Original ledger status
QGEO.ENERGY.02
THE ENERGY-COMMUTATOR CONTRACT (external-review proposal 2026-06-14; 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 -- rho the carrier clock, Lambda_Sigma the Dirichlet-to-Neumann (Calderon) energy form of the RAW RP seam double. Three sub-claims make it non-circular: (1) [E] rho is defined from the CARRIER algebra (the Coxeter element of W(A3)=S4, order 4=|mu4|, v117/HOR.CLOCK), NOT imported from the seam geometry; (2) [O] THE NON-CIRCULARITY HINGE -- Lambda_Sigma must be the DtN of the RAW RP seam (definable from RP data alone), NOT the mu4 normal-form operator; if 'raw seam' requires P^1\\mu4 assumed it is circular, if from RP data alone the commutator is a genuine rigidity statement (this is the ONE honestly open premise); (3) [E] [rho,Lambda_Sigma]=0 forces the mu4 character grading (1,2,3) on H^1, already established (QGEO.COHOM.01, eigenforms z^{k-1}dz/(z^4-1), v177). EXACT [I] consistency anchor: the induced EH coefficient k=(ln m)/(12 pi) (v150) reproduces c3/2=1/(16 pi) iff ln m=q(A3)=3/4=N_fam/|mu4| (FAMILY glue norm), and q(D5)=5/4=g_car/|mu4| (carrier) gives the WRONG k (off by g_car/N_fam=5/3) -- so the energy form MUST reproduce the family norm, NOT the carrier norm (gravity is family-geometry-induced). VERDICT: a PROOF TARGET, strictly sharper and non-circular IFF sub-claim 2 holds; if proven, QGEO.SYM.01 turns from definitional bedrock into an ENERGY-RIGIDITY THEOREM. Not a proof -- the bedrock stays [O]. Python-only (the q(A3) rigidity is exact and Wolfram-mirrored)
Recorded status: Geometric premise [O] (energy-commutator proof target; sharper, non-circular IFF sub-claim 2)
Dependencies, status detail and source
Canonical status: Geometric [O]: QGEO.ENERGY.02 = [rho,Lambda_Sigma]=0 (energy commutator); non-circular proof target for QGEO.SYM.01 IFF the raw-seam DtN (sub-claim 2) is RP-definable; exact rider: the EH coefficient selects q(A3)=3/4 (family), not q(D5)=5/4 (carrier)
Dependencies: QGEO.ENERGY.01; QGEO.SYM.01; QGEO.COHOM.01; HOR.CLOCK.01; v150_replica_eh_model.py; v192_qgeo_energy_clock.py
Scripts: v193_qgeo_energy_commutator.py
Location: tfpt_research_contracts
Original CSV, line 267Original ledger status
QGEO.DTN.01
SUBCLAIM 2 OF QGEO.ENERGY.02 TACKLED -- 'the raw RP-seam DtN is RP-definable' (external-review next-lever 2026-06-14): a REDUCTION, not a closure. (1) FINITE CORE [E]: rho:z|->iz on H^1(P^1\\mu4)=<w1,w2,w3>, wk=z^{k-1}dz/(z^4-1), acts as rho* wk = i^k wk (characters i,-1,-i DISTINCT; (iz)^4-1=z^4-1 invariant) -- verified symbolically; distinct eigenvalues => any operator commuting with rho is DIAGONAL on H^1, and the harmonic structure IS rho-equivariant there (v177/COHOM core) => [rho,Lambda_Sigma]=0 on H^1 is [E]. (2) HINGE MET [I]: the DtN/transfer operator is RP-canonical -- Osterwalder-Schrader reconstruction yields T=e^{-H} from the RP state + reflection alone (no coordinates; v54/SEAM.HORIZON.01, tfpt_2 (4)) -- on a quasi-free seam state (v155: the boundary marginal of a Gaussian bulk is the compression P Gamma P, a contraction). So Lambda_Sigma is RP-definable WITHOUT choosing P^1\\mu4: subclaim 2's non-circularity hinge is essentially met at the canonical level. (3) RESIDUAL RELOCATES [O]: [rho,Lambda_Sigma]=0 on the FULL boundary L^2 (beyond the 3-dim H^1) <=> the quasi-free seam state's modular flow is GEOMETRIC (Bisognano-Wichmann) -- a named, standard (hard) AQFT property, strictly sharper than 'the seam IS P^1\\mu4 (definitional)'. CIRCULARITY WATCH: BW normally presupposes a conformal/Poincare-covariant net (part of what is derived), so the BW step must be established INTRINSICALLY (modular geometricity from RP+quasi-free alone) or it re-circulates -- that intrinsic geometricity is the irreducible analytic content. VERDICT: subclaim 2's hinge met + finite core [E] => QGEO.SYM.01 REDUCES from a definitional postulate to 'intrinsic BW geometric modular covariance of the quasi-free seam state' -- one notch sharper, STILL [O]; the last structural fog point sharpens, does NOT close to [E]. Python-only (H^1 commutation is exact Moebius-symbolic like v177; reduction is structural)
Recorded status: Geometric premise [O] (reduction: subclaim-2 hinge met, finite core [E], bedrock -> BW modular)
Dependencies, status detail and source
Canonical status: Geometric [O]: subclaim 2 (raw-seam DtN RP-definability) met at the canonical OS level + finite core [E]; QGEO.SYM.01 reduces to intrinsic Bisognano-Wichmann modular covariance of the quasi-free seam state -- sharper, still [O], not closed
Dependencies: QGEO.ENERGY.02; QGEO.SYM.01; QGEO.TREE.01; SEAM.HORIZON.01; v177_seam_marking_kernel.py; v155_quasifree_boundary.py; v54_seam_horizon_keystones.py
Scripts: v194_raw_seam_dtn_rp.py
Location: tfpt_research_contracts
Original CSV, line 268Original ledger status
QGEO.MARKS.02
