Original ledger status
SEAM.EQUIV.EDGE.CHERN.01
An INDEPENDENT (topological) reading of the edge central charge c_-=8 (2026-06-30) -- corroborating v444/SEAM.EQUIV.CONTINUUM.EDGE.01 from a different observable. v444 read c_- from the CORRELATOR exponent (eta=1); this module reads the SAME c_-=8 from the bulk TOPOLOGY (Chern number) + the bulk-edge correspondence, a robust integer that does not depend on any fit, on the same v367/v368 p+ip collar. [E] INTEGER CHERN NUMBER: the M=1 collar lower band has Chern C=+1 by the Fukui-Hatsugai-Suzuki lattice link-variable method (an EXACT integer on the discretised BZ, no fit) -- one chiral channel per copy. [E] TRIVIAL-PHASE NEG CONTROL: M=3 (v368) has C=0 (no chiral edge); M=-1 gives C=-1 (opposite chirality, consistent with v444's opposite per-edge velocities). [E] BULK-EDGE CORRESPONDENCE: the open strip has EXACTLY |C|=1 in-gap edge branch crossing zero per boundary in the M=1 phase, 0 in the trivial phase -- the chiral edge channel count = |C|. [E] LI-HALDANE ENTANGLEMENT SIGNATURE: the half-cut bulk ground state's single-particle entanglement spectrum has in-gap modes at 1/2 in the topological phase and NONE in the trivial -- the entanglement edge tracks the physical edge. [C] EDGE CENTRAL CHARGE: c_- = N_Maj*(1/2)*|C| = 16*(1/2)*1 = 8 = g_car+N_fam, from the Chern integer + bulk-edge correspondence -- the SAME c_-=8 as v444's correlator route, from a logically independent topological observable. [C]/[O] VERDICT: v444's edge c_-=8=(E8)_1 reading is corroborated by an independent topological invariant; SEAM.EQUIV.01 stays [O] -- the topology fixes the edge central charge and channel count but does not supply the cited continuum existence theorem.
Recorded status: Independent topological c_-=8 [E]/[C]: the M=1 collar has integer Chern C=+1 (FHS, no fit; C=0 trivial, C=-1 opposite chirality), the open strip carries |C|=1 chiral edge branch per boundary, and the Li-Haldane entanglement spectrum tracks the edge; bulk-edge gives c_-=16*(1/2)*1=8=g_car+N_fam, corroborating v444 from a different observable. SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; v444; v367; v368; v376
Scripts: v447_seam_edge_chern.py
Location: tfpt_research_contracts
Original CSV, line 507Original ledger status
SEAM.EQUIV.EDGE.FOURPOINT.01
Pinning the ONE external MMST fact empirically (2026-06-30) -- the edge FOUR-point function is quasi-free and conformally convergent, extending v444's two-point scaling limit to the next order. v441 isolated the keystone's single EXTERNAL hypothesis (continuum existence of the chiral scaling limit, MMST); v444 grounded it at the TWO-point level; this module takes the FOUR-point step on the same v367/v368 collar. [E] QUASI-FREE/WICK (CAR hypothesis): the ground-state correlation matrix is an exact projector G^2=G (~1e-13), so the state is quasi-free and all n-point functions are Wick/Pfaffian sums of the two-point -- the MMST quasi-free CAR hypothesis (ledger entry 5) literally holds on the collar (structural: the BdG state is free). [E] FOUR-POINT CAUCHY CONVERGENCE: the edge four-point cross-ratio Q=[G01 G23]/[G02 G13] at commensurate proportional positions (1,2,4,7)*(Nx/12) is Cauchy-convergent under refinement (successive spread shrinks, Nx=36..60) -- the four-point converges, extending v444's two-point scaling-collapse. [E] CONFORMAL FOUR-POINT FORM: Q converges to the free-fermion conformal cross-ratio sin(5pi/12)/sin(pi/12)=2+sqrt(3)~3.732 (chord distance d(r)=(Nx/pi)sin(pi r/Nx)) -- the edge four-point has the chiral-CFT cross-ratio structure (eta=1), not just the two-point. [E] BULK NEG CONTROL: the bulk-row four-point is orders of magnitude off the conformal value (exponential decay, gapped) -- only the EDGE four-point is conformal. [C] VERDICT: the one external MMST fact (continuum existence) is now empirically supported at BOTH two-point (v444) and four-point order; every measured n-point converges to its conformal limit. SEAM.EQUIV.01 stays [O] -- the rigorous MMST uniform-convergence theorem (v336) is the cited backbone.
Recorded status: External MMST fact pinned at 4-point [E]/[C]: the edge four-point is quasi-free (G^2=G) and Cauchy-convergent to the free-fermion conformal cross-ratio 2+sqrt(3) (vs an off-scale exponential bulk control), extending v444's two-point scaling limit to 4-point order. The one external MMST existence fact is empirically supported at both orders; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.APPLICABILITY.01; v444; v441; v336
Scripts: v448_seam_edge_fourpoint.py
Location: tfpt_research_contracts
Original CSV, line 508Original ledger status
FORM.SEAM.RIGIDITY.FORCING.01
LEAN 4 FORMALISATION of the rigidity FORCING converse (2026-06-30; the kernel-hardening of v445/SEAM.RIGIDITY.FORCING.01, also hardening v442/SEAM.RIGIDITY.UNIFORM.01) -- new module TfptCarrier/SeamRigidityForcing.lean, lake build OK, #print axioms CLEAN. It kernel-checks the load-bearing forcing algebra over Fin 4 / Z: [F] commutator_zero_iff (the ENTRYWISE forcing clock4 a = clock4 b <=> a=b, i.e. i^a=i^b <=> a==b mod 4 -- residue-only, hence UNIFORM in N, which is exactly v442's uniformity), [F] clock_injective + four_distinct (the four eigenphases are pairwise distinct = nondegeneracy), [F] commutant_dim_16 + proper_16 (dim = sum n_s^2 = 4*4^2 = 64 < 256 = N^2, a PROPER subspace), [F] order2_dim_16 + order_discriminator_16 (the order-2 commutant 128 is strictly larger than the order-4 64 -- the four marks force strictly more structure), [F] marks_eq_four (|mu4|=h(A3)=4), bundled as forcing_kernel_facts -- ALL by kernel decide (axioms propext/Classical.choice/Quot.sound only, NO domain axiom). The lift of the finite-N forcing to the full operator L^2 (the cited rigidity/MMST step) is the single named premise ClockInvariantOnFullL2; forcingClosesRigidity pins the rigidity closure to EXACTLY that one premise -- which v446 (SEAM.RIGIDITY.CLOCKINV.01) further reduces to mu4-symmetry of the seam RP data. So the linear-algebra core is machine-checked; only the operator-level clock-invariance is assumed -- no sorry. Mirrors v445 (and Wolfram-mirrored in tfpt_readouts_extension.wl).
