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Jusang Lee edited this page Jun 5, 2026 · 16 revisions

Pillar 11 — Dark-sector phenomenology

Tier: T4|T2-split-PROVISIONAL — SPLIT POST-MATH409-ADDH-ADDA-ADDA (B2 1-loop STRONG NEGATIVE INDICATOR): Pillar 11.A (TECT textures as dominant DM via Path B2 = TECT-Brazovskii asymptotic safety) = T2 INTERIM NEGATIVE, exact-RG pending. Math409-AddH-AddA-AddA explicit 1-loop running-G calculation gives omega_TECT = 4.42 (band [1.5, 13.3]), G_eff(k_*)/G_obs = 0.572 vs required threshold 0.0385 (factor 15 shortfall, structural — well outside scheme-dependence band). Required omega = 148, gap factor 33 above 1-loop estimate (well beyond typical higher-order corrections O(2-3)). Effective Schwarzschild with G_eff: r_tex/r_Sch^eff = 0.067 < 1 (still INSIDE Schwarzschild even with full B2 1-loop credit). PROVISIONAL qualifier RETAINED — formal removal is premature until Math409-AddH-AddA-AddA-AddA exact-RG Wetterich verdict (target 2026-12-31). Most-likely outcome (70-80%) confirms B2 FAIL -> Pillar 11.A T1 OPEN terminal -> TECT dominant DM mechanism defaults to Pillar 11.B (subdominant relic + bulk DM from nu_R via Math408 LRSM cascade, T4 STRONG EVIDENCE UNCHANGED). 11.B is now the LIKELY PRIMARY DM PATH for TECT.
Epistemic category: C3 — Phenomenological add-on (TECT does NOT predict from internal axioms)
Last sync: 2026-06-05

TECT framework positioning (binding 2026-05-26): TECT is an emergent vacuum-and-gravity framework (7 C1 pillars at T6/T7) with an unresolved gauge-sector completion (Pillar 6 standard routes EXHAUSTED) and phenomenological dark-matter/quantum-origin pillars open. TOE-level claim SUSPENDED until non-standard rescue independently derived (per Math411-AddB §10).

Statement

Dark-sector phenomenology from BCC-channel topological-defect (Reading H: KZ-frozen defects of fluctuation regime) spectrum; relic abundance + GW signatures.

Status semantics

This pillar is currently classified at tier T4|T2-split-PROVISIONAL (SPLIT POST-MATH409-ADDH-ADDA-ADDA (B2 1-loop STRONG NEGATIVE INDICATOR): Pillar 11.A (TECT textures as dominant DM via Path B2 = TECT-Brazovskii asymptotic safety) = T2 INTERIM NEGATIVE, exact-RG pending. Math409-AddH-AddA-AddA explicit 1-loop running-G calculation gives omega_TECT = 4.42 (band [1.5, 13.3]), G_eff(k_*)/G_obs = 0.572 vs required threshold 0.0385 (factor 15 shortfall, structural — well outside scheme-dependence band). Required omega = 148, gap factor 33 above 1-loop estimate (well beyond typical higher-order corrections O(2-3)). Effective Schwarzschild with G_eff: r_tex/r_Sch^eff = 0.067 < 1 (still INSIDE Schwarzschild even with full B2 1-loop credit). PROVISIONAL qualifier RETAINED — formal removal is premature until Math409-AddH-AddA-AddA-AddA exact-RG Wetterich verdict (target 2026-12-31). Most-likely outcome (70-80%) confirms B2 FAIL -> Pillar 11.A T1 OPEN terminal -> TECT dominant DM mechanism defaults to Pillar 11.B (subdominant relic + bulk DM from nu_R via Math408 LRSM cascade, T4 STRONG EVIDENCE UNCHANGED). 11.B is now the LIKELY PRIMARY DM PATH for TECT.) per the canonical TOE-FACT-SHEET.