LEFSCHETZ/CHARACTER DERIVATION OF THE FOUR SEAM MARKS (external-review path 2/priority 2, 2026-06-14): instead of POSITING D4 boundary marks (the old QGEO.MARKS.01 finite core), the marks are FORCED to be a free mu4 orbit by orbit-counting plus the H^1 character. (1) [E] rho:z|->iz has order 4 and fixes EXACTLY {0,infinity}; rho^2:z|->-z also fixes only {0,infinity}; so any point off the fixed locus has a FREE order-4 orbit {z,iz,-z,-iz}. (2) [E] NO TRIVIAL CHARACTER: the carrier datum H^1=chi_1+chi_2+chi_3 (the A3 exponents (1,2,3)=Spec Q_+, v177/COHOM) has no trivial component, so no puncture sits at a rho-fixed point. (3) [E] rank H_1(P^1 minus n)=n-1=3 => n=4 punctures, all off the fixed locus => exactly ONE free order-4 orbit => the marks are a free mu4 orbit, Moebius-equivalent to {1,i,-1,-i} (cross-ratio 2). (4) [E] LEFSCHETZ CONSISTENCY: Tr(rho|H^1)=i+i^2+i^3=-1 and on closed P^1 L(rho)=Tr(H^0)+Tr(H^2)=1+1=2=#fixed points. RESIDUAL [O]: the marks no longer rest on a geometric posit (D4 marks) but on the milder representation premise 'the raw seam H^1 carries the A3-exponent rep chi_1+chi_2+chi_3' -- a carrier-algebra statement (A3 exponents are algebraic, independent of the seam geometry). So QGEO.MARKS reduces from geometry to an H^1-rep premise; less circular, still [O]. Exact; Wolfram-mirrored
Recorded status: Identity [E] (orbit/character core) + premise [O] (H^1-rep)
Dependencies, status detail and source
Canonical status: Identity [E]: orbit-counting (order-4 rho fixes {0,inf}; no trivial H^1 char; rank 3 => 4 punctures) forces the marks to a free mu4 orbit; residual [O] relocates from 'seam is P^1\\mu4' to 'H^1 carries the A3-exponent rep' -- less circular
Dependencies: QGEO.MARKS.01; QGEO.TREE.01; QGEO.SYM.01; v177_seam_marking_kernel.py; v178_marks_kernel_attempt.py
Scripts: v195_marks_lefschetz_character.py
Location: tfpt_research_contracts
Original CSV, line 269Original ledger status
QGEO.VARI.01
THE E_fail VARIATIONAL FUNCTIONAL (external-review path 1, 2026-06-14): a concrete, numerically-testable sharpening of the v194/QGEO.ENERGY.02 target. E_fail(Sigma,rho) = ||[rho,Lambda_Sigma]||^2_HS + ||rho^4-1||^2 + ||Theta rho Theta - rho^{-1}||^2, with E_fail=0 <=> the QGEO.SYM.01 conditions hold (energy commutator + order-4 clock + RP-reflection twist). (1) [E] on the finite H^1 block (rho=diag(i,-1,-i), Lambda diagonal=mu4-equivariant, Theta=complex conjugation) all three terms vanish: [rho,Lambda]=0, rho^4=I, conj(rho)=rho^{-1} => E_fail=0. (2) [N] a mu4-breaking off-diagonal Lambda gives [rho,Lambda]!=0 and a non-order-4 rho gives rho^4!=I, both raising E_fail>0 => the mu4 configuration is a true minimiser (verified numerically). (3) [I] the zero locus is exactly the seam-deck conditions of QGEO.SYM.01. (4) [O] minimising E_fail over the FULL raw-seam DtN (beyond the finite H^1) is exactly the v194 statement that the quasi-free seam state's modular flow is geometric (Bisognano-Wichmann) -- so E_fail gives a discretisable variational handle on the open bedrock, a TARGET not a closure. Python-only (finite-block linear algebra; finite vanishing exact, Wolfram-mirrored)
Recorded status: Numerical/identity [N]/[I] (finite vanishing exact) + target [O] (full-operator min = BW)
Dependencies, status detail and source
Canonical status: Numerical/identity [N]/[I]: E_fail=0 exactly at the mu4 config and >0 for perturbations (mu4 minimiser); full-operator minimisation = the v194 Bisognano-Wichmann residual [O] -- a concrete discretisable handle on the bedrock
Dependencies: QGEO.ENERGY.02; QGEO.DTN.01; QGEO.SYM.01; v193_qgeo_energy_commutator.py; v194_raw_seam_dtn_rp.py
Scripts: v196_seam_energy_functional.py
Location: tfpt_research_contracts
Original CSV, line 270Original ledger status
QGEO.MODULAR.01
CRACKING THE BEDROCK TO A STATE-INVARIANCE (2026-06-14): the v194 residual -- the full-L^2 commutator [rho,Lambda_Sigma]=0, framed as Bisognano-Wichmann geometric modular flow with a circularity worry -- is (a) made EXACT at the principal-symbol level and (b) reduced via Tomita-Takesaki to a clean state-invariance, removing the circularity. (1) PRINCIPAL SYMBOL [E]: the DtN principal symbol |k|=sqrt(-d^2/dtheta^2) (Lee-Uhlmann, v156) is diag(|n|) and the clock rho:z|->iz (rotation by pi/2) is diag(i^n) in the Fourier basis e^{in theta} -- both DIAGONAL => [rho,|k|]=0 EXACTLY on ALL of L^2 (verified on 33 modes), not just the 3-dim H^1; the leading-order commutation is FREE on the whole boundary, so the open part was never the principal symbol. (2) TOMITA-TAKESAKI [E]: the modular operator Delta_omega is INTRINSIC to (M,Omega); a state-preserving algebra symmetry (rho Omega=Omega, rho M rho*=M) commutes with Delta, hence with the modular Hamiltonian log Delta ~ Lambda_Sigma. So [rho,Lambda_Sigma]=0 FOLLOWS from omega o rho = omega, needing ONLY the state + reflection (which RP supplies) and NO conformal/Poincare covariance -- this REMOVES the v194 Bisognano-Wichmann circularity worry. (3) FINITE BLOCK [E]: on H^1 the covariance is mu4-equivariant (rho* w_k = i^k w_k, v177), so omega o rho = omega already holds there; with (1) the open part is neither the principal symbol nor the finite block. (4) IRREDUCIBLE RESIDUAL [O]: the FULL raw quasi-free seam state is rho-invariant (omega o rho = omega) <= the Gaussian bulk MEASURE is rho(deck)-invariant = the mu4 deck is a symmetry of the raw seam -- the maximally-operational, non-circular form of QGEO.SYM.01 (a STATE/MEASURE invariance, checkable, not a geometric or modular-flow statement). A foundational symmetry postulate cannot be derived from nothing (like c=const); it is now in its sharpest falsifiable form. VERDICT: bedrock cracked to omega o rho = omega -- principal symbol EXACT, Tomita-Takesaki NON-CIRCULAR, finite block invariant; NOT a full closure but the sharpest form, circularity gone. Python-only (principal-symbol Fourier commutation exact, Wolfram-mirrored; Tomita-Takesaki is a standard AQFT theorem)
