 0.5 sigma, i.e. Delta a_mu converged OUTSIDE 2.879e-9 +- 0.5e-9: the seam-vertex identification is excluded in its present form (the dated kill (5) fired exactly as preregistered). The bridge is dissolved; the exact compiler number (1) [E] and the compiler core are untouched -- the watchdog delivered
Recorded status: Physical bridge KILLED: dated kill test [X] fired 2026-08-04 (exact value stays [E]; the [C] vertex identification is excluded in its present form -- bridge dissolved, compiler core untouched)
Dependencies, status detail and source
Canonical status: Physical bridge KILLED [X] fired 2026-08-04: a_mu^seam = delta_2/(2 pi) = 45/(524288 pi^9) = 2.879e-9 stays an exact value [E], but the WP25+BMW lattice-HVP consensus (Experiment - SM = 0.5 sigma) puts the converged Delta a_mu outside 2.879e-9 +- 0.5e-9 -- the [C] vertex identification is excluded in its present form; bridge dissolved, compiler core untouched; not a compiler power
Dependencies: v2_carrier_pascal.py; v23_anchor_generator.py; v51_boundary_half_step.py
Scripts: v204_muon_seam_g2.py
Location: tfpt_4_frontier
Original CSV, line 289Original ledger status
GRAV.XI.01
GRAVITATIONAL 3/4 OVER-DETERMINATION (2026-06-15, archive integration of the old Eliminating-K note): the torsion-compression ansatz kappa^2 = xi phi0/c3^2 with the Einstein limit kappa^2=8 pi G fixes xi = 8 pi c3^2/phi0 = c3/phi0 (since 8 pi c3^2 = c3), and at tree level phi_tree=1/(6 pi) this is EXACTLY 3/4. (1) [E] tree identity xi = c3/phi_tree = (1/8pi)/(1/6pi) = 3/4. (2) [E] Einstein-limit reduction 8 pi c3^2 = c3 => G drops out, G is an output not an input. (3) [E] OVER-DETERMINATION: 3/4 = q(A3) = N_fam/|mu4| = ln(m/mu) (v152 k=c3/2) -- the torsion-compression xi and the gapped EH replica are two structurally independent appearances of the same gravitational 3/4. (4) [N] with the retained seed xi=c3/phi0=0.748303 (-0.226%). (5) [C] the kappa^2=xi phi0/c3^2 ansatz is the [C] bridge; the modern derivation of 3/4 is v152/v68, xi is no new free parameter, 1/G stays the one declared anchor (v60/v153). Wolfram-mirrored (tree identity + over-determination exact)
Recorded status: Identity [E] (tree 3/4 + over-determination) + physical bridge [C] (the kappa^2 ansatz)
Dependencies, status detail and source
Canonical status: Identity [E]: xi = c3/phi_tree = 3/4 = q(A3) = ln(m/mu) -- an independent route to the gravitational 3/4; the kappa^2=xi phi0/c3^2 ansatz is the [C] bridge, G is a metrology output, 1/G the one anchor
Dependencies: v152_norm_is_anchor.py; v68_seeley_dewitt_residual.py; v60_lambda_metrology_branch.py
Scripts: v205_xi_threequarter.py
Location: tfpt_4_frontier
Original CSV, line 290Original ledger status
GRAV.H0.01
COSMOLOGICAL-CONSTANT BRANCH H0 = 66.5..67.1 (2026-06-15, archive integration of the tfpt-31 CC note; sharpens the qualitative H0-vs-Lambda row): the seam number delta_Sigma = m_1 e^{-2 pi alpha^-1/3} with the lowest-block m_1=Omega_adm=48. (1) [N] delta_Sigma^lead = 48 e^{-2 pi*137.036/3} ~ 1.085e-123. (2) [E] convention split M_Pl^4/Mbar^4 = (8 pi)^2 = 631.65 (unreduced vs reduced Planck density -- must be displayed). (3) [N] rho_Lambda = M_Pl^4 delta_Sigma ~ 2.4e-47 GeV^4 (observed vacuum order). (4) [N] 123-orders cross-link |log10 delta_Sigma| ~ 122.96, consistent with the v60 split 122.948. (5) [C] under the flatness closure H0^lead=66.51, H0^disc=67.10 km/s/Mpc -- 1.6..0.5 sigma BELOW Planck (67.36+-0.54), ~6 sigma below SH0ES (73.0+-1.0): HONEST, the Hubble tension is NOT relieved by raising the vacuum scale; H0 uses imported Omega_b, omega_DM (external cosmology), so it is a [C] readout, not a from-nothing output. Python-only (the (8pi)^2 ratio is on the Wolfram path via v60)
Recorded status: Numerical [N] (delta_Sigma, 123-orders) + physical bridge [C] (H0 flatness readout)
Dependencies, status detail and source
Canonical status: Numerical/bridge [N]/[C]: delta_Sigma=48 e^{-2pi alpha^-1/3}~1.085e-123, (8pi)^2 convention split, |log10|~122.96 (v60 consistent); H0=66.5..67.1 below Planck, does NOT relieve the Hubble tension; [C] readout (imported Omega_b, omega_DM)
Dependencies: v60_lambda_metrology_branch.py; v55_coxeter_cycle.py
Scripts: v206_h0_lambda_branch.py
Location: tfpt_4_frontier
Original CSV, line 291Original ledger status
GRAV.ASYMP.01
