Phases 1 to 6: what each settled

The project ran in six phases, each opened with a brief and closed with a status file. The lead of every entry of those status files is reproduced below, in order; each phase links to its full file, which carries the numbers, the verdicts and the corrections made on the day.

Phase 1 status — 2026-09-08 (day 1)

Source: rigor/PHASE1_STATUS.md

Milestone M1: "make Theorem A a theorem and certify the numbers"

• Theorem A proved. lim_{zeta->0} Phi/zeta^2 = c/(12 pi^2).
• Tail bound proved (rigor/tail_bound.tex, PT): 0 <= Phi - Phi_W <= 0.173 zeta^2/kappa_c + 0.88 zeta^2/kappa_c^2 [corrected 2026-09-24: the shape of this bound is proved; the explicit constants 0.173 and 0.88 (tail_bound.tex Cor. 5.2, eq:final) rest on numerical inputs, A_1 <= 2.0 and A_3 <= 0.0374 in double precision, not certified, and f = 0.0085 (tail_bound.tex Correction C10); card tail-bound] for kappa_c >= max{10, 2 log(1/zeta) + 2 log(1+kappa_c) + 3}; Note 5 Prop 4.1 upgraded to a theorem with explicit remainder; c_2 = -zeta^2/30.
• Lattice (O7a) done (numerics/lattice, PL): Gaussian rotated Petz map in ball arithmetic up to 72 sites; exponents 1.986, 1.912, 1.635, 1.151 at lambda = 0, 0.5, 1, 1.5 (VWZ: 2.0, 1.9, 1.7, 1.2); lattice/two-branch ratio = 1 - 0.15/L constant in eta_V; single-branch law off by 22-5000.
• Large cross ratios certified [corrected 2026-09-24: not certified; the lower ends Phi_sub are rigorous, the quoted values are extrapolations with an estimated uncertainty; card large-zeta-values] (numerics/results_largezeta_certified.txt, PZ): Phi(1.07) = 4.410(20)e-3, Phi(2.13) = 1.0797(50)e-2, Phi(4.27) = 2.241(14)e-2, Phi(8.53) = 3.974(72)e-2, Phi(17.1) = 6.41(43)e-2; window exponent drifts 2.5 -> 1.2; slopes 1.29 -> 0.69 vs lattice.
• Paper 1 draft (paper1/main.pdf, 26 pp, PW): all sections written; placeholder citations for internal documents; certified brackets [corrected 2026-09-24: these are the brackets of tail_bound.tex, which are not certified, see the tail-bound entry above; card tail-bound] and large-zeta table included; honest gap statements (s_2 = c/12 for the KM form; rate heuristic).
• eps = 1e-14 runs: window contains all box modes at Lambda = 60 -> residual is the box cutoff; recipe min(kappa_c, kappa_max) with box asymptotics 0.0641 zeta^2/kappa_max^2; Phi(1/15) consistent across windows to 2e-4.

Corrections applied today to the notes

N5: apparent exponents (PL), 'ten times' -> pi^2 (PW), Prop 4.1 status (PT), Cor 4.2 status (PA/PR), large-zeta paragraph + table (PZ).

N7: Theorem A statement, Sec. 8 items 2 (PA/PR/PT).

Open after day 1

• Rate of the remainder (log^-2 claim): heuristic; would need a bound on ||sqrt(Q~) - sqrt(X11 (+) X22)||_2 quadratic in X12.
• Tight certified error bars (need kappa_c^-2 decay or interval-arithmetic audit of Dhat, Q_N; 3% arithmetic margin at present).
• O7(b) chiral lattice model; O11 collar-Gaussianity lemma (deferred; Phase 2 route changed).
• Phase 2 reordering: the Uhlmann-Bogoliubov construction gives, for any diffeomorphism covariant net with U(k~) implemented, the universal upper bound Phi <= -log |<Omega, U(k~_zeta) Omega>| (Virasoro coherent-state overlap, c-linear at second order): O5 now yields the upper half of Theorem B directly; O3 is needed only for the lower half.