Recorded status: Geometric premise [O] (bedrock reduced to a non-circular state-invariance; principal symbol exact)
Dependencies, status detail and source
Canonical status: Geometric [O]: bedrock cracked to omega o rho = omega (raw seam state mu4-invariant) -- principal symbol |k| commutes exactly (Fourier diag), Tomita-Takesaki removes the BW circularity, finite block holds; the irreducible residual is the mu4-symmetry of the raw seam measure (operational QGEO.SYM.01), still [O]
Dependencies: QGEO.DTN.01; QGEO.ENERGY.02; QGEO.SYM.01; QGEO.VARI.01; v194_raw_seam_dtn_rp.py; v177_seam_marking_kernel.py
Scripts: v198_modular_commutator_reduction.py
Location: tfpt_research_contracts
Original CSV, line 271Original ledger status
QGEO.STATE.01
ATTACKING omega o rho = omega DIRECTLY ON THE RAW SEAM (work package A, external-review priority 1, 2026-06-14): a further localising reduction, not a closure. (1) SETUP NON-CIRCULAR [E]: rho is the carrier clock = the Coxeter element of W(A3)=S4 (order 4, rho^4=1, v117), an automorphism of the RAW CAR algebra (16 Majoranas = the carrier net, v156/v175); both rho and the algebra are defined from the CARRIER side with NO seam geometry, so 'rho preserves the seam state' is a genuine, non-circular question. (2) GROUND-STATE REDUCTION [E]: for a quasi-free RP state the covariance is the positive-frequency projection C_Sigma=1/2(1+sgn H_1) of the one-particle Hamiltonian H_1 (~ the DtN), so omega o rho=omega <=> rho preserves C_Sigma <=> [rho,H_1]=0. (3) CHARACTER-BLOCK-DIAGONAL CRITERION [E]: rho has order 4, so the one-particle space splits into the four mu4-character classes {n=r mod 4}; [rho,H_1]=0 <=> H_1 is block-diagonal in these classes -- verified numerically (a character-block-diagonal H commutes, an off-character entry breaks it). (4) [E] the principal symbol |k|=diag(|n|) is diagonal, hence trivially character-block-diagonal, so [rho,|k|]=0 exactly (v198). (5) SHARPENED RESIDUAL [O]: what remains is purely 'the BOUNDED sub-principal symbol of the raw seam DtN has NO off-character (mod 4) matrix elements' -- a bounded, finite-character-class operator condition, not a diffuse geometry; holds on the finite H^1 block (mu4-eigenforms, v177), open on the continuum. VERDICT: omega o rho=omega is confirmed non-circular and reduced to a bounded character-block-diagonal residual; NOT a closure (the foundational symmetry persists), but the sharpest typed form. Python-only
Recorded status: Identity [E] (non-circular setup + reduction) + premise [O] (bounded sub-principal residual)
Dependencies, status detail and source
Canonical status: Identity/premise [E]/[O]: omega o rho=omega is non-circular (rho a carrier automorphism, raw CAR algebra) and reduces to [rho,H_1]=0 <=> H_1 is mu4-character-block-diagonal; residual = the bounded sub-principal symbol has no off-character matrix elements; not a closure
Dependencies: QGEO.MODULAR.01; QGEO.SYM.01; QGEO.DTN.01; v198_modular_commutator_reduction.py; v177_seam_marking_kernel.py
Scripts: v199_seam_state_invariance.py
Location: tfpt_research_contracts
Original CSV, line 272Original ledger status
QGEO.SUBPRIN.01
THE v199 BEDROCK RESIDUAL REDUCED ONE GENUINE STEP FURTHER (2026-06-15): block-diagonality of the sub-principal symbol is NOT an independent postulate. (1) [E] the seam DtN is Lambda=|k|+M_f, M_f = multiplication by a curvature function f(theta) on the seam circle (standard Steklov sub-principal structure), so <n|M_f|n'>=f_{n-n'}. (2) [E] M_f is mu4-character-block-diagonal <=> f has Fourier support only on modes ≡0 (mod 4) <=> f is Z4-invariant (the principal |k| already commutes, v198). (3) [E] a mu4-MARK SUM f(theta)=sum_{j=0}^3 g(theta-2pi j/4) has f_m=4 g_m [m≡0 mod 4] for ANY profile g, automatically Z4-invariant, so (v195 marks=mu4 orbit) FORCES the sub-principal symbol block-diagonal. (4) [E] negative controls: 3 equally-spaced marks (Z3) and 4 generic unequal marks both break the commutator -- specific to mu4. (5) [O] NEW RESIDUAL: omega o rho=omega reduces to 'the DtN sub-principal symbol is MARK-LOCAL (seam flat away from the mu4 marks = the conformal-deck structure, v177)'; given it, (v195 marks=mu4 orbit) => Z4-invariant sub-principal => block-diagonal => [rho,H_1]=0 => omega o rho=omega. A genuine reduction; the residual is now structurally definitional (conformal-deck flatness), the narrowest form yet of QGEO.SYM.01. Python-only (exact mark-sum Fourier identity + block-diagonal criterion; Wolfram-mirrored)
Recorded status: Identity [E] (block-diagonality = image of the mu4 marks) + premise [O] (mark-local DtN residual)
Dependencies, status detail and source
Canonical status: Identity/premise [E]/[O]: the v199 sub-principal block-diagonality is FORCED by the v195 mu4-mark orbit under a mark-local DtN (a mark-sum over mu4 points is Z4-invariant => Fourier support on modes ≡0 mod 4 => block-diagonal); residual reduces to DtN mark-locality (conformal-deck flatness), the narrowest form of QGEO.SYM.01; not a closure
Dependencies: QGEO.STATE.01; QGEO.MARKS.02; QGEO.SYM.01; v199_seam_state_invariance.py; v195_marks_lefschetz_character.py; v198_modular_commutator_reduction.py
Scripts: v201_seam_subprincipal_marks.py
Location: tfpt_research_contracts
Original CSV, line 273Original ledger status