ASYMPTOTIC SAFETY -- A SECOND [P] ROUTE TO THE METRIC SECTOR (2026-06-15, archive integration of the old Five-Problems note; NOT load-bearing, does NOT close G_metric): the current theory closes QG via the discrete seam-net/AQFT programme (v175; open premise QGEO.REALIZE.01); this records a COMPLEMENTARY continuum-FRG argument that a non-Gaussian UV fixed point exists from the same axioms. Setup d_t g = (2 + eta_R(g)) g, eta_R = -a1 g (torsion K^2 positive-definite => a1>0). (1) [E] the axion coupling is fixed and small: y^2 = 16 c3^2 = 1/(4 pi^2) = 0.025330 exactly (g_agg=-4 c3 from the cubic, scale-invariant). (2) [N] existence by IVT: 2+eta_R is +2>0 as g->0 and <0 at large g (torsion damping) => a root g_*>0 (non-Gaussian UV fixed point). (3) [N] g_*=2/a1>0 finite, Reuter-type O(1), coordinates set by {c3,phi0}+field content, no new free input. (4) [O] HONEST: a continuum-FRG ansatz the discrete-compiler theory did NOT adopt; a second plausibility route only; the primary machine-checked route stays the seam-net (v175); G_metric open either way. Python-only
Recorded status: Open/plausibility [O] (a [P]-typed second QG route; not load-bearing) + [E] (y^2=16 c3^2)
Dependencies, status detail and source
Canonical status: Open/plausibility [O]: a complementary continuum-FRG argument (non-Gaussian UV fixed point exists from {c3,phi0}); y^2=16 c3^2 exact; NOT load-bearing, the seam-net (v175) stays primary, G_metric open
Dependencies: v175_net_existence_full_cone.py; v28_gravity_fR.py
Scripts: v207_asymptotic_safety.py
Location: tfpt_4_frontier
Original CSV, line 292Original ledger status
HOR.BHTHERMO.01
BLACK-HOLE THERMODYNAMICS FROM THE SEAM: modular Hawking temperature + scalaron-corrected Wald entropy (2026-06-15, archive integration of the dropped tfpt-42 stationary-horizons/compact-objects section, re-typed through the Tomita-Takesaki seam work v198/v199). (1) [E] MODULAR T_H: the modular flow Delta^{it}=e^{-2 pi t K_H} (Bisognano-Wichmann/Tomita-Takesaki) gives KMS at beta=2 pi and T_H = kappa/(2 pi); the 2 pi IS the seam unit 1/(4 c3) (1/2pi=4c3, v58), reproducing T_H=c3/M (v57); the seam<->horizon modular identification is [P] (ties to [rho,Lambda]=0, v198/v199). (2) [E] SCALARON WALD ENTROPY: with the induced f(R)=R+R^2/(6 M_s^2) (M_s=c3^{7/2} Mbar, v28), S_W=(f_R A)/(4 G)=(A/4G)(1+R_h/3M_s^2), f_R=1+R/(3M_s^2) -- leading A/4G is the c3 area law (1/4=1/|mu4|), the correction an EXACT scalaron consequence; BH application [P]. (3) [E] SdS T_H=(1/4pi r_h)(1-Lambda r_h^2), 1/4pi=2c3. (4) [E] area law S_BH=M^2/(2c3)=4 pi M^2=A/4G, same 1/(4c3)=2pi factor as the modular beta. Wolfram-mirrored (the f_R Wald identity + modular 2pi=1/4c3 are exact)
Recorded status: Identity [E] (modular 2pi=1/4c3, scalaron Wald factor, area law) + physical bridge [C] (the BH/modular identification)
Dependencies, status detail and source
Canonical status: Identity/bridge [E]/[C]: T_H=kappa/2pi from modular flow (2pi=1/4c3); scalaron Wald entropy S_W=(A/4G)(1+R_h/3M_s^2) exact from v28; area law S_BH=A/4G; the BH/modular identification is [C]
Dependencies: AX.P1.01; HOR.01; v57_horizon_crosslinks.py; v28_gravity_fR.py; v198_modular_commutator_reduction.py
Scripts: v208_bh_thermodynamics.py
Location: tfpt_horizon_readouts
Original CSV, line 293Original ledger status
HOR.BHCORE.01
BLACK HOLE AS A SEAM FIXED POINT (topological defect), NOT a singularity (2026-06-15, archive integration of the old Five-Problems BH picture; the SUPERSEDED torsion/RN-vortex metric is DROPPED -- RN-type metric superseded by Nariai/seam=horizon v101-v104/v190, cf. v203). Built from established suite atoms. (1) [E] NO SINGULARITY = SEAM FIXED POINT: the gapped attractor multiplier lambda_2=(2/3)^6=64/729 with gap Delta=6 log(3/2)>0 (v56) -- the core is a fixed point (d phi/dt=0), rho->inf replaced by phi->phi_* [P]. (2) [E] MAXIMAL BH = ANCHOR: Nariai cubic t^3-3t+2=(t-1)^2(t+2), roots (1,1,-2) = the traceless projection of the anchor a=(1,1,2) (a-(p1/3)1=-(1/3)(1,1,-2), p1=4; v101/v190). (3) [E] INFORMATION stays in the boundary field: Page recovery decays at lambda_2=(2/3)^6 per step (v54), no interior loss [P]. (4) [E] holographic remnant S_BH=A/4G, 1/4=1/|mu4|, a Planck-scale floor not a point singularity [P]. (5) [P] HONEST: a qualitative structural synthesis (reuses v54/v56/v57/v101/v190), NOT load-bearing, NOT a metric, the old RN/torsion metric NOT resurrected. Python-only
Recorded status: Open/synthesis [O] (a [P]-typed structural reading; not load-bearing) + [E] (the underlying anchor/attractor atoms)
Dependencies, status detail and source
Canonical status: Open/synthesis [O]: BH = seam fixed point (no singularity, phi->phi_*), maximal BH = anchor (Nariai roots (1,1,-2)=traceless a), info via Page recovery (2/3)^6, Planck-scale floor; [P] reading, NOT load-bearing, RN metric not resurrected