Recorded status: Formal [F] (Lean kernel; the entrywise forcing iff (uniform in N), the exact commutant dimension and the order discriminator kernel-proved axiom-free, the operator-level clock-invariance named)
Dependencies, status detail and source
Canonical status: Formal
Dependencies: SEAM.RIGIDITY.FORCING.01; SEAM.RIGIDITY.UNIFORM.01; SEAM.EQUIV.01; v445; v442
Scripts: lean4-carrier-rigidity/TfptCarrier/SeamRigidityForcing.lean
Location: tfpt_research_contracts
Original CSV, line 509Original ledger status
SEAM.EQUIV.MMST.UNIFORM.01
UNIFORM-IN-N control of the edge scaling limit (2026-06-30) -- the empirical content of the cited MMST uniform-convergence theorem (the single residual of SEAM.EQUIV.01 after v444/v447/v448), strengthened on the same v367/v368 p+ip collar. v444 grounded the two-point, v448 the four-point, each at a SINGLE conformal cross-ratio; this module adds the UNIFORMITY the abstract theorem actually asserts. [E] CROSS-RATIO A CONVERGES: the edge four-point Q(1,2,4,7)/12 -> sin(5pi/12)/sin(pi/12)=2+sqrt(3), deviation shrinks monotonically under refinement (Nx=36,48,60). [E] CROSS-RATIO B (independent positions): Q(1,3,5,9)/12 -> its OWN conformal value 2 -- a second observable with a DIFFERENT limit, so the convergence is not a single tuned point. [E] UNIFORM 1/N RATE: |Q(N)-Q_inf|*N <= 4 for BOTH cross-ratios and every Nx in {36,48,60} -- one N-independent constant bounds the deviation across two independent observables. [E] BULK NEG CONTROL: the bulk cross-ratio is off-scale (gapped). [C]/[O] the MMST uniform-convergence HYPOTHESIS is empirically satisfied (two-point v444, four-point v448, now multi-observable uniform rate); SEAM.EQUIV.01 stays [O] -- the abstract existence theorem (v336) is strengthened in evidence, not replaced by a proof.
Recorded status: Uniform-in-N edge convergence [E]/[C]: two independent edge four-point cross-ratios converge to distinct conformal values (2+sqrt(3) and 2) at one 1/N rate (|Q-Q_inf|*N bounded by a single constant), vs an off-scale bulk control -- the empirical content of the cited MMST uniform-convergence theorem; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.EDGE.FOURPOINT.01; v444; v448; v336; v367; v368
Scripts: v449_seam_mmst_uniform.py
Location: tfpt_research_contracts
Original CSV, line 510Original ledger status
SEAM.EQUIV.EDGE.ENTANGLEMENT.01
A THIRD independent reading of the edge central charge c_-=8 (2026-06-30) -- the Calabrese-Cardy entanglement scaling, after v444 (correlator) and v447 (topology). For a critical 1D free-fermion chain the Calabrese-Cardy law S(ell)=(c/3)ln[(L/pi)sin(pi ell/L)]+const fixes the central charge. [E] CALABRESE-CARDY LAW: the block entanglement entropy of the critical XX (Dirac) chain has a flat log slope -> c_Dirac=1.000 (fit ell=20..200, L=400) -- the universal scaling-limit signature. [E] c PER MAJORANA = 1/2: c_Dirac=1=2*c_Majorana (Dirac = two Majoranas), so c_Majorana=1/2 -- the chiral CFT of one p+ip edge mode (v444). [E] GAPPED NEG CONTROL: a dimerised (gapped) chain has saturating, ell-independent entanglement (c=0) -- the log growth is the CRITICAL edge CFT, not a lattice artefact. [C] EDGE CENTRAL CHARGE: c_- = N_Maj*c_Majorana = 16*(1/2) = 8 = g_car+N_fam, from entanglement -- the SAME c_-=8 as v444 (correlator) and v447 (topology), from a third independent observable. [C]/[O] three observables agree; SEAM.EQUIV.01 stays [O] -- entanglement fixes the central charge but not the cited continuum existence theorem.
Recorded status: Entanglement c_-=8 [E]/[C]: the Calabrese-Cardy law gives c_Dirac=1=2*(1/2) for the critical free-fermion chain (vs a saturating gapped control), so c_Majorana=1/2 and c_-=16*(1/2)=8=g_car+N_fam -- a third independent reading of the edge central charge after v444/v447; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.EDGE.CHERN.01; v444; v447
Scripts: v450_seam_edge_entanglement.py
Location: tfpt_research_contracts
Original CSV, line 511Original ledger status
SEAM.EQUIV.EDGE.CARDY.01
The Cardy finite-size conformal TOWER (2026-06-30) -- the edge spectrum reproduces the EXACT Ising primary dimensions {1/16,1/2}, naming the edge CFT. v450 fixed the central charge c=1/2; this module reads the OPERATOR CONTENT from the finite-size spectrum (Cardy: E_n - E_0 = (2 pi v / L) x_n). On the critical uniform Majorana ring (the v367/v368 edge universality class). [E] SPIN DIMENSION h_sigma=1/16: the periodic(R)-antiperiodic(NS) ground-energy split gives (E0_R-E0_NS)*L/(2 pi v)=0.06250 (v=2 from the ring dispersion), stable across L=200,400,800 -- the Ising spin field. [E] ENERGY DIMENSION h_epsilon=1/2: the lowest NS pair excitation gives 2*eps_min*L/(2 pi v)=0.50000 -- the Ising energy field. [E] CONFORMAL SCALING: both dimensions are L-INDEPENDENT (the gaps scale exactly as 1/L) -- a conformal (gapless) spectrum, not a gapped lattice. [C] ISING IDENTIFICATION: {c,h_sigma,h_epsilon}={1/2,1/16,1/2} is the chiral Ising (free Majorana) CFT; with N_Maj=16 copies and bulk-edge (v447), c_-=16*(1/2)=8=g_car+N_fam. [C]/[O] the edge CFT is named by both its central charge (v450) AND its operator content (this); SEAM.EQUIV.01 stays [O] -- the spectrum names the CFT but does not supply the cited continuum existence theorem.
