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The finite mirror projector candidate

Original ledger classification: Candidate [C] (exact finite-cluster gapping; open: dynamical gauge fields, domain-wall geometry, 4D volume theorem) | Round-3 v1023 is a separate finite block lemma and does not upgrade DET16.

Recorded statement: THE DET16 RANK-ONE MIRROR PROJECTOR CANDIDATE (registered 2026-08-29, review wave 5, v1002). Candidate [C]: exact finite-cluster gapping mechanism on the full 2^16 chiral Spin(10) Fock space. [E-finite] Omega^dag Omega = P_16, Omega…

Statement, assumptions and sources

Paper context: TFPT — Introduction and reading guide

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CHIRAL4D.MIRROR.DET16.01

Original ledger classification: Candidate [C] (exact finite-cluster gapping; open: dynamical gauge fields, domain-wall geometry, 4D volume theorem) | Round-3 v1023 is a separate finite block lemma and does not upgrade DET16.

Original claim text

THE DET16 RANK-ONE MIRROR PROJECTOR CANDIDATE (registered 2026-08-29, review wave 5, v1002). Candidate [C]: exact finite-cluster gapping mechanism on the full 2^16 chiral Spin(10) Fock space. [E-finite] Omega^dag Omega = P_16, Omega Omega^dag = P_0, defect 0; P_phi hermitian idempotent (2.8e-18), rank 1; h_mir spectrum {0^1, 1^65535}, gap exactly 1; all 45 so(10) Fock commutators = 0 (det rho_16 = 1); parity commutator 0; |0> and |F> singlets. [N] hopping stability on the 4-mode analogue: gap 0.90 (2 clusters) / 0.86 (3 clusters) at t = 0.2; spectators untouched. OPEN: dynamical gauge fields, domain-wall geometry, 4D volume theorem. The number-preserving Casimir projector route is a SEPARATE kill recorded on CHIRAL4D.NOMIRROR.01 (not this candidate). Display Candidate [C]; no upgrade of NOMIRROR. EXECUTED UPDATE (2026-08-30, review wave 7, v1007): T1/T2 proved; T3 Michalakis-Zwolak cited-verified (LTQO, t*>0, gap>=1/2). [N] counting line n=2,3; n=2 closing scan through t=4 (no closing); dynamical Z2/Z4 uniquely gapped (full-probe min 0.728854416 pinned; compressed grid coarser). SW 1-alpha t^2 + O(t^4); c_geo/L bounded. 1/192 remains the uncertified counting threshold (ordinary hopping fails the relative-form block hypothesis). BDL supplies no volume-uniform gap remainder (boxed). Display stays Candidate [C]; NOMIRROR unmoved. MONDAY MORNING (2026-08-31, v1016): Froehlich-Pizzo hypotheses cited-verified for Z2/Z4 DET open chains -- volume-uniform spectral gap >= Delta/2 in the parametric window tau_FP (a=0.0233, a/16=0.00146). Dynamical-link leg closed at that finite-group level. Remaining: rotor links, wall mixing, 3+1D placement. Display stays Candidate [C]. EXECUTED UPDATE (2026-09-05, v1023): the separate derivative-filtered wall block preserves ker(D_GW) in a finite quasi-free singular-value model, but it does not realize the DET16/DET32 interaction. The bare mirror response is not gapped and the low branch is quadratic. DET16 remains Candidate [C]; NOMIRROR/T4 remain Open [O].

Original classification detail

Candidate [C] (v1002/v1007): T1/T2 proved; T3 MZ cited-verified (t*>0, gap>=1/2); dynamical [N] + boxed BDL/1/192. Open: compact-group dynamical links, domain-wall geometry, 4D volume theorem. No marker upgrade of NOMIRROR. WAVE-7 2026-08-30. | MONDAY MORNING (2026-08-31 v1016): DFP volume-uniform gap cited-verified for finite-group open chains; dynamical-link leg closed at that level; remaining rotor/wall/3+1D. Display stays [C]. | ROUND3 2026-09-05: derivative-filtered witness is not a DET16/DET32 realization and supplies no 4D theorem. Candidate stays [C].