QGEO.PILLOW.01
PILLOWCASE REDUCTION OF QGEO.SYM.01 (2026-06-15): the two separately-tracked open QGEO residuals -- QGEO.ISO.01 (v180; the carrier clock is an order-4 ISOMETRY of the seam metric) and QGEO.SUBPRIN.01 (v201; the DtN sub-principal symbol is MARK-LOCAL, the seam flat away from the mu4 marks) -- are ONE statement: 'the seam carries the uniformising flat orbifold (pillowcase) metric'; and the order-4 clock is DERIVED from cross-ratio 2 (v168), not assumed. (1) [E] EUCLIDEAN ORBIFOLD: chi_orb(S^2 with four order-2 cone points)=2-4(1-1/2)=0 => flat (Troyanov cone-metric uniformisation, unique up to scale); Gauss-Bonnet four cone deficits pi sum to 4pi=2pi chi(S^2) => int K_smooth=0 (flat away from the marks = the v201 conformal-deck residual). (2) [E] NEW LINK cross-ratio 2 => j=1728: j(lambda)=256(l^2-l+1)^3/(l^2(l-1)^2); j(2)=1728 and all six harmonic cross-ratios {2,-1,1/2} give 1728, so the square modulus (hence the order-4 clock) is a CONSEQUENCE of the cross-ratio 2 that v168 already proved. (3) [E]/[L] j=1728 <=> tau=i (square torus) <=> CM by Z[i] <=> Aut=Z/4; the only other exceptional modulus j=0 (tau=rho) gives Z/6, generic moduli only Z/2. (4) [E] explicit order-4 CM: on y^2=x^3-x the map (x,y)->(-x,iy) negates the defining polynomial (same zero locus) and has order 4 = the z->iz isometry. (5) [E] NEGATIVE CONTROL: a generic cross-ratio lambda=3 gives j=21952/9 not in {0,1728} => Aut=Z/2, no order-4 clock -- specific to mu4. (6) [I] UNIFICATION: 'seam = uniformising flat pillowcase metric' implies BOTH v201 mark-locality (flat away from marks) AND v180 QGEO.ISO.01 (z->iz isometric, from the square modulus), so the two open residuals collapse to ONE canonical-metric statement. (7) [O] does NOT close QGEO.SYM.01: the premise 'RP seam metric = the uniformising constant-curvature representative of its conformal class' stays open -- milder and more canonical than the bare isometry premise, and it SUBSUMES v201; net existence + full-cone RP remain v175's [E]. Wolfram-mirrored (chi_orb=0, cross-ratio 2 => j=1728, harmonic orbit, lemniscatic CM order 4, neg control); Klein-J modular values mpmath-numerical
Recorded status: Identity/Lattice [E]/[L] (cross-ratio 2 => j=1728 => order-4 CM clock) + premise [O] (canonical constant-curvature metric)
Dependencies, status detail and source
Canonical status: Identity/Lattice [E]/[L] + premise [O]: cross-ratio 2 (v168) => j=1728 => square modulus => order-4 CM clock (DERIVED, not assumed); unifies the two open QGEO residuals (v180 ISO + v201 SUBPRIN) into ONE canonical flat-pillowcase-metric statement; does NOT close QGEO.SYM.01 (the constant-curvature representative choice stays open, milder/more canonical than the bare isometry premise)
Dependencies: QGEO.ISO.01; QGEO.SUBPRIN.01; QGEO.RIGID.01; v168_qgeo_rigidity.py; v180_clock_is_mobius.py; v201_seam_subprincipal_marks.py
Scripts: v214_seam_pillowcase.py
Location: tfpt_research_contracts
Original CSV, line 274Original ledger status
QGEO.KILL.01
BEDROCK AS A FALSIFIABLE KILL-TEST (2026-06-15): QGEO.SYM.01 (omega o rho = omega) is recast as four independently-falsifiable predictions about the raw seam DtN, NOT a proof (a foundational symmetry cannot be derived from nothing, v181/v153; Bisognano-Wichmann would presuppose the very covariance it should produce). Non-circular reduction (v198/v199/v201): omega o rho=omega <=> [rho,C]=0 for the quasi-free covariance C=1/2(1+sgn H_1); the principal symbol |k|=diag(|n|) commutes with rho=diag(i^n) exactly, so the whole residual is the bounded sub-principal M_f, and [rho,M_f]=0 <=> f has Fourier support only on modes ≡0 (mod 4); measure form = the raw Gaussian bulk quadratic form A is mu4-deck-invariant. FOUR LEVERS (carrier-side [E] green, seam-side [O] deferred): K1 EXACTLY 4 MARKS (the order-4 clock = Coxeter of W(A3)=S4 has free orbit mu4, and rank H^1 = #marks-1 = 3 = N_fam => #marks = 4 = |mu4| = e1(a); KILL if #marks!=4, b1!=3); K2 SQUARE CONFIG (cross-ratio(mu4)=2 => j=1728, Aut=Z/4; only j in {0,1728} give Aut>Z2, and clock order h(A3)=4 selects Z/4 NOT the j=0 Z/6; a generic mark set (cross-ratio 3 -> j=21952/9) has only Z/2; KILL if non-square); K3 MOD-4 SUB-PRINCIPAL SUPPORT (a mu4-mark sum has f_m=4 g_m[m≡0 mod4] => M_f block-diagonal => [rho,M_f]=0; off-character breaks it; KILL if a first-principles seam DtN has off-character weight above noise); K4 TRANSFER GAP (cusp weights {0,1/3,2/3} => transfer eigenvalues (1-w)^6={1,(2/3)^6,(1/3)^6}, gap (2/3)^6=64/729; KILL if gap!=64/729). FREEZE-BOUND: freeze_file.csv row 'seam_deck_symmetry' names the four levers (prediction+test locked, v84/v100 discipline). The DECISIVE kill is DEFERRED (QGEO.REALIZE.01): the raw P1 seam-collar DtN must yield exactly 4 square marks WITHOUT inputting them; no first-principles raw-seam construction exists yet (deferred, like axion_DM 'no test possible until the scale is closed'). Carrier-side K1-K4 are green regression guards; a KILL-TEST, NOT a closure of QGEO.SYM.01. Python-only (falsification layer like v65/v100; exact sub-parts j=1728 + gap (2/3)^6 already Wolfram-mirrored via v214/v54/v56)
Recorded status: Open (falsifiable kill-test; carrier-side K1-K4 [E], seam-side emergence deferred [O])
Dependencies, status detail and source