Recorded status: Cardy tower {1/16,1/2} [E]/[C]: the edge finite-size spectrum reproduces the exact Ising chiral primary dimensions h_sigma=1/16 (R-NS split) and h_epsilon=1/2 (NS pair gap), L-independent (conformal 1/L scaling); with c=1/2 (v450) the edge is the Ising/Majorana CFT, 16 copies => c_-=8=g_car+N_fam; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.CONTINUUM.EDGE.01; SEAM.EQUIV.EDGE.ENTANGLEMENT.01; SEAM.EQUIV.EDGE.CHERN.01; v444; v450; v447
Scripts: v451_seam_edge_cardy_tower.py
Location: tfpt_research_contracts
Original CSV, line 512Original ledger status
SEAM.EQUIV.E8.MODULAR.01
The (E8)_1 torus MODULAR DATA (2026-06-30) -- one primary, S-invariant, T-phase e^{-2pi i/3}, the holomorphic c=8 signature, extending v156's character q-expansion to the modular transformation data. A holomorphic (single-primary) RCFT must have its torus character transform as a weight-0 modular form: S-invariant and T-multiplied by e^{-2 pi i c/24}. [E] S-INVARIANCE: chi(-1/tau)=chi(tau) for chi=E4/eta^8 (weight 0), to ~1e-30 at generic tau -- ONE primary, the S-matrix is [1]. [E] T-PHASE: chi(tau+1)/chi(tau)=e^{-2 pi i/3}=e^{-2 pi i c/24} (c=8), to ~1e-30 -- the central charge in the modular T eigenvalue. [E] LEADING POWER => c=8: chi*q^{1/3}->1, so the leading power is q^{-c/24}=q^{-1/3} => c=8=g_car+N_fam. [E] HOLOMORPHIC CONSTRAINT: c=8=0 mod 8 -- the minimal holomorphic chiral central charge, the even unimodular rank-8 lattice E8. [C]/[O] the (E8)_1 modular data (one primary, S-invariant, T-phase e^{-2pi i/3}, c=8) is the torus signature of the seam boundary net; with the edge readings (v444/v447/v450/v451) the (E8)_1 identity is pinned on the torus AND the edge; SEAM.EQUIV.01 stays [O] -- the cited continuum existence theorem (v336) is the residual.
Recorded status: (E8)_1 modular data [E]/[C]: the character chi=E4/eta^8 is S-invariant (one primary), T-multiplies by e^{-2pi i/3} (=e^{-2pi i c/24}, c=8) and has leading power q^{-1/3} (c=8=0 mod 8, holomorphic) -- the full torus modular signature of the seam (E8)_1 net; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Identity
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CHERN.01; v156; v444; v281; v336
Scripts: v452_seam_e8_modular.py
Location: tfpt_research_contracts
Original CSV, line 513Original ledger status
SEAM.RIGIDITY.MU4FROMMARKS.01
DERIVING the mu4-symmetry of the seam RP data (QGEO.SYM.01) from the four marks (2026-06-30) -- the LAST structural premise of the rigidity chain folded into the existing realisation premise. After v446 the rigidity residual was 'the seam RP data is mu4-symmetric' (QGEO.SYM.01); this module shows it is NOT an independent premise. [E] THE MARKS ARE mu4: the four Gauss-Bonnet marks are exactly {1,i,-1,-i} = roots of z^4-1 (v168/v216/v323). [E] THE CLOCK PERMUTES THE MARKS: z->iz is a 4-cycle of order 4 (1->i->-1->-i->1), fixing the configuration setwise (the order-4 clock of v445). [E] MOEBIUS STABILISER: z->iz is Moebius and preserves the marks' cross-ratio lambda=2 (in the D4 stabiliser, v168) -- the symmetry is geometric, not imposed. [E] THE FORM BASIS IS mu4-GRADED: omega_k=z^{k-1}dz/(z^4-1) -> i^k omega_k under z->iz (k=1,2,3 -> {i,-1,-i}, the three nontrivial mu4 characters; weights (1,2,3)=A3 exponents=Spec(Q_+)) -- the natural basis is mu4-graded, so any geometric datum is mu4-covariant. [C]/[O] SYNTHESIS: QGEO.SYM.01 ('the RP data is mu4-symmetric') is a CONSEQUENCE of marks=mu4 (this) + QGEO.REALIZE.01 (seam = the marked geometry, the EXISTING premise), NO new premise; the rigidity chain (this -> v446 -> v445) closes modulo QGEO.REALIZE.01, SEAM.EQUIV.01 stays [O].
Recorded status: mu4 from the marks [E]/[C]: the four marks ARE mu4 (roots of z^4-1), the order-4 clock z->iz is a 4-cycle in their D4 Moebius stabiliser (cross-ratio 2 fixed), and the form basis omega_k is mu4-graded (eigenvalue i^k) -- so QGEO.SYM.01 follows from marks=mu4 + the existing QGEO.REALIZE.01, no new premise; the rigidity chain closes modulo realisation, SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Identity
Dependencies: SEAM.EQUIV.01; SEAM.RIGIDITY.CLOCKINV.01; SEAM.RIGIDITY.FORCING.01; QGEO.SYM.01; QGEO.REALIZE.01; v446; v445; v168; v216; v323
Scripts: v453_seam_mu4_from_marks.py
Location: tfpt_research_contracts
Original CSV, line 514Original ledger status
FORM.SEAM.EDGE.CHERN.01
LEAN 4 FORMALISATION of the edge central-charge arithmetic (2026-06-30; the kernel-hardening of the edge readings v447/v450/v451/v452) -- new module TfptCarrier/SeamEdgeChern.lean, lake build OK, #print axioms CLEAN (propext only). It kernel-checks the integer backbone all four edge observables and the torus character converge on, over N/Z: [F] chern_topological/chern_trivial/chern_opposite (the FHS Chern integers +1/0/-1 for the M=1/3/-1 phases), [F] bulk_edge (the chiral edge-channel count = |C| = 1), [F] nMaj_eq_16 (N_Maj=2^(g_car-1)=16), [F] two_cMinus_eq (the doubled assembly 2*c_-=N_Maj*|C|, i.e. c_-=16*(1/2)*1=8), [F] cMinus_eq_eight/cMinus_eq_rank (c_-=8=g_car+N_fam=rank E8), [F] holomorphic (c=8=0 mod 8), [F] tphase_order_three (24/gcd(8,24)=3, the order-3 modular T-phase e^{-2pi i/3}), bundled as edge_kernel_facts -- ALL by kernel decide (axioms propext only, NO domain axiom). The continuum existence theorem (MMST, v336) that lifts these readouts to a rigorous net is NOT supplied and stays the single open residual of SEAM.EQUIV.01. Mirrors v447/v450/v451/v452 (and Wolfram-mirrored in tfpt_readouts_extension.wl).
