HOUSE

WHAT THIS PAGE IS

TSO has involved retiring and downgrading several of its own claims after scrutiny. This page documents those retirements alongside the results that have held up. It is the honesty page — where failures live alongside successes.

As of May 24, 2026, the framework has retired, downgraded, or narrowed ten specific claims across multiple rounds of adversarial testing. The retirements are documented in detail below. One of them (the April 4 Γ_c retirement) was narrowed in scope on April 6 after a notation clarification — what was actually wrong was a specific numerical value, not the underlying concept. Another (the April 5 tension asymmetry "downgrade") turned out on reflection to be more nuanced: TSO's dynamics map onto the Lindblad master equation, and a later compliance test (June 2026) verified they are completely-positive, trace-preserving, and — on the enlarged (bond + stored-energy) state, given that γ_o latches — Markovian, so the Lindblad form is forced rather than merely chosen. The genuine TSO-specific results from this thread are that charge conservation falls out structurally for pair-correlated operators and that the Lindblad anticommutator plays the Faddeev-Popov ghost role; the Lindblad equation itself is the standard GKSL form, inherited not invented, so "reproducing Lindblad" is consistency, not a novel prediction. The April 10 null test round added narrowing #7 — the spatial-cluster-enumeration reading is not Z = 7-specific on cubic lattices — and introduced three new structural commitments (bidirectional X₁ bonds, 1/3-per-bond charge rescaling, and an algebraic Lindblad derivation of charge conservation), documented in the v11.3 additions section. A subsequent session (April 18–19, 2026) ran a projection falsification suite that derived Heisenberg Uncertainty from partial dimensional projection rather than treating it as an axiom, confirmed TSO is numerically identical to standard QM (double-slit Pearson r = 1.0000000000), identified the Space Roar as Larmor radiation from W-space bobbing, and generated four new testable predictions — including two new Rydberg observables (predictions 29 and 30) extractable from the same sweep as the existing decisive test. Documented in the projection suite section. The May 19–20 session completed the σ derivation from K3 + tetrahedron geometry and ran six stress tests of TSO's geometric primitives; documented in the v11.6 section. The most recent round (May 24, 2026) added narrowing #10 — magnitude predictions on quantum-computer hardware — based on a cluster of four pre-registered hardware tests showing structural-yes / magnitude-no across the board. Documented as the v11.7 empirical scope finding on predictions.html.

JULY 13, 2026 — GAUGE-GROUP RANK CORRECTION (and related fixes)

A computational session built the octonion derivation algebra Der(𝕆) directly and measured its dimension and rank: dim = 14, rank = 2 (the rank taken as the dimension of the centralizer of a regular element). The Standard Model gauge group has rank 4 (SU(3): 2, SU(2): 1, U(1): 1). Because a subgroup cannot exceed its parent's rank, the full SM group cannot embed in G2 (rank 2) — nor in SU(4) (rank 3), which was the other form the claim took on foundation. Both statements are retracted.

What is legitimate. Colour SU(3) has rank 2 and is realized concretely as the stabilizer of ∅ (HERE) inside G2 — an 8-dimensional su(3) subalgebra, computed as the kernel of D ↦ D(e_∅) within Der(𝕆). This matches the established octonionic-Standard-Model programs (Furey; Dixon), in which Der(𝕆) = G2 supplies only SU(3). The electroweak SU(2)×U(1) requires a separate rank-≥4 source — the octonion left-multiplication / triality structure (up to SO(8)/Spin(8)), not G2 — and constructing it explicitly is open. The corrected claim is: colour SU(3) from G2 at p_c [proposed]; electroweak from the larger multiplication structure [open].

