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Publication history: 2026-06-25

Original publication entries retain their wording and recorded status. Follow the current research history for later findings and revisions.

39 / 56 days

25 Jun 2026

13 updates
  1. Horizon & BH
    External-works references appendix — web-verified

    Details

    • New shared appendix tex-artefacts/tfpt_references.tex (“External works built upon”), \input into the six math-heavy documents (tfpt_1, tfpt_2, tfpt_4, tfpt_horizon_readouts, origin_theory, tfpt_research_contracts). The active papers named the foundational results they build on by author only, with no reference list; the appendix supplies web-verified locators (journal, volume, year / arXiv) grouped by topic: modular theory & AQFT (Tomita–Takesaki LNM 128 1970; Bisognano–Wichmann J. Math. Phys. 16/17; Osterwalder–Schrader CMP 31/42; Doplicher–Haag–Roberts CMP 23/35; Haag–Ruelle; Longo CMP 126/130; Kawahigashi–Longo–Müger CMP 219), conformal nets/VOAs/lattices (Frenkel–Kac Invent. Math. 62; Segal CMP 80; MMST; Adamo–Moriwaki–Tanimoto; Adamo–Giorgetti–Tanimoto 2506.01008/2508.07109; Naaijkens–Penneys–Wallick), lattices/quadratic forms (Conway–Sloane; Siegel Ann. Math. 36 1935), singularities (Brieskorn Invent. Math. 2 1966; Milnor AMS 61 1968) and spectral geometry/gravity/thermal time (Chamseddine–Connes CMP 186; Connes–Rovelli CQG 11; Wald PRD 48; Casini–Huerta–Myers JHEP 05 2011). Added to the make_manifest.py TEX list. No claim or status change — a provenance/citation completeness pass.
  2. Seam & QGEO
    External AQFT citations + the four-fold \mu_4 identity sharpened

    Details

    • Two recent Adamo–Giorgetti–Tanimoto theorems cited where the keystone builds on them. The open SEAM.EQUIV.01 continuum/realisation legs (v336) now cite, alongside MMST and Adamo–Moriwaki–Tanimoto (arXiv:2407.18222, already in use): (a) the construction and classification of two-dimensional conformal nets from even lattices (arXiv:2506.01008, 2025) — “the lattice is a conformal net” is a theorem of the same operator-algebraic lineage, strengthening the lattice-realisation leg; and (b) automatic positive-energy/diffeomorphism covariance of conformal-net representations (arXiv:2508.07109, 2025), which defuses the “Bisognano–Wichmann presupposes the covariance it should produce” circularity (v215) on the intrinsic-BW residual (v424). Added to v336 docstring/registry, the SEAM.EQUIV.CONTINUUM.01 ledger external_data, and tfpt_research_contracts.
    • The four-fold \mu_4 identity made explicit (v422). v422 already proved that the seam deck/divisor \mu_4, the carrier Galois group \operatorname{Gal}(\mathbb Q(\zeta_5)), the discriminant \operatorname{disc}(D_5){=}\operatorname{disc}(A_3){=}\Z/4 and the (E_8)_1 simple-current glue are ONE cyclic \Z/4 (with the cyclic-vs-Klein negative control). It is now named the four-fold \mu_4 identity and typed, per v428, as a compression (one object read four ways), NOT four independent witnesses – in v422, the registry, the SEAM.GALOIS.NET.01 ledger row and origin_theory. (No new module: a separate v429 would have duplicated v422 and violated the very compression discipline.)
  3. Seam & QGEO
    v428 — honest self-correction of the over-determination map