Dependencies: v54_seam_horizon_keystones.py; v56_unique_attractor.py; v57_horizon_crosslinks.py; v101_horizon_anchor.py; v190_nariai_entropy_bound.py
Scripts: v209_bh_regular_core.py
Location: tfpt_horizon_readouts
Original CSV, line 294Original ledger status
QGEO.MARKLOCAL.01
MARK-LOCAL DtN CERTIFIED ON REALISTIC STEKLOV PROFILES + AT THE STATE LEVEL (2026-06-15): a numerical SHARPENING of QGEO.SUBPRIN.01/v201, carried to the quasi-free STATE (the actual omega o rho = omega, not just [rho,Lambda]=0); NOT a closure of QGEO.SYM.01. v201 argued at the abstract mode level; here a real von-Mises curvature bump g(theta) localised at a puncture, summed over the four mu4 marks f(theta)=sum_{j=0}^3 g(theta-j pi/2), builds the Steklov DtN Lambda=|D_theta|+M_f. (1) [E] every off-(mod 4) Fourier coefficient of the REAL f vanishes to <1e-16 (exact 4-fold cancellation), although g itself has all modes. (2) [N] ||[rho,Lambda]||<1e-15 (principal |k| diagonal, v198; sub-principal M_f Z4-invariant). (3) [N] STATE INVARIANCE (the genuinely new step vs v201): the quasi-free covariance C=1/2(1+sgn H_1) satisfies rho C rho^dag = C to <1e-13 and [rho,C]<1e-13 -- the STATE, not just the operator, is clock-invariant. (4) [N] negative controls (a single off-centre bump, a Z3 3-mark source, 4 generic unequal marks) each break BOTH the operator and the state by O(1) -- specific to the mu4 orbit. (5) [N] convergence: mark-local commutator stays <1e-10 for N in {16,32,64}, the Z3 control stays O(1) -- not a truncation artefact. (6) [O] VERDICT: certifies the implication 'mark-local DtN => omega o rho = omega' for concrete Steklov profiles AND at the state level (the operational content of QGEO.SUBPRIN.01); the PREMISE that the physical seam DtN IS mark-local / conformal-deck flat stays [O] -- the narrowest definitional form of QGEO.SYM.01, NOT its closure (net existence + full-cone RP remain the [E] of v175). Python-only (numerical; the exact mark-sum identity is Wolfram-mirrored via v201)
Recorded status: Numerical certification [N] (implication on realistic profiles + state level) + premise [O] (the physical seam being mark-local)
Dependencies, status detail and source
Canonical status: Numerical/premise [N]/[O]: mark-local DtN => omega o rho = omega certified for realistic von-Mises Steklov profiles AND at the quasi-free state level (rho C rho^dag = C to 1e-13); negative controls (off-centre, Z3, generic) break by O(1); the seam being mark-local stays [O], the narrowest form of QGEO.SYM.01, not a closure
Dependencies: QGEO.SUBPRIN.01; QGEO.STATE.01; QGEO.MODULAR.01; v201_seam_subprincipal_marks.py; v198_modular_commutator_reduction.py
Scripts: v210_mark_local_dtn.py
Location: tfpt_research_contracts
Original CSV, line 295Original ledger status
FORM.QGEO.01
LEAN 4 FORMALISATION of the QGEO.SYM.01 CONDITIONAL THEOREM (2026-06-15; the algebraic core of v201/v210, machine-proved) -- AUDIT: PASS, no sorry/admit, no domain axioms (#print axioms = propext, Classical.choice, Quot.sound only; the two block-diagonality theorems even use only propext+Quot.sound). New module TfptCarrier/SeamDeckClosure.lean proves: (1) [F] geom_sum_fourth_root: for ζ:ℂ with ζ^4=1, Σ_{j<4} ζ^j = (if ζ=1 then 4 else 0) -- the 4th-root character orthogonality (finite geometric sum) that makes a μ4-mark sum vanish off mod 4; (2) [F] clock_gen_pow_four + mark_sum_residue_nonzero: the clock generator -i has order 4 and the three non-trivial residues give a vanishing mark sum, so a μ4-mark-sourced curvature is supported only on modes ≡0 (mod 4); (3) [F] markLocal_blockDiagonal: a mark-local Toeplitz symbol (support ⊆ {m : (m:ZMod 4)=0}) connects only equal clock-characters (f_{n-n'}≠0 ⇒ (n:ZMod4)=(n':ZMod4)), i.e. [ρ,M_f]=0; (4) [O] SeamDeckPremise (structure): the PHYSICAL premise that the raw seam DtN sub-principal symbol is mark-local, encoded as a typed target exactly like CalderonProjector -- consumed, NOT proved; (5) [F] SeamDeckPremise.clock_invariant: GIVEN the premise, the clock commutes with the DtN at the character level => the quasi-free covariance C=1/2(1+sgn H1) is clock-invariant => omega o rho = omega. So the IMPLICATION mark-local => omega o rho = omega is now [F] (machine-proved); the PREMISE (the physical seam IS mark-local) stays [O] -- the one fundamental seam-identification postulate, NOT closed. Complements FORM.GLUE.01/FORM.LADDER.01; mirrors v201/v210.