Related fixes made the same day:

• sin²θ_W = 3/13 — scale caveat added. The weak mixing angle runs with energy (≈ 0.2312 at M_Z, ≈ 0.2386 as Q→0), so a fixed integer ratio can match at most one scale; the 0.231 match implicitly selects M_Z, and why that scale is privileged is unexplained. Treat 3/13 as the value at M_Z, not a scale-independent derivation. The "13 gauge degrees" count is also downstream of the gauge-group reading corrected above. [caveat, not retirement]
• "Two-qubit" node clarification. Fixing ∅/HERE decomposes the node's C⁴ as C¹⊕C³ — a direct sum (1+3, the SU(3) branching), not a two-qubit tensor product C²⊗C². The S⁷ dimension count (dim = 7 = Z) is unaffected; only the tensor-structure reading is retracted. Consequently entanglement is inherited via the tensor-product composition rule (same tier as "inherits Born via Gleason"), not derived, and is an inter-node property, not a single-path property. [clarification]

Verification notebook (self-checking, numpy-only): the Der(𝕆) construction, the rank computation, and the su(3) stabilizer are all computed in-cell. These corrections tighten the framework's alignment with the octonionic-SM literature rather than removing content: colour SU(3) from the octonions is a genuine, and now correctly-stated, result.

APRIL 5 NULL TEST — TOPOLOGY CHARGE SPECTRUM

The v11.0 headline result was that enumerating spanning clusters on the Z = 7 lattice with symmetry reduction (SO(3) × X₁↔X₂ swap × conjugation) produces the complete SM fermion charge spectrum {0, ±1/3, ±2/3, ±1} with three generations and color triplets. This was one of the strongest-looking results in the framework.

The null test swept 750 parameter combinations across Z ∈ {5, 6, 7, 8, 9, 10} with multiple symmetry groups and several sector-combination schemes, checking how often each combination produced an exact match to {0, ±1/3, ±2/3, ±1}. Results:

Translation: "the SM charge spectrum from Z=7 combinatorics with zero free parameters" is technically true but strongly misleading. Once you fix n_spatial = 3 (an observational input — we live in 3D space) and combine the A and B sectors, almost any reasonable lattice size plus symmetry scheme will give you {0, ±1/3, ±2/3, ±1}. The factor of 1/3 in the charge formula Q = χ_spatial/3 is doing almost all of the work.

What was downgraded: the "zero free parameters SM charge derivation" talking point is removed from the framework. It has been replaced with the more honest "Q = χ_spatial/3, combined with sectors A and B, produces the SM charge spectrum. This is a consequence of having three spatial dimensions, not a Z=7-specific derivation."

What survives: four things carry forward untouched.

(1) The Z = 7 geometric anchor — 6 cardinal directions in 3D plus "stay" — remains clean and independent of the charge result.

(2) The detailed class structure (three classes per charge value = three generations; three color choices for |χ_spatial| = 1 = quark color triplets; one class with |χ_spatial| = 3 = leptons; total 17 fermion classes) has NOT been tested across Z values, and may still be Z=7-specific. This is the natural next null test.

(3) All non-particle results — the baryon asymmetry calculation, life at p_c, kBT ≈ δW, JWST consistency — are untouched.

(4) The tension asymmetry and percolation sigmoid prediction are untouched.

Honest next step: build a second null test on the detailed class structure. If Z = 7 gives all four of (3-per-charge, 3-fold color, total 17, proper sector assignment) while Z = 5, 6, 8, 9 fail at least one, the framework survives in stronger form. If Z = 7 is not distinguished, the topology result needs to be retired entirely, not just downgraded.

Topology null test notebook (Colab)

APRIL 10, 2026 — v11.3 STRUCTURAL ADDITIONS

1. Z null test — spatial cluster geometry is generic (narrowing #7)

Result: polycube enumeration on three cubic-family lattices — simple cubic (Z = 6), body-centered cubic (Z = 8), and face-centered cubic (Z = 12) — produced qualitatively similar chirality and stability patterns. FCC actually fits framework intuitions better than SC on every metric tested. Conclusion: spatial cluster combinatorics on a cubic lattice are not Z = 7-specific.

This narrows what spatial-cluster enumeration work can claim. It is not a retirement of the framework — the Z = 7 geometric anchor (6 cardinal directions plus stay, forced by dimensionality of space) is untouched, and several independent consistency checks at Z = 7 (Ω_m, δW ≈ k_BT, minimal genome size) remain. What it narrows is specifically the claim that cluster-shape patterns encode particle content via their cubic-lattice combinatorics. The positive implication: the framework's TSO-specific content must live in the path identity space {X₁, X₂, x, y, z, T, ∅}, where the seven functions are categorically distinct and not interchangeable. This is where future derivation work should operate.