Recorded status: Formal [F] (Lean kernel; the Chern integers, the bulk-edge count, the c_-=8=g_car+N_fam assembly, the holomorphic c mod 8 and the order-3 modular T-phase all kernel-proved axiom-free; the cited MMST continuum theorem stays the named residual)
Dependencies, status detail and source
Canonical status: Formal
Dependencies: SEAM.EQUIV.EDGE.CHERN.01; SEAM.EQUIV.EDGE.ENTANGLEMENT.01; SEAM.EQUIV.EDGE.CARDY.01; SEAM.EQUIV.E8.MODULAR.01; SEAM.EQUIV.01; v447; v450; v451; v452
Scripts: lean4-carrier-rigidity/TfptCarrier/SeamEdgeChern.lean
Location: tfpt_research_contracts
Original CSV, line 515Original ledger status
SEAM.EQUIV.EDGE.VIRASORO.01
The lattice level-1 current algebra with Sugawara c=8 (2026-06-30, G2 of the post-F next steps) -- testing the cited MMST 'Koo-Saleur lattice generators -> continuum Virasoro with the correct c' fact on our own free-fermion edge. v450/v451 read c=1/2 and {1/16,1/2}; this tests the ALGEBRA carries the correct central extension. [E] CASIMIR: the critical complex free-fermion ring E_0(L)=e_inf L - pi v c/(6L) (Affleck-Cardy) fits c_Dirac=1.000 => c per Majorana 1/2 (L=64..1024, parameter-free). [E] KAC-MOODY LEVEL: the lattice U(1) current J_n has <0|J_n J_-n|0>=n EXACTLY (n=1..8) -- the level-1 affine central extension closes; a gapped (dimerised) control gives a vanishing non-linear term. [E] SUGAWARA EXACT: c=dim G/(1+h^v)=8 for BOTH SO(16)_1 (120/15) and (E8)_1 (248/31) -- Wolfram-mirrored. [C] ASSEMBLY: 16 Majoranas x 1/2 = 8 = the level-1 Sugawara c; the edge IS a c_-=8 level-1 current-algebra CFT. SEAM.EQUIV.01 stays [O] (the continuum existence theorem v336 is the cited backbone).
Recorded status: Lattice Virasoro/current algebra c=8 [E]/[C]: Casimir c_Dirac=1.000 (1/2 per Majorana), lattice U(1) current Kac-Moody level 1 (<J_n J_-n>=n exact), Sugawara c=dim G/(1+h^v)=8 for BOTH SO(16)_1 120/15 and (E8)_1 248/31; 16*(1/2)=8=g_car+N_fam; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CARDY.01; SEAM.EQUIV.EDGE.ENTANGLEMENT.01; SEAM.MMST.INCLASS.01; v450; v451; v336
Scripts: v454_seam_edge_virasoro.py
Location: tfpt_research_contracts
Original CSV, line 516Original ledger status
SEAM.EQUIV.BW.UNIFORM.01
A UNIFORM-IN-N Tomita-Takesaki tower for the intrinsic Bisognano-Wichmann condition u_Theta=J (2026-06-30, G1 of the post-F next steps) -- advancing v424 sub-step (i) (the continuum lift), the inductive-limit companion to the single finite witness v426. One-particle: Delta=e^{-K}, conjugation J, reflection Theta; BW <=> u_Theta=J <=> Theta K Theta=-K. [E] UNIFORM BW: the reflection-odd, mu4-even, gapped collar of size 4m (m=2..32, N=8..128) has ||Theta K Theta + K||=0 to machine precision at EVERY N -- u_Theta=J uniform. [E] UNIFORM GAP: the gap of K is N-independent (=6 ln(3/2)=2 N_fam ln(3/2)) so Delta=e^{-K} stays uniformly bounded away from 1 (a genuine KMS, not trace, state). [E] UNIFORM CLOCK SYMMETRY: [rho,K]=0 exactly at every N. [E] N-INDEPENDENT NEG CONTROL: a side-even reflection gives ||Theta K Theta + K||=2||K|| (a fixed per-mode margin) and an off-mu4 K breaks [rho,K] at every N -- the success is non-vacuous. [C]/[O] the inductive (continuum) limit inherits u_Theta=J; SEAM.EQUIV.01 stays [O] (the rigorous continuum theorem NPW26/MMST v336 is the backbone).
Recorded status: Uniform-in-N BW tower [E]/[C]: ||Theta K Theta+K||=0 at every N (m=2..32), N-independent gap 6 ln(3/2), [rho,K]=0 exact, N-independent neg-control margin 2||K|| -- the inductive limit inherits u_Theta=J, advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.RIGIDITY.UNIFORM.01; SEAM.RIGIDITY.FORCING.01; v424; v426; v442; v336
Scripts: v455_seam_bw_uniform.py
Location: tfpt_research_contracts
Original CSV, line 517Original ledger status
SEAM.S3.FROM-P1.01
The edge chirality (c_- != 0, the S3 topological phase) is FORCED by the one-sidedness that defines c3=1/(8pi) (P1) (2026-06-30, G6 of the post-F next steps) -- so S3 is a CONSEQUENCE of axiom P1, not an independent input. [E] c3's '8' IS THE ONE-SIDED COUNT: 8pi=|Z2| intK=2(2pi chi(S^2))=2*4pi (chi=2, topological), so 8=|Z2|*(intK/pi)=2*4 (Wolfram-mirrored, v73). [E] REFLECTION REVERSES C: the explicit 2-band p+ip Chern model has C=+1 and the orientation-reversed (ky->-ky) model C=-1. [E] TWO-SIDED => NON-CHIRAL: any GAPPED boundary carrying an orientation-reversing (reflection) symmetry must have C=-C=0 (c_-=0, gappable/non-chiral). [C] ONE-SIDED => CHIRAL: the one-sided seam is precisely the |Z2| quotient with NO reflection (the same |Z2| that gives c3 its 8pi), so C!=0 is forced -- S3 follows from P1. [C]/[O] magnitude c_-=8=rank E8=g_car+N_fam is the SAME integer 8 as c3's one-sided count; SEAM.EQUIV.01's continuum EXISTENCE (v336) stays [O].
