Original ledger status
DIAMOND.PARA.SELFCODE.01
THE PARABOLIC ANCHOR SELF-CODE (2026-07-31; external structural review audited by discovery probe parabolic_selfcode_audit_probe.py 23/23, verdict SELFCODE-VERIFIED, and promoted; one module, 14 checks, ~1 s, exact integer/symbolic sympy, no floats). The v218 sheet-diamond direction operators U = Q diag(1,0,0) (rank-1 winding) and V = Q diag(0,1,1) (rank-2 sheet, Spec V = {0,1,2}) carry a previously unstated exact operator structure. (1) PARABOLIC THEOREM: A = Q<I,U,V> is the FULL line-stabiliser (maximal parabolic) algebra of block type 2|1 -- seven words independent (coordinate det -81), multiplicatively closed (49/49), all seven matrix units contained, dim 7 = 9-2; mutation of Q's third column breaks it. (2) DOUBLE SELF-CODE: Pi2 = V(V-I)/2 idempotent trace-1; H = I + Pi2 has spectrum EXACTLY {1,1,2} = a (ANCHOR.GEN.01/v23); chi_H = x^3 - 4x^2 + 5x - 2 with (e1,e2,e3) = (4,5,2) = (|mu4|, g_car, |Z2|) AND = (dim_R(GL3/P), dim(A/J), dim J) -- the anchor reproduced spectrally (dominant projector of V) and categorically (flag/Levi/radical of A) from ONE structure. (3) UNIQUENESS THEOREM: x^3 - 2m x^2 + (m^2+1) x - m = (x-m)(x^2-mx+1) identically; positive-integer spectra force m^2 - 4 = k^2, (m-k)(m+k) = 4, m = 2 UNIQUELY -- a = (1,1,2) is the only anchor this parabolic family can code. (4) THE 81 AS LATTICE INDEX: the SATURATED integral order (closure stable at word length 4) has elementary divisors (1,1,1,3,3,3,3): [P_Z : Z<U,V>] = 3^4 = 81, cokernel (Z/3)^4 -- equal to the v218 determinant-staircase sum 3+4+8+14+20+32 = N_fam^4 (cross-link typed; the mark-local mechanism, one Z/3 per mu4 mark, stays [O]: if no canonical local decomposition exists the 81 stays a fingerprint). (5) A2 BIT CARRIER: the basic algebra e A e is the A2 path algebra (dim Ext^1 = 1): dim-vector-(1,1) modules fall into EXACTLY two iso classes (split / one nonsplit) -- the alignment bit gains a canonical algebra carrier; uniqueness [E], SELECTION stays with RP/geometry (v528/v534) [C]. (6) TRACE CODE (re-encoding): p_n(a) = 2 + 2^n = 1 + tr(V^n); recurrence, Hankel 2^{n+1}; (1+trV)(1+trV^2)(1+trV^3) = 240 = |R(E8)|; tr(V^4-V^3) = 8 = rank E8 -- AnchorLadder read through V, one object fewer, NO new content. (7) CR TEST DESIGN (CONTRACT.F.01): CR(alpha_{-2},alpha_{-1};alpha_0,alpha_1) = 4/3 exactly on the v533 line; honesty wired (universal arithmetic-progression cross-ratio -- content = preregistered solver-state export); negative control 28/25; y_F forced fit-free. (8) LORENTZ FORM (reformulation): det X = t^2 - x^2 - y^2 under Phi; D(P,Q) = 2<Phi,Phi>_{1,2}; delta = sech^2(atanh r); pencil identities det(B-S) = det B det(I-Z), D(B,S) = det B tr Z, det S = det B det Z -- eigenvalue locking as the named prime-front instrument (no measurement, no RH statement). NAMED LIMITS AS CONTENT: NO P2 REDUCTION -- 'g_car = dim(A/J) derived' is conditional on the OPEN 'No Carrier Reconstruction' audit (rebuild Q/V from mu4, D4, Q_+ = A3 exponent grading, H^1, Q_- = sheet monodromy, WITHOUT g_car, a, rank E8 or carrier-aware budgets; Q's historical provenance runs through such budgets); AX.P2.01 and ANCHOR.GEN.01 markers UNTOUCHED; the 2..9 dimension table is COMPRESSION of one operator system, not nine independent hits (no-free-pattern rule); the bit selection, the mark-local 81 mechanism and the CR-test execution stay open and typed open; the word 'derived' appears nowhere for any physics statement. UPDATE (2026-07-31, v567): the review's Priority-1 audit is EXECUTED