Canonical status: Open / falsifiable: bedrock omega o rho=omega recast as four levers (exactly 4 square marks, mod-4 sub-principal support, transfer gap (2/3)^6), freeze-bound via seam_deck_symmetry; carrier-side [E] regression guards, decisive emergence test deferred to QGEO.REALIZE.01; not a closure and not a Bisognano-Wichmann proof
Dependencies: QGEO.SYM.01; QGEO.PILLOW.01; QGEO.SUBPRIN.01; v168_qgeo_rigidity.py; v201_seam_subprincipal_marks.py; v214_seam_pillowcase.py
Scripts: v215_seam_deck_killtest.py
Location: tfpt_research_contracts
Original CSV, line 275Original ledger status
QGEO.MARKS.03
THE FOUR MARKS EMERGE FROM GAUSS-BONNET (2026-06-15; executes the K1 lever of QGEO.KILL.01): an independent Euclidean-orbifold derivation of the seam-mark structure. (1) [E] MARK COUNT = 2 chi: Z2 branch points (cone angle pi, deficit pi) on a flat sphere (chi=2) give n*pi = 2pi*chi(S^2) = 4pi => n = 2 chi = 4 = |mu4| = e1(a) = N_fam+1; the SAME chi=2 that supplies the 8 in c3 = 1/(|Z2| 2pi chi) = 1/(8pi). (2) [E] the Euclidean sphere 2-orbifolds (chi_orb = 2-sum(1-1/m_i) = 0) are exactly (2,3,6),(2,4,4),(3,3,3),(2,2,2,2) (n<=4 since sum 1/m_i = n-2 <= n/2). (3) [E] ALL-ORDER-2 (= the |Z2| sheet branch) selects (2,2,2,2) uniquely. (4) [E] N_fam=3 selects it: rank H^1 = #marks-1 = 3 only for (2,2,2,2); the three 3-mark orbifolds give 2 families => the three generations pick the 4-mark square over the 3-mark hexagonal (3,3,3); family count and mark count are ONE fact (N_fam = #marks-1). (5) [E] the free order-4 deck selects it: only (2,2,2,2) admits a free order-4 rotation of the marks ((2,4,4) order-4 FIXES its order-4 cone points, (3,3,3)=Z3, (2,3,6)=Z6) -- three independent criteria converge. (6) [O] residual: (2,2,2,2) carries a 1-parameter modulus (the mark cross-ratio); the SQUARE (cross-ratio 2 => j=1728, v214) is selected by the order-4 carrier clock (the one irreducible, over-determined as |mu4|=h(A3)=N_fam+1). So K1 (mark count + equality) is now DERIVED; only the square modulus stays input -- does NOT close QGEO.REALIZE.01, but removes the mark COUNT from the inputs. Exact (sympy); Wolfram-mirrored
Recorded status: Identity/Lattice [E] (Gauss-Bonnet count + Euclidean-orbifold uniqueness) + premise [O] (square modulus)
Dependencies, status detail and source
Canonical status: Identity/Lattice [E] + premise [O]: the mark COUNT (4 = 2 chi) and EQUALITY (all order-2) are DERIVED from Gauss-Bonnet + Z2 branch + flatness (three independent selectors: all-order-2, N_fam=3, free order-4); only the square modulus (cross-ratio 2, the order-4 clock) stays the carrier input; executes K1 of QGEO.KILL.01, does NOT close QGEO.REALIZE.01
Dependencies: QGEO.MARKS.02; QGEO.PILLOW.01; QGEO.KILL.01; AX.P1.01; v195_marks_lefschetz_character.py; v214_seam_pillowcase.py
Scripts: v216_marks_gauss_bonnet.py
Location: tfpt_research_contracts
Original CSV, line 276Original ledger status
QGEO.EMERGE.01
NUMERICAL EMERGENCE SCAN (2026-06-15; the numerical sibling of QGEO.MARKS.03, as v210 is to v201): with the NUMBER of marks n AND their POSITIONS free, the 4-equally-spaced (square) configuration is the joint minimiser of {clock-invariance + 3-family cohomology + transfer gap (2/3)^6} on the actual seam DtN/state. (1) [N] n=4 equally-spaced marks give a clock-invariant quasi-free state ||rho_4 C rho_4^dag - C|| ~ 1e-14. (2) [N] family count selects n=4: scanning n in {2..8}, only n-1 = 3 = N_fam (n=3 -> 2 families = the hexagonal alternative, excluded; n=5 -> 4) -- the number is not input. (3) [N] the gap: a mu4-equivariant operator is forced diagonal in the cusp basis (deck-average off-diagonal ~1e-16 for V=diag(i,-1,-i)), cusp weights {0,1/3,2/3} => transfer (1-w)^6 = {1,(2/3)^6,(1/3)^6}, gap (2/3)^6 = 64/729. (4) [N] position scan: among 4-mark configs the square minimises the defect (square ~1e-14 vs off-square ~0.4) -- the operator-level v200 theta=pi/2. (5) [N] negative controls: n=3 (Z3/hexagonal) -> 2 families + breaks the order-4 clock; n=5,6 wrong count. (6) [O] numerically confirms the emergence on the real DtN (number scanned, positions free); selector = the 3-family/gap-(2/3)^6 structure (carrier input); executes K1 numerically but does NOT close QGEO.REALIZE.01 (the order-4/N_fam input stays irreducible, v153/v181); complements v216. Python-only
Recorded status: Numerical [N] (emergence confirmation on the real DtN/state) + premise [O] (selector is carrier input)
Dependencies, status detail and source
Canonical status: Numerical [N] + premise [O]: free-n + free-positions scan on the actual DtN/state confirms 4-square is the unique joint minimiser of {clock-invariance, 3 families, gap (2/3)^6}; executes K1 of QGEO.KILL.01 numerically, does NOT close QGEO.REALIZE.01; Python-only
Dependencies: QGEO.MARKS.03; QGEO.KILL.01; v200_seam_variational_scan.py; v210_mark_local_dtn.py; v216_marks_gauss_bonnet.py
Scripts: v217_marks_emergence_scan.py
Location: tfpt_research_contracts
Original CSV, line 277Original ledger status
FR.RAREKAON.01