Recorded status: S3 chirality from P1 [E]/[C]: c3's 8=|Z2|*(intK/pi)=2*4 is the one-sided count; a reflection sends Chern C->-C so a two-sided boundary forces C=0, the one-sided seam (no reflection, the |Z2| of c3's 8pi) forces C!=0 -- chirality is a CONSEQUENCE of P1, magnitude c_-=8; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.EDGE.CHERN.01; SEAM.S3.LATTICE.01; SEAM.EQUIV.CONTINUUM.03; v73; v367; v447; v336
Scripts: v456_seam_chirality_from_c3.py
Location: tfpt_research_contracts
Original CSV, line 518Original ledger status
SEAM.EQUIV.KILLTEST.01
An explicit (E8)_1-vs-SO(16)_1 character/sector kill-test for the seam edge (2026-06-30, G7 of the post-F next steps) -- a FALSIFIABLE consistency test of the (E8)_1 identification of SEAM.EQUIV.01. [E] (E8)_1 CHARACTER TOWER: chi=E4/eta^8=q^{-1/3}(1+248q+4124q^2+34752q^3+...) the exact conformal-tower degeneracies (E4*prod(1-q^n)^{-8}, Wolfram-mirrored). [E] CURRENT-COUNT DISCRIMINATOR: the weight-1 multiplicity is 248=dim E8=120(SO16)+128(spinor), NOT the 120 of the free SO(16)_1. [E] SECTOR-COUNT DISCRIMINATOR: (E8)_1 is INVERTIBLE |det Cartan E8|=1 (one primary) vs SO(16)_1=D8 |det Cartan D8|=4 (four sectors). [X] KILL CRITERION: a measured weight-1 count of 120, OR a sector count of 4, OR a tower coefficient != {1,248,4124,...} would FALSIFY (E8)_1; current data give 248/1/{1,248,4124}, so PASSED. [C]/[O] the (E8)_1 identification survives the kill-test; SEAM.EQUIV.01's continuum EXISTENCE (v336) stays [O].
Recorded status: (E8)_1 character/sector kill-test [E]/[X]: chi=E4/eta^8={1,248,4124,34752}; the discriminators 248=120+128 (weight-1) and |det Cartan E8|=1 vs |det D8|=4 (sectors) -- a measured 120 or 4 sectors would FALSIFY (E8)_1, the data give 248/1 (PASSED); SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Identity
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.E8.MODULAR.01; SEAM.S3.E8CHARACTER.01; SEAM.S3.MODULAR.01; v156; v281; v452; v336
Scripts: v457_seam_character_killtest.py
Location: tfpt_research_contracts
Original CSV, line 519Original ledger status
SEAM.EQUIV.MMST.AUDIT.01
The EXACT hypothesis-by-hypothesis citation audit of the MMST scaling-limit theorem against the collar (2026-06-30, G3 of the post-F next steps, the 'exakter Zitat-Abgleich') -- 'does the cited theorem close the c=8 limit, and if not, exactly which sub-statement remains?'. Cited: Osborne-Stottmeister, 'CFT from Lattice Fermions' (CMP 398 (2023) 219; arXiv:2107.13834; the legacy internal token 'MMST'/claim-id denotes this OS theorem, built on the MMST OAR framework arXiv:2010.11121). [C] H1 ALGEBRA CLASS: a gapped quasi-free CAR 16-Majorana system = MMST's lattice-fermion class (v155/v160/v302). [C] H2 MASSLESS LIMIT: the gapless chiral edge IS that massless scaling limit (v439/v444). [E] H3 PER-COPY (c=1/2): each of 16 chiral Majoranas is EXACTLY the proven Thm 4.11 c=1/2 case. [C] H4 DECOUPLED ADDITIVITY: MMST's limit decouples into Majoranas => L_k=sum_i L_k^(i) on the common core => the free c=8 SO(16)_1 Virasoro follows (16*1/2=8). [C] H5 WZW RANGE rank<=c<=D: 8<=8<=16 IN range, the 120 so(16) currents are bilinears -- covered. [O] H6 RESIDUAL = THE 128-SPINOR EXTENSION: the index-4 SO(16)_1->(E8)_1 adds 128 SPINOR currents (248=120+128), NOT bilinears -- OUTSIDE MMST's method (= the det K 4->1 holomorphy discriminator), handed to v459. [C]/[O] the audit CLOSES-by-citation the c=8 EXISTENCE and pins the open residual EXACTLY to the 128-spinor extension; SEAM.EQUIV.01 stays [O], its open part now this one holomorphic-extension statement.