Z null test notebook (SC / BCC / FCC) — Colab

2. Bidirectional X₁ bonds — charge conjugation via direction reversal (working position)

Proposal: X₁ functions in their classical (bond) mode carry a direction (±1). X₂ does not. Wave-operational functions and propagating disturbances are direction-blind. Under this commitment, charge magnitude comes from bond count, charge sign comes from direction.

Three consequences follow cleanly:

The proposal passed 5 of 6 stress tests in session. The 6th was fractional quark charges, which are handled by the 1/3-per-bond rescaling (next section). This is a working commitment, not a derived result. The framework does not yet have a principled rule selecting which paths in which configurations correspond to which specific particles; specific bond-count assignments are demonstrated for consistency, not predicted from first principles.

3. 1/3-per-bond charge rescaling

Each classical X₁ bond carries ±1 framework charge. The mapping to Standard Model charges is simply:

Under the framework's natural unit (the down quark = 1), the SM particle content accommodates as:

All SM charges become non-negative integer counts in framework units. The factor of 3 is historical rather than fundamental: the electron was discovered in 1897 and assigned charge 1; quarks were found in 1964 and the SM chose to keep the electron at 1, which forces quarks to 1/3 and 2/3. If the quark had been discovered first, modern charge tables would have integer values and the electron would have charge 3.

Relationship to the retired Q = χ_spatial / 3 reading. These are different mechanisms. The old /3 was "three spatial paths" under an SO(3) chirality argument; the new /3 is a historical unit rescaling on bond counts. They are not compatible and should not be combined. The old reading is retired on foundation; the new reading is the one documented here.

4. Algebraic Lindblad connection — charge conservation as a structural consequence

The April 5 session established that TSO's γ-dynamics reproduce the Lindblad (GKSL) master equation numerically across four observables. The April 10 follow-up examined this at the operator level: on a 2-bond toy state space with γ events built as matrices and a charge operator Q defined from the bidirectional-bond proposal, charge conservation was checked by computing the commutator norm ‖[L, Q]‖. (What is derived is the charge-conservation consequence — not the Lindblad equation itself, which is the standard GKSL form.)

Interpretation: charge conservation is not a separate postulate in TSO. It is a structural consequence of the jump operators implementing physical γ events being pair-correlated. Single-bond operators are ruled out algebraically, not by hand.

Three implications worth flagging:

(a) Coupling rules become derivable in principle from path-subset structure plus Lindblad selection. Which paths can couple to which is, in principle, a question the formalism can answer, though it does not yet produce specific numerical answers.

(b) Contextuality acquires a structural mechanism. Measurement outcomes are created by L_k events whose form depends on measurement context. This is not yet a quantitative prediction; it is the first time the framework has had a concrete mechanism for where contextuality could come from.

(c) Quantitative predictions remain pending. Lindblad matrix elements for specific particles would enable predicted decay rates and coupling strengths that could be tested against observation. The algebraic skeleton exists; the matrix elements do not.

Bidirectional paths + 1/3 rescaling + Lindblad connection notebook (Colab)

5. Three-regime phase structure — clarification, not retirement

Older framework language described classical matter as sitting at the 2/7 floor. This was wrong, but in a specific and correctable way: the evidence at p_c (minimal cell at W = 0.315, Ω_m ≈ p_c, scale-free networks) had always been there. What was wrong was the description of where that evidence placed classical matter, not the evidence itself.

The corrected picture has three regimes:

The minimal-cell result, the Ω_m ≈ p_c match, and the life-at-p_c arguments all become stronger under the clarification because they now all live at the same point (p_c) through different mechanisms, rather than being awkwardly split between "at the floor" and "at criticality." The δW ≈ k_BT match also sharpens: it is the thermodynamic climb energy from the 2/7 dead floor up to p_c, not a coincidence of band widths.

The predicted intermediate dead zone (rocks, planets, dust showing no scale-free internal structure) matches observation and becomes a structural claim of the framework. So does the finite-size scaling prediction δp ≈ N^−1/ν ≈ N^−1.14, which gives a passive-criticality limit of ~27 nodes at 300 K — matching observed biological module sizes (ribosome modules 15–40, respiratory chain complexes 15–40, nuclear pore ~30).

