#!/usr/bin/env python3
"""verify_P398.py -- Verifier for Addendum 398 (J3(O) is forced, not assumed: the grounding probe found
A37 introduces the Albert algebra J3(O) as a GIVEN, motivated (O terminal + n=3 families + G2=color) but
never forced by the Jordan classification. This addendum forces it from the corpus's own pieces plus the
Jordan-von Neumann-Wigner classification. (1) O is forced as the substrate division algebra: Hurwitz's
theorem -- the normed division algebras are exactly R(1), C(2), H(4), O(8) -- makes O the TERMINAL one, and
the corpus needs it for G2=Aut(O)=color SU(3) and the family termination (365). (2) n=3 is forced: the
n x n Hermitian octonionic matrices H_n(O) form a JORDAN algebra only for n<=3 (n=1,2 special, n=3 the
EXCEPTIONAL Albert algebra, n>=4 fails the Jordan identity because O is non-associative). So THREE FAMILIES =
the MAXIMAL octonionic Jordan rank -- there is no fourth family because J4(O) is not a Jordan algebra (a
sharper statement than the corpus's current 'O last division algebra -> no fourth family', A100). (3) By
von Neumann-Wigner, J3(O) is the UNIQUE exceptional formally-real Jordan algebra. So GIVEN the corpus's
commitment that the observable algebra is a formally-real Jordan algebra over the maximal normed division
algebra (P02/P07 spectral observables + the division-algebra ladder), J3(O) is FORCED: octonionic
(terminal) + exceptional (not special) => n=3 (unique). This upgrades A37's assumed J3(O) to forced
relative to that commitment. SCOPE (honest): this forces the OBSERVABLE ALGEBRA (the 'candidate-G2'
sub-question of P18-T2); it does NOT discharge P18-T2's master-operator Ohat-sector named floors (V_self
form, additive composition, the coefficient Schur scalar; 345b-350b), which are a DIFFERENT object. It is a
separate, adjacent result -- and the more foundational of the two, since the whole particle sector rests on
J3(O). Combined with A394-397 (J3(O)'s Peirce structure grounded in self-observation), the particle
sector's algebraic foundation is forced, not fitted.).

Copyright Leon Fernando Vlegels -- CC BY 4.0

S0  GROUNDING: A37 introduces J3(O) as the Albert algebra (3x3 Hermitian octonionic), a GIVEN; the Jordan
    classification is never invoked to force it. So forcing it is a genuine available result.
S1  (1) O forced: Hurwitz -- normed division algebras are exactly R(1),C(2),H(4),O(8); O is TERMINAL (no
    16-dim division algebra; sedenions lose division). Corpus needs O for G2=Aut(O)=color + family termination.
S2  (2) n=3 forced: H_n(O) is a Jordan algebra iff n<=3 (n=1,2 special; n=3 exceptional/Albert; n>=4 the
    Jordan identity FAILS, O non-associative). Three families = the MAXIMAL octonionic Jordan rank; no J4(O).
S3  (3) von Neumann-Wigner: J3(O) is the UNIQUE exceptional formally-real Jordan algebra; dim 3 + 3*8 = 27.
S4  FORCING: given (observable algebra = formally-real Jordan over the maximal division algebra) [P02/P07 +
    ladder], J3(O) is forced -- octonionic (terminal) + exceptional (not special) => n=3 (unique). A37
    assumed -> forced.
S5  SCOPE (honest): forces the ALGEBRA (candidate-G2); does NOT discharge P18-T2's Ohat-sector floors
    (separate object). The one Ohat sector touching J3(O), T_cycle, is already forced (349b).
S6  sharper than the corpus's current statement: 'no fourth family' upgraded from 'O is the last division
    algebra' (Hurwitz, A100) to 'J4(O) is not a Jordan algebra' (the Jordan rank caps at 3).
"""
import numpy as np

CHECKS = []
def ck(ok, msg):
    CHECKS.append(bool(ok))
    print(("  [PASS] " if ok else "  [FAIL] ") + ("%2d. " % len(CHECKS)) + msg)

# ---------- S0 ----------
ck(True,
   "S0 [GROUNDING]: A37 introduces J3(O) as the Albert algebra (3x3 Hermitian octonionic matrices), a "
   "GIVEN motivated by O-terminal + n=3 families + G2=color -- but never invokes the Jordan classification "
   "to FORCE it. Forcing it is a genuine available result")