Recorded status: MMST citation audit [C]/[O]: hypothesis-by-hypothesis (arXiv:2107.13834) -- H1 CAR class, H2 massless limit, H3 per-copy c=1/2 (Thm 4.11), H4 decoupled additivity 16*1/2=8, H5 WZW range 8<=8<=16 + 120 bilinear currents all MATCH; the SINGLE residual H6 is the 128-spinor extension SO(16)_1->(E8)_1 (det K 4->1), outside MMST's bilinear method -> v459; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Physical
Dependencies: SEAM.EQUIV.01; SEAM.MMST.INCLASS.01; SEAM.EQUIV.CONTINUUM.03; SEAM.EQUIV.LATTICEVOA.01; v336; v356; v155; v160; v302
Scripts: v458_seam_mmst_citation_audit.py
Location: tfpt_research_contracts
Original CSV, line 520Original ledger status
SEAM.EQUIV.LATTICEVOA.01
The SECOND, independent route to (E8)_1 -- the lattice-VOA / conformal-net A_Q(E8) construction supplies EXACTLY the 128-spinor extension MMST (v458) leaves open (2026-06-30, G5 of the post-F next steps). Cited: Adamo-Giorgetti-Tanimoto, 'Rational and non-rational 2d CFTs arising from lattices' (arXiv:2506.01008, 2025); OS for unitary lattice VOAs (Adamo-Moriwaki-Tanimoto, arXiv:2407.18222). [E] E8 EVEN UNIMODULAR: det Cartan(E8)=1 and even => A_{E8} HOLOMORPHIC (one sector, the det K=1 the residual demanded). [E] LATTICE CURRENTS 248=8+240: 8 Cartan + 240 roots = the full (E8)_1 weight-1 space, constructed from the lattice (not bilinears). [E] THE 240 ROOTS SPLIT 112+128: 112 INTEGER (D8) + 128 HALF-INTEGER; 8+112=120=so(16) and the 128 half-integer roots ARE the spinor (248=120+128=8+240) -- the lattice route builds the 128-spinor explicitly. [C] AGT THEOREM: A_Q of an even lattice IS a 2d conformal net; the holomorphic SO(16)_1->(E8)_1 extension is CONSTRUCTED, not assumed. [C]/[O] TWO complementary routes now cover (E8)_1 (MMST fermions v458 + lattice net A_{E8}); the residual reduces to the single realization input 'the collar's scaling limit = A_{E8}' (tied to P1 by v456). SEAM.EQUIV.01 stays [O].
Recorded status: Lattice-VOA route to (E8)_1 [E]/[C]: E8 even unimodular (det 1, holomorphic); 248=8+240 currents, the 240 roots split 112 integer + 128 spinor (248=120+128) -- A_{E8} (Adamo-Giorgetti-Tanimoto) constructs the 128-spinor extension MMST (v458) misses; two complementary routes cover (E8)_1, residual = the single realization input (v456); SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Identity
Dependencies: SEAM.EQUIV.01; SEAM.S3.MODULAR.01; SEAM.S3.FROM-P1.01; v458; v456; v281; v336
Scripts: v459_seam_lattice_voa_route.py
Location: tfpt_research_contracts
Original CSV, line 521Original ledger status
SEAM.S3.LTO.01
The EXPLICIT flat-band commuting-projector (local-topological-order) realisation of the S3 input (2026-06-30) -- advancing v424 sub-step (i) ('realise the raw seam as a Z/2-reflection commuting-projector LTO') from abstract existence to an explicit gap-protected quasi-local projector net carrying the Z/2 reflection and the mu4 clock. NPW26 (arXiv:2605.10693) prove the intrinsic Bisognano-Wichmann lemma for COMMUTING-PROJECTOR (LTO) models with a Z/2 reflection; this attacks the commuting-projector side of sub-step (i) on the same v367 collar. [E] FLAT-BAND PARENT: the flattened Bloch Q=h/|h| has Q^2=I (flat +-1 bands) and P=(I-Q)/2 is an exact projector EQUAL to v367's occupied ground-state projector -- a frustration-free commuting-projector parent with the same gapped ground state (machine precision over a k-grid). [E] GAP-PROTECTED QUASI-LOCALITY: the gapped projector kernel |P(n,0)| decays rapidly (xi~1.14 at M=1, exp R^2=0.995, |P(20,0)|<1e-6) and xi GROWS as the gap closes (2.7 at gap 0.02); at the GAPLESS point the kernel decays as a clean power law |R|^{-2.1} (R^2=0.999, tail ~2e-4) -- the projector net is quasi-local BECAUSE the bulk is gapped. [E] Z/2 REFLECTION + mu4 INHERITED: the flattened modular Hamiltonian K_flat=sign(K) inherits Theta K_flat Theta=-K_flat (BW, u_Theta=J) and [rho,K_flat]=0 (mu4) EXACTLY (sign is odd; rho commutes with any f(K)); side-even and off-mu4 neg controls fire. [C]/[O] CHIRAL OBSTRUCTION: the chiral c_-=8!=0 (FHS Chern |C|=1, trivial control 0) admits NO strictly finite-range commuting-projector Hamiltonian (Kapustin-Fidkowski arXiv:1810.07756; C!=0 forbids exp-localised Wannier, Thouless), so the realisation is the QUASI-LOCAL LTO net NPW26 use. Advances v424 sub-step (i); does NOT supply the continuum theorem; SEAM.EQUIV.01 stays [O].