This is classified as a clarification rather than a retirement: the at-p_c evidence was always being collected; only the language describing what it showed was wrong.

APRIL 10, 2026 — THE BRUTAL FALSIFICATION TEST

What the test did

The bidirectional-bond proposal assigns classical X₁ bonds a direction (±1) and defines framework charge as a signed sum of bond directions, with SM_charge = framework_charge / 3. On its face this looks like a derivation of the {0, ±1/3, ±2/3, ±1} spectrum: framework charges are non-negative integer counts; divide by 3; get the SM values. The brutal test asked the question that derivation-talk tends to hide: if I vary which path subsets I declare to be particles, does the framework still reproduce the SM, or does it only reproduce the SM under a specific assignment I picked because I knew the answer?

It varied the assignment. It found that the framework has astronomical flexibility in which path-subset configurations can be interpreted as which particles. There are many internally consistent ways to assign bonds to particles, and the current formalism contains no rule selecting the one that reproduces observed masses, cross-sections, or decay rates. The {0, ±1/3, ±2/3, ±1} spectrum is accommodated by the proposal; it is not predicted by it.

What the test changed

Three things were tightened across the framework as a direct consequence:

(1) The 1/3-per-bond rescaling is documented throughout the site as a consistency demonstration, with explicit language that the framework does not yet derive why specific particles have specific bond counts. The red caveat box on the 1/3 rescaling section and the matching box on foundation.html both carry this warning.

(2) The bidirectional X₁ bond proposal is labeled a working commitment, not a derived result. The three consequences it does produce cleanly (photon self-conjugacy, electron↔positron symmetry, automatic pair-production charge conservation) are real structural results. Specific particle-to-bond-count assignments beyond those three are not.

(3) The topology-enumeration tables, already downgraded on April 5 by the charge-spectrum null test, were further caveated to make clear that the assignment shown is one of many possible consistent assignments, not the one the framework selects from first principles. See the red box on the topology enumeration section.

Why this matters

Working on a framework you believe in tends to drift toward flattering the framework — partial results get framed as complete ones, accommodations get framed as derivations, and the language tightens around the parts that work while loosening around the parts that don't. The brutal test is an explicit countermeasure: a procedural rule that before publishing any v11.3 commitment, an adversarial version of the same session would try to break it. In this case the countermeasure worked — the overclaim was caught and corrected before it reached the site.

The test does not prove the framework is right. It proves that a specific class of overclaim was ruled out at a specific moment. That is the only kind of credibility claim a pre-empirical framework can honestly make, and it is the claim this page tries to make consistently.

Open question surfaced by the test. Why does the electron have 3 X₁ bonds rather than 6, 9, or any other multiple of 3? The 1/3 rescaling accommodates any of these; the framework does not yet select between them. This is an explicit open theoretical task alongside the principled-function-subset-selection question and the Lindblad-matrix-elements question.

APRIL 5 NULL TEST — TENSION ASYMMETRY

The v11.1 tension asymmetry (γ_c closing, γ_o opening, Γ_C/Γ_O/Γ_net running sums) was introduced as a reframing of how TSO describes coupling channels. On April 5, a second null test asked the harder question: does the asymmetry, by itself and separate from the percolation sigmoid, make quantitative predictions that standard decoherence theory does not?

Five tests were run:

Tests 1–4 returned exactly zero numerical difference because TSO dynamics reproduce standard Lindblad dynamics under the direct mapping γ_c → dissipator, γ_o,active → Hamiltonian drive, γ_o,stored → protected subspace. Test 5 showed that theoretical Pips/joule estimates across different γ_o sources (laser, strong force, chemical bond, RF) span many orders of magnitude, indicating the Pip unit is not a universal physical constant in its current form.

What this means — and why it's a strength, not a downgrade. The tension asymmetry is a reformulation of the Lindblad master equation in TSO vocabulary. That is not a failure; it is a significant structural result:

A reformulation of established mathematics is historically how new physics sometimes arrives — Lagrangian mechanics is a reformulation of Newtonian mechanics, and it enabled a century of new physics by revealing structure that the Newtonian form hid. TSO's Lindblad-compatible framework means every theorem proven for quantum master equations over the last sixty years is inherited automatically. A critic who accepts Lindblad cannot reject TSO's microscopic dynamics.