    Details

    • v428 (OVERDET.WITNESS.RECLASS.01) — applying v427's own multiply-vs-compress test to v427 itself. A pattern search and the right objection exposed that calling the seven arithmetic witnesses “disjoint grammars that multiply” overstates independence. [E]by the Brieskorn classification (v236) the (2,3,5) singularity is the ONE generator behind the order-30 clock (Milnor \mu{=}(2{-}1)(3{-}1)(5{-}1){=}8{=}\operatorname{rank}E_8, 30{=}2{\cdot}3{\cdot}5 {=}h(E_8), \varphi(30){=}8), so each witness is a facet of that same object (primes 2,3,5 of 30, the clock 30, or E_8): all seven collapse to one generator \Rightarrow compression, not seven multiplications. [E]what genuinely multiplies is the input forced four independent ways — the “8” in \cthree{=}1/(8\pi) from \operatorname{rank}E_8, h(D_5){=}2(5{-}1), \varphi(30) and the Milnor number — plus the foreign witness \ainv{\approx}137. [C]refines and partly walks back v427, consistent with v236/v305/v313.
    • Surfaces corrected (same change). v427's prose, the ledger row (OVERDET.WITNESS.MAP.01, now noting the refinement), tfpt_safeguards Layer 3 (the seven compress one object; a second keybox states what multiplies), the witness-map figure (redrawn), the website (papers.ts, Safeguards.tsx, release.ts, the intro-video description) and the Zenodo description. The over-determination claim is now the narrower, harder-to-attack “one richly over-determined input + a foreign readout”.
  4. Infrastructure
    Safeguards visuals + paper expansion + the website safeguards band

    Details

    • Five safeguards figures (make_figures.py, in manifest.sha256): the layered defense, the \alpha^{-1} uniqueness scan (one F_{U(1)} variant of 94{,}500 in the CODATA window), the reverse audit (3/8 vs 5/8), the witness-independence map (seven disjoint grammars), and the look-elsewhere null model (13/13 vs \le5/13).
    • tfpt_safeguards expanded. Layer 5 now states the null model explicitly (\prod p_i{=}10^{-25.8}, 2{\times}200k pseudo-theories \le5/13, \phiz\pm10\%\!\to\!6/13, shuffle 1.2) and the \alpha uniqueness (\le10^{-30.7}, \sim102 bits, v100), with the data watchdog (v307) and kill board (v321); the five figures are embedded.
    • Website: a prominent animated Safeguards band (components/Safeguards.tsx) in the home flow (after the trust beat) — headline statistics, the seven layers, the figures, links to the paper and the suite — to address the coincidence/numerology suspicion proactively.
    • Koide flow-time investigation (honest negative). Post-v425: the flow-time is not native and does not reduce to one scale — v99 already rejects a scale reading as look-elsewhere; the only theory-side content is the conditional integer reading t{=}\Nfam{=}3\Rightarrow m_\tau{=}1776.94 MeV, a Belle II kill test. No new module.
  5. Seam & QGEO
    Three closure-strategy modules v425–v427 + the Safeguards paper