RARE KAON SECTOR K -> pi nu nubar FROM THE TFPT CKM POINT (2026-06-15, archive integration): the kernel fixes the full CKM matrix (s12=lambda_C, s23=phi0/(1+lambda_C)=|V_cb|, s13=lambda_C^3/3, delta=pi/3+3 lambda_C^2=1.19823 rad; v18/v88) with NO flavor fit; feeding the exact point into the standard Brod-Gorbahn-Stamou short-distance formulas gives BR(K+ -> pi+ nu nubar)=9.45e-11 and BR(K_L -> pi0 nu nubar)=3.33e-11. (1) [I] the CKM point + exact apex (rhobar,etabar)=(0.1374,0.3509) and lambda_t=V_ts* V_td. (2) [C] the branching ratios -- NOT parameter-free: X_t=1.462, P_c, dP_cu, kappa_pm are external electroweak/QCD input (recomputed from the CURRENT CKM point; an earlier scaffold quoted BR(K_L)=3.47e-11 from an Im(lambda_t)/lambda^5 slip, corrected here to 3.33e-11). (3) [N] BR(K+)=9.45e-11 lands ON the NA62 full 2016-2024 combination (9.6 +1.9/-1.8)e-11 (La Thuile 2026), +0.08 sigma (the earlier 2016-2022 obs was (13.0 +3.3/-3.0)e-11, ~1.2 sigma above); a strong consistency hit for the CKM point, but an F_transfer bridge -- NOT a unique TFPT-vs-SM discriminator (the SM (8.6 +- 0.4)e-11 also sits inside the NA62 error). (4) [I] Grossman-Nir ratio BR(K_L)/BR(K+)=0.352 << 4.3 (model-independent isospin corridor). (5) [X] dated kill test: a stable NA62 BR(K+) outside [7,12]e-11, or a KOTO-II BR(K_L) off the GN-plane point, breaks the TFPT flavor bridge for this sector (compiler core untouched); the ~2 sigma delta_CKM tension (v88) makes the corridor discriminating. A downstream FLAVOR readout, guarded [C] by v187 -- NOT a compiler power. Python-only
Recorded status: Physical bridge [C] (CKM point [I], branching ratios external-SD [C], dated kill test [X])
Dependencies, status detail and source
Canonical status: Physical bridge [C]: the TFPT CKM point (no flavor fit) + standard short-distance input give BR(K+)=9.45e-11 and BR(K_L)=3.33e-11; consistent with NA62 at ~1.2 sigma; dated kill test = stable NA62 outside [7,12]e-11 or KOTO-II off the GN point; not a compiler power
Dependencies: v18_quark_yukawa.py; v88_cp_phase_audit.py
Scripts: v202_rare_kaon.py
Location: tfpt_2_standard_model
Original CSV, line 278Original ledger status
HOR.EHT.01
EHT ACHROMATIC POLARIZATION-ROTATION INTERCEPT (2026-06-15, archive integration): beta_BH(r) = Q_e^eff Q_m^eff/(256 pi^4 r^2) = 16 c3^4 (Q_e Q_m / r^2) around a horizon-scale black hole. (1) [E] the coupling 16 c3^4 = 1/(256 pi^4) EXACTLY (c3=1/8pi) -- no free coupling. (2) [E] alpha-kernel cross-link: 16 c3^4 = delta_top/3 with delta_top = 48 c3^4 = 3/(256 pi^4), the SAME top-form coefficient that fixes the alpha-kernel precision-zone correction (one row from the cosmic-birefringence seed beta_rad=phi0/4pi) -- EHT polarimetry probes the same admissibility data as the alpha row, projected onto the local horizon collar. (3) [C] TFPT fixes the structured 1/r^2 profile, the achromaticity (no lambda^2 dependence), and the sign-flip under effective E.B reversal; the amplitude Q_e^eff Q_m^eff (local effective E.B structure) is an MHD/GR weight, NOT a compiler output -- so the channel is [C] (shape/sign), never a parameter-free pixel amplitude. (4) [X] dated kill test: the residual intercept chi_0^res = chi_0^obs - chi_0^GRMHD must pass THREE nulls simultaneously (frequency/achromatic, spatial 1/r^2, sign-flip under E.B reversal); a residual statistically consistent with zero after honest GRMHD subtraction, or failing any null, falsifies THIS channel -- the compiler core (the determinant-line coupling shared with alpha/beta_rad) is untouched. Scope: the old UFE closed RN-type metric (D/r^4, regular cores, shadow shift) is NOT imported -- superseded by the Nariai/seam=horizon readouts (v101-v104, v190); only the polarization signature survives. Wolfram-mirrored (the coefficient identity 16 c3^4 = 1/(256 pi^4) = delta_top/3 is exact)
Recorded status: Identity [E] (coupling 16 c3^4=1/(256 pi^4)=delta_top/3) + physical bridge [C] (1/r^2 shape, achromaticity, sign-flip) + dated kill test [X] (three nulls)
Dependencies, status detail and source
Canonical status: Identity [E]: the EHT polarization coupling 16 c3^4 = 1/(256 pi^4) = delta_top/3 is the SAME compiler number as the alpha-kernel correction; the 1/r^2 shape + achromaticity + sign-flip are TFPT-fixed [C], the amplitude an MHD/GR weight; dated [X] kill test = three nulls after honest GRMHD subtraction; NOT the UFE metric (Nariai/seam=horizon supersedes it)
Dependencies: AX.P1.01; HOR.01; EM.FP.01
Scripts: v203_eht_achromatic.py
Location: tfpt_horizon_readouts
Original CSV, line 279Original ledger status
QGEO.VARI.02
THE DISCRETISED VARIATIONAL TEST (work package B, external-review, 2026-06-14): v196 showed E_fail=0 at the mu4 config on the fixed H^1 block; here the clock angle is a FREE parameter and E_fail is MINIMISED over it. (1) [N] taking rho(theta)=diag(e^{i m theta}) on the H^1 generators m=(1,2,3) and E_fail(theta)=||rho(theta)^4-1||^2 + faithfulness penalty (penalising coincident characters), a scan over theta in (0,pi) has its UNIQUE minimum at theta=pi/2 (E_fail=0): there rho^4=1 AND the three characters i,-1,-i are distinct = the (1,2,3)=A3-exponent grading. (2) [N] decoys lose: theta->0 is trivial (one character), theta->pi gives only two distinct characters (m=1,3 coincide at -1), generic pi/3 and 2pi/3 give E_fail~6 -- so argmin E_fail = the mu4 normal form, no other clock reproduces the faithful order-4 mu4 grading. (3) [E] consistent with v196/v199: theta=pi/2 is the clock z|->iz whose principal-symbol commutator vanishes (v198) and whose H^1 covariance is mu4-equivariant (v177); E_fail=0 there reproduces the v196 finite-block vanishing. (4) [O] this is a numerical sign on the finite H^1 grading, not the full operator-valued minimisation over the raw-seam DtN on the continuum (the v199 bounded-sub-principal residual); strengthens the case that the variational principle selects mu4 but does not close it. Python-only (numerical variational scan)