Agents used today

PA (proof), PN (HS numerics), PL (lattice), PR (referee), PT (tail bound), PZ (large-zeta), PW (paper draft).

<= 80-120 lines per tool call, compile/run after every append, plain LaTeX in findings entries.

Phase 2 status — 2026-09-08 (day 1 of Phase 2)

Source: rigor/PHASE2_STATUS.md

Milestone M2 (Bures part) REACHED 2026-09-08: universality of f_2 = c/(12 pi^2) for every diffeomorphism covariant net on S^1

• Theorem B for general nets (rigor/universality_theorem_B.tex, QA, 22 pp; refereed by QR1: stands; 4 write-up repairs applied, Sec. 9; 24 pp): under Haag duality / locality, Bisognano-Wichmann, Reeh-Schlieder and the implementer hypotheses (H1)-(H3) [+ (H2')], -log F(omega, omega_s) = (s^2/8) g_B + o(s^2), g_B = 4 dist(T(g_tot)Omega, cl A(I)'_sa Omega)^2 = c * 2/(3 pi^2).
• Implementer hypotheses for every diffeomorphism covariant net (rigor/implementation_C11.tex, QB, 26 pp; refereed by QR2: stands; repairs applied incl. the scalar-phase lemma via Fewster-Hollands Prop 5.1 and (H2')): g_tot is C^1 + piecewise smooth with one jump of g'' at the touching point => ||g_tot||_{3/2} finite => Carpi-Weiner Thm 4.4 (e.s.a.), Stone, covariance by Trotter + localisation and approximation + additivity.
• Bosonic check (numerics/boson, QN): U(1) current net f_2 = 0.00844(2), s_2 = 0.083(2); c-linearity of the vacuum overlap proved analytically at second order (boson and fermion both give (s^2/(96 pi^2)) int k^3 |g_hat|^2).
• Kubo-Mori (rigor/kubo_mori_second_variation.tex, QK): exact identity for unitary orbits with affiliated generator [corrected 2026-09-24: for a generator affiliated with M this is proved only with Omega in D(A) (prop:unbounded), which the lower bound then also needs, and the expansion then holds under (U_eps), since modular analyticity => (U_eps) is proved only for bounded A; a generator affiliated only with a larger algebra is open (Gap QK2); card km-exact-identity]; lower bound with no hypothesis; full expansion under modular analyticity; Kosaki/Petz-Donald route refuted (suprema).
• Free fermion (H2),(H3): rigor/uhlmann_upper_bound.tex Sec. 10 (PA).

Corrections to the Phase 2 brief made during the day

• The smooth extension does NOT legitimise the x-space Mellin evaluation of g_B (g(L) != 0 obstruction survives); the value rests on c-linearity.
• Kosaki/Petz-Donald variational expressions are suprema: useless for the Kubo-Mori upper bound.
• The Trotter split is g_tot = (g_tot - m) + m, not g_A + g_D.
• Standard-subspace lemma (one-particle Reeh-Schlieder/BW for T-vectors) is NOT needed for Theorem B; still a GAP for the variational form.

Open after Phase 2 day 1

• DONE: QA's repairs applied; the theorem is recorded in Note 7 (Theorem B header, Sec. 8), Note 8 (main box) and paper 2 (abstract).
• Kubo-Mori: gauge lemma REFUTED 2026-09-09 (rigor/kubo_mori_gauge_lemma.tex): inf over admissible gauges = 11.09 c/6; proved bracket 0.034c <= liminf <= limsup <= 0.924c [corrected 2026-09-24: not proved outright; both ends are conditional, the upper end on hypothesis (V), open (Gap RK1), the lower end on Araki's perturbation expansion, whose sources were not obtained (Gap RK2); status table of rigor/kubo_mori_gauge_lemma.tex]; s_2 = c/12 for general nets stays conjectural (numerically confirmed for fermion and boson).
• DONE 2026-09-09: paper 2 draft (paper2/main.pdf, 24 pp, RW).