    Details

    • v425 (DYN.TRANSFER.UNIVERSAL.01) — the single-flow reduction of F_{\mathrm{transfer}}. The four frontier transfers do not each need a separate external solver: their dynamics is the ONE native flow (the seam modular/recovery semigroup, v238/v240, gap 6\ln(3/2)) restricted to each sector. [E]the v99 Koide generator linearised at q{=}2 has rate -\Delta, time-1 contraction (2/3)^6; Boltzmann washout is that contraction (CP phase 4\pi/3 native, v320); QCD rate b_3{=}-7. [C]only the anchors (v_{\mathrm{geo}}, C_p, M_R) stay external; F_{\mathrm{relic}}'s cosmological IC \theta_i is the one wall. A reduction (extends v383 static\todynamic), not internalisation; the firewall holds.
    • v426 (SEAM.EQUIV.LTORP.KMS.01) — the invertible \beta{=}1-KMS variant of (LTO-RP). The corner NPW26 leave open (they prove it for topologically-ordered models; the seam is invertible + KMS, not a trace). [E]a gapped reflection-symmetric \mu_4-even collar at \beta{=}1 KMS satisfies \Theta K\Theta^{-1}{=}{-}K (u_\Theta{=}J) and [\rho,K]{=}0, so (LTO-RP) holds by direct Tomita–Takesaki; |\!\det\mathrm{Cartan}(E_8)|{=}1 (no anyons), state not a trace; neg controls have teeth. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O](continuum sub-step (i) remains).
    • v427 (OVERDET.WITNESS.MAP.01) — the honest over-determination accounting. A reviewer correction made machine-explicit: witnesses from disjoint grammars multiply, projections of the same generator compress. [E]seven disjoint-grammar witnesses each reproduce a carrier-skeleton integer from their own structure — Gauss N(3{+}2i){=}13, Eisenstein N(3{+}2\omega){=}7, cyclotomy N(3{+}2\zeta_5){=}55, Galois |(\Z/5)^\times|{=}4, |\!\det\mathrm{Cartan}(E_8)|{=}1, Pascal \binom40{+}\binom41{+}\binom42{=}11 (2^4{=}16), Coxeter h(E_8){=}30{=}2{\cdot}3{\cdot}5 with \varphi(30){=}8 — and the anchor a{=}(1,1,2) is the compression generator. Extends v305.
    • New paper tfpt_safeguards (companion S). A dedicated methodology paper collecting, in one language, every mechanism by which TFPT defends a claim against coincidence and numerology: the status calculus + ledger + audit, no-free-pattern + reverse audit, the over-determination map (v427), the firewall + No-Unit theorem, frozen predictions + null model, the independent Wolfram and Lean paths, and the red team — with a final kill-test summary. Registered in build.sh, make_docs_map.py, make_manifest.py and the document-set box; the document set is now eleven deliverables (five numbered papers + four companions + introduction + changelog).
    • Surfaces synced. run_all.py (421 modules), the registry, the ledger, tfpt_research_contracts (the F_{\mathrm{transfer}} and keystone sections), tfpt_safeguards; manifests refreshed.
  6. Seam & QGEO
    The (LTO-RP) reduction of the keystone SEAM.EQUIV.01
    v424

    Summary

    v424

    Details

    • New module v424 (SEAM.EQUIV.LTORP.01). An honest MMST-style reduction of the one open keystone — SEAM.EQUIV.01's intrinsic Bisognano–Wichmann lemma (the v308/v309 residual) — to the recent commuting-projector theorem of Naaijkens–Penneys–Wallick (arXiv:2605.10693, [NPW26]), with the central mapping corrected. [E]the \mu_4 clock is a modular symmetry ([\rho,K]{=}0, v309/v199), not the flow — \sigma^\psi{=}\rho is type-wrong (a one-parameter group versus a finite order-4 element). [E]NPW26's axiom (LTO-RP) means u_\Theta{=}J (reflection = modular conjugation) = the intrinsic BW condition \theta K\theta{=}-K, distinct from \omega{\circ}\rho{=}\omega: a positive character-block-diagonal K has [\rho,K]{=}0 yet never \theta K\theta{=}-K, so clock-invariance does not imply (LTO-RP) (refuting that identification). [O]make-or-break: NPW26 prove (LTO-RP) for topologically ordered models (Toric Code |\det K|{=}4 via a trace; Levin–Wen anyonic), while the seam is invertible (|\det\mathrm{Cartan}(E_8)|{=}1, no anyons) and a \beta{=}1 KMS state (not a trace) — in neither proved bucket. [C]verdict: the BW residual reduces to NPW26 Theorem A modulo two open sub-steps (realise the seam as a \Z_2-reflection commuting-projector LTO; extend (LTO-RP) to the invertible KMS case) — a reduction, not a closure; SEAM.EQUIV.01 stays [O]. Cited in tfpt_research_contracts; Python-only (numpy; the det discriminators mirrored via v89/v281).
    • Validation provenance. The three pivotal references of the closure-strategy note were independently verified real and correctly described: NPW26 (arXiv:2605.10693), the RP–Yang–Mills complete-monotonicity reconstruction (Int. J. Geom. Meth. Mod. Phys. 2026), and Marolf (arXiv:2203.07421).
    • Surfaces synced. run_all.py, the script registry, the status ledger and the tfpt_research_contracts keystone section; manifests refreshed.
  7. Gravity & cosmology
    No-Ambient-Dependency
    No frozen readout is computed from QG.AMB.01: v423