Recorded status: Numerical sign [N] (argmin = mu4 clock) + open full minimisation [O]
Dependencies, status detail and source
Canonical status: Numerical [N]: argmin E_fail over the clock angle = theta=pi/2 (the mu4 clock z|->iz), decoys lose; a numerical sign that the variational principle selects mu4; full operator-valued minimisation stays the v199 residual [O]
Dependencies: QGEO.VARI.01; QGEO.STATE.01; v196_seam_energy_functional.py; v199_seam_state_invariance.py
Scripts: v200_seam_variational_scan.py
Location: tfpt_research_contracts
Original CSV, line 280Original ledger status
REVIEW.MAP.01
THE FOUR EXTERNAL-REVIEW CONCERNS REDUCE TO THE TWO ALREADY-NAMED RESIDUALS + ONE DATA-ONLY ITEM (a careful 2026-06-14 review raised A mapping 2D->4D, B UV-gravity abstractness, C grammar-tuning/meta-look-elsewhere, D Koide Mobius flow; this maps them honestly): three reduce by the same discipline as v175-v181 onto {QGEO.SYM.01 (the seam realisation), F_transfer (the continuous transfer functor / missing dynamical principle)}, and the fourth is experiment-only. (A) MAPPING [I]/[C]: the readout integer is exact -- the Plucker norm 11 = sum_{k<=2}C(4,k) = 1+4+6 = dim S+ - g_car = 16-5 (the QBL-protected boundary state count, v174); the 2D->4D step is the holographic reduction (bulk measure -> seam-boundary measure, gap-decoupled G5, Delta_eff>0); so A = QGEO.SYM.01 + F_transfer, NOT a new gate. (B) UV GRAVITY [I]: the TFPT answer 'gravity is the geometric readout of the boundary' IS QGEO.SYM.01 (the seam conformal deck); B is the bedrock postulate itself, an identity not a new premise. (C) GRAMMAR TUNING [N]/[C]: the strongest honest defence -- the grammar is FROZEN pre-data (v84/REG.FREEZE.01), minimal (anchor a + pi), and OVER-DETERMINED (the same anchor a=(1,1,2) lands in >=6 mutually-independent not-selected-for structures: Nariai roots (1,1,-2) HOR.NARIAI.01; Coxeter-30=|Z2|N_fam g_car SEAM.CYCLE.01; spine tetrahedron {2,3,4,5} ARCH.SPINE.01; EM budget 41=a^T K a; seed 4/3=|mu4|/N_fam; c3=1/(2 e1(a) pi)); a look-elsewhere-resistant signature, but -- the review is correct -- the residual meta-risk falls only to out-of-sample data (JUNO/CMB-S4), never to argument. (D) KOIDE MOBIUS FLOW [I]/[C]: the Mobius nature is not arbitrary -- Aut(P^1)=PSL(2,C) is the Mobius group and the Koide/cusp flow lives on the deck P^1 minus mu4, so its generator is Mobius BY THE GEOMETRY; the non-integer flow time t~2.84 (FR.KOIDE.06) only locates the physical point; the missing QFT RG generator (the beta-function realising the orbit) IS F_transfer; so D = QGEO.SYM.01 + F_transfer, not a new gate. NET: A/B/D collapse onto {QGEO.SYM.01, F_transfer}; C is the single experiment-only meta-question. A unification of the concern structure, NOT a closure of the underlying premises (which stay [O]/[C], honestly)
Recorded status: Reduction map [I]/[C] (A/B/D reduce to the two named residuals; C is experiment-only)
Dependencies, status detail and source
Canonical status: Reduction [I]/[C]: external-review concerns A/B/D reduce to {QGEO.SYM.01; F_transfer}; C is data-only (JUNO/CMB-S4)
Dependencies: QGEO.SYM.01; FR.KOIDE.06; FR.ETAB.02; AUDIT.NULLMC.01; REG.FREEZE.01; QFT.FOCK.01; HOR.NARIAI.01; SEAM.CYCLE.01
Scripts: v182_reviewer_residual_map.py
Location: tfpt_5_redteam
Original CSV, line 281Original ledger status
FR.ETAB.02
ETA_B LEPTOGENESIS TRANSFER OPERATIONALISED (the Tier-3 F_transfer Boltzmann path made falsifiable; NOT a new structural class, NOT a compiler theorem): type-I thermal leptogenesis eta_B ~ 0.96e-2 eps_1 kappa_f (sphaleron a_sph=28/79) fed by TFPT's neutrino sector -- normal ordering (m2,m3 from dm2_sol,dm2_atm) and the predicted phase delta_CP=4pi/3=240deg (|sin|=0.866). (1) Davidson-Ibarra eps_1=(3/16pi)M1 m3/v^2 ~ 8.5e-7 for M1=1e10 GeV [scenario M1]; (2) efficiency kappa_f(m~1) (Buchmueller-Di Bari-Pluemacher) ~ 0.03-0.19 for m~1 in [m_*, m_atm] [scenario washout]; (3) KILL TEST: over natural M1 in [3e9,3e10] GeV and m~1 in [m_*,m_atm], eta_B spans [8.4e-11, 4.7e-9] which BRACKETS the observed 6.1e-10 -- the canonical point (M1=1e10, m~1=5e-3, delta=240deg) gives 6.0e-10 (NOT tuned). So the leptogenesis route fed by TFPT's neutrino sector REACHES the observed asymmetry; but M1 and the washout are SCENARIO inputs (not TFPT outputs), so eta_B stays [C] -- a falsifiable transfer interface, NOT promoted. If a precise Boltzmann/RGE computation excluded the window, the leptogenesis ROUTE falls, NOT the theory (the downstream Omega_b readout FR.ETAB.01 is independent and already hit). Seesaw RGE running (PyR@TE) = an O(1) correction, flagged. Python-only
Recorded status: Physical [C] (viable falsifiable scenario; M1/washout scenario inputs)
Dependencies, status detail and source
Canonical status: Physical [C]: thermal-leptogenesis route viable (band brackets observed 6.1e-10); M1/washout scenario inputs, eta_B stays [C]
Dependencies: FR.ETAB.01; FLAV.RGSTAB.01; v9_neutrino_texture.py
Scripts: v169_etaB_boltzmann_interface.py
Location: tfpt_4_frontier
Original CSV, line 282Original ledger status
E8.SLICE.01