Recorded status: Formal [F] (conditional theorem) + [O] (the seam-deck premise)
Dependencies, status detail and source
Canonical status: Formal [F]/[O]: the QGEO.SYM.01 conditional theorem (mark-local DtN => omega o rho = omega, via 4th-root orthogonality + character block-diagonality) is Lean-proved (AUDIT: PASS, clean axioms); the seam-deck premise stays [O], a typed target like CalderonProjector -- not a closure of QGEO.SYM.01
Dependencies: QGEO.MARKLOCAL.01; QGEO.SUBPRIN.01; QGEO.MARKS.01; FORM.GLUE.01
Scripts: lean4-carrier-rigidity
Location: tfpt_research_contracts
Original CSV, line 296Original ledger status
FORM.QGEO.02
LEAN 4 FORMALISATION of QGEO.UNIFORM.01 (the geometric normal form of v177; the UNIFORM node of the QGEO proof tree) -- AUDIT: PASS, no sorry/admit, no domain axioms (only propext, Classical.choice, Quot.sound). New module TfptCarrier/MobiusUniformisation.lean machine-proves the constructive uniformisation content: (1) [F] rho_pow_four: the clock rho:z->iz has rho^4=id; (2) [F] rho_order_exactly_four: rho^2 != id (rho^2(1)=-1), so the clock has order EXACTLY 4; (3) [F] sigma_invol: the reflection sigma:z->1/z is an involution; (4) [F] sigma_rho_sigma: sigma.rho.sigma = rho^{-1} (the dihedral relation => <rho,sigma>=D4 of order 8); (5) [F] orbit_scales_to_mu4: a non-fixed point a!=0 has orbit {a,ia,-a,-ia} that scales (z->z/a) exactly to mu4={1,i,-1,-i}; (6) [F] sigma_perm_mu4: sigma permutes mu4 (fixes 1,-1; swaps i,-i); (7) [F] mult_order_four_iff: a multiplier map z->zeta z has order exactly 4 (zeta^4=1, zeta^2!=1) IFF zeta=i or zeta=-i (the two primitive 4th roots; the inversion conjugates one to the other => canonical rep z->iz). Bundled as uniformisation_normal_form; signature-locked in AuditContract.lean. SCOPE: this is the constructive normal form GIVEN the four marks + clock (the [E] UNIFORM node); the RAW-seam obligation QGEO.MARKS.01 (the seam canonically producing a genus-0 four-marked D4 boundary) stays [O], NOT closed. Mirrors v177 (QGEO.UNIFORM.01); complements FORM.QGEO.01.
Recorded status: Formal [F] (uniformisation normal form) + [O] (the raw-seam marking obligation QGEO.MARKS.01)
Dependencies, status detail and source
Canonical status: Formal [F]/[O]: QGEO.UNIFORM.01 (order-4 Moebius normal form z->iz, D4 relation sigma.rho.sigma=rho^{-1}, orbit->mu4, multiplier classification zeta=+-i) Lean-proved (AUDIT: PASS, clean axioms, signature-locked); the raw-seam marking obligation QGEO.MARKS.01 stays [O]
Dependencies: QGEO.TREE.01; QGEO.MARKS.01; FORM.QGEO.01; FORM.GLUE.01
Scripts: lean4-carrier-rigidity
Location: tfpt_research_contracts
Original CSV, line 297Original ledger status
FORM.QGEO.03
LEAN 4 FORMALISATION of QGEO.COHOM.01 (the character node of v177) + the MODULE reflection parity -- AUDIT: PASS, no sorry/admit, no domain axioms (only propext, Classical.choice, Quot.sound). New module TfptCarrier/CohomologyGrading.lean machine-proves: (1) [F] the three H^1(P^1 minus mu4) eigenforms omega_k = z^{k-1}/(z^4-1) (k=1,2,3) have pullback character rho* omega_k = i^k omega_k under the clock rho:z->iz -- omega1_pullback (i^1=i), omega2_pullback (i^2=-1), omega3_pullback (i^3=-i), each an exact rational-function identity (the denominator is clock-invariant, (iz)^4-1=z^4-1 since i^4=1); (2) [F] character_grading: the eigenvalues (i,-1,-i) = the characters of weights (1,2,3) = the A3 exponents = Spec(Q_+), rank 3 = N_fam; (3) [F] the MODULE reflection parity: the reflection sigma:z->1/z (pullback g(1/z)*(-1/z^2)) satisfies sigma* omega_1 = omega_3, sigma* omega_2 = omega_2, sigma* omega_3 = omega_1 (omega1_reflection/omega2_reflection/omega3_reflection) -- the integer-model parity w1<->w3, w2 fixed (v141/v146/v177); bundled cohom_grading, signature-locked in AuditContract.lean. SCOPE: this is the geometric COHOM/MODULE-parity content GIVEN the marked curve (P^1, mu4) and the clock (the [E] middle nodes); it does NOT close QGEO.MARKS.01/QGEO.KERNEL.01, and the MODULE *uniqueness* (multiplicity-free + residue normalisation) stays the symbolic v177 reading. Mirrors v177 (QGEO.COHOM.01); complements FORM.QGEO.01/02.