Conditional on:

  • Tier demotion 2026-05-15 (T6 → T4) per Math409 honest evaluation: Math402's PRE-ANCHOR closure window (T6 baseline) was OBSOLETE under Math404 Planck anchor. Three Math409 pathways (I/II/III) each yield T2 CONJECTURE or T4 STRONG EVIDENCE, none reaches T5/T6. Honest conservative tier = T4 (matches Pillar 6 post-Math407/Math408 floor; symmetric audit posture).
  • Internal Λ inconsistency repair (Q-2026-05-02-Pillar11-Lambda-Numerical-Repair)
  • E_3'(cosm) sub-task (post-Math391, 2026-05-11): KZ-defect cosmological abundance vs Omega_DM ≈ 0.27 matching; T2 CONJECTURE; Math393 defect-as-DM realisation queued; reformulated from former Pillar 4 Lemma E_3' obstruction per Math388 acceptance.
  • KZ-defect Pillar 11 absorption (post-Math400-AddE/AddF, 2026-05-11): under Reading H fluctuation-stabilized disordered vacuum, cosmological quench through the order-disorder transition produces Kibble-Zurek frozen topological defects with density n ~ xi(tau_Q)^{-3}. Estimated abundance ~10^{58} excess over Omega_DM ~ 0.27 — requires dilution mechanism (inflationary stretching, defect annihilation, etc.); Math402-AddB queued.
  • Math402 + Math404 KZ closure analysis (post-Math404, 2026-05-12): under Pillar 9 Planck-scale anchoring, KZ frozen-defect excess deepens to ~10^114 (vs Math402 estimate 10^58). Closure requires N_e ≈ 88 e-folds of inflation after defect formation — beyond standard slow-roll (50-70). Three options: (i) extreme inflation non-standard model, (ii) revised Pillar 9 anchoring, (iii) sub-dominant defect-DM. Math402-AddA explicit inflation-model verification queued. T2 CONJECTURE retained with explicit closure-pathway constraint.
  • ADVERSARIAL AUDIT NOTE 2026-05-12 (operator review of Math402): Math402 N_e ≈ 45 closure window is PRE-MATH404-ANCHOR (used 10^58 baseline). Under Math404 Planck-scale anchor, KZ excess deepens to 10^114 requiring N_e ≈ 88 — beyond standard slow-roll. Math402 is RECLASSIFIED as PRE-ANCHOR EXPLORATORY note; the four mechanism categories (inflationary, annihilation, mass-dilution, restricted-phase-volume) remain valid framework but the quantitative N_e ~ 45 closure claim is OBSOLETE. Math409 (Math402-AddA under Math404) queued for canonical N_e ~ 88 regime calculation.
  • Math409 outcomes (2026-05-15, post-Math404 anchor): three Pillar 11 closure pathways evaluated against Math404's N_e ≈ 87.6 requirement. (I) Single-field N_e=88: φ² inflation gives r=0.090 (BICEP/Keck excludes), φ⁴ gives r=0.180 (strongly excluded). Requires TECT-derivable low-tensor inflaton (Starobinsky / α-attractor / D-brane class). T2 CONJECTURE. (II) Two-stage TECT-natural inflation: pre-BCC hill-top (low-r) + post-BCC quasi-flat A_1g breathing direction. Structurally appealing; T2 CONJECTURE pending explicit V_1+V_2 derivation from TECT effective Lagrangian. (III) KZ defects sub-dominant (f_def < 1) + bulk DM from gravitationally-reheated ν_R via Math408 LRSM cascade. T4 STRONG EVIDENCE for f_def ∈ [10⁻⁵⁰, 10⁻³⁰] and standard slow-roll N_e ∈ [50,70]. Pre-registered falsification gates: G1 (TECT-derivable inflaton with r<0.036 at N_e=88), G2 (two-stage potential derivation), G3 (ν_R relic abundance via gravitational reheating at T_reh~10¹⁵ GeV), G4 (horizon-tempered defect-mass recalc — may reduce 10¹¹⁴ excess by orders of magnitude, making Path I viable with standard inflation). Math402 PRE-ANCHOR EXPLORATORY status preserved; Math409 supplies the explicit-pathway evaluation Math407 demanded.
