TECT -- Topological Energy Condensate Theory

Last curated: 2026-06-14 (UTC) -- auto-generated mirror of the canonical research repository.

Cite as: Lee, J. (2026). TECT — Topological Energy Condensate Theory (Version v0.1.0-cascade-2026-05-26) [Software]. Zenodo. https://doi.org/10.5281/zenodo.20410377

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[Legacy Archive Repository ]

This repository is preserved as a legacy archive as of 2026-06-05.

The current TECT preview repository is being cleaned up here:

https://github.com/TECT-OpenPhysics/TECT-Preview

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Stage-1 Pillar Scoreboard (post-2026-05-26 C1/C2/C3 reclassification)

TECT positioning (canonical, 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 a non-standard rescue is independently derived (per Math411-AddB §10 operator binding).

Aggregate $S_1$ qualification: 4 × T7 PROVED + 2 × T6 PROVED CONDITIONAL + 1 × T5 CLOSED@1-loop + 3 × T4 STRONG EVIDENCE + 1 × T0/T2 hybrid + 1 × T4|T2-split. Categories: 7 × C1 core / 2 × C2 secondary / 2 × C3 phenomenological (Math411-AddB §10 binding 2026-05-26).

#	Pillar (subject)	Tier	Cat.	Conditional inputs / verdict
1	BCC condensate uniqueness (mass $m^*$)	T6	C1	Math60-Stage 1 hypothesis list (regime-split: shallow T4, deep T0 post-Math350)
2	Lorentz invariance + emergent metric	T6	C1	matter-coupling hierarchy
3	Diffeomorphism invariance / gravity	T5@1-loop	C1	higher-loop pending
4	Gauge group emergence ($U(1)\times SU(2)\times SU(3)$)	T4 aggregate	C2	atomic T6 post-Math268; sub-task 3 T1 INSUFFICIENT post-Math411-AddB cascade (standard routes EXHAUSTED)
5	Standard-model fermion content (chirality)	T7	C1	—
6	GUT unification (SO(10)) / Higgs mechanism	T4 — standard routes EXHAUSTED 2026-05-26	C2	all 4 standard promotion routes REFUTED/INSUFFICIENT (Math410-AddA Pathway B SO(4), Math410 §4 $T_{2u}$ doubling, Math411-AddA Stueckelberg-direct, Math411-AddB Pathway A SO(10)). T5 promotion requires axiom extension (fermion-condensate aux Math411-AddB-AddA, holographic Math411-AddB-AddB)
7	Spin-statistics / quantum consistency	T7	C1	—
8	Equivalence principle / Lorentz emergence	T7	C1	—
9	Geodesic motion	T7	C1	—
10	Quantum completion ($\hbar$ origin)	T0 + T2 hybrid	C3	classical no-go (T0) + phase-transition origin programme (T2)
11	Cosmological / dark-sector phenomenology	T4|T2-split	C3	11.A T2 INTERIM NEG (compactness, Math409-AddH AUDIT); 11.B T4 but DOUBLY BLOCKED at TECT-natural (G3-A Math412-AddA: $T_{\rm reh}$ factor 10 below window; G3-B Math412-AddB: $\nu_R$ stability requires $y\sim 10^{-20}$)

Category legend (Math411-AddB §10 binding): C1 Core emergence (TECT-internal, rigorously derived from Reading-H axioms; 7 pillars at T5/T6/T7) — C2 Secondary emergence (TECT-internal with residual gates; Pillar 6 standard routes EXHAUSTED) — C3 Phenomenological add-on (TECT does NOT predict from internal axioms; matching only).

Detailed per-pillar status, evidence chains, Math note links, and forbidden-phrase audit: status/TOE-FACT-SHEET.md. Forbidden phrases ("TECT is the TOE", "completes the GUT programme", "all 11 pillars closed", "final theory", "complete theory", "completed framework", "completed TOE") are §6.3.5(a) violations and are not used anywhere in this repo; the canonical positioning sentence above is the single allowed framing.

Cascade status (2026-05-26 EOD): Pillar 6 standard promotion programme EXHAUSTED (4 routes all REFUTED/INSUFFICIENT); Pillar 11.B DOUBLY BLOCKED (both abundance G3-A and stability G3-B fail at TECT-natural parameters); TOE-level claim SUSPENDED. See status/research-log.md (2026-05-26 EOD entry) for the full cascade record and status/TOE-FACT-SHEET.md Pillar 11 banner for the canonical positioning sentence.