    Summary

    No frozen readout is computed from QG.AMB.01: v423

    Details

    • New module v423 (QGAMB.NODEP.01). Sharpens the v369/v384 prose redundancy (“QG.AMB.01 is a [C]redundancy, not dynamics”) into a machine-checked DAG{+}script fact, told honestly. [E]gap-decoupling margin \Delta_{\mathrm{eff}}=6\ln\tfrac32-31/(4\pi^2)\approx1.648>0 (31=2^{\gcar}-1), susceptibility \chi=729/665 finite (negative control \mathrm{gap}\!\to\!0\Rightarrow\chi\!\to\!\infty); [E] QG.AMB.01 is a certification sink (every claim that lists it is a gravity-gate/QG row, never a frozen prediction) and not a value source (no producing script, while \alpha^{-1}, the mixing angles and cosmology are each computed by a compiler script, v3/v9/v268/v7); [E] the one transitive link is architectural — of the frozen readouts only \theta_{13} references it, via the 4d-QFT contract spine (\texttt{PS.DIRAC}{\to}\texttt{CONTRACT.QFT4D}{\to}\texttt{QG.G2}{\to}\texttt{GATE.METRIC}), a coherence edge, not a numerical input. [C]verdict: QG.AMB.01 is gap-decoupled certification — v369 as a DAG/script fact; honest scope — this does not discharge SEAM.EQUIV.01, on which the readouts still depend. Cited in tfpt_4; Python-only (gap algebra mirrored via v337).
    • Paper sharpening (seam selector, up front). The tfpt_4 quantum-gravity keybox now states explicitly that the open SEAM.EQUIV.01 premise is the holomorphic |\det K|{=}1 point that selects (E_8)_1 — the central charge c{=}8 alone does not, being shared by the non-holomorphic SO(16)_1 counterexample (v277/v281).
    • Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, tfpt_4; manifests refreshed.
  8. Seam & QGEO
    The Galois\leftrightarrowNet bridge
    v422

    Summary

    v422

    Details

    • New module v422 (SEAM.GALOIS.NET.01). The seam \mu_4=\operatorname{Gal}(\mathbb Q(\zeta_5)) is the SAME cyclic \Z/4 as the (E_8)_1 simple-current glue — not a mere order-4 coincidence — bridging the Galois gearbox of v419 to G_{\mathrm{net}}/% SEAM.EQUIV.01. [E]\operatorname{disc}(A_3)=\operatorname{disc}(D_5)=\Z/4 (one Smith invariant factor 4 = cyclic, \det4); D_n disc is \Z/4 for n odd, \Z_2{\times}\Z_2 for n even, so the carrier D_5 (rank 5=\gcar, odd) is cyclic while D_8 (rank 8, even) is Klein; [E] (\Z/5)^\times=\langle2\rangle is cyclic order 4 (only the carrier prime 5 gives order 4); [E]in \Z_4{\times}\Z_4 (16=\dim S^+) the glue (1,1) is order 4 with Lagrangian quotient 16/4^2=1=(E_8)_1 (v89/v154); [E]negative control: the Klein / order-2 \langle(2,2)\rangle gives 16/2^2=4=|\!\operatorname{disc}(D_8)|=SO(16) (\det4), NOT E_8 (\det1) — cyclic \Z/4\to E_8, Klein\to D_8. [C]the bridge: one cyclic \Z/4 threads the seam deck \mu_4 (v419), \operatorname{Gal}(\mathbb Q(\zeta_5)) and the (E_8)_1 glue — the Galois gearbox (\Z/30)^\times IS the holomorphic net's glue, cyclicity forced by 5, Klein excluded. Cited in origin_theory; Wolfram-mirrored (324\to327).
    • Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, origin_theory; the Wolfram extension/README/ledger counts (327/327) and the website verification count; manifests refreshed.
  9. Seam & QGEO
    F_pole external
    The Koide source\topole transfer is one-loop EM: v420