  • Math409-AddA outcomes (2026-05-15, gate G4 PARTIAL): horizon-tempered defect mass recalculation. Math404 §5's m_def ≈ 5×10²⁰ GeV ≈ 41 M_Pl is super-Planckian and physically forbidden by horizon-mass bound. Two independent corrections (length-scale consistency: ξ_KZ vs a_cond gives factor 20.1; horizon bound: m_def ≤ M_Pl gives factor 41) coincide. Corrected m_def = M_Pl ≈ 1.22×10¹⁹ GeV. Revised excess: 1.30×10¹¹⁴ → 3.13×10¹¹². Revised N_e: 87.6 → 86.8 e-folds (only ΔN_e ≈ 0.8). Insufficient alone — gap to standard slow-roll (50-70 e-folds) is ~26 e-folds = factor 10³⁴⁺ in cumulative reduction needed. Identifies four higher-leverage follow-ups (Math409-AddC: defect topology π_n(O(8)/H_BCC); AddD: KZ applicability under Reading H adiabatic-vs-quench; AddE: scaling-regime vs at-formation density; AddF: TECT τ_Q derivation; AddG: multi-stage BCC condensation timeline). G4 PARTIAL: structurally valid correction but insufficient single-pathway resolution. Pillar 11 tier T4 UNCHANGED.
  • Math407-AddA outcomes (2026-05-15, operator Round-2 audit, BINDING): three decisions accepted. (1) Pillar 11 SPLIT into 11.A (dominant-DM, T2) + 11.B (subdominant-relic, T4) — see tier_split structured field. (2) Decisive-test priority REORDER: Math409-AddD (KZ applicability under Reading H adiabatic vs quench) > AddC (defect topology) > AddF (τ_Q derivation). Justification: KZ inapplicability would invalidate ALL downstream (Math402/Math404/Math409/AddA/AddC); prerequisite question must be resolved first. (3) Math409-AddA Hawking-evaporation hidden cost FOREGROUNDED: m_def → M_Pl correction makes 11.A WORSE (Planck-mass primordial black holes evaporate in t_Pl), but STRENGTHENS 11.B (small m_def is exactly what subdominant relic should have). Net effect: horizon-tempering is asymmetric — bad for 11.A, good for 11.B.
  • Math409-AddD outcomes (2026-05-15, Q-2026-05-15-KZ-Applicability-Under-Reading-H CLOSED): BCC condensation under Reading H is a Brazovskii (1975) fluctuation-induced WEAK FIRST-ORDER transition. KZ applies in modified bubble-nucleation form, NOT continuous-quench form. Continuous-quench scenario REJECTED; smooth crossover scenario REJECTED; first-order scenario CONFIRMED. Quantitative consequences: (i) n_bubble^completion ~ H^3 ≈ 2.4×10¹⁰⁵/m³ (factor ~200 higher than Math404 §5's continuous-quench 1.33×10¹⁰³, but same order of magnitude); (ii) dominant defect type: WALLS between misaligned BCC orientation domains (NOT monopoles); sub-leading monopoles at triple-bubble junctions; (iii) Math404/Math409/Math409-AddA quantitative chain SURVIVES at order-of-magnitude level (Brazovskii first-order does NOT invalidate the framework, only revises defect-type assumption). NEW CONSTRAINT identified: wall-DM CMB compatibility (Friedland-Murayama-Perelstein 2003) further weakens 11.A by ~T3 → T2 effective; net 11.A status UNCHANGED at T2 (KZ-applicability +T promotion offset by wall-DM constraint -T demotion). 11.B robust under either defect type. Self-adversarial review revised initial promotion claim T2→T3 for 11.A back to T2 UNCHANGED. New Q-2026-05-15-Wall-DM-Bias-Mechanism Active OQ registered.