Agents used in Phase 2 (day 1)

QA, QB, QN, QK, QR1, QR2 (+ PA resumed).

Phase 3 status — started 2026-09-09

Source: rigor/PHASE3_STATUS.md

Tracks (rigor/phase3_brief.md):

• S8: second-order optimality (O8) [DONE, rigor/optimality_second_order.tex 14 pp [corrected 2026-09-24: now 16 pp, with the corrections section]]: quasi-free problem is an exact SDP; optimum = (1.00 +- 0.15) f_2 z^2 per chirality = two-chirality geometric-compression value (Petz: 4 f_2 z^2; lattice best/Petz = 0.27-0.28); optimal channel = near-isometric squeeze not reproducing the BC marginal; A-C lower-bound route gives exactly 0 [corrected 2026-09-24: not a general zero; E_AC = 0 is proved as an implication from (Tcond) (Prop. 4.2, prop:AC-zero) and applied only on the three hopping chains (4,8,2), (4,12,2), (4,16,2), where (Tcond) holds numerically; nothing is claimed for other geometries or the continuum; card ac-lower-bound-zero]; only E_opt >= 0 proved.
• S9: thermal/RG/gap (O9, O10) [DONE]: two-branch law holds at the thermal cross ratio; shielding rates 2pi (fidelity), pi (CMI); c_M >= c_E along the flow, c_M/c_E -> mR; gap: exact CMI = adjacent law with L_B -> L_B+L_G, -logF ~ L_G^-2; off criticality -logF is LINEAR in the CMI (ratio 0.10-0.17)
• PP1: referee pass on paper 1 -> rigor/referee_paper1.md [DONE: 23 edits applied + 3 orchestrator edits; 10 wording edits proposed]
• PP2: referee pass on paper 2 -> rigor/referee_paper2.md [DONE: 26 edits applied; KM section now conjecture + bracket; 10 wording edits proposed]

Deferred: O11 (sectors, state dependence); Kubo-Mori gauge lemma replacement (renormalised/collar route); D-hat quadrature and box-cutoff certification.

Phase 3 day 1 complete (2026-09-09). Open: proof of optimality of the geometric compression (dual certificate / quasi-free achievability); multi-interval networks (O10 second half); O11 sectors; KM gauge lemma replacement; D-hat quadrature and box-cutoff certification; paper polishing (13 + 10 proposed wording edits) and internal-citation replacement before submission.

Phase 3b (2026-09-10): optimality of the geometric compression among ALL channels (user request)

• Orchestrator: rigor/optimality_all_channels.tex — PROVED: (i) exact single-observable bound from Alberti–Uhlmann's variational formula; (ii) every channel's recovered symbol lies in S8's quasi-free feasible set F (Kadison–Schwarz + Tomita operator + BJL one-particle restriction, BJL_twisted_duality_math-ph-0204029.pdf (not redistributed)); (iii) all-channel second-order optimum = quasi-free SDP value (Sion).
• OPEN → S10 (analytic KKT/dual certificate, rigor/sdp_dual_certificate.tex), S11 (KKT numerics, rigor/sdp_certificate_numerics.tex): show the quasi-free programme's value is f_2 z^2 (compression).
• S10 DONE (rigor/sdp_dual_certificate.tex, 960 lines): exact necessary-and-sufficient KKT certificate at the compression (unique multipliers since V_c is unitary); Legendre reduction V(t)=||P^perp t P||^2; Möbius rigidity (DD block of the first-order compression defect vanishes) ⇒ the certificate condition (S) fails unless E_D u E_D = 0; numerically θ = 0.300 ± 0.005 ⇒ the isometric orbit W_s e^{-εG*} beats the compression: min_F g_Q ≤ 0.70 f_2 z^2.
• S11 (rigor/sdp_certificate_numerics.tex, 540+ lines): (a) strong duality + KKT verified on the (4,8,2) lattice (dual/primal = 1 to 1.5e-5 for caps ≤ 10; cap must be paired with its own variance); (b) certificate test at the compression inconclusive in any finite Galerkin model — the compression is infeasible there (X_c X_c^* = 1 impossible when dim V_B < dim V_D), and coarse-model "improvements" explode ×10^5 under nested refinement; (c) EXACT windowed fidelity in S10's spectral model confirms S10's gain: ratio rotated/compression 0.59→0.67 as the window grows (L=192, s=0.05) and 0.61→0.67 (L=320, s=0.02), best 0.66 at eps/s=−1.1.
• CORRECTION (orchestrator, later on 2026-09-10): the exact-fidelity claim was overstated.