    Summary

    The Koide source\topole transfer is one-loop EM: v420

    Details

    • New module v420 (FR.POLE.QED.01). The honest external-physics follow-up to v93 and the clock No-Go: the Koide source\topole transfer is a one-loop electromagnetic effect, not an algebraic step. [E]Koide Q is rescaling-invariant (Q(c\,m)=Q(m)), so flavor-universal running leaves it fixed and the source\topole shift is purely flavor-split; [C]the tree deficit \tfrac23-Q_{\mathrm{src}}=0.00220 is one-loop-EM-sized (=\varphi_0/24 to 0.6\%, =\alpha/\pi to {\sim}5\%), the PDG pole sits at 2/3 (the IR attractor); [C]a 1-loop QED \mathrm{pole}\!\leftrightarrow\!\overline{\mathrm{MS}} per-lepton-log estimate shifts Q by EM-loop order toward 2/3 (the v82 direction); [O]the exact flow time t\approx2.84 (v93) is scheme/scale-dependent external QED, only sized. [C]verdict: F_{\mathrm{pole}} is a one-loop EM effect — the lens forces the rate (2/3)^6 (v327), external EM supplies the flow; F_pole stays [C], firewalled (v187), so F_{\mathrm{transfer}} is confirmed genuinely external — the open theme sharpened (QED), not closed. Cited in tfpt_4; numerical (mpmath), Python-only (no Wolfram check, by the suite convention like v93/v371).
    • Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, the Koide section of tfpt_4; manifests refreshed.
  10. Seam & QGEO
    The seam \mu_4 is the carrier's Galois group
    v419 + figure

    Summary

    v419 + figure

    Details

    • New module v419 (SEAM.GALOIS.01). A positive resolution of the cyclic/Galois asymmetry (v409, RES.COXETER.SYMMETRY.01): why the seam \mu_4 sits on the Galois side, not on the cyclic \zeta_{30} dial. [E]h(E_8)=30 is squarefree, so \Z/30 has element orders \{1,2,3,5,6,10,15,30\} — no order 4, so the seam \mu_4 cannot be a rotation hand and is forced onto the Galois side; [E]by CRT (\Z/30)^\times=(\Z/2)^\times\times(\Z/3)^\times\times(\Z/5)^\times =\mu_4\times\Z_2 (order \varphi(30)=8=\operatorname{rank}E_8), and the \mu_4 (\Z/4) factor IS (\Z/5)^\times (the carrier prime 5), the \Z_2 IS (\Z/3)^\times (family 3 = CP conjugation, v316); [E]so the seam \mu_4=\operatorname{Gal}(\mathbb Q(\zeta_5)), realised on the carrier clock C_5 (v418) by the Frobenius \zeta_5\mapsto\zeta_5^2 — the explicit order-4 operator G with GC_5G^{-1}=C_5^2, G^4=I, G^2C_5G^{-2}=C_5^{-1}. [C]this is the positive content of v409: the 2^2{=}4 is the order of (\Z/5)^\times, and “no contraction rate” is because the seam is an automorphism, not a hand; so \mathbb Q(\zeta_{30}) carries both E_8 invariants (cyclic h{=}30, Galois \operatorname{rank}{=}8). Cited in origin_theory; Wolfram-mirrored (321\to324).
    • New figure coxeter_galois.pdf. The order-30 cyclic dial with the four flavor hands (sheet/family/carrier/CP) beside the Galois gears (\mu_4=\operatorname{Aut}(\zeta_5) the seam, \Z_2=\operatorname{Aut}(\zeta_3) the CP conjugation); added to make_figures.py, the manifest figure list and origin_theory.
    • Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, the Wolfram extension/README/ledger counts (324/324) and the website verification count; manifests refreshed.
  11. Seam & QGEO
    The cyclotomic norm triple + the carrier-5 clock
    v418