  • Math413-AddA outcomes (2026-05-15, merges Math409-AddC defect-type classification with Math413 Direction B fermion-mass gate F-B): π_n(O(8)/O_h) homotopy enumerated. Topologically protected: walls (π_0), strings (π_1), textures (π_3). NOT protected: monopoles (π_2 = 0 bare; junction-monopoles are geometric only, no topological charge). This CONFIRMS Math409-AddD §4 dominant-defect-type identification (walls). Bare soliton mass scales: wall ~ 4e19 GeV, string ~ 1e20, monopole ~ 1.6e20, texture ~ 4e21 — all super-Planckian within O(100). Pillar 11 lever (i) CLOSED at topology level. Direction B (mass = dissolution resistance) gate F-B: SM fermion masses span 5e-4 to 173 GeV; Planck-to-top gap = 17 orders of magnitude. BARE Direction B (TECT soliton = SM fermion) REFUTED. Direction B with Higgs-cascade dressing (LRSM v_R ~ 1e14, v_EW ~ 246, combined factor 10^-16.7 matches gap within order) T2 CONJECTURE only — Math408 LRSM cascade gives gauge boson masses NOT fermion Yukawa values; mass-mapping mechanism not supplied. Pillar 11.A T2 UNCHANGED; Pillar 11.B T4 UNCHANGED. Two new follow-ups queued: Math413-AddA-AddA (explicit Higgs-cascade fermion-dressing mechanism) + Math413-AddB (Frogatt-Nielsen flavour-symmetry investigation).
  • Math409-AddD-AddC outcomes (2026-05-15, Q-2026-05-15-Wall-DM-Bias-Mechanism CLOSED): Friedland-Murayama-Perelstein 2003 wall-DM CMB constraint quantitatively evaluated for TECT canonical params. Wall-DM bare hypothesis REFUTED by factor 10^60 (verified by Math409_AddD_AddC_wall_DM_constraint.py). Three bias mechanisms evaluated: (a) cubic-sym-breaking requires epsilon_bias ~ 10^-93 GeV^-1, rho_bias ~ 6e-17 GeV^4 = 10^31 above Lambda — severe fine-tuning; (b) inflation dilution NOT a bias (walls re-dominate later); (c) sector coupling to LRSM W_R requires g ~ 10^-13.5 (vs g_EW ~ 0.65) — extreme fine-tuning. NO mechanism reaches T3. Pillar 11.A T2 -> T1 OPEN. Operator's Math407-AddA Round-2 audit foreseeing 11.A's difficulty CONFIRMED. Math409-AddD-AddD (texture-DM alternative) queued — last chance to rescue 11.A.
  • Math409-AddD-AddD outcomes (2026-05-26, Q-2026-05-15-Texture-DM-Alternative CLOSED PROVISIONAL): texture-DM evaluated as Pillar 11.A rescue alternative to refuted wall-DM. Verdict: T1 -> T2 PROVISIONAL pending Math409-AddH. Four-axis test result: (i) Topology PASS — pi_3(O(8)/O_h) = Z (Math413-AddA §2 confirmed), textures are codim-4 Hopf-instanton defects in 3+1D, topologically protected as Belavin-Polyakov class. (ii) Derrick PASS — TECT Lagrangian intrinsically contains Brazovskii (k^2-k_*^2)^2 dispersion which generates a Skyrme-class (∇^2 phi)^2 term cancelling Derrick scaling (rescaling argument: 0 = (D-2)E_2 + (D-6)E_S admits stable solitons at D=3 with finite r_tex). (iii) FMP CMB constraint inapplicable — Friedland-Murayama-Perelstein 2003 is a wall-domain CMB constraint (sigma_wall H bound); textures are point-like codim-4 defects with cold-matter scaling rho ~ a^-3 not sigma * H scaling. (iv) Late-time relic PASS — cold-matter dilution avoids the FMP-class overproduction failure of walls. Quantitative (Math409_AddD_AddD_texture_DM_constraints.py): r_tex = 2.43 ell_Pl < r_Sch = 63.2 ell_Pl (M_tex = 3.86e20 GeV ~ 31.6 M_Pl, no Schwarzschild collapse); required N_e = 95.6 e-folds (extreme but feasible: 10^49.4 more permissive than walls); texture-DM dilution requirement is 10^49.4 less stringent than wall-DM. PROVISIONAL caveat: M_tex ~ 31.6 M_Pl makes textures behave as primordial super-Planck-mass objects; Hawking-stability (or whatever replaces it in the Brazovskii regime) MUST be verified before T2 -> T3 PROOF SKETCH. Math409-AddH queued as critical dependency. Operator audit decision pending.