Recorded status: Formal [F] (COHOM character grading + MODULE reflection parity) + [O] (the raw-seam obligations)
Dependencies, status detail and source
Canonical status: Formal [F]/[O]: QGEO.COHOM.01 (pullback character rho* omega_k = i^k omega_k, grading (1,2,3)=A3 exponents) + MODULE reflection parity (sigma swaps omega1<->omega3, fixes omega2) Lean-proved (AUDIT: PASS, clean axioms, signature-locked); the MARKS/KERNEL obligations and MODULE uniqueness stay [O]
Dependencies: QGEO.TREE.01; QGEO.MARKS.01; FORM.QGEO.01; FORM.QGEO.02
Scripts: lean4-carrier-rigidity
Location: tfpt_research_contracts
Original CSV, line 298Original ledger status
DM.AXION.SPINE.01
AXION DM SPINE-ANGLE BRANCH theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (2026-06-15): an ALTERNATIVE, more ROBUST misalignment angle than the determinant-line hilltop theta_i ~ 170 deg (FR.DM.03/v185, which over-produces Omega_a h^2 ~ 0.66 ~ 5.5x). (1) [I] RATIONAL MOTIVE: theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg EXACTLY (the central spine quotient 3/5 = N_fam/g_car, a rational multiple of pi, NOT tuned to the relic). (2) [N] ON TARGET: the v185 converged finite-T solver (normalised theta=1 -> Omega_a h^2 ~ 0.03) reaches Omega_DM h^2 = 0.12 at theta ~ 106 deg; the spine angle 108 deg sits essentially there (~2 deg away), harmonic estimate 0.03 theta^2 = 0.107 confirms the order, a mild anharmonic factor lifts to ~0.12. (3) [C] ROBUST (the genuine advantage over v185): 108 deg lies 62 deg BELOW the 170 deg hilltop, in the MILD-anharmonic regime, so the relic is NOT exponentially sensitive -- a far more robust landing on Omega_DM; IF theta_i is the spine quotient, the determinant-line axion is ALL of dark matter without hilltop tuning. (4) [O] HONEST -- AN ALTERNATIVE ANSATZ: theta_i = 3 pi/5 (spine motive) and theta_i = pi(1-phi_seam) ~ 170 deg (seam-value motive, v185) are MUTUALLY EXCLUSIVE readings of the same angle; neither is forced -- the full finite-T solver decides. A sharper SCENARIO, not a derivation; abundance is cosmology, never a compiler output. (5) [X] KILL TEST: a full finite-T solve at theta_i = 3 pi/5, f_a = M_scal/128, N_DW = 1 -- Omega_a h^2 outside ~[0.08,0.16] demotes the spine branch (compiler core untouched; the no-WIMP E8 scan is independent). Python-only. POST-INFLATIONARY REFRAMING [C] (2026-07-10, FR.DM.02): given H_inf/f_a~66>1 (R^2 branch) a FIXED pre-inflationary spine angle carries CDM isocurvature >> Planck beta_iso<0.038; in the natural POST-INFLATIONARY scenario theta_i is a HORIZON AVERAGE, NOT observable -- so 108 deg = 3 pi/5 is retained as a STRUCTURAL RATIONAL (3/5=N_fam/g_car), NOT a relic prediction; the relic follows from f_a + N_DW=|C|=1 (v470/v472). The [X] angle kill test is SUPERSEDED by the post-inflationary reading (angle not measurable).
Recorded status: Physical [C] (spine angle = structural rational / NOT observable post-inflationary; relic from f_a + N_DW=1; reframed by FR.DM.02)
Dependencies, status detail and source
Canonical status: Physical [C]: theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (spine quotient) lands on Omega_DM ~106 deg ROBUSTLY (mild-anharmonic, far below the 170 deg hilltop) -- a more robust DM branch than v185; an alternative ansatz, the full solver decides; kill test outside ~[0.08,0.16] demotes the branch
Dependencies: FR.DM.01; FR.DM.02; FR.DM.03; v185_axion_relic_solver.py
Scripts: v211_axion_spine_angle.py
Location: tfpt_4_frontier
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DM.AXION.PENTAGON.01
THE 'UNMAPPED' ICOSAHEDRAL/GOLDEN E8 STRUCTURE SURFACES IN THE ONE EXTERNAL COSMOLOGICAL INPUT theta_i (2026-06-26, v429): a geometric re-reading answering the parsimony question 'why does so much idle E8 structure exist?' for the one case where the idle structure has a job. [E] THETA_I = g_car-GON INTERIOR ANGLE: because N_fam = g_car-2, the spine quotient theta_i = pi N_fam/g_car = (g_car-2)pi/g_car is EXACTLY the interior angle of the regular g_car-gon; for g_car=5 the PENTAGON, theta_i = 3 pi/5 = 108 deg (the carrier polygon, not a bare rational multiple of pi). [E] COS IS GOLDEN: cos(3 pi/5) = (1-sqrt5)/4 = -1/(2 phi), partner 2 cos(2 pi/5) = 1/phi = phi-1, phi = 2 cos(pi/5) = (1+sqrt5)/2 -- the golden ratio v354 called 'unmapped' (and v313/GOLD.ATOMS.01 named the g_car=5 signature) IS theta_i's angular data. [E] GOLDEN CHARACTER UNIQUE TO g_car=5: among small regular n-gons only n=5 has an irrational interior-angle cosine (n=3,4,6 give 1/2,0,-1/2, all rational); so theta_i is golden BECAUSE the carrier is 5 (P2). [C] BRIDGE (NO STATUS CHANGE): connects E8.REVERSE.AUDIT.01/GOLD.ATOMS.01 (phi = the unmapped g_car=5 signature) to DM.AXION.SPINE.01/v211 (theta_i = 3 pi/5 the one external input); REFINES v354 from 'phi in NO physical readout' to 'phi touches ONLY the [C] spine misalignment angle' (conditional, branch-level, never a frozen prediction); a geometric MOTIVE for the spine over the hilltop theta_i ~ 170 deg (whose cosine is not a clean pentagon value), consistent with -- not replacing -- the finite-T solver that decides the branch (v185/v211). Does NOT derive theta_i, does NOT upgrade DM.AXION.SPINE.01 (stays [C]), closes no gate. Wolfram-mirrored, Python (sympy)
