A registered result in its source context

GRAV.ENTROPIC.HODGE.01

Original ledger classification: Formal/Identity [E] (operator-level Clifford/weights/enumeration, exact) + [C] (physical Q choice) + [O] (AP2 + R^2 unchanged)

Original claim text

THE OPERATOR LEVEL OF THE ENTROPIC-ACTION BRIDGE (2026-07-13; work packages 1 + 4-algebra of GRAV.ENTROPIC.ACTION.01/v473) -- upgrades the v473 vector-space identity to the operator level on the finite carrier Fock space and DECIDES the Q-block conjecture by exact enumeration; closes NO gate. [E] CAR => CLIFFORD: on Lambda^* C^5 (dim 2^g_car = 32) the ten Gamma_{2k} = a_k^dag + a_k, Gamma_{2k+1} = i(a_k^dag - a_k) satisfy {Gamma_a,Gamma_b} = 2 delta_ab exactly (all 100 pairs, integer matrices) -- the so(10) = D5 Clifford algebra acts on the carrier exterior algebra. [E] S+ = LAMBDA^EVEN AS D5-MODULE: the 45 bilinears [Gamma_a,Gamma_b]/4 commute with gamma = (-1)^N, span exactly 45 = dim so(10), and leak nothing from the 16-dim even subspace -- the carrier half-spinor IS the even exterior algebra WITH its spinor structure (the operator level of v2/v44/v197). [E] HODGE-DIRAC SYMBOL = CLIFFORD MULTIPLICATION: c(xi) = a^dag(xi) - a(xi), the principal symbol of Bianconi's D = d + delta, satisfies c(xi)^2 = -|xi|^2 Id exactly for symbolic real xi. [E] THE FOLD IS THE 5 -> 5bar CONJUGATION: the traceless u(5) weights of Lambda^4 are exactly the negatives of those of Lambda^1, so the Hodge fold *: Lambda^1 ~ Lambda^4 conjugates the vector block -- Bianconi's fiber 1+5+10 becomes the GUT half-spinor 16 = 1 + 5bar + 10, the SM carrier decomposition (not a mere dimension match). [E] HONEST OBSTRUCTION: Omega^{<=2} is NOT Clifford-invariant (c(xi) maps Lambda^2 into Lambda^3), while Lambda^even is invariant under the even Clifford algebra -- the bridge identification must go through the Hodge fold, never through naive degree truncation (a recorded constraint on the AP2 compression). [E] Q-TARGET ENUMERATION (the v473 [C] conjecture DECIDED at the algebraic level): the uniform-block ansatz Q = k c3^2 P_d solves Tr Q^2 = d k^2 c3^4 = 32 c3^4 with integer k for EXACTLY three supports (d,k) = (2,4), (8,2), (32,1) -- the sheet pair |Z2|, the rank g_car + N_fam = rank E8 = 8, and the full Fock space 2^g_car = 32 = dim Lambda^* C^5 (minimal uniform solution: eps_* = e^{-1/alpha} c3^2 per Fock mode, q = c3^2 = 1/(64 pi^2)); the two-form block d = 10 (the problem-note 'pair sector' conjecture) and the fiber d = 16 require irrational multipliers sqrt(16/5) and sqrt(2) -- the naive pair-block reading is KILLED; bonus identity Tr Q^2 / delta_top = 32/48 = 2/3 = dim Lambda^*/Omega_adm (the 2/3 in the v473 target is a mode-count ratio). [C] the physical choice among the three admissible supports (uniform-32 is minimal, not derived). [O] UNCHANGED: the AP2 compression P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} and the R^2 mechanism (v473 pre-registered kill test) stay open; no status marker moves anywhere

Original classification detail

Operator-level Hodge bridge [E]/[C]/[O]: the D5 Clifford algebra exhibited on the carrier Fock space (10 gammas exact, 45-dim so(10) preserving Lambda^even = 16), Hodge-Dirac symbol = Clifford multiplication, the fold = 5 -> 5bar conjugation (1+5+10 -> GUT 16), Omega^{<=2} not Clifford-invariant (honest constraint on AP2); Q-target d k^2 = 32 solved exactly by supports {2, 8, 32} with minimal uniform q = c3^2 and the 10/16 blocks killed; AP2 compression + R^2 mechanism stay [O]

Dependencies as recorded

GRAV.ENTROPIC.ACTION.01; v2; v44; v60; v197; v359; v473