  • Math409-AddH outcomes (2026-05-26, Q-2026-05-26-Math409-AddH-Hawking-Stability CLOSED): three decay-channel analysis for super-Planck-mass (M_tex ~ 31.6 M_Pl) topological textures. (a) Hawking-class radiation INAPPLICABLE — r_Sch(M_tex)/r_tex = 26.0 >> 1 means no event horizon; Schwinger-analog Brazovskii pair-production additionally inapplicable since Goldstones are neutral scalars (no Schwinger U(1) coupling) and Compton wavelength comparable to texture size (violates slow-varying-background assumption). Verdict: tau^(a) = infty under standard-physics null hypothesis; T3 with explicit QG gap. (b) Quantum tunnelling SUPPRESSED — Coleman-de Luccia bounce action S_bounce = 2pi^2 * M_tex * r_tex / hbar = 1516; tunnelling rate ~ omega_* * exp(-1516) ~ 10^(-614) Hz; lifetime tau^(b) ~ 10^614 s ~ 10^596 * tau_0. Even with factor-10 prefactor uncertainty (S_bounce -> 150), tau^(b) ~ 10^60 s ~ 10^42 tau_0 — still overwhelmingly stable. T3 (prefactor uncertainty). (c) Defect-antidefect annihilation NEGLIGIBLE — sigma_ann ~ pi r_tex^2 ~ 4.8e-69 m^2; n_tex = Omega_DM rho_crit / M_tex ~ 3.3e-21/m^3; Gamma = n sigma v_rel ~ 3.2e-84/s; tau^(c) ~ 3e83 s ~ 7e65 tau_0. Robust even with 10^16 enhancement from halo concentration + topological-attractive forces. T6 (rigorous kinematics). Composite: min over channels = tau^(c) ~ 10^66 tau_0; tier_composite = T3 PROOF SKETCH. Pillar 11.A: T2 PROVISIONAL -> T3 PROOF SKETCH; PROVISIONAL qualifier LIFTED. Remaining OPEN GAP alpha (blocking T3 -> T4): Wheeler-DeWitt or LQG-analog stability theorem for super-Planck topological solitons in the Brazovskii regime. Math409-AddH-AddA queued (target 2026-07-15). Math409-AddH-AddB queued for Hopf-instanton CMB temperature anisotropy spectrum vs Planck 2018 (target 2026-08-15).
  • AUDIT ROLLBACK 2026-05-26 (operator adversarial review of Math409-AddH): Math409-AddH T2 PROVISIONAL -> T3 PROOF SKETCH promotion REVERSED. Operator identified that Math409-AddH (and upstream Math409-AddD-AddD) mis-read the compactness inequality r_tex < r_Sch as "no horizon" — the CORRECT classical-GR reading is that texture lies INSIDE its Schwarzschild radius, so horizon forms and standard black-hole collapse + Hawking evaporation (t_evap ~ 1.7e-39 s for M_tex = 31.6 M_Pl) is the expected fate. Channel (a) is therefore NOT INAPPLICABLE but REFUTING. Compactness inequality (correct form): M_tex < r_tex/(2 ell_Pl) M_Pl, requires M_tex < 1.215 M_Pl for current r_tex = 2.43 ell_Pl; actual M_tex = 31.6 M_Pl is over-compact by factor 26. Pillar 11.A tier ROLLED BACK from T3 to T2 PROVISIONAL. Math409-AddH-AddA scope REVISED from "Wheeler-DeWitt QG stability theorem" to "compactness-corrected stability theorem" — the decisive test is either (i) demonstrate TECT-internal mechanism preventing horizon formation for sub-Schwarzschild topological solitons, or (ii) re-derive (r_tex, M_tex) from Brazovskii physics yielding r_tex >= r_Sch. Verified by Codes/supplementary/Math409_AddH_compactness_audit.py (5/5 self-test asserts PASS). NEGATIVE-RESULTS entry: R-2026-05-26-Math409-AddH-CompactnessSignError. Both Math409-AddD-AddD and Math409-AddH carry AUDIT-FLAG headers; cited "(no Schwarzschild collapse)" / "no horizon exists" phrases must NOT be cited; use compactness audit script verdict instead. Operator audit successful; CLAUDE.md §6.3.4 quantitative-sanity-check (sign-direction physics) and §6.3.5(a) self-adversarial review failed to catch this defect upstream.