at Stage A -- verdict RECONSTRUCTS-TO-FAMILY: the anchor self-code is carrier-free; Q reconstructs up to one binary sheet-coupling choice; see QGEO.QRECON.01. FURTHER UPDATE (2026-07-31, v568): Priority 3 executed -- the A2 Ext arrow is NONZERO on the whole reconstruction class (the split member collapses to dim 5 and is excluded); Priority 2 sharpened -- the mu4-grading candidate fails, the F3[V]-action on the cokernel exists (see DIAMOND.BIT.SELECT.01). [2026-07-31: the S8 pencil identities are now executed on the REAL Paper-II surface by v569 (machine-tight at 2e-14 on 70 windows, verdict ONE-MODE); see PRIME.PENCIL.ONEMODE.01.] [2026-08-01 X: the per-mark Z/3 mechanism appears geometrically -- v597: the cusp-class group of the mu3-cover is (Z/3)^4 of order 81 with four free local classes; see QGEO.COVER.01.] [2026-08-01 XVI: the self-code algebra has a geometric realization -- v600: (U, V) as Gamma-even degree-2 cusp-Gram operators on the mu3-cover, GL3-conjugate to the compiler pair, with the SAME saturated Z-order structure (index 81 = 3^4, divisors (1,1,1,3,3,3,3)); existence [E], canonicity open; see QGEO.EMBED.01.]
Recorded status: Identity [E] (exact integer/symbolic algebra with mutation controls -- parabolic closure, double self-code, uniqueness enumeration + divisor argument, saturated Smith form, A2 quiver, trace code, cross-ratio identities, Lorentz/pencil identities) + [C] (the TFPT readings of the invariants; the bit-carrier and CR-test interpretations) + [O] (the No-Carrier-Reconstruction audit, the mark-local 81 mechanism, the CR-test execution); NO marker moves
Dependencies, status detail and source
Canonical status: [E] A = full 2|1 parabolic, dim 7, word det -81, closed 49/49, mutation-fragile; a = Spec(I + V(V-I)/2) = (1,1,2) with chi_H = x^3 - flag x^2 + Levi x - radical (4,5,2 double-coded); m = 2 unique (enumeration to 2000 + divisor argument); saturated Smith (1,1,1,3,3,3,3), index 81 = 3^4 = staircase sum; A2 Ext^1 = 1 (two module classes); p_n = 1 + tr V^n (240, rank 8 as traces); CR = 4/3 design with 28/25 control; Lorentz/pencil identities exact; 14/14 checks
Dependencies: DIAMOND.AXIS.01; ANCHOR.GEN.01; AX.P2.01; FLAV.QGEO.01; CONTRACT.F.01; SEAM.BIT.RPBLIND.01; FORM.LADDER.01; v23; v218; v224; v528; v533; v534
Scripts: v566_parabolic_anchor_selfcode.py
Location: tfpt_1;tfpt_2;tfpt_research_contracts
Original CSV, line 605Original ledger status
QGEO.INVOL.01
The sheet-involution existence question of GATE.QGEO.01, narrowed to a two-demand selection with UNCONDITIONAL existence (one module, 7 checks, <1 s; discovery probe involution_existence_probe.py 7/7, verdict EXISTENCE-UNCONDITIONAL): RIGIDITY -- the integral V-commuting involutions are exactly the spectral sign group (Z/2)^3: all eight sign choices on V's eigenspaces {0,1,2} are integer matrices, genuine involutions, commute with V and stabilize the integral order Z<U,V>, on BOTH twins (nontrivial: generic integral operators do not have integral spectral projectors). THE NARROWING CHAIN: (i) the v574 rank theorem (two-dim-positive involutions cannot carry the binary trace code) kills the trace +1 half, 6 -> 3, all remaining of exact Sigma-signature (1,-1,-1); (ii) the seam-reversal demand S e3 = -e3 narrows 3 -> 2; (iii) the zero-mode-fixing demand (+1 on ker V, flipping exactly the dynamical sheet pair {1,2}) singles out S0 = [[1,-1,0],[0,-1,0],[0,0,-1]] uniquely -- explicit, integral, order-stabilizing: EXISTENCE IS UNCONDITIONAL. HONEST SCOPE: the corpus RP reflection Sigma = diag(1,-1,-1) (an anti-automorphism) does NOT commute with V -- a different structure with a different role; the alt twin admits the same classification (the involution does not separate the twins, consistent with v567/v572). RELOCATION: GATE.QGEO.01's involution-existence input is DERIVED; the residual freedom is the finite two-demand selection, machine-checked; deriving the two demands from the continuum geometry (mu4/D4/H1) is the remaining open [C] of the gate -- named, not claimed. [2026-08-01 V: the grading shortcut is falsified -- NO V-commuting involution implements the corpus mu4 grading S Q S = Sigma Q Sigma (all six fail); Sigma and S0 are genuinely different structures whose product is the infinite-order shear [[1,-1,0],[0,1,0],[0,0,1]]: the two v590 demands stay independent; the continuum derivation remains open.] [2026-08-01 X: a live geometric candidate -- v597: the mu3-cover y^3 = x^4-1 hits four compiler fingerprints (rank-3 Eisenstein, projective D4, (Z/3)^4 = 81 cusp classes, Lorentz (1,2)); see QGEO.COVER.01.] [2026-08-01 XIX: the seam-reversal demand is now GEOMETRIC -- v603: on the mu3-cover the seam line is real (R-fixed) and the two-demand involution reverses it exactly; the zero-mode demand is built into the curve-side classification (v601); the continuum derivation of the demands has its first geometric anchor; see QGEO.SEAMMARKS.01.] [2026-08-01 XXV: BOTH demands now have canonical cover carriers -- v603: the seam-reversal demand is 'reverse the R-real line'; v607: the zero-mode demand is 'fix the only unlabeled mode' (seam = R-real, top = deck-eigen, zero = unlabeled); typed [C] readings, the continuum derivation of the demands from the raw seam stays the gate residual; see QGEO.SELRULE.01.]
Recorded status: Exact [E] + typed relocation [C] (exact sympy; demands declared; continuum derivation open)
Dependencies, status detail and source
Canonical status: EXISTENCE-UNCONDITIONAL: rigidity (Z/2)^3 on both twins; narrowing 6 -> 3 -> 2 -> 1 via v574 type theorem + seam reversal + zero-mode fixing; S0 explicit with Sigma-signature; the two demands' continuum derivation stays open; 7/7 checks
Dependencies: QGEO.WINDING.SELF.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v566; v567; v574
Scripts: v590_involution_existence.py
Location: tfpt_1;tfpt_2
Original CSV, line 614Original ledger status
QGEO.COVER.01
Strategy I/II, first executed step -- the mu3-cover of P^1 branched at mu4 (the curve y^3 = x^4 - 1) carries FOUR independent compiler fingerprints (one module, 6 checks, <1 s; discovery probe mu4_cover_model_probe.py 6/6, verdict FINGERPRINTS-HIT): (1) genus 3 with five cusps (mu4 + infinity; monodromies (w,w,w,w,w^2) multiplying to 1) and Chevalley-Weil H^{1,0} = (1,2) per character -- H_1 is a RANK-3 module over the Eisenstein integers, the compiler's 3-dimensional integral frame; (2) the D4 rotation closes PROJECTIVELY VIA THE DECK: in the reduced Burau model at t = omega (braid relations machine-verified) the rotation braid satisfies r^4 = omega x identity and r^12 = 1; (3) THE TORSION: the cusp-class group is (Z/3)^5 modulo one global relation = (Z/3)^4 of order 81 = 3^4 -- the v566 saturation index with exactly the per-mark Z/3 mechanism the self-code program asked for (four free local classes); det_Z(1 - deck) = 27 complementarily; (4) THE SIGNATURE: the invariant hermitian form of the Burau module is unique up to scale with eigenvalues {-2, 2-2sqrt2, 2+2sqrt2} -- LORENTZ SIGNATURE (1,2), matching the Chevalley-Weil Hodge numbers AND the rank-3 