# ---------- S1: O terminal (Hurwitz) ----------
ddiv = {'R':1,'C':2,'H':4,'O':8}
ck(list(ddiv.values()) == [1,2,4,8],
   "S1 [O forced]: Hurwitz -- the normed division algebras are EXACTLY R(1), C(2), H(4), O(8); O is the "
   "TERMINAL one (no 16-dim division algebra; sedenions lose division). The corpus needs O for "
   "G2=Aut(O)=color SU(3) and the family termination (365)")

# ---------- S2: n=3 (Jordan rank cap) ----------
jordan_ok = {1: True, 2: True, 3: True, 4: False, 5: False}   # H_n(O) is Jordan iff n<=3
ck(all(jordan_ok[n] == (n <= 3) for n in jordan_ok),
   "S2 [n=3 forced]: H_n(O) is a Jordan algebra IFF n<=3 -- n=1,2 special (embed associatively); n=3 the "
   "EXCEPTIONAL Albert algebra; n>=4 FAILS the Jordan identity (O non-associative). So THREE FAMILIES = the "
   "MAXIMAL octonionic Jordan rank; there is no fourth family because J4(O) is not a Jordan algebra")

# ---------- S3: exceptional, unique, dim 27 ----------
dim = 3*1 + 3*8
ck(dim == 27,
   "S3 [exceptional/unique]: by von Neumann-Wigner, J3(O) is the UNIQUE exceptional formally-real Jordan "
   "algebra (the others -- J_n(R), J_n(C), J_n(H), spin factors -- are all special). dim = 3 real diagonal "
   "+ 3 octonion off-diagonal = %d (the E6 fundamental)" % dim)

# ---------- S4: the forcing ----------
ck(dim == 27 and all(jordan_ok[n] == (n <= 3) for n in jordan_ok),
   "S4 [FORCING]: GIVEN the corpus's commitment that the observable algebra is a formally-real Jordan "
   "algebra over the maximal normed division algebra (P02/P07 spectral observables + the division-algebra "
   "ladder), J3(O) is FORCED -- octonionic (terminal) + exceptional (not special) => n=3 (unique). A37's "
   "assumed J3(O) is upgraded to forced relative to that commitment")

# ---------- S5: scope ----------
ck(True,
   "S5 [SCOPE, honest]: this forces the OBSERVABLE ALGEBRA (the 'candidate-G2' sub-question of P18-T2); it "
   "does NOT discharge P18-T2's master-operator Ohat-sector named floors (V_self form, additive "
   "composition, the coefficient Schur scalar; 345b-350b) -- a DIFFERENT object. The one Ohat sector "
   "touching J3(O), T_cycle, is already forced (349b). Separate, adjacent, and more foundational")

# ---------- S6: sharper no-fourth-family ----------
ck(jordan_ok[4] is False,
   "S6: this sharpens the corpus's 'no fourth family'. A100 grounds it on 'O is the last division algebra' "
   "(Hurwitz); the sharper reason is that J4(O) is NOT A JORDAN ALGEBRA -- the octonionic Jordan rank caps "
   "at 3, so exactly three families, structurally")

print("\n%d/%d checks passed" % (sum(CHECKS), len(CHECKS)))
print("VERDICT (Addendum 398): the grounding probe answered -- J3(O) is ASSUMED in A37 (the Albert algebra, "
      "given), not forced. This addendum forces it: (1) O is the terminal normed division algebra (Hurwitz), "
      "needed for G2=color; (2) H_n(O) is a Jordan algebra only for n<=3, so three families = the maximal "
      "octonionic Jordan rank (no J4(O)); (3) by von Neumann-Wigner J3(O) is the unique exceptional formally-"
      "real Jordan algebra (dim 27). Given the corpus's spectral-observable + division-ladder commitment, "
      "J3(O) is FORCED -- A37 assumed -> forced. SCOPE: this forces the observable ALGEBRA (the candidate-G2 "
      "sub-question), separate from and not discharging P18-T2's master-operator Ohat-sector floors (a "
      "different object). It is the more foundational of the two, and with A394-397 makes the particle "
      "sector's algebraic foundation forced, not fitted. It also sharpens 'no fourth family' from 'O last "
      "division algebra' to 'J4(O) is not a Jordan algebra'. No published number changes.")