  • Math409-AddH-AddA outcomes (2026-05-26, Q-Math409-AddH-AddA-Compactness PARTIAL CLOSE): five-sub-path compactness investigation post-audit-rollback. (A1) spherical Hopf at canonical TECT: structurally FAIL (compactness violation factor 26 at N=1; higher N makes it worse since M ~ N^2 grows faster than r ~ N^(1/3)). (A2) parameter-space r_R reduction: FAIL (requires r_R < 0.023 vs canonical 0.42 = factor 18 reduction; TECT axioms A0+A1 fix mu^2 = +0.005, Brazovskii Hartree floor ~0.4 is far above target). (A3) cylindrical hoop-conjecture exploit: FAIL (longitudinal hoop escape requires L_parallel > 2pi r_Sch = 397 ell_Pl; resulting L_perp = 0.226 ell_Pl is sub-Planck, classical-field description breaks down). (B1) topological obstruction / Wheeler geon: NO THEOREM (speculative; 70+ years of GR research has not produced concrete Hopf-class geon construction). (B2) Brazovskii-medium running-G / asymptotic safety: IDENTIFIED as principal viable path. Standard Reuter ansatz (omega=1) at k_* = 1/r_tex ~ 0.41 M_Pl gives G_eff/G_obs = 0.855 (only 14.5% suppression); required suppression < 1/26 = 0.0385. TECT-Brazovskii q_*-scale enhancement needs additional factor 22x over standard Reuter. Math409-AddH-AddA-AddA queued (explicit TECT-Brazovskii running-G calculation from Math110-AddI condensate elastic modulus + Math200-AddC 1-loop running, target 2026-08-15). Verified by Math409_AddH_AddA_path_analysis.py (8/8 self-test asserts PASS including mandatory POSTMORTEM §8.7 Schwarzschild check). Pillar 11.A tier UNCHANGED at T2 PROVISIONAL; promotion path now constrained to single explicit calculation. Confidence T2->T3 within 6 months: 20-25%; T2 -> T1 OPEN terminal (B2 FAIL): ~50%.
  • Math412 outcomes (2026-05-26, NEUTRAL post-operator-correction 2026-05-26 UTC): RH sector + U(1){B-L} origin + nu_R relic abundance via gravitational reheating. U(1){B-L} TECT-derived from SO(10) -> SU(5) x U(1)_chi chain (Math157 16-rep anomaly-free); nu_R = SM singlet with (B-L)=+1. M_nu_R = y_nu_R * v_R ~ 10^14 GeV at natural Yukawa. Required T_reh for Omega_nu_R h^2 = 0.12: T_reh in [3e12, 2e13] GeV (closure window). TECT inflation T_reh achievable via inflaton-matter Yukawa y_inf-matter ~ 1e-4 to 7e-4 (Math412-AddA explicit derivation queued; G3-A abundance gate). EXISTENCE-AS-DM GATE G3-B (Math412-AddB): nu_R cosmological stability at M10^14 GeV catastrophically fails — standard active-sterile mixing y_mix1e-12 gives tau_nu_R ~ 4e-38 s << tau_0; warm-DM keV regime requires extreme y_nu_R1e-23 tuning. The note is structurally favourable on abundance + origin but NEUTRAL overall because stability is unresolved and is the dominant constraint. Pillar 11.B tier UNCHANGED at T4 STRONG EVIDENCE. T5 promotion conditional on BOTH (G3-A) AND (G3-B) PASS — joint confidence 20-25% (dominated by stability gate). Verified by Math412_nuR_relic_abundance.py 8/8 self-test asserts PASS. Operator adversarial review 2026-05-26 UTC corrected: "abundance-compatible but not yet DM-compatible because cosmological stability is unresolved" is the canonical wording; AddA-only closure is insufficient.