Dependencies as recorded

CHIRAL4D.NOMIRROR.01; DIMENSION.UPLIFT.FIREWALL.01

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Registered verification modules

  • v1002_spin10_mirror_projector.pyRegistered verification module · frontier
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    CHIRAL4D.MIRROR.DET16.01 Kandidat [C] (Review-Welle 5, 2026-08-29; genaues endliches-Cluster-Gapping; offen: dynamische Eichfelder, Domain-Wall-Geometrie, 4D-Volumentheorem): DET16 Rang-1-Projektor auf 2^16, Omega^dag Omega=P_16, P_phi hermitesch idempotent 2.8e-18, h_mir {0^1, 1^65535} Gap 1, 45 so(10)-Kommutatoren 0; Hopping-Stabilitaet Gap 0.90/0.86 bei t=0.2; Casimir-Route AUSGESCHLOSSEN (Zensus N=8:0, k=2:27, k=3:1463, |W(D5)|=1920). Python-only / Wolfram-Spiegel zurueckgestellt.

  • v1007_det16_stability_battery.pyRegistered verification module · frontier
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    CHIRAL4D.MIRROR.DET16.01 Kandidat [C] (Review-Welle 7, 2026-08-30): T1/T2 bewiesen (v1002); T3 Michalakis-Zwolak zitiert-verifiziert (LTQO, t*>0, Gap>=1/2). [N] Zaehlgerade n=2,3; n=2-Scan bis t=4; dynamische Z2/Z4 eindeutig gelueckt (Vollsonden-Min 0.728854416); SW 1-alpha t^2 + O(t^4); c_geo/L beschraenkt. [X] gewoehnliches Hopping nicht blockdiagonal, daher 1/192 boxed; BDL liefert keinen volumen-uniformen Gap-Rest. Bleibt [C]. Python-only / Wolfram-Spiegel zurueckgestellt.

  • v1016_state_gap_batteries.pyRegistered verification module · frontier
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    Zustands-/Luecken-Batterien (Montagmorgen 2026-08-31): R4 Naht-Modular-Kappe einzigartiger endlicher nicht-entropischer Selektor (D_tr 0.1434, SK-RMS 3.038e-3, Sattel erhalten; Finite-Proxy-Caveat). TEL-B zertifizierter Schwanz BLOCKIERT (N^2 D_N 530->19032); skalares dyadisches A_R<43 mit Faktor-350-Headroom; N-uniforme CAR-Nuklearitaet bewiesen (C_beta 67.97/29.73/12.15); ALG-EXH unveraendert. DFP volumenuniforme Luecke zitiert-verifiziert fuer Z2/Z4-offene Ketten (a=0.0233, a/16=0.00146); Rest Rotor/Wand/3+1D. FTRANSFER.SK.RHO0.01 [O] / SEAM.MMST.TYPEIII.CHARGED.01 [O] / CHIRAL4D.MIRROR.DET16.01 [C] unbewegt. Python-only / Wolfram zurueckgestellt.

  • v1023_derivative_filtered_wall.pyRegistered verification module · frontier
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    Ableitungsgefilterter Wall-Block (2026-09-05): H_df erhaelt ker(D_GW) exakt; det Block=lambda g^2 Delta s^2; das adiabatisch spiegel-kontinuierte hohe Band liegt >=Delta+lambda; der niedrige Ast ist quadratisch. Der nackte Spiegeloperator ist nicht invariant und hat fuer s>0 Niedrigbandgewicht -- kein Spiegel-Antwortgap und kein Weyl-Kegel. Gauge-Defekt 2.41e-15, Spektraldefekt 1.78e-15. Endliches quasi-freies Blocklemma; kanonische DET16/DET32-Realisierung, Vielteilchenparent, Zulässigkeit/Quasilokalitaet, Mass/Anomalie/Index, Universalitaet und 4D-Limes bleiben offen. Keine Markerbewegung. KEIN RH-CLAIM.

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