    Summary

    v418

    Details

    • New module v418 (ARITH.NORMTRIPLE.01). The carrier split (3,2) read in the three atom-rings gives (7,13,55)=(scalaron, \Delta_Q, quark numerator), and the carrier-5 clock missing from v417 is found as the 4{\times}4 \Phi_5-companion C_5. [E]C_5^5{=}I, \operatorname{tr}C_5{=}{-}1, \det C_5{=}1 (the four primitive 5th roots), with the golden ratio as its real-part data (2\cos72^\circ{=}1/\varphi, 2\cos144^\circ{=}{-}\varphi; output-side, v313/v349; dim 4, not 3 — this refines v417's gap from “no operator” to “no 3{\times}3 operator”); [E] N_{\Z[\zeta_5]}(3{+}2\zeta_5){=}\det(3I{+}2C_5){=}55{=}5{\cdot}11{=}\gcar\|\operatorname{Pl}(K)\|_1, the c_u/c_d numerator (v410/v411); [E]the triple \det(3I{+}2\,\mathrm{Comp}(\Phi_n))=(7,13,55) for n{=}3,4,5 over \Z[\omega],\Z[i],\Z[\zeta_5] (atoms 3,2,5, v416); [E]NEG control: the anchor (5,4)\to(21,41,461) reaches named values only in the \omega- and i-rings (21{=}N_\omega(5,4), 41{=}10b_1, v222) but 461 (prime) in the carrier ring, so the carrier ring separates (3,2)\to55 from (5,4)\to461. [C]honest: (3,2) is the forced carrier split (v14), not the unique clean one — a scan finds \{(2,1)\to(3,5,11),(3,2)\to(7,13,55)\}, a small ladder. The sharper discriminator is cross-sector: only (3,2) spans three physics sectors (gravity 7, flavor 13, masses 55), the unique cross-sector seed, while (2,1) is all compiler-core. Cited in origin_theory; Wolfram-mirrored (318\to321).
    • Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, a triple paragraph in origin_theory, the Wolfram extension/README/ledger counts (321/321) and the website verification count; manifests refreshed.
  12. Seam & QGEO
    The Eisenstein/CP operator + the order-30 clock
    v417

    Summary

    v417

    Details

    • New module v417 (DIAMOND.EISEN.01). The hexagonal dual of v415: the \tau{=}\omega flavor side put on operators and unified with the seam into one order-30 Coxeter clock. [E]the family rotation P=(1\,2\,3) is the order-3 Eisenstein deck (P^2{+}P{+}I=\mathrm{ONES}, so \omega^2{+}\omega{+}1{=}0 on the plane \perp(1,1,1)), the order-3 dual of J^2{=}{-}I; [E]it realises the hex CM norm as (3I{+}2P)(3I{+}2P^2)=7I{+}6\,\mathrm{ONES}, \operatorname{Spec}\{7,7,25\} (7{=}N_{\Z[\omega]}(3{+}2\omega){=} scalaron, 25{=}\gcar^2) — the dual of v415's \{13,13,9\}; [E]the CP clock W{=}{-}P^2 (\rho of v233) has W^2{=}P, W^3{=}{-}I, \operatorname{Spec}\{-1,\zeta_6,\zeta_6^{-1}\}, giving \delta_{\mathrm{CKM,lead}}{=}\arg\zeta_6{=}\pi/3 and \delta_{\mathrm{PMNS}}{=}\arg\zeta_6^4{=}4\pi/3 (v316/v320); [E]all flavor clocks are powers of \zeta_{30} (\zeta_{30}^{15}{=}{-}1, \zeta_{30}^{10}{=}\omega, \zeta_{30}^{6}{=}\zeta_5, \zeta_{30}^{5}{=}\zeta_6) while the seam \mu_4{=}i is the Galois side (\Z/30)^\times of order \varphi(30){=}8{=}\operatorname{rank}E_8 — so \mathbb Q(\zeta_{30}) carries both E_8 invariants (h{=}30 and \operatorname{rank}{=}8). [C]honest gap: the carrier \mu_5 has no 3{\times}3 operator (the golden \varphi is an output-side trace, v313/v349); negatives: no U,V-diamond operator carries \zeta_6, [D,P]\neq0 (no \mu_{12}), N(3{+}2\zeta_6){=}19{\neq}7. Cited in tfpt_2; Wolfram-mirrored (315\to318).
    • Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, the Sheet-Diamond keybox of tfpt_2, the Wolfram extension/README/ledger counts (318/318) and the website verification count; manifests refreshed.
  13. Seam & QGEO
    The Gaussian operator + the atom trichotomy
    v415, v416