Recorded status: Identity [E] (theta_i = regular g_car-gon interior angle; cos = -1/(2 phi); golden character unique to g_car=5) + [C] bridge re-reading (no status change)
Dependencies, status detail and source
Canonical status: Identity [E]: theta_i = 3 pi/5 is the regular g_car-gon (pentagon) interior angle, cos theta_i = -1/(2 phi), and the golden character is unique to g_car=5 -- so the 'unmapped' golden/icosahedral E8 structure (v354/v313) is the geometry of the one external cosmological input theta_i (v211); [C] a re-reading that refines v354 and gives a geometric motive for the spine branch, but does NOT upgrade DM.AXION.SPINE.01 (stays [C]) or close any gate
Dependencies: DM.AXION.SPINE.01; E8.REVERSE.AUDIT.01; GOLD.ATOMS.01
Scripts: v429_axion_pentagon_phi.py
Location: tfpt_4_frontier
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FR.ETAB.04
LEPTOGENESIS SCALARON-DECUPLE BRANCH (2026-06-15): BOTH Boltzmann inputs share the decuple A_Lambda = 10 = |E(K5)| -- M_1 = M_scal phi0^2/A_Lambda ~ 8.65e9 GeV and m~1 = m_3/A_Lambda ~ 5 meV. A cleaner [C] route than FR.ETAB.03/v184's M_1 = M_R phi0^4, which only RELOCATED the free input to the un-pinned seesaw scale M_R. (1) [I] SHARED DECUPLE: A_Lambda = 10 = |E(K5)| (action-grammar atom) divides BOTH inputs -- M_1 = M_scal phi0^2/A_Lambda, m~1 = m_3/A_Lambda; leptogenesis = scalaron reheating dressed by two seed insertions (phi0^2), over the pair sector (A_Lambda). (2) [N] HEAVY MASS in the window: M_1 = M_scal phi0^2/A_Lambda = 8.65e9 GeV (M_scal = c3^{7/2} Mbar ~ 3.06e13 GeV, phi0 ~ 0.053172), inside the natural thermal window [3e9, 3e10] GeV (v169) and above the Davidson-Ibarra floor ~1e9 GeV. (3) [N] WASHOUT anchor: m~1 = m_3/A_Lambda ~ 5 meV (m_3 ~ sqrt(dm2_atm) ~ 0.05 eV), K = m~1/m_* ~ 4.6 (strong washout, m_* ~ 1.08 meV) -- the SAME A_Lambda = 10 as M_1. (4) [N] eta_B LANDS: Davidson-Ibarra eps_1 = (3/16 pi) M_1 m_3/v^2 ~ 8.5e-7; with kappa_f ~ 0.15 (strong washout) and sphaleron 28/79, eta_B ~ 0.96e-2 eps_1 kappa_f ~ 1.2e-9 -- O(1e-9), bracketing the observed 6.1e-10 (v169 band), no tuning. (5) [C] HONEST: unlike v184's M_1 = M_R phi0^4 (relocates the free input to the un-pinned M_R), here M_1 uses ONLY {M_scal, phi0, A_Lambda} -- no hidden seesaw scale, both inputs share A_Lambda; but the coupling MECHANISM (why M_1 = M_scal phi0^2/A_Lambda) is posited, not derived, and kappa_f is washout-dependent -- so eta_B stays [C], a sharper scenario, not a derivation. (6) [X] KILL TEST: a precise flavored-Boltzmann/RGE solve at M_1 = M_scal phi0^2/A_Lambda, m~1 = m_3/A_Lambda -- an eta_B outside the observed band falls the ROUTE, not the theory (Route A / Omega_b is independent, FR.ETAB.01). Python-only
Recorded status: Physical [C] (sharper scenario; both Boltzmann inputs share A_Lambda=10, no hidden seesaw scale) + [X] kill test
Dependencies, status detail and source
Canonical status: Physical [C]: M_1 = M_scal phi0^2/A_Lambda ~ 8.65e9 GeV and m~1 = m_3/A_Lambda ~ 5 meV both use the decuple A_Lambda=10 (no hidden seesaw scale) -- cleaner than v184's M_R phi0^4; eta_B brackets 6.1e-10 but stays [C] (coupling mechanism posited, not derived)
Dependencies: FR.ETAB.01; FR.ETAB.02; FR.ETAB.03; v169_etaB_boltzmann_interface.py
Scripts: v212_leptogenesis_decuple.py
Location: tfpt_4_frontier
Original CSV, line 301Original ledger status
CONTRACT.F.01
THE F_transfer FUNCTOR CONTRACT (2026-06-15): the four frontier transfers are ONE typed functor F_transfer = F_observable o F_threshold o F_RG satisfying four structural axioms, each instance a DISCRETE compiler kernel [E/I] composed with an EXTERNAL continuous solver [C/O]. This is the third research contract (alongside U_wall and G_metric), the structural complement of the v187 typing GUARD: v187 enforces the [C]/[O] markers, CONTRACT.F.01 verifies the functor structure. The four instances are F_pole (Koide, v183), F_Boltzmann (eta_B, v212), F_relic (axion, v211), F_QCD (m_p/m_e, v164/FR.MPME). (1) [I] AXIOM 1 mu4-DECK EQUIVARIANCE: the discrete kernels are carrier/deck invariants {|mu4|=4, N_fam=3, g_car=5, A_Lambda=10}; the Koide relaxation multiplier lambda_2 = (2/3)^6 = 64/729 IS the mu4-deck transfer eigenvalue (the deck map fixes the branch divisor, v82/v54), so F_pole commutes with the deck. (2) [I] AXIOM 2 PLUCKER PRESERVATION: where the observable is dimensionless the discrete readout is an Anchor-Plane Plucker reading -- Koide 53 = a^T(R+Q)1 is the F-corner reading (Pl(F)=(1,22,30)), the quark ratio uses ||Pl(K)||_1 = 11; the functor carries these integer Plucker invariants unchanged. (3) [I] AXIOM 3 POSITIVITY/STOCHASTICITY: where Boltzmann/relic densities enter the kernel is positive -- spec(T) = {1,(2/3)^6,(1/3)^6} is positive (RP/OS, v54), the washout kappa_f in (0,1), Omega_a > 0. (4) [I] AXIOM 4 EXTERNAL MODULES EXPLICIT: the continuous content of each instance is a named external module (pole transfer; flavored Davidson-Nardi-Nir Boltzmann; finite-T misalignment ODE with chi(T); non-perturbative lattice C_N), never a compiler number; b_3 = -7 is a carrier output (|b_3| = 11 - (2/3)(2 N_fam) = Omega_adm - 10 b_1 = 48 - 41 = 7). (5) [C] VERDICT: F_transfer is a typed FUNCTOR with four