Recorded status: S3 commuting-projector realisation [E]/[C]: v367's gapped collar has an exact flat-band parent (Q^2=I, P^2=P=P_gs) that is gap-protected quasi-local (xi~1.14 at M=1, power-law |R|^{-2.1} at the gapless point) and whose flattened modular Hamiltonian inherits Theta K_flat Theta=-K_flat and [rho,K_flat]=0 exactly; the chiral c_-=8 obstructs STRICT commuting-projector locality (Kapustin-Fidkowski), so the realisation is the quasi-local LTO net of NPW26 -- advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.S3.LATTICE.01; SEAM.EQUIV.BW.UNIFORM.01; v424; v426; v367; v455; v336
Scripts: v460_seam_s3_lto.py
Location: tfpt_research_contracts
Original CSV, line 522Original ledger status
FORM.SEAM.RESIDUAL.01
LEAN 4 FORMALISATION reducing the SEAM.EQUIV.01 residual to a SINGLE named realization axiom (2026-06-30, G4 of the post-F next steps) -- new module TfptCarrier/SeamResidualAxiom.lean, lake build OK. It kernel-checks the arithmetic backbone of the whole G-block over Nat/Int: [F] c3_eight_one_sided (the one-sided count |Z2|*(intK/pi)=2*4=8, v456), [F] two_sided_nonchiral (a reflection sends C->-C so a two-sided boundary forces C=0), [F] cMinus_eq_eight (c_-=8=g_car+N_fam=rank E8), [F] mmst_range (rank<=c<=D, 8<=8<=16), [F] decoupled_additivity (16*1/2=8), [F] currents_fermionic (120 so(16) bilinears) + currents_lattice (248=8+240), [F] roots_split (240=112+128, 248=120+128), [F] holomorphic_discriminator (det K_E8=1 vs det K_SO16=4) -- ALL by kernel decide. UPDATE (2026-06-30, v461/v462 round): the two cited existence theorems are MERGED into ONE combined external axiom seam_realisation_theorem (dropping the intermediate ChiralCFT_c8), so #print axioms seamResidualClosed now lists exactly FOUR axioms (CollarRealizedAsLatticePhase, SeamIsE8, collar_realizes, seam_realisation_theorem), down from six; and two more kernel facts are added -- [F] strict_locality_forbidden (v461: the Wannier winding=|C|=1!=0 and c_-=8!=0, so a strict finite-range commuting projector is topologically forbidden) and [F] fock_sectors (v462: C(16,2)=120 currents, 2^8=256 Ramond states, spinor 128, 120+128=248). residual_arithmetic (now incl. v461/v462) depends only on [propext]. So the residual is reduced to ONE TFPT-internal realization axiom + ONE combined cited theorem. UPDATE (2026-06-30, v463/v464 round): two more kernel facts -- [F] c8_three_candidates (A8/D8/E8 each dim=8*(1+h^v), so c=8 is not unique at level 1) and [F] holomorphy_selects_e8 (the forced dim V_1=248=dim E8 != 80(A8) != 120(D8), so holomorphy promotes 'c=8' to 'E8' uniquely, v463); v464 (the one-particle realization rigidity) is NUMERICAL (symbol idempotency + Cauchy convergence + entanglement c-fit), so it sharpens the meaning of collar_realizes ('the unique quasi-free realization, modulo Araki/Shale-Stinespring') but is not a kernel fact. seamResidualClosed stays exactly FOUR axioms; residual_arithmetic (now incl. v463) depends only on [propext]. UPDATE (2026-07-02, v469 round): a PARALLEL peer-reviewed derivation seamResidualClosed' is added -- the invariant-level axiom collar_invariants (R1': quasi-free [C] + gap [E] + class D + c_-=8 [E from P1], replacing the model-level fiat) plus ONE combined 1995-2001 package crossedproduct_route_theorem (Kitaev/Kapustin-Spodyneiko phase classification o MMST+Boeckenhauer scaling limit o Longo-Rehren/BE local Z2 crossed product + KLM o Dong-Mason uniqueness); its joints are NEW kernel facts with NO axioms -- [F] lr_locality_integer (h_s = 16/16 in Z), [F] glue_locality_integers (mu4 h(J^k) = {1,1,1} in Z), [F] klm_holomorphic (4/2^2 = 1 and 16/4^2 = 1), [F] sixteenfold_pin (nu = 16 = 0 mod 16, c_- = nu/2 = 8), bundled as crossedproduct_arithmetic [propext only]. #print axioms seamResidualClosed' = exactly FOUR (CollarHasInvariants, SeamIsE8, collar_invariants, crossedproduct_route_theorem); the original seamResidualClosed (AGT/AMT witness) is kept unchanged as the independent second route. lake build OK (3285 jobs).
Recorded status: Formal [F] (Lean kernel): the G-block arithmetic (one-sided 8=|Z2|*intK/pi, reflection C->-C, c_-=8, MMST range 8<=8<=16, decoupled additivity, 248=8+240=120+128, 240=112+128, det K 1 vs 4, + v461 winding=|C|=1!=0 obstruction + v462 Fock counts 120/128/256 + v463 holomorphy selector forced dimV1=248=E8) all kernel-proved by decide; the two cited theorems MERGED into ONE combined axiom seam_realisation_theorem so seamResidualClosed depends on FOUR axioms (down from six); v464 one-particle rigidity is numerical; residual reduced to ONE realization axiom + ONE combined cited theorem; PLUS the v469 parallel peer-reviewed route seamResidualClosed' (invariant-level collar_invariants + crossedproduct_route_theorem; new kernel facts lr_locality_integer, glue_locality_integers, klm_holomorphic, sixteenfold_pin, all [propext only]); SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Formal
Dependencies: SEAM.EQUIV.01; SEAM.S3.FROM-P1.01; SEAM.S3.LOCALITY.01; SEAM.EQUIV.KILLTEST.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.SPINOR.01; SEAM.EQUIV.UNIQ.01; SEAM.EQUIV.ONEPARTICLE.01; SEAM.EQUIV.CROSSEDPRODUCT.01; v456; v457; v458; v459; v461; v462; v463; v464; v469
Scripts: lean4-carrier-rigidity/TfptCarrier/SeamResidualAxiom.lean
Location: tfpt_research_contracts
Original CSV, line 523Original ledger status
SEAM.S3.LOCALITY.01
The Kapustin-Fidkowski/Thouless strict-locality OBSTRUCTION made EXPLICIT (2026-06-30) -- sharpening v460's verdict from a cited statement to an exhibited topological obstruction on the same v367 p+ip collar. v460 realised the S3 input as a flat-band commuting-projector net and noted BY CITATION that the chiral c_-!=0 forbids strict finite-range locality; this module EXHIBITS that obstruction. [E] CHERN INTEGER: the FHS Chern is C=+1 (chiral M=1), 0 (trivial M=3), -1 (opposite mass) -- the invertible-phase invariant. [E] WILSON-LOOP / WANNIER-CENTRE WINDING = CHERN: the hybrid Wannier centre theta(ky) winds by |C|=1 (chiral) and 0 (trivial) over the BZ; exp-localised Wannier exist IFF the winding is 0 (Brouder-Panati et al. PRL 98 046402; Thouless), so the chiral collar has NONE. [E] OBSTRUCTION INTEGERS TIED: winding=|C|=1!=0 and c_-=16/2=8=g_car+N_fam=rank E8!=0; winding!=0 <=> C!=0 <=> c_-!=0 (the same 8 as c3's one-sided count, v456). [E] NEG CONTROL: BOTH phases are gapped, yet only the trivial (winding 0) admits exp-localised Wannier and a strict finite-range commuting projector -- it is the CHIRALITY, not the gap. [C]/[O] Kapustin-Fidkowski: a strictly finite-range commuting projector would force winding 0, C=0; since C=1!=0 NONE exists, so v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net (strict locality topologically forbidden, not a missing premise). SEAM.EQUIV.01 stays [O] (continuum theorem MMST v336 the backbone).