E8 SLICE COMPRESSION LEMMA (the seven E8 audit slices are ONE projection of two small alphabets; anti-numerology compression, NOT new physics): (1) the anchor power sums p_n(a)=2+2^n give P=(3,4,6,10) whose K4 edge products are all carrier blocks (12=|R(A3)|, 18=N_fam|R+(A3)|, 30=h(E8), 24=|W(A3)|, 40=|R(D5)|, 60=D_start; triples 120,240,720) [I]; (2) the six Sheet-Diamond operators Q,K,R,C,L,F have determinants (3,4,8,14,20,32) summing to 81=N_fam^4 -- the total determinant charge of the flavour space [I]; (3) all SEVEN E8 248-slices (D8, D5xA3, E6xA2, E7xA1, F4xG2, A4xA4, A8) are written from P, D, Delta=p0+p3=13 and (e1,e2,dim S+) [I]; (4) one affine map row(C+sU+tV)=(7,11,13)+s(3,3,3)+t(1,2,3) generates every flavour row budget (democratic winding + A3 exponents) [I]; (5) dim E6 = 78 = p2*Delta = |R+(A3)|(N_fam+A_Lambda) [I]. The E8 atlas is thus a closed grammar over two typed alphabets, not seven trophy lists; the physical reading is AUDIT
Recorded status: Identity/Lattice [I]/[L] (exact integer/lattice identities)
Dependencies, status detail and source
Canonical status: Identity/Lattice [I]/[L]: E8 atlas = projection of power-sum + determinant alphabets; audit bridge, anti-numerology
Dependencies: ARCH.SPINE.01; FLAV.KQ.01; CAR.SM.01
Scripts: v170_diamond_slice_compression.py
Location: tfpt_3
Original CSV, line 283Original ledger status
QFT.OSMOMENT.01
ATOMIC OS MOMENT LEMMA + SUGAWARA GAP SAFETY (the admissible transfer sector's OS positivity as a positive moment problem, and its gap as an E8 Sugawara inequality): (1) spec(T)={1,(2/3)^6,(1/3)^6}={1,64/729,1/729} makes every scalar two-point correlator C(n)=A+B r^n+C s^n the moment sequence of the positive atomic measure A d1+B dr+C ds, so every OS Hankel matrix factorises as a Vandermonde Gram H=W diag(A,B,C) W^T => positive semidefinite (positive spectrum => moment measure => reflection positivity; a clean proof, not a numerical Gram test) [I]; (2) H=-log T gives a positive discrete Kallen-Lehmann spectrum {0, 6log(3/2), 6log3} with masses m1=2.43279, m2=6.59167 [I]; (3) the exact cluster constant 1/(1-r)=729/665 and correlation length xi=1/(6log(3/2))=0.41105 [I]; (4) SUGAWARA GAP SAFETY: 31=1+h^v(E8)=1+30 and c((E8)1)=248/31=8, so Delta_eff=6log(3/2)-31/(4pi^2)=1.64755>0 reads as |R+(A3)|log(N_fam/|Z2|) > (1+h^v(E8))/(4pi^2) -- the A3 transfer gap beats the E8 Sugawara metric budget
Recorded status: Identity [I] (moment/spectral/Sugawara exact) + [C] (declared V_metric norm)
Dependencies, status detail and source
Canonical status: Identity [I]: OS positivity = positive moment problem; Sugawara gap safety 31=1+h^v(E8); cluster 729/665, xi=0.411
Dependencies: GATE.METRIC.XI.01; SEAM.CYCLE.01; GATE.METRIC.07
Scripts: v171_os_moment_cluster.py
Location: tfpt_research_contracts
Original CSV, line 284Original ledger status
SEED.ANOMALY.01
TRACE-ANOMALY ORIGIN OF THE SEED PREFACTOR 4/3 + the shared integer 7 (b3 = scalaron exponent): (1) the retained seed phi0=(4/3)c3+48 c3^4 has base (4/3)c3=1/(6pi); the prefactor 4/3 is the conformal Weyl anomaly of the seam CFT -- over S^2 (chi=2) the (E8)1 (c=8) trace anomaly is c*chi/6=8/3, halved by the one-sided |Z2| Moebius identification to 4/3 = dim S+/dim g_SM = 16/12; the Cabibbo seed prefactor IS the halved (E8)1 Weyl anomaly [I]; (2) the QCD one-loop coefficient |b3|=11-(2/3)N_f=7 (N_f=2 N_fam=6) equals the scalaron exponent 7=Omega_adm-10 b1=48-41 (M_scal=c3^{7/2} Mbar) -- confinement (Lambda_QCD) and inflation (scalaron) are driven by ONE integer (the carrier occupancy deficit) [I]; (3) AUDIT: the external Volkov-Wipf Nariai graviton/ghost coefficient -98/45 (v138) = -2*7^2/45 = -2 (scalaron exp)^2/dim(D5) (dim so(10)=45). Gives the seed prefactor and the QCD/scalaron integer a first-principles CFT/QFT origin, not a fit
Recorded status: Identity [I] (trace anomaly + b3 + VW factorisation exact)
Dependencies, status detail and source
Canonical status: Identity [I]: seed 4/3 = halved (E8)1 Weyl anomaly on S^2; b3=7=scalaron exponent (confinement<->inflation)
Dependencies: CAR.SM.01; FR.MPME.01; ANCHOR.GEN.01; v138_vw_firewall.py
Scripts: v172_trace_anomaly_seed.py
Location: tfpt_2
Original CSV, line 285Original ledger status
FLAV.STRONGCP.02
STRONG CP AS PFAFFIAN REALITY (theta_eff=0 as a model-level CAR/Pfaffian theorem; formalises the tfpt_2 structural chain): in the self-dual CAR (16-Majorana) seam model, (1) the sheet-odd Calderon involution {D_Sigma, Sigma}=0 forces the spectrum into +-lambda pairs [I] (verified: eigenvalues symmetric about 0 to 1e-15); (2) gamma5-Hermiticity D^dag=gamma5 D gamma5 makes conj(det D)=det D, so the fermion-measure Pfaffian is real, arg in {0, pi} [I] (verified: Im(det D)~1e-9); (3) a theta-term gives Z(theta)=e^{i theta}|det D|; reflection positivity (Z>0, OS axiom) excludes theta=pi (Re Z<0) and every theta not in {0,pi} (Im Z != 0), selecting theta_eff=0 [I]. So theta_eff=0 is a topological null theorem -- the antisymmetric Calderon pairing makes the Pfaffian real and RP picks the positive branch; the CAR/Pfaffian form of the polar+gamma5+sheet+RP derivation
Recorded status: Formalised [F] (machine-checked model demonstration)
Dependencies, status detail and source
Canonical status: Formalised [F]: theta_eff=0 via Pfaffian reality (sheet pairing + gamma5-Hermiticity + RP positive branch)
Dependencies: FLAV.QGEO.02; CAR.QFREE.01; GATE.METRIC.13
Scripts: v173_pfaffian_cp_car.py
Location: tfpt_2
Original CSV, line 286