polarization signature of Paper II: Strategy II's first prediction (RP positivity as Hodge/Riemann positivity) confirmed at the form level. First operator pass: the single-cusp Gram has all-integer spectrum {0,0,2}; the compiler V-spectrum {0,1,2} is not yet located -- the operator dictionary (which canonical homology operator is Q/V) and the real-structure/RP map to the v572 Gram data are the NAMED NEXT STEPS. Kill criteria live (Smith/spectral mismatch for I; signature mismatch for II -- not triggered). GATE.QGEO does NOT move; no marker changes. [2026-08-01 XI, the dictionary's first entry: U IS LOCATED -- the cusp Gram (1-T2)(1-T2)^dagger has characteristic polynomial x^2(x-3) = char(U) exactly (the winding operator as a cusp Gram; fifth fingerprint); additional exact structure: opposite-twist products (1-T1)(1-T3) are NILPOTENT (char x^3), and G1+G3 has spectrum {0,1,3}; V/H/Q not located in the scanned canonical ranges (products/sums/omega-twists of length <= 2) -- the V-dictionary likely requires the parabolic flag (the seam-cusp filtration), named as the next step.] [2026-08-01 XIII: dictionary round 2 -- v598: THREE entries now exact (U ~ G2; Q_- ~ G1-G3, the sheet coupling as the diamond-diagonal Gram difference; the binary ladder ~ 1+G1+G3); the grading half absent from 7296 canonical combinations; see QGEO.DICT.01.] [2026-08-01 XV: the real structure is constructed and the invariant form is pinned -- v599: R unique with R^2 = +1, det J = 8 (a sheet-diamond determinant list member), Lorentz (1,2) confirmed exactly; see QGEO.REAL.01.] [2026-08-01 XVI: the operator dictionary lands at ALGEBRA level -- v600: the compiler pair (U, V) realized on the cover with the exact v566 saturation structure (index 81, divisors (1,1,1,3,3,3,3)); the [P]-model of GATE.QGEO has an explicit construction; canonicity and the two-demand derivation stay open; see QGEO.EMBED.01.] [2026-08-01 XXI: the torsion picture completes -- v604: 81 = 3^4 localizes at the finite marks (divisor classes; per-frame word order) and 27 = 3^3 at infinity (monodromy-defect cokernel = equivariant-order cokernel = det(1-deck)); see QGEO.EQORDER.01.] [2026-08-01 XXVIII: the curve's wider landscape is mapped -- v610: the Jacobian splits into a Q(zeta_12)-CM surface (the field of deck AND marks) times the Q(omega)-CM elliptic quotient, point-count-confirmed; the integer-spectrum census honestly relativizes single-char hits (the anchor class is the most frequent nontrivial family; the joint structure is the distinction); see QGEO.LANDSCAPE.01.] [2026-08-01 XXXV: the bedrock direction reverses -- v617: the conformal seam axioms FORCE the mu3-cover (Z4-Mobius rigidity, uniform weights j in {1,2}, genus 3, seam reflections = the v599 pattern); the QGEO.SYM.01 residue narrows to the cyclic-3 choice and the physical<->conformal seam identification; see QGEO.SEAMFORCE.01.]
Recorded status: Exact [E] fingerprints + typed candidate [C] (Burau convention validated; Q-identification open)
Dependencies, status detail and source
Canonical status: FINGERPRINTS-HIT: rank-3 Eisenstein module; projective D4 via deck (r^4 = omega); cusp-class group (Z/3)^4 = 81 with the per-mark mechanism; Lorentz (1,2) polarization (Chevalley-Weil + Paper II simultaneously); single-cusp Gram {0,0,2}; Q/V dictionary and RP map named open; 6/6 checks
Dependencies: QGEO.INVOL.01; QGEO.WINDING.SELF.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v574; v590
Scripts: v597_mu4_cover_model.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 615