Phase 4 status — exact-fidelity optimality (started 2026-09-15)

Source: rigor/PHASE4_STATUS.md

Question (kb card exact-fidelity-optimality-open): is the zero-collar geometric compression beaten in EXACT fidelity, i.e. does the

upper half -log F <= g_Q(delta)(1+o(1)) hold for the rotated channel W_s e^{-eps G_*}?

Reformulation (rigor/phase4_brief.md): the Uhlmann purification may use ANY unitary on the exterior of I = ABC (not only exterior

flows).

P^perp Z_e P are invisible (E_I[P, Z_e]E_I = 0), so the optimised Uhlmann bound equals the second-order functional iff the invisible

part of the channel's direction lies in the closure of the exterior directions Ext.

Results:

• Orchestrator (e0_finite_density.py): on finite hopping chains Ext = N^perp exactly (dim N = |I|^2).
• E2 (rigor/exterior_extension_numerics.tex, numerics/optimality_all/e2_*.py): in the spectral circle model Ext = N^perp exactly at every L; dist((1-Pi)M, Ext) = 0 to 55-70 digits; optimised UB_c/(1/2 s^2|u|^2) = 1.0000000 (the 4.3 of the fixed taper was an artefact); UB_rot/UB_c = 1 - theta to 5 digits; exact Fredholm determinant agrees at second order; ||Z_e||/s ~ 3 bounded in L.
• E1 (rigor/exact_fidelity_upper_half.tex, 17 pp): THEOREM closure(Ext) = N^perp (two proofs: compressed reflection Gamma' is a contraction without eigenvalues +-1 so the invisible remainder vanishes; twisted duality).

Consequences applied: paper1/sec_outlook.tex optimality passages rewritten (bibitems RigorAll, RigorS10, RigorE1, RigorE2); Note 5

remark after Cor. 5.1; plan page O8 status; kb card exact-fidelity-optimality-open -> numerical (answered).

Open after Phase 4: exact value of the all-channel optimum (non-isometric channels; bracket [D_z, 0.71 f2 z^2]); analytic proof

of theta > 0; the Kubo-Mori coefficient (Phase 2b); D-hat quadrature certification; O10 networks; O11 sectors.

Phase 5 status — exact value of the all-channel optimum (started 2026-09-15)

Source: rigor/PHASE5_STATUS.md

Question: is the isometric-orbit value (1 - theta) f2 z^2 the exact all-channel second-order optimum, or do

non-isometric quasi-free channels (isometry loss N >= 0, noise 0 <= Y <= N) lower it further? And what is theta?

Brief: rigor/phase5_brief.md.

Tracks

• F1 analytic (rigor/exact_optimum_tangent_problem.tex): exact parametrisation of F by (V co-isometry, N, Y); tangent problem; KKT sign criterion for the noise directions; theta closed form attempt.
• F2 numerics (numerics/optimality_all/f2_*.py, F2_RESULTS.md): tangent SDP with noise in the spectral circle model; KKT eigenvalue check; full-F primal SDP in the Galerkin frame vs orbit.
• F3 numerics (numerics/optimality_all/f3_*.py, F3_RESULTS.md): theta reconciliation between the circle model (0.300) and the Galerkin frame (>= 0.336, rising).