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A candidate Theory of Everything built on an effectively two-axiom foundation (Math195): one physics axiom — a $\mathbb Z_2$-symmetric continuum order parameter $\Psi$ at the Brazovskii weak-crystallisation point — and one cosmological axiom (the cooling history $T(t)$). Under the post-Math401 Reading H interpretive framework (Math400-AddE Path α confirmed), the BCC structure emerges as the cosmologically-relevant stable fluctuation channel within the Brazovskii fluctuation-stabilised disordered ensemble vacuum; both the channel selection and the disordered vacuum are derivations, not assumptions. Every law of physics in scope must follow as a mathematical or thermodynamic necessity from these two inputs.

What is TECT?

TECT proposes that the Standard Model, General Relativity, and the quantum and cosmological structure of nature all emerge as low-energy consequences of a single object: the order parameter $\Psi$ of an emergent three-dimensional BCC topological condensate, governed by the Brazovskii free energy $\mathcal F[\Psi]$ on its first-order locked branch. Particles are defect excitations of the condensate, gauge fields are Berry connections on the emergent band structure, gravity is a long-wavelength elastic response, and quantum mechanics is the thermodynamic language of fluctuations around the locked vacuum.

The framework distinguishing claim is axiom minimality: after the Math195 reduction, two foundational axioms (one physics + one cosmological), against the much larger publicly stated input lists of SM, GR, GUT, String, LQG, and Wolfram Physics. The price of this minimalism is that essentially every empirical observation must be derived rather than postulated.

The BCC ordering itself is a derivation, not an assumption. TECT postulates only a $\mathbb Z_2$-symmetric continuum field with a Brazovskii spectral minimum on the shell $|\mathbf k|=q_0$; under the post-Math401 Reading H interpretive framework (Math400-AddE one-loop self-consistency confirms Path α at TECT canonical $\mu^2=+0.005$: $r_R = +0.4193 > 0$), the canonical vacuum is the Brazovskii fluctuation-stabilised disordered phase ($\langle\Psi angle=0$, $\langle\Psi^2 angle eq 0$). Within this disordered ensemble vacuum, BCC emerges as the cosmologically-relevant stable fluctuation channel (Math400-AddF: $n_{ m neg}=0$ at canonical, $N=64$ tighter convergence; FCC saddle-6 rejected; lamellar/HEX uniaxial-isotropy-rejected). This is the original Brazovskii (1975) main result, not the commonly-misread mean-field broken-symmetry interpretation. The Theory page makes this emergence explicit. The framework design principle is "minimum assumption + maximally reasonable initial datum", never ad hoc fitting.

This site presents TECT as it actually stands. Proved parts are clearly marked as proved, conditional parts as conditional, retracted parts as retracted. Every status claim links to its evidence file in Docs/math/ and every numerical result carries a unique $R$-tag tied to a code fingerprint. The append-only History page records every retraction.

Latest state snapshot (2026-05-26 EOD — post-Pillar 6/11 cascade + C1/C2/C3 reclassification)

Canonical 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 a non-standard rescue is independently derived (per Math411-AddB §10 operator binding).

Epistemic-category reclassification (Math411-AddB §10 binding):

Category	Definition	Pillars (count)
C1 Core emergence	TECT-internal mechanism rigorously derived from Reading-H axioms	7 — Pillars 1, 2, 3, 5, 7, 8, 9 (T5/T6/T7)
C2 Secondary emergence	TECT-internal mechanism with residual gates	2 — Pillar 4 (mass spectrum); Pillar 6 (gauge group, standard routes EXHAUSTED 2026-05-26)
C3 Phenomenological add-on	TECT does NOT predict from internal axioms	2 — Pillar 10 (ℏ-origin); Pillar 11 (dark matter, DOUBLY BLOCKED at TECT-natural)
Aggregate TOE qualification predicates:

Predicate	State	Verdict
$S_1$ — eleven pillars	PARTIAL	T5+: 7 C1 pillars at T5/T6/T7; C2 Pillar 6 T4 with standard routes EXHAUSTED; C3 Pillars 10 + 11 do not contribute to internal-axiom $S_1$.
$S_2$ — Global Closure + 4 quantum gates	PARTIAL	Pillar 6 standard routes EXHAUSTED ⇒ GAP-2/GAP-3 unconditional closures blocked at standard-framework level; non-standard rescues queued.
$S_3$ — external phenomenology	PROVISIONAL prediction	Math172 Kibble–Zurek GW $\Omega_{\rm GW}\sim 10^{-15}$ at PTA, observable by SKA 2028–2030.
Pillar 6 closure programme — FINAL (all 4 standard promotion routes evaluated, all REFUTED/INSUFFICIENT):