Recorded status: Strict-locality obstruction exhibited [E]/[C]: the chiral collar's Wilson-loop/Wannier-centre winding = Chern = 1!=0 (vs 0 for the gapped trivial control), the exact obstruction to exp-localised Wannier (Brouder et al./Thouless); so no strictly finite-range commuting projector exists for c_-=8!=0 (Kapustin-Fidkowski) -- v424 sub-step (i)'s realisation is PROVABLY the quasi-local NPW26 LTO net, sharpening v460 from citation to exhibited obstruction; SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Numerical
Dependencies: SEAM.EQUIV.01; SEAM.S3.LTO.01; SEAM.S3.FROM-P1.01; SEAM.EQUIV.EDGE.CHERN.01; v460; v447; v456; v367
Scripts: v461_seam_strict_locality.py
Location: tfpt_research_contracts
Original CSV, line 524Original ledger status
SEAM.EQUIV.SPINOR.01
The 128-spinor extension SO(16)_1 -> (E8)_1 exhibited at CHARACTER level and as a finite-L -> continuum convergence (2026-06-30) -- the constructive companion of the v458 (MMST audit) / v459 (lattice-VOA) residual. v458 isolated the residual to the 128 spinor currents; v459 named the lattice-VOA route; this module EXHIBITS the extension three ways. [E] (E8)_1 TOWER FROM SO(16)_1 SECTORS: the exact Jacobi/E8 identity theta2^8+theta3^8+theta4^8=2E4 (~1e-30) gives chi_{(E8)_1}=E4/eta^8 = (1/2)[(t3/eta)^8+(t4/eta)^8] + (1/2)(t2/eta)^8 = chi_o^{SO16}+chi_s^{SO16} -- the holomorphic (E8)_1 character IS the SO(16)_1 vacuum+spinor sector sum (the conformal embedding SO(16)_1 in (E8)_1). [E] 128 SPINOR WEIGHT-1 STATES FILL 120 -> 248: chi_o gives weight-1 multiplicity 120 (=dim so(16)), chi_s gives 128 at weight 1, summing to the exact (E8)_1 tower {1,248,4124,34752} -- the 128 spinor currents ARE the (E8)_1 - SO(16)_1 deficit (248=120+128). [E] FINITE 16-MAJORANA FOCK: C(16,2)=120 NS bilinear (so(16)) currents and 2^{16/2}=256 Ramond ground states = 128 spinor + 128 cospinor (split by fermion parity) -- the spinor present already at finite N. [E] FINITE-L -> CONTINUUM CONVERGENCE: the critical free-fermion ring Casimir E_0(L)=e_inf L - pi v c/(6L) fits c_Dirac -> 1 (c_-=16*(1/2)=8) and the lowest scaled NS gap x_1 -> the weight-1 dimension 1 with |x_1-1| ~ 1/L^2 (L=64..1024) -- the lattice flows to the conformal tower. [C]/[O] the rigorous holomorphic extension (the simple-current chi_o+chi_s projection) is supplied by AGT/AMT (v459); SEAM.EQUIV.01's continuum existence (v336) stays [O] -- this constructs and converges, it does not prove the uniform scaling-limit theorem.
Recorded status: 128-spinor extension exhibited [E]/[C]: the Jacobi/E8 identity theta2^8+theta3^8+theta4^8=2E4 makes chi_{(E8)_1}=E4/eta^8=chi_o+chi_s (vacuum 120 + spinor 128 = 248); the finite 16-Majorana Fock space carries C(16,2)=120 currents and 2^8=256=128+128 Ramond states; the critical ring converges c_- -> 8 and x_1 -> 1 as L grows; AGT/AMT (v459) supply the rigorous extension; SEAM.EQUIV.01 existence stays [O]
Dependencies, status detail and source
Canonical status: Identity
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.KILLTEST.01; v458; v459; v457; v336
Scripts: v462_seam_spinor_continuum.py
Location: tfpt_research_contracts
Original CSV, line 525Original ledger status
SEAM.EQUIV.UNIQ.01
The c=8 HOLOMORPHIC-UNIQUENESS pin (2026-06-30, B of the A+B 'actually close it' round) -- the positive lever making the IDENTIFICATION half of SEAM.EQUIV.01 classification-forced rather than assumed. v458/v459 close the continuum EXISTENCE; this pins WHY the limit must be (E8)_1 and not a same-c rival. [E] c=8 NOT UNIQUE AT LEVEL 1: c=dim g/(1+h^v)=8 has THREE simple solutions A8 (dim 80), D8=so(16) (dim 120), E8 (dim 248) -- c=8 is necessary but NOT sufficient (the v457 det K=4 SO(16) rival is a genuine gleich-c competitor). [E] HOLOMORPHY FORCES dim V_1=248: a single-sector c=8 chiral CFT has an SL(2,Z)-covariant character; the only weight-0 form with leading q^{-1/3} and integer coeffs is j^{1/3}=E4/eta^8, q^1 coeff 248 (1-dim candidate space, vacuum a_0=1) -- forced. [E] dim 248 + level-1 c=8 => E8 uniquely (only E8 has dim 248; 248/(1+30)=8); A8/D8 excluded. [E] tower {1,248,4124,34752,213126}=E4/eta^8; E4/eta^8=j^{1/3} (numeric <1e-25). [C] Dong-Mason/Schellekens: the holomorphic VOA at c=8 is UNIQUE = V_{E8}; the even unimodular rank-8 lattice is unique = E8 (Mordell-Witt). So 'c=8 holomorphic limit = (E8)_1' is classification-forced; with v458/v459 the continuum/identification HALF of SEAM.EQUIV.01 is [C], leaving the realisation R1 (v464). SEAM.EQUIV.01 stays [O].
Recorded status: c=8 holomorphic-uniqueness pin [E]/[C]: c=8 has three level-1 candidates (A8/80, D8/120, E8/248) so c=8 is not sufficient; holomorphy forces the character E4/eta^8 => dim V_1=248 => E8 uniquely; Dong-Mason/Schellekens give holomorphic c=8 VOA unique = V_{E8}, so the identification half of SEAM.EQUIV.01 is [C] (classification-forced); SEAM.EQUIV.01 stays [O]
Dependencies, status detail and source
Canonical status: Identity
Dependencies: SEAM.EQUIV.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.KILLTEST.01; v457; v458; v459
Scripts: v463_seam_c8_holomorphic_uniqueness.py
Location: tfpt_research_contracts
Original CSV, line 526