Integer constraints from the E₈ invariant degrees
Original ledger classification: E8 degree joint-lock [E] (the two structural integers g_car=5, N_fam=3 = unique roots of x^2-8x+15 with coeffs rank E8=8 and h/2=15; all E8 combinatorics fixed by its degrees; consolidation of v6/v66/v355/v431, no new mining)
Recorded statement: THE TWO STRUCTURAL INTEGERS ARE JOINTLY FORCED BY E8 DEGREE-INVARIANTS, AND E8 IS FIXED BY ITS DEGREES (2026-06-29, v437): a CONSOLIDATION of v6/v66/v355/v431 deriving more from the E8 Casimir degrees WITHOUT crossing the v354/v355…
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Paper context: Red-team audit
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E8.DEGREE.JOINT.01
Original ledger classification: E8 degree joint-lock [E] (the two structural integers g_car=5, N_fam=3 = unique roots of x^2-8x+15 with coeffs rank E8=8 and h/2=15; all E8 combinatorics fixed by its degrees; consolidation of v6/v66/v355/v431, no new mining)
Original claim text
THE TWO STRUCTURAL INTEGERS ARE JOINTLY FORCED BY E8 DEGREE-INVARIANTS, AND E8 IS FIXED BY ITS DEGREES (2026-06-29, v437): a CONSOLIDATION of v6/v66/v355/v431 deriving more from the E8 Casimir degrees WITHOUT crossing the v354/v355 anti-numerology line -- no new per-degree coincidence, only a joint lock. [E] E8 IS FIXED BY ITS DEGREE MULTISET: from deg(E8)={2,8,12,14,18,20,24,30} alone -- rank=#degrees=8 (=2nd degree); h=max(deg)=30; #positive roots=sum(exponents)=120; #roots=240; dim E8=#roots+rank=248; |W(E8)|=prod(deg)=696729600; sum(deg)=128=2^7=dim S^+ (the spinor half of 248=120+128). [E] THE JOINT QUADRATIC (the new lock): the two TFPT structural integers (g_car,N_fam) are the UNIQUE root pair of x^2-(rank E8)x+(h/2)=x^2-8x+15=(x-3)(x-5); sum of roots=rank E8=8 (the rank-fill, v6), product=h/2=15=g_car*N_fam (2 g_car N_fam=h, v431); {3,5} is the unique positive-integer factor pair of 15 summing to 8, so BOTH structural integers are forced TOGETHER by rank and Coxeter number. [E] THREE MAPPED DEGREES ARE CANONICAL: the 3 primary-readout degrees (v354) {2,8,30}={min degree (quadratic Casimir), rank, max degree (Coxeter h)} -- the invariants every simple Lie algebra distinguishes; g_car=5=max prime of h=30, N_fam=3=h/(2 g_car). [E] DISCIPLINE UPHELD: no new per-degree mining beyond the joint quadratic (the solution of two pre-existing constraints v6+v431); the five non-primary degrees keep their v431 family homes; the v354/v355 line is not crossed.
Original classification detail
[E] the joint quadratic lock (both structural integers as the unique roots of one degree-invariant quadratic) and the 'E8 fixed by its degrees' reconstruction; a consolidation of v6/v66/v355/v431 that respects the v354/v355 anti-numerology discipline (no new per-degree coincidence). Closes no gate; strengthens the over-determination of g_car and N_fam.
Dependencies as recorded
E8.REVERSE.AUDIT.01; E8.DEGREE.LADDER.01; v6; v66; v355; v431
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- Red-team audit
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- 2026-06-29 (discoverability — an SEO/GEO pass on the public website: a generated /llms.txt, explicit AI-crawler robots.txt, query-targeted metadata, and the first real backlinks from the README and the Zenodo deposit)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- OutcomeRed-team audit
Registered verification modules
- v437_e8_degree_joint.pyRegistered verification module · registry
Read the description recorded in the module registry
E8.DEGREE.JOINT.01: a CONSOLIDATION of v6/v66/v355/v431 deriving more from the E8 Casimir degrees WITHOUT crossing the v354/v355 anti-numerology line -- no new per-degree coincidence, only a joint lock. [E] E8 IS FIXED BY ITS DEGREE MULTISET: from deg(E8)={2,8,12,14,18,20,24,30} alone -- rank=#deg=8 (=2nd deg), h=max=30, #pos roots=Σexp=120, #roots=240, dim=248, |W|=∏deg=696729600, Σdeg=128=2^7=dim S^+. [E] THE JOINT QUADRATIC: the two structural integers (g_car,N_fam) are the UNIQUE root pair of x²-(rank E8)x+(h/2)=x²-8x+15=(x-3)(x-5) -- sum=rank E8=8 (rank-fill, v6), product=h/2=15=g_car·N_fam (2 g_car N_fam=h, v431); {3,5} the unique factor pair of 15 summing to 8, so BOTH are forced together by two degree-invariants. [E] the 3 mapped degrees {2,8,30}=min/rank/max (the canonical invariants); g_car=5=max prime of 30. [E] DISCIPLINE: no new per-degree mining; the 5 non-primary degrees keep their v431 homes. Exact integer/lattice [E], mirrored in wolfram/tfpt_readouts_extension.wl.
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