A registered result in its source context
Original ledger classification: E8 character CM-duality [E]/[C] (chi_E8(i)=12 vs chi_E8(omega)=0: the (E8)_1 character maximal at the seam tau=i, zero at the family tau=omega; the two are the only elliptic points so the values are forced) -- the seam/flavor rigidity duality, extends v282
Recorded statement: The (E8)_1 vacuum character at the two CM points (2026-06-24): chi_E8 = E4/eta^8 = j^{1/3} equals 12 at the seam CM point tau=i and 0 at the family CM point tau=omega, extending v282 (chi_E8(i)=12) to the second elliptic point. [E]…
Statement, assumptions and sources
Paper context: Origin Theory
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E8.CMDUAL.01
Original ledger classification: E8 character CM-duality [E]/[C] (chi_E8(i)=12 vs chi_E8(omega)=0: the (E8)_1 character maximal at the seam tau=i, zero at the family tau=omega; the two are the only elliptic points so the values are forced) -- the seam/flavor rigidity duality, extends v282
Original claim text
The (E8)_1 vacuum character at the two CM points (2026-06-24): chi_E8 = E4/eta^8 = j^{1/3} equals 12 at the seam CM point tau=i and 0 at the family CM point tau=omega, extending v282 (chi_E8(i)=12) to the second elliptic point. [E] chi_E8^3=j (q-series). [E] SEAM tau=i (Q(i), order 4): cross-ratio 2 => j=1728 => chi_E8(i)=1728^{1/3}=12 (E8 present, v282/v214). [E] FAMILY tau=omega (Q(sqrt-3), order 6): the equianharmonic lambda (lambda^2-lambda+1=0) => j=0 => chi_E8(omega)=0 (E4 vanishes, v220/v267). [E] DUALITY: the SAME function chi_E8 takes its two CM values {12,0} at the two atoms' CM points -- maximal at the seam, zero at the family. [E] tau=i (order 4) and tau=omega (order 3/6) are the ONLY elliptic points of PSL(2,Z), so chi^3=j forces chi(i)=12 and chi(omega)=0 (not chosen). [C] STRUCTURAL READING: the seam is rigid where E8 is present (chi=12, SEAM.EQUIV.01 closes) while the flavor/CP sector is the soft residual where E8 degenerates (chi=0, the U_wall amplitude / v_geo / the CP texture). [C] DUAL-KEYSTONE CANDIDATE: the flavor sector = the j=0 degeneration of (E8)_1 at tau=omega, dual to the seam = (E8)_1 at tau=i.
Original classification detail
[E] the modular arithmetic (chi^3=j; j(i)=1728, chi(i)=12; j(omega)=0, chi(omega)=0; the two-elliptic-point uniqueness); [C] the seam/flavor rigidity reading and the dual-keystone candidate. Extends v282 to the second CM point; core values j=1728/j=0 already mirrored (v214/v220/v267/v282), so Python-only by the synthesis convention; no new number.
Dependencies as recorded
QGAMB.UNIFY.01; v282; v214; v220; v267; v390
Open the original ledger source, row 474Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-24 (The arithmetic hull + the [formula in source] -character CM duality: v403, v404)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- 2026-06-24 (The [formula in source] dual keystone + the flavor residuals it homes: v405– v407)Research changelog; TFPT — Introduction and reading guide
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- The unique attractor: a gapped transport selects ONE fixed pointOrigin Theory
Registered verification modules
- v404_e8_char_cm_duality.pyRegistered verification module · frontier
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E8.CMDUAL.01: the (E8)_1 vacuum character chi_E8=E4/eta^8=j^{1/3} at the two CM points -- 12 at the seam (tau=i), 0 at the family (tau=omega), extending v282. [E] chi_E8^3=j (q-series). [E] SEAM tau=i (Q(i), order 4): cross-ratio 2 => j=1728 => chi_E8(i)=12 (E8 present, v214/v282). [E] FAMILY tau=omega (Q(sqrt-3), order 6): equianharmonic lambda (lambda^2-lambda+1=0) => j=0 => chi_E8(omega)=0 (E4 vanishes, v220/v267). [E] tau=i,tau=omega are the only elliptic points of PSL(2,Z) so the values are forced. [C] the seam/flavor rigidity duality: seam rigid where E8 present (chi=12), flavor the soft residual where E8 degenerates (chi=0); dual-keystone candidate. Core values j=1728/j=0 already mirrored (v214/v220/v267/v282).
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