What the asymmetry adds that standard Lindblad notation does not is structural. The second law and the arrow of time drop out of γ_c being one-directional — Γ_C grows monotonically in any isolated system, which is invisible in the symmetric standard form of the master equation. The tardigrade case is resolved naturally because γ_o,stored (latched configurations) is a distinct category from γ_o,active (metabolism). Whether the asymmetry reveals deeper structure inside Lindblad dynamics that standard notation obscures is an open question — worth exploring.

The empirical novelty of TSO — the one thing that, if confirmed, would distinguish the framework from standard decoherence — is the percolation sigmoid at threshold (κ ≈ 4/3). That comes from the percolation layer on top of the Lindblad/asymmetry framework. The Rydberg sweep and the IBM gate-depth sweep are both tests of the percolation part.

On test 5 (Pips/joule non-universality): this is a real result, and it changes how the Pip catalog is presented rather than retiring it. Only γ_BS = 522 Pips from Quandela is anchored to an independent measurement; biological Pip values that were estimated by analogy have been removed. To make the Pip unit cross-platform requires independent calibrations from ≥3 hardware platforms. The IBM gate-depth experiment is the first attempt. See the predictions page.

Tension asymmetry null test (PDF)

NOTATION FOR COUPLING TENSIONS

TSO uses a consistent case distinction for coupling tensions. This was always the intent; it was not always written down clearly.

In the original TSO documents and in several notebooks, "Γ_c" was used as the aggregate closing tension acting on a cluster. There are multiple kinds of γ_c contributions — thermal, Casimir, photon, gravitational, impact — and each adds to the total. When the aggregate Γ_c = Σγ_c,i drives the path closures across the percolation threshold, the phase transition occurs. That picture is foundational and has not changed.

What was retired on April 4, 2026 was not the concept — it was a specific numerical value and derivation attached to Γ_c during a later session. Details in the next section.

MASS AS STORED CLOSING TENSION

TSO's working position on rest mass: the rest mass of a particle is the aggregate count of non-operational (classical) functions stored in its spanning cluster. Total energy partitions as E² = (active γ_o)² + (stored Γ_C)², which identifies pc with the active part and m₀c² with the stored part. This recovers the standard relativistic energy-momentum relation as a Pythagorean partition between the two tension types — not as a postulate, but as a consequence of the active/stored distinction the framework already had.

Three immediate consequences:

Photons have zero rest mass by construction. Photons sit at W = 1, Γ_C = 0 (the pure-wave bookend). With zero stored closing tension, their predicted rest mass is zero. No extra rule needed.

Black holes preserve mass across the horizon. When the spanning cluster fragments at r_c2 (the inner phase boundary), the closure count is preserved — closures become local rather than relational. The connectivity is destroyed but the stored tension count is conserved. This is the microscopic reason BH mass is conserved across the horizon under the position.

Matter and antimatter annihilate cleanly. Particle and antiparticle have identical stored Γ_C (so identical masses) and opposite topological invariants (so opposite quantum numbers). When they meet, the topologies cancel and the stored Γ_C is released as active γ_o. For e⁻ + e⁺ at rest, the released energy is exactly 2m_ec² = 1022 keV, split symmetrically into two 511 keV photons. Standard annihilation kinematics, with a geometric reason for why the energy comes out at exactly 2m_ec².

The position is internally consistent with all standard physics in the limits tested, and the cluster enumeration it requires is concretely computable. See the particle structure notebooks for the verification, including the photon limit, the relativistic partition, the annihilation balance, the small-cluster enumeration on a 3D simple cubic lattice (verified against OEIS A001931), and the fragmentation/handshake-lemma check for BH mass conservation.

What this position does not yet do: it does not predict specific particle masses from first principles. The bigger empirical tests — whether the SM fermion mass spectrum matches the predicted bond-count distribution, whether topology classes match SM quantum numbers, and whether composite particles like the proton can be accommodated under this framing — are pending. The first of these is the proton binding-energy test (linked separately).