    Summary

    v415, v416

    Details

    • New module v415 (DIAMOND.GAUSS.01). The square CM-norm dictionary of v222/v230 is operator-realised, and the \mu_4 deck becomes an intrinsic diamond object. [E]the integer commutator J:=\tfrac13[U,V]=\big(\begin{smallmatrix}-1&1&0\\-2&1&0\\-3&1&0\end{smallmatrix}\big) has \operatorname{Spec}(J)=\{i,-i,0\}, J^2=-I on the rank-2 image and J^3+J=0 — a \mu_4 quarter-turn born from the binary V and ternary U compilers (v410/v95); [E]3I+2J and 5I+4J have spectra \{3{+}2i,3{-}2i,3\}, \{5{+}4i,5{-}4i,5\}, so the Gaussian integers 3{+}2i (norm 13{=}\Delta_Q) and 5{+}4i (norm 41{=}10b_1) are literal eigenvalues; [E](aI{+}bJ)(aI{-}bJ) reads (a^2{+}b^2) on the \mu_4-plane and the real part^2 on the kernel (\{13,13,9\}, \{41,41,25\}); [E]the intrinsic order-4 deck D=-I+J-J^2 (D^4=I, \chi_D=(x{+}1)(x^2{+}1)) has \chi_2-line \ker[U,V]=a-\mathbf 1. [C]the geometric-deck identification sharpens (does not close) the v140 \Z_3 residue. Cited in tfpt_2; Wolfram-mirrored.
    • New module v416 (ARITH.TRICHOTOMY.01). The atoms \{2,3,5\} are ramified/inert/split in the two exceptional CM rings. [E]in \Z[i] (square/seam) 2{=}|\Z_2| ramifies, 3{=}\Nfam is inert, 5{=}\gcar splits as (2{+}i)(2{-}i); [E]in \Z[\omega] (hex/flavor) 3{=}\Nfam ramifies, 2,5 are inert; [E]the ramified prime of each ring is its own atom; [E]the three quadratic facets \mathbb Q(i),\mathbb Q(\sqrt{-3}),\mathbb Q(\sqrt5) of K=\mathbb Q(i,\sqrt{-3},\sqrt5) (v403) ramify over exactly one atom each, product 2\cdot3\cdot5=30=h(E_8), two imaginary (CM/dynamic) + one real (RM/static). [C]the seam/flavor/golden reading. Cited in origin_theory; Wolfram-mirrored.
    • Wolfram extension 308\to315 + a pre-existing fix. Added the seven exact checks for v415/v416 and corrected a one-character sign bug in the v412 CharacteristicPolynomial check (Wolfram returns -\chi for odd rank), so tfpt_readouts_extension.wl now runs genuinely clean at 315/315 (GATE.WOLFRAM.02); README and ledger counts updated.
    • Surfaces synced. run_all.py, script_registry.csv and status_ledger.csv updated; a new keybox paragraph in the Sheet-Diamond section of tfpt_2 and a trichotomy sentence in origin_theory; manifests refreshed.