axioms, NOT a bag of open topics; the discrete kernels are [E]/[I] (exact compiler invariants), the external solvers [C]/[O] (standard physics). An architectural consolidation, NOT a closure -- each measured observable still needs its external solver. [2026-08-05 LXII, THE JET ROUTE CLOSES + THE FIBERED REFINEMENT (v777): the executed external-clock contract FTRANSFER.CLOCKS.01 fires its preregistered K1 kill sympy-exactly (transported Schwarzians {-Delta^2/2, -Delta^2/2, 0}; Delta = 6 log(3/2)) -- NO common continuous clock exists for the four jets; the surviving architecture is measured to be a FIBERED functor: one shared DISCRETE PGL2 base (anchor cross-ratio 4/3 exact -- symbolic AND through the frozen g_*s table at dev 6.5e-4; deck unit |Delta sigma| = 1; the deck generator is the unit parabolic translation tr^2 - 4 det = 0, the v632 lattice-explicit discrete half) + channel-specific projective connections whose obstruction is the CONSTANT seam cocycle with values EXACTLY {0, +-Delta^2/2} (antisymmetry + cocycle identity exact on all 27 triples, d/dsigma identically zero); classification matches the predeclared marker: thermal/proper-time clocks {tau, beta} -> coset -Delta^2/2, RG clock {log mu} -> coset 0, relic degenerate NOT classified; torsor difference = Delta^2/2 exactly. The contract's four functor axioms stand; the jet/continuous-clock unification demand is REPLACED by the fibered form, with the Schwarzian differences stored as the obstruction cocycle -- not fitted away. The one named proof-grade next step: derive the coset assignment from KMS/modular clock typing. The four observables stay [C]/[O]; no marker move.] Python-only UPDATE (2026-07-31, v566): a preregistered fit-free cross-ratio check is added to the contract surface -- if the four external solvers export canonical projective states (declared BEFORE evaluation), Moebius transport along the v533 disc -7 norm line forces CR(y_J, y_K; y_C, y_F) = 4/3 exactly (the universal arithmetic-progression cross-ratio; negative control: the raw determinant path gives 28/25 -- the known norm sequence cannot pass); the fourth state is forced from the first three (y_F reconstruction identity). Test design only; no solver states are exported yet; contract status unchanged. [2026-07-31: the preregistered cross-ratio check is EXECUTED on its discrete half by v571 (CR-PASSES; y_F = 53 Moebius-forced by the upstream corners); the continuous-solver half stays open; see FTR.CRDISC.01.] [2026-07-31: the continuous half of the cross-ratio check executed by v575 (UNIFLOW-DEGENERATE -- native exports cannot form four independent projective states, by the v425 single-flow theorem); both halves of the check are now executed; see FTR.CRCONT.01.] NOTE (2026-08-06, v792 -- marker UNCHANGED): the fibered refinement's clock-fiber classification is completed at theorem level -- the Schwarzian coset carried by a clock chart is a FUNCTION of its KMS/modular typing datum alone (KMS-COSET-THEOREM: modular/thermal clocks carry -Delta^2/2, autonomous RG charts carry 0), with the non-circularity of the classifier gated on its own source; see FTRANSFER.KMSCOSET.01.
Recorded status: Contract [C]/[O] (typed functor: discrete kernels [E], external solvers [C]; not a closure)
Dependencies, status detail and source
Canonical status: Contract [C]/[O]: F_transfer = F_observable o F_threshold o F_RG, four axioms (mu4-deck equivariant, Plucker-preserving, positive/stochastic, external-module-explicit), four instances v183/v212/v211/v164; discrete kernels [E], external solvers [C]; the third research contract alongside U_wall/G_metric, guarded by v187, NOT a closure; 2026-08-05 (v777): the jet route is closed by the executed K1 kill of FTRANSFER.CLOCKS.01 -- the surviving form is the FIBERED functor (discrete PGL2 base, constant seam obstruction cocycle {0, +-Delta^2/2}, thermal-vs-RG coset split, relic degenerate); next named step: coset assignment from KMS/modular clock typing
Dependencies: FR.KOIDE.04; FR.ETAB.04; DM.AXION.SPINE.01; FR.MPME.01; v187_ftransfer_laws.py
Scripts: v213_ftransfer_functor.py
Location: tfpt_research_contracts
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DIAMOND.AXIS.01
DIAMOND AXIS CURVATURE LEMMA (2026-06-15): along the two canonical axes of the centered cross (C center, U winding, V sheet, v95) the Sheet-Diamond determinant is LINEAR on the winding axis and QUADRATIC on the sheet axis: det(C+xU)=14+6x (flat, slope 6=|R^+(A_3)|), det(C+yV)=14+14y+4y^2 (2nd difference 8=rank E_8); the anchor block satisfies det B(C+xU)=28+12x=2 det(C+xU) and det B(C+yV)=28+21y+3y^2 (2nd difference 6=|R^+(A_3)|). So winding flows linearly (A_3-driven) and the sheet split curves, with curvatures (8 det, 6 anchor)=(rank E_8, |R^+(A_3)|) -- a discrete differential geometry of the diamond. Exact identity UPDATE (2026-07-31, v566, DIAMOND.PARA.SELFCODE.01): the direction operators U, V generate the full 2|1 parabolic algebra (dim 7); the anchor a = (1,1,2) is the spectrum of I + V(V-I)/2 with chi_H coefficients (4,5,2) = (flag, Levi, radical) -- the double self-code; the staircase-sum 81 is simultaneously the saturated lattice index [P_Z : Z<U,V>] = 3^4 (cokernel (Z/3)^4). Markers unchanged.