  • Math412-AddB outcomes (2026-05-26, gate G3-B INTERIM NEGATIVE): nu_R cosmological stability calculation across mass regimes (keV/MeV/GeV/TeV/10^14 GeV) using seesaw-I active-sterile mixing sin^2(2θ) ≈ 4 m_active / M_R. At TECT-natural LRSM scale M_R = 10^14 GeV: sin^2(2θ) = 2e-24, Γ = 4.6e32 GeV, τ = 1.4e-57 s — CATASTROPHIC FAIL by factor 3e74 vs τ_0. At keV seesaw: lifetime stable (τ=1.4e23 s) but X-ray bound EXCLUDES mixing (2e-4 >> 1e-8). νMSM-tuned regime (M=1 keV, sin^2(2θ)=1e-9): STABLE (τ=2.9e28 s > τ_0) but requires y_νR = 1e-20 at v_R=10^14 GeV — factor 1e20 below natural O(1), TECT does NOT naturally provide this. Verdict: G3-B FAIL at standard analysis level; νMSM PASS conditional on extreme TECT-internal tuning. Pillar 11.B bulk-DM via ν_R REFUTED at TECT-natural mass; subdominant defect relic survives but cannot serve as dominant DM. Math412-AddB-AddA (inverse-seesaw rescue) + Math412-AddB-AddB (topological decay-protection via π_1 cosmic string winding) queued. Pillar 11.B tier UNCHANGED at T4 STRONG EVIDENCE but DM-prediction capability structurally weakened. Verified by Math412_AddB_nuR_stability.py 8/8 self-test asserts PASS. CONFIRMS operator adversarial prediction 2026-05-26 UTC.
  • Math412-AddA outcomes (2026-05-26, gate G3-A INTERIM NEGATIVE per operator correction): TECT-derived inflaton-matter coupling derivation from BCC channel structure. A_1g breathing mode identified as inflaton candidate; corrected dimensional analysis (§3 honest re-derivation) gives y_inf-matter ~ λ_3^BCC × m_inf/M_Pl^3 ~ 1.5e-7 (NOT the 1e-4 to 1e-3 of initial draft — wrong dimensional denominator). T_reh^Yukawa ~ 5e9 GeV; gravitational T_reh^grav ~ 2.9e11 GeV; combined ~2.9e11 GeV. Closure window [3e12, 2e13] GeV; shortfall factor 10. Gate G3-A INTERIM NEGATIVE. Combined with G3-B INTERIM NEGATIVE (Math412-AddB): Pillar 11.B ν_R-bulk-DM standard route DOUBLY BLOCKED — both abundance and stability fail at TECT-natural parameters. Joint closure requires simultaneous retrofit (Math412-AddA-AddA explicit LRSM-mediated coupling + Math412-AddB-AddA inverse-seesaw rescue) — structurally unlikely without non-TECT-natural inputs. Pillar 11.B framework retained at T4 only via subdominant defect relic (~10^-30, not dominant); ν_R-as-dominant-bulk-DM REFUTED at standard analysis. Verified by Math412_AddA_T_reh.py 7/7 self-test asserts PASS. This is a negative-diagnostic note. Composite Pillar 11.B verdict post-Math412-AddA + AddB: NEGATIVE-NEGATIVE on ν_R-bulk-DM, T4 only via subdominant relic.
  • Math412-AddB-AddB-AddA: Path A (BCC-channel + Delta_R neutral + H_R charged) Z_2 charge assignment STRUCTURALLY VIABLE; Path B (SO(10) chirality embedding) REJECTED. Pillar 11.B T5 promotion gate: Math412-AddB-AddB-AddA-AddA queued for explicit Math408 LRSM cascade Phi-bidoublet Z_2 decomposition verification (~50% confidence). Joint closure probability (mechanism + assignment + cosmological): 25-35% -> 35-45%.

Key Math notes

Open questions

See Open-questions for active items tagged with P11. GitHub Issues with the same label provide discussion threads: issues with P11 label.

Source of truth

Canonical pillar tier: Codes/config/pillar_status.json (raw). Per-pillar narrative: Docs/status/TOE-FACT-SHEET.md Stage-1 scorecard (raw). Updates propagate through the cascade pipeline (CLAUDE.md §18) — do not edit this wiki page directly.

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