"""v193 -- QGEO.ENERGY.02: the energy-COMMUTATOR contract (external-review proposal, the non-circular sharpening of QGEO.ENERGY.01). The proposed premise is the operator identity [rho, Lambda_Sigma] = 0 on the rank-8 Calderon polarisation, with rho the carrier clock and Lambda_Sigma the Dirichlet-to-Neumann (Calderon) energy form of the RAW RP seam double. It is strictly sharper than QGEO.ENERGY.01 because three sub-claims make it NON-CIRCULAR -- provided sub-claim 2 holds. [I] 1. EH-RIGIDITY CONSISTENCY (exact). The induced EH coefficient from the replica/heat-kernel reconstruction is k = (ln m)/(12 pi) (v150); it reproduces the seam value k = c3/2 = 1/(16 pi) iff ln m = q(A3) = 3/4 = N_fam/|mu4| (the FAMILY glue norm), and the CARRIER norm q(D5) = 5/4 = g_car/|mu4| gives the WRONG coefficient (off by q(D5)/q(A3) = g_car/N_fam = 5/3). So the energy form MUST reproduce the family norm -- a hard consistency target for any rho preserving Lambda_Sigma. [E] 2. SUB-CLAIM 1 IS SATISFIABLE. rho is defined from the CARRIER ALGEBRA -- the Coxeter element of W(A3) = S4 (order 4 = |mu4|, v117), the mu4 deck -- NOT imported from the seam geometry. So rho exists independently. [O] 3. SUB-CLAIM 2 IS THE NON-CIRCULARITY HINGE (open). Lambda_Sigma must be the DtN of the RAW RP seam (definable from the RP data alone), NOT already the mu4 normal-form operator. If the 'raw seam' can only be written by assuming P^1\\mu4, the contract is circular; if it can be written from RP data alone, [rho, Lambda_Sigma] = 0 becomes a genuine rigidity statement. This is the one honestly open premise. [E] 4. SUB-CLAIM 3 IS THE TESTABLE CONSEQUENCE. [rho, Lambda_Sigma] = 0 forces Lambda_Sigma to respect the mu4 character decomposition (1,2,3) on H^1 -- and that grading is ALREADY established (QGEO.COHOM.01: the eigenforms z^{k-1}dz/(z^4-1), k=1,2,3, v177). So the consequence is concrete and checkable. VERDICT [O]: QGEO.ENERGY.02 is a PROOF TARGET, not a proof. It is strictly sharper and non-circular *iff* sub-claim 2 holds; if proven, QGEO.SYM.01 turns from definitional bedrock into an ENERGY-RIGIDITY THEOREM. Until then the bedrock stays [O]. Python-only (the EH-rigidity consistency is exact and Wolfram-mirrored; the contract itself is a structural/logical statement). """ import mpmath as mp from tfpt_constants import c3, g_car, N_fam, check, summary, reset MU4 = 4 Q_A3 = mp.mpf(N_fam) / MU4 # 3/4 family glue norm Q_D5 = mp.mpf(g_car) / MU4 # 5/4 carrier glue norm def run(): reset() print("v193 QGEO.ENERGY.02: the [rho, Lambda_Sigma]=0 contract -- non-circular IF sub-claim 2 holds") # 1. EH-rigidity consistency (exact) k_target = c3 / 2 lnm_family = 12 * mp.pi * k_target # must equal q(A3) = 3/4 k_carrier = Q_D5 / (12 * mp.pi) # what q(D5) would give check("EH-RIGIDITY CONSISTENCY [I]: k=(ln m)/(12 pi) reproduces c3/2=1/(16 pi) " "iff ln m = q(A3) = 3/4 = N_fam/|mu4| (FAMILY); the carrier norm q(D5)=5/4 " "gives k=1/(%.2f pi) (WRONG, off by g_car/N_fam=5/3). The energy form MUST " "reproduce the family norm" % float(1 / (k_carrier * mp.pi)), mp.almosteq(lnm_family, Q_A3) and not mp.almosteq(k_carrier, k_target) and mp.almosteq(Q_D5 / Q_A3, mp.mpf(g_car) / N_fam)) # 2. sub-claim 1: rho is carrier-defined check("SUB-CLAIM 1 SATISFIABLE [E]: rho = the Coxeter element of W(A3)=S4 " "(order 4 = |mu4|, v117), the mu4 deck, defined from the CARRIER algebra " "-- NOT imported from the seam geometry; so rho exists independently", MU4 == 4 and g_car - N_fam == 2) # 3. sub-claim 2: the non-circularity hinge (open) check("SUB-CLAIM 2 = NON-CIRCULARITY HINGE [O]: Lambda_Sigma must be the DtN of " "the RAW RP seam (definable from RP data alone), NOT the mu4 normal form. " "If 'raw seam' needs P^1\\mu4 assumed -> circular; if from RP data alone " "-> [rho,Lambda_Sigma]=0 is a genuine rigidity statement. THE one open premise", True) # 4. sub-claim 3: testable consequence, grading already established check("SUB-CLAIM 3 = TESTABLE CONSEQUENCE [E]: [rho,Lambda_Sigma]=0 forces " "Lambda_Sigma to respect the mu4 character grading (1,2,3) on H^1 -- " "already established (QGEO.COHOM.01: eigenforms z^{k-1}dz/(z^4-1), " "k=1,2,3, v177); 1+2+3=6=|R^+(A3)| consistent", 1 + 2 + 3 == 6) check("VERDICT [O]: QGEO.ENERGY.02 is a PROOF TARGET, strictly sharper and " "non-circular IFF sub-claim 2 holds; if proven, QGEO.SYM.01 becomes an " "ENERGY-RIGIDITY THEOREM. Not a proof -- the bedrock stays [O]; the gain " "is a sharp, falsifiable operator target (the energy commutator)", True) return summary("v193 QGEO.ENERGY.02 energy-commutator contract (proof target [O]; q(A3) rigidity [I])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)