Recorded status: Identity (exact)
Dependencies, status detail and source
Canonical status: [E] det(C+xU)=14+6x linear, det(C+yV)=14+14y+4y^2 quadratic; curvatures (8,6)=(rank E_8, |R^+(A_3)|)
Dependencies: v94_sheet_diamond.py; v95_centered_diamond.py
Scripts: v218_diamond_axis_geometry.py
Location: tfpt_2_standard_model
Original CSV, line 303Original ledger status
DIAMOND.PLUCKER.01
PLUCKER TRANSFER LADDER (2026-06-15): the anchor-Plucker coordinates lift K->C->F in two exact steps: Pl(K)=(-1,6,4), Pl(C)=(0,14,14), Pl(F)=(1,22,30); Pl(C)-Pl(K)=(1,8,10)=(N_Phi, rank E_8, A_Lambda) and Pl(F)-Pl(C)=(1,8,16)=(N_Phi, rank E_8, dim S^+). Each step adds (1,8) in the first two coordinates; the third adds first the decuple A_Lambda=10 then the full spinor generation dim S^+=16. The Plucker identity is [E]; the F_transfer source->pole reading (a two-stage sheet lift in the Anchor Plane) stays [C]
Recorded status: Identity (exact)
Dependencies, status detail and source
Canonical status: [E] Pl ladder K->C->F: steps (1,8,10) and (1,8,16) = decuple then full generation; transfer reading [C]
Dependencies: v94_sheet_diamond.py; v213_ftransfer_functor.py
Scripts: v218_diamond_axis_geometry.py
Location: tfpt_2_standard_model
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DIAMOND.SPECTRAL.01
SPECTRAL RAMIFICATION LADDER (2026-06-15): for Q,K,C,F the characteristic polynomial splits as (t-r)(t^2-bt+c) with r rational, and the full cubic discriminant factors as Disc(chi)=q(r)^2 Disc(q): squares |q(r)| in {1,3,4,6}=(N_Phi,N_fam,|mu4|,|R^+(A_3)|), kernels Disc(q) in {13,48,65,105}=(Delta_Q, Omega_adm, g_car Delta_Q, N_fam g_car 7), with Delta_Q=13=b^2+s^2=|R(A_3)|+1 the carrier-split norm (v14). The missing corner F carries the kernel 105=3*5*7=family*carrier*scalaron. Exact identity. (Audit blocks in the same module: the anchor-defect spectrum Delta_a(Q,K,R,C,L,F)=(8,16,18,21,24,26) and the pair-sum atlas/determinant staircase are AUDIT fingerprints -- the 28=2 dim G_2, 52=dim F_4 labels stay audit-only, not spine; F as transfer-completion corner is a HEURISTIC, not a hard filter)
Recorded status: Identity (exact)
Dependencies, status detail and source
Canonical status: [E] Disc(chi)=q(r)^2 Disc(q) for Q,K,C,F: squares (1,3,4,6), kernels (13,48,65,105); audit/heuristic blocks honestly typed
Dependencies: v14_carrier_uniqueness.py; v94_sheet_diamond.py
Scripts: v218_diamond_axis_geometry.py
Location: tfpt_2_standard_model
Original CSV, line 305Original ledger status
MCKAY.E8.01
McKAY BEDROCK (2026-06-16): WHY the three atoms are {2,3,5}. E8 is the exceptional TOP of the McKay tower of finite SU(2) subgroups (2T->affine E6 h=12, 2O->affine E7 h=18, 2I binary icosahedral order 120 -> affine E8 h=30=2*3*5). Built FROM the group, not asserted: (1) the 120 unit-quaternion icosians close to 2I with element orders {1,2,3,4,5,6,10} (it CONTAINS the 2,3,5 rotation-axis orders), 9 conjugacy classes; (2) the 9 irreducible character DEGREES from the class algebra (Dixon) are {1,2,2,3,3,4,4,5,6}, sum=30=h(E8), sum of squares=120=|R^+(E8)|; (3) the McKay adjacency A_ij=<chi_V chi_j,chi_i> from the natural 2-dim rep IS the affine E8 graph (tree, top eigenvalue 2), with Kac marks = the irrep degrees; (4) negative controls 2T->E6, 2O->E7 -- only the icosahedral (2,3,5) gives E8. A BACKWARD CERTIFICATE of the already-closed E8, NOT an independent P2 proof.
Recorded status: Lattice (McKay correspondence, exact)
Dependencies, status detail and source
Canonical status: [E] 2I (order 120) irrep degrees {1,2,2,3,3,4,4,5,6} = affine E8 Kac marks, sum 30=h(E8), sum^2 120=|R^+(E8)|; the McKay graph built from the group; E8 = the (2,3,5) icosahedral top. Backward certificate [E], 'explains (2,3,5)' [C], raw-seam origin [O]
Dependencies: AX.P2.01; E8.GLU.01; ARCH.CORE.01
Scripts: v219_icosahedral_mckay.py
Location: origin_theory
Original CSV, line 306