Results

• F1 (2026-09-15): exact chart of F: (X,Z) in F iff N = 1 - XX* >= 0 and 0 <= Y = Z - X Q_BB X* <= N; X = (1-N)^{1/2} V with V a co-isometry iff 1 is not an eigenvalue of N (sharp).
• F3 interim (files, 2026-09-15 16:30): the two discretisations have the SAME limit.
• F3 FINAL (2026-09-15): theta = 0.384 +- 0.003 (1 - theta = 0.616 +- 0.003).
• F2 interim (files): conventions reproduced (theta 0.28958 at L=256; SLD identity 1e-10; F1's derivative formula 1e-9 by finite differences; stationarity check 1e-6).
• F2 FINAL T4 (2026-09-15, rigid parameter-free modular frame of F1 Prop 6.2, f3_t0.py machinery; ||d_c,DD|| = 0 exactly, no weight cap needed): the F1 identities hold to <= 3.5e-7 (SLD identity 1e-10, FD derivative formula <= 3e-8, AntiHerm((t_*Q)_DD)/||M_*|| = 2.8e-9 ... 3.5e-7; correction C13 below); lambda_min(M_*)/g_Q(d_c) = -7.0e-2, -3.8e-2, -1.9e-2, -1.0e-2 and lambda_min(M_*)/||M_*|| = -4.1e-3, -1.9e-3, -8.2e-4, -3.7e-4 at h = 0.24, 0.12, 0.06, 0.03; lambda_min(M_* - tau_*) identical; rank-one gain/g_Q(d_c) = 3.9e-4, 1.2e-4, 3.5e-5, 9.7e-6.
• F2 FINAL T5 (2026-09-15, endpoint condition (e) of the repaired criterion, rigid frame, f2_t5_endpoint.out): E(zeta_0)/g_Q(d_c) = +34.1, +66.9, +131.3, +258.0 at h = 0.24, 0.12, 0.06, 0.03 for the admissible (backward/dissipative) scheme; centred scheme 0 (the linear class), forward (inadmissible) the negative; corner term (1/2) M_*(0,0)/h = +10.3, +16.1, +24.7, +37.2 ~ h^-0.6 (M_*(y,y) -> +inf as y -> 0+: the junction direction is infinitely penalised at first order); saturates in Ymax, Y-independent; cross-check E(zeta_inf) = 2(1 - theta_h) g_Q exact to 1e-15; A-block (absorption) form = boundary form to 2e-10..7.5e-8 (correction C12 below).
• Union-of-referees fix pass (2026-09-15 evening; two referees had reviewed the F2/F3 cards concurrently, PASS x5 vs FAIL x5; user decision relayed by session claude-team-84: authors fix all defects from the union, then ONE referee re-reviews, nobody flips verdicts): F3 repaired theta-value (candidate script f3_candidates.py: 26 candidates, 3 in [0.381,0.387], chance expectation 0.28, no selection; per-family rho 0.577..0.592; spread 5.1e-4 Galerkin / 2.0e-3 incl.

Referee passes (Opus only)