• Math410-AddA (2026-05-26) — Pathway B SO(4) cubic-irrelevance STRUCTURALLY REFUTED at exact-RG ($A^* = +0.018 M_{ m Pl}^2$ across 4 regulators; no NGFP with $A^* < 0$).
• Math410 §4 — Pathway B $T_{2u}$ doubling STRUCTURALLY ABSENT in BCC shells 1–3.
• Math411-AddA — Stueckelberg-direct REFUTED via $\Lambda^2 T_{2g} = T_{1g} \neq T_{2g}$ Lie-algebra closure failure + $A^{\rm asym}{A{2g}} = 0$ via $\pi_1(M_{ m BCC}) = {e}$.
• Math411-AddB — Pathway A direct SO(10) GUT revival STRUCTURALLY INSUFFICIENT within standard BCC/Brazovskii channel framework (dimensional gap shells 1-3 = 20 modes vs SO(10) adj 45; 44 vs 45 NOT a near-miss because Lie-algebra closure requires irrep closure, not scalar dimensional matching; $T_{1g}$ structurally absent from shells 1–3).
• Non-standard rescues queued: Math411-AddB-AddA (fermion-condensate aux channels), Math411-AddB-AddB (holographic / dim-reduction emergence) — both require substantial axiom extension.

Pillar 11.B closure programme — DOUBLY BLOCKED at TECT-natural:

• Math412-AddA (2026-05-26) — G3-A INTERIM NEGATIVE: TECT-natural $T_{ m reh} \sim 3 \times 10^{11}$ GeV is factor 10 below the abundance window $[3 \times 10^{12}, 2 \times 10^{13}]$ GeV.
• Math412-AddB — G3-B INTERIM NEGATIVE: heavy seesaw ($M \sim 10^{14}$ GeV) gives $\tau \sim 10^{-57}$ s catastrophic; νMSM warm-DM keV requires $y \sim 10^{-20}$ extreme tuning.

Forbidden phrases going forward (§6.3.5(a) violations): "TECT is the TOE", "completes the GUT programme", "all 11 pillars closed", "final theory", "complete theory", "completed framework", "completed TOE". The canonical positioning sentence above is the single allowed framing.

For full per-pillar status, the cascade evidence chain, and the C1/C2/C3 framework derivation see Status, Results, and the canonical Math411-AddB §10 archive.

Comparison vs other candidate frameworks (compact)

One-glance positioning of TECT against three reference frameworks: the Standard Model (SM), Superstring / M-theory, and Loop Quantum Gravity (LQG). "Predicts" means the framework derives the listed item from its axioms rather than postulating it externally.

Axis	TECT	Standard Model	Superstring / M	Loop Quantum Gravity
Foundational axioms	2 ($\mathbb Z_2$-symmetric Brazovskii continuum + cooling history $T(t)$); BCC ordering derived; Math195	~26 parameters + chosen gauge group + 3 generations	5–10 (10/11-dim spacetime, branes, compactification, flux, SUSY, vacuum)	4–5 (spin-network kinematics, holonomy/flux algebra, dynamics, semiclassical limit)
Predicts SM gauge group	PARTIAL — standard routes EXHAUSTED 2026-05-26; SO(10) anomaly cancellation persists (Math157); dynamical realisation blocked — Math411-AddB §10	No — postulated	Landscape — no unique selection	No — gauge content imported
Predicts gravity	Yes — emergent spin-2 graviton (Pillar 3 CLOSED@1-loop)	No	Yes — closed-string graviton mode	Yes — quantises GR by construction
Predicts $\hbar$	CONDITIONAL — $\hbar = c^3 a_{\rm BCC}^2/(16\pi G)$ (Math110-AddI)	No	No	No
Predicts $\Lambda$	CONDITIONAL — Math58-v7 four-sector cancellation	No	No	No
Quantum-consistent	Partial — anomaly cancellation conditional on Pillar 4	Yes	Yes (perturbative)	Partial
Falsifiable predictions	3 pre-registered + KZ GW (Math172)	Many	Few	Few
Empirical agreement	Partial — classical sector agrees	Excellent within scope	Untested	Untested
Reading the table: TECT distinguishing claim is parameter compression — the smallest published foundational input list among comparable candidates.

How to navigate this repository

• note/ — TECT Math notes (proof-level derivations).
• paper/ — papers and supplementary material.
• code/ — numerical implementation (PDE solvers, supplementary).
• status/ — canonical scoreboard (TOE-FACT-SHEET, OPEN-QUESTIONS, NEGATIVE-RESULTS).
• site/ — website source mirror.
• CHANGELOG.md — full chronological change log.

All artefacts in this repository are written in English per the output-language policy.

License & maintainer

• License: Creative Commons Attribution 4.0 International (CC BY 4.0) -- see LICENSE for the verbatim legal code. SPDX: CC-BY-4.0.
• Maintainer: Jusang Lee (jtkor@outlook.com).
• Issues / contributions: open an Issue on GitHub or contact via email.
• Citation: see CITATION.cff.