THE Γ_c NUMERICAL VALUE RETIREMENT (APRIL 4, 2026)

The error. In v10 and v11.0 drafts, a specific frequency-valued Γ_c = 1.5 × 10¹⁵ Hz appeared in the framework, claimed to derive from α³ m_ec² / (2ℏ). Two things were wrong with that. First, computing α³ m_ec² / (2ℏ) directly gives 1.508 × 10¹⁴ Hz — a factor of ten off from the claimed value. Second, neither the original Two State Ontology paper nor the Fire Model paper contains this specific frequency. It was introduced during a later session and retrofitted with a derivation that did not survive verification.

How it was caught. During the April 2026 rebuild of the γ catalog in Pips, the Casimir-at-1-μm calculation turned out to depend on the spurious frequency value for its anchoring. The catalog was supposed to be dimensionless and lattice-anchored, so this dependence was a red flag. Tracing back found the missing derivation and the factor-of-ten error.

What replaced it. Nothing in frequency units. The transition condition is expressed directly in the dimensionless form the framework always used conceptually: Γ_net < −p_c, where Γ_net = Γ_O − Γ_C. The Pip unit (1 Pip = p_c/1000) gives clean integer-range values; the crystallization threshold is exactly 1000 Pips by construction. Each specific experiment produces its own rate when these dimensionless values are multiplied by the event frequencies appropriate for that experiment.

What it costs the framework. Nothing quantitative. No downstream result used the spurious frequency value numerically — it was load-bearing only for the Casimir-anchoring calculation, and that calculation is now re-anchored in lattice constants directly. The framework is cleaner without the bad number, and the notation clarification above should prevent the concept-vs-value confusion from recurring.

Pip catalog notebook (Colab) — contains the verification of the factor-of-ten error.

THE IBM QPU UN-RETIREMENT (APRIL 5, 2026)

The original retirement: in mid-2025, IBM was dropped as a TSO test platform on the argument that QPUs are engineered to stay deep in the wave phase (W well above p_c), so a single qubit at rest cannot cross Γ_net = −p_c. The argument was based on idle T2 measurements. It is correct as far as it goes.

The correction: Quandela's γ_BS measurement did not sweep temperature or a single element. It swept the number of sequential beam splitters. Each beam splitter contributed γ_BS to cumulative coupling, and the transition was crossed at approximately N = 2 events. The same logic applies to IBM: cumulative γ_gate across a long sequence of entangling gates can cross p_c even though a single resting qubit cannot.

The proposed experiment: Run sequential two-qubit entangling gate sequences (CNOTs or similar) on a small number of qubits. For each depth N, measure state fidelity against the ideal output. Fit the resulting fidelity curve to both exponential decay and a percolation sigmoid. Extract γ_gate from the sigmoid fit.

This experiment does three things at once:

(1) Provides a second independent Pip calibration from a different hardware class. If γ_gate converted to J/Pip agrees with the photonic value (~5 × 10⁻²² J/Pip) to within an order of magnitude, that is a real cross-platform universality result.

(2) Provides a second sigmoid test on hardware complementary to the pending Rydberg experiment.

(3) Runs on credits already held, without requiring anyone else's permission or collaboration.

Possible outcomes: If a clean sigmoid appears at the predicted cumulative-γ location, it provides the second calibration point and the Pip framework gains predictive power. If pure exponential decay appears across all accessible depths with no sigmoid knee, the original IBM retirement is confirmed — IBM noise (1/f, charge noise, amplifier) dominates before cumulative γ_gate approaches p_c. Either outcome is publishable and useful.

THE TENSION ASYMMETRY INTERNAL TEST SUITE

A runnable Colab notebook implementing nine numerical consistency tests of the tension framework. Each test is an executable experiment producing a PASS/FAIL verdict. All nine pass. Note that seven of the nine are internal self-consistency checks — they verify that the code implementing the axioms behaves the way the axioms say it should. Tests 3, 5, 6, and 7 make partial contact with external physics.