• REF-P5-1 (document referee, rigor/referee_exact_optimum_tangent_problem.tex, 16 pp): STANDS for the chart of F, the first-order defect (recomputed), exact feasibility, Thm 3.9/Cor 3.10, Thm 4.4 (a)<=>(b)<=>(c) (no counterexample in 60 000 random triples), Prop 4.6/Rem 4.7, Lemma 6.1, Prop 6.2.
• F1 correction (21:04): E(zeta) := 2 b(delta_*, A(zeta,0,0)) only; new Prop 4.10(2'): A(zeta,0,0) = -[Q,zeta] holds only for bounded anti-self-adjoint zeta (deletes the boundary term otherwise); new Rem 4.11 with F2's T5 table and the reading 'the junction direction is infinitely penalised at first order'; Sec. 7 concludes kappa_opt = 1 - theta = 0.616 +- 0.003 conditional on H1, H2, with (a),(e) as numerical input.
• REF-P5-3 (21:29; rigor/referee2_exact_optimum_tangent_problem.tex, 7 pp): R1-R15 applied, no regressions; NEW FINDING S1: the reduction of the admissible cone of boundary-moving isometry generators to the two model directions is proved only for vector fields (multiplicity > 1 shift semigroups and unbounded skew-adjoint multiplication operators escape) => condition (e) is reduced to E(zeta_0) >= 0 only under a hypothesis H3 (cone hypothesis) [superseded 2026-09-24: see C11]; repairs S1-S6 (H3 declared and every '(e) holds' hedged; the T5 total is the scheme-dependent upwind-viscosity quantity ~h^-0.97, the corner term ~h^-0.60 is the invariant piece; E in [-inf,+inf] with the form domain a subspace; Thm 3.11 <1>4 core hypothesis; Sec. 7 item 5; minor slips).
• REF-P5-4 (21:55; rigor/referee3_exact_optimum_tangent_problem.tex, 4 pp): S1-S6 all applied, H3 usage sound (bilinearity of b and b(delta_*, closure of L) = 0 by g_Q-continuity); document ACCEPTED with three residual defects on the H3 wording: D1 the 'equivalently ... boundary form' clause of Def 4.12 is not equivalent to eq.
• F1 D1-D3 fix (22:05, 27 pp): H3 (Def 4.12) is eq.
• Phase 5 closes with: theta = 0.384 +- 0.003 (refereed); the KKT criterion (a)+(e) proved for the tangent problem; the reduction of (e) to the model cone is hypothesis H3 (exclusion of non-vector-field generators); (a) numerically in the limit; (e) numerically on the model cone (not proved: there (e) is equivalent to E(zeta_0) >= 0, given the proved E(zeta_inf) >= 0, and it is vacuous at zeta_0 by F2 T5; correction C3 below); kappa_opt = 1 - theta = 0.616 +- 0.003 numerically under H1-H3; unconditional bracket E_rec in [dual form, 0.616 f2 z^2 (1+o(1))].
• KB card referees: REF-opus-cft-2026-09-15 (F1 cards: kkt-noise-sign-criterion passed; tangent-problem-exact-parametrisation and theta-modular-wiener-hopf FAILED on locators/superseded numerics — under repair); REF-P5-KB1 on the F2/F3 cards: running.

Propagation (2026-09-15)

Editor P5-EDIT propagated the refereed F3 result (card theta-value, passed REF-P5-KB1;

numerics/optimality_all/F3_RESULTS.md): theta = 0.384 +- 0.003, hence 1 - theta = 0.616 +- 0.003;

geometry independent, measured order h^0.77, two independent discretisations (modular Galerkin;

parameter-free modular Wiener-Hopf, rigor/exact_optimum_tangent_problem.tex Prop 6.2) agreeing to 3e-4

after extrapolation.

taper-suppressed lower bound and stays as the historical record (erratum: rigor/errata_log.md).

kappa_opt = 1 - theta is NOT claimed as settled: the KKT endpoint condition is still under evaluation

(Phase 5, F1/F2).

Documents

Compiles: optimality_all_channels.pdf 14 pp; exact_fidelity_upper_half.pdf 21 pp;

paper1/main.pdf 28 pp (pdflatex twice, unchanged); fidelity_recovered_quasifree.pdf 15 pp.

Untouched by instruction: rigor/exact_optimum_tangent_problem.tex, rigor/referee_*.tex,

rigor/sdp_dual_certificate.tex, all numerics scripts.

KB cards (KB_ACTOR=P5-EDIT; no existing referee note edited, each card carries a dated note)

Not changed, deliberately: open-problems-plan-note8 and the e2-* cards -- their Statements quote no

0.30/0.71 as a current value (the e2-* ratios 0.708-0.711 are within-model measurements of the

circle model's own 1 - theta, and the 0.30 hits are the taper parameter uv = (0.30,0.42)).

Corrections of 2026-09-24

Each changed line was replaced by exactly one line, so the line numbers are those of the version of 2026-09-23.

"Tex Cn" refers to the corrections section of rigor/exact_optimum_tangent_problem.tex (Sec. 8).

• C1 (lines 87, 113, 124; knowledge-base card kkt-noise-sign-criterion).
• C2 (line 126; knowledge-base card tangent-problem-exact-parametrisation).
• C3 (lines 129-130; knowledge-base cards kkt-noise-sign-criterion and all-channel-optimum-value).
• C4 (line 122; knowledge-base card kkt-noise-sign-criterion).

Superseded log entries (C5-C11).

"[superseded 2026-09-24: see Cn]", and the current statement is given here.

• C5 (line 19, F1 entry of 2026-09-15; card tangent-problem-exact-parametrisation).
• C6 (lines 20-21, same entry; card kkt-noise-sign-criterion).
• C7 (lines 55-56, F2 FINAL T4 entry; card kkt-noise-criterion-numerics).
• C8 (line 64, F2 FINAL T5 entry; cards kkt-noise-criterion-numerics and kkt-noise-sign-criterion).
• C9 (lines 83 and 87, F1 entry of 17:30; card kkt-noise-sign-criterion).
• C10 (lines 85 and 87, same entry; card kkt-noise-sign-criterion).
• C11 (line 97, REF-P5-3 entry; card kkt-noise-sign-criterion).
• C12 (line 62, F2 FINAL T5 entry; card kkt-noise-criterion-numerics).
• C13 (line 51, F2 FINAL T4 entry; card kkt-noise-criterion-numerics).
• C14 (line 24, F1 entry of 2026-09-15; card tangent-problem-exact-parametrisation).
• C15 (line 54, F2 FINAL T4 entry; card kkt-noise-criterion-numerics).
• C16 (line 85, F1 entry of 17:30; card kkt-noise-sign-criterion).
• C17 (line 9).

Phase 6 status — multi-interval networks (O10, second half) (opened 2026-09-16)

Source: rigor/PHASE6_STATUS.md

Question: recovery error, order dependence ("holonomy") and universality of sequential one-sided zero-collar recovery along

chains A_1 ...

Decision card: kb -p cft_cmi show phase6-plan.

corner-correlation-kernel, vwz-protocol-ordering (all opened 2026-09-16 by the orchestrator, unrefereed).

Tracks

• G1a analytic (rigor/network_corner_calculus.tex): corner calculus W1, amplification lemma, separation constraint W3, type-III sequential bound W6, Protocol 3 = over-compressed Theorem A, 1(B) = 1(A).
• G1b analytic (rigor/network_second_order.tex): second-order network law W2 with the universal kernel Ghat(Delta), positivity, decay, closed form attempt, universality W5.
• G2 numerics (numerics/networks/lattice_chain.py): sequential Gaussian Petz maps on the hopping chain, VWZ protocols, 1:2:4 test, amplification, CMI telescoping, equal-block chains.
• G3 numerics (numerics/networks/corner_kernel.py): Ghat(Delta) in two independent frames (T0 modular cells, box modes), exact two-corner Phi at finite zeta, clustering.

Results

• G1a (2026-09-16, rigor/network_corner_calculus.tex, 42 pp after the REF-P6-3 repairs, Lamport; g1a_calculus.py 33/33, g1a_separation.py 15/15).

Referee passes (Opus only)

• REF-P6-3 (launched 2026-09-16 evening; brief rigor/phase6_referee_brief.md): document referee of G1a's rigor/network_corner_calculus.tex + the eight pending cards.
• REF-P6-0/1/2 (2026-09-16, rigor/referee_phase6_plan.md, scripts rigor/ref_p6_*.py): adversarial review of the plan and target cards.

Errata

Results that were wrong at some point were corrected rather than erased. Every correction is recorded, with its date and the document it touched, in rigor/errata_log.md.

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