Honest caveat: test 3 is the single test that most closely approximates the real decoherence-shape question, but it uses synthetic data generated from the TSO sigmoid and then fits the TSO sigmoid back — it verifies that if a real experiment produces TSO-shaped data, a standard fit procedure would detect it. It is not evidence that TSO is correct; it is evidence that the detection procedure works. The decisive observational test is still the Rydberg sweep (or the IBM gate-depth sweep).

Tension Asymmetry Test Suite (Colab)

TOPOLOGY ENUMERATION (downgraded, then further narrowed)

The computational enumeration of equivalence classes on the Z = 7 lattice still runs and still produces the same classes it always did. What changed on April 5 is the interpretation: the charge-spectrum match is not specific to Z = 7 (see topology null test above). What changed further on April 10 is that the cluster-shape patterns underlying the enumeration are also not Z = 7-specific. The computed class structure is documented here for completeness, and because it may still be a useful organizing scaffolding even after the narrowings.

Sector A (all 5 rotatable functions non-operational, W = 2/7)

Raw configurations: 32. After symmetry reduction: 6 equivalence classes. Three classes at Q = −1 (charged leptons e, μ, τ) and three at Q = −1/3 (down-type quarks d, s, b). The three-per-charge structure suggests three generations.

Sector B (4 rotatable functions non-operational, W = 3/7)

Raw configurations: 80. After symmetry reduction: 9 equivalence classes. Split by whether the operational function is spatial (charges 0, ±2/3 → neutrinos, up-type quarks) or quantum (charges −1, −1/3 → overlaps with Sector A). Contains one self-conjugate class at χ = 0 that would be a Majorana neutrino candidate if Majorana is confirmed experimentally.

Sector C (0 rotatable functions non-operational, W = 1)

One configuration: all rotatable functions operational, no chirality, no charge, no mass. The massless gauge bosons (photon, gluon).

Grand total

The extra Sector B class compared to the Standard Model is either an artifact of the enumeration method or a prediction of an unobserved particle (candidate: sterile neutrino). W, Z, and Higgs are not placed in the current enumeration. Whether the class structure itself (generation count, color multiplicity, total count) is Z=7-specific or also generic across Z values is an open question — not yet tested in the same way as the charge spectrum.

Full enumeration notebook (Colab)

BARYON ASYMMETRY WITHIN 19% (retrodiction)

Walton-Chalmers / Avrami crystallization mathematics applied to TSO cosmic solidification:

Observed: 6.1 × 10⁻¹⁰. Ratio: 1.19×. Every parameter pre-fixed from lattice geometry or cosmological observation. Labeled as retrodiction — the calculation was run after the observed η was known, not before, so this is not a pre-registered prediction. No parameter was adjusted to fit. This is evidence that 150-year-old metallurgical mathematics applied to TSO parameters produces the observed value to within the precision of the method — a consistency check rather than a test. Unaffected by the April 5 topology null test.

Baryon asymmetry notebook (Colab)

LIFE AT p_c — CONSISTENCY CHECKS

All retrodictive consistency checks on published data, not pre-registered predictions. The weakest is the single-number fraction match (149/473 ≈ p_c), a coincidence on its own. The connectivity enrichment was downgraded after a bias audit: an earlier 73% estimate was hand-classified by someone who knew the prediction; a frozen mechanical classifier gives ~48% vs ~26% baseline (odds ratio 2.6) — about half the original, but still a real enrichment, and the scale-free PPI topology is the robust piece. syn3.0 annotation audit notebook

EXTERNAL LITERATURE CONSISTENT WITH TSO

Several recent papers are consistent with TSO's predictions, though none was written with TSO in mind. None is a dedicated TSO test.

Entanglement formation time: 232 attoseconds (TU Wien, October 2024). Burgdörfer, Březinová et al. (Physical Review Letters, October 2024) measured the formation time of quantum entanglement during helium photoionization using attosecond laser pulses. Key finding: entanglement does not form instantaneously — it unfolds over approximately 232 attoseconds, with the timing correlated to the energy state of the bound electron. "The electron doesn't just jump out of the atom. It is a wave that spills out — and that takes a certain amount of time." This is directly consistent with TSO: entanglement formation is the physical process of γ_o,stored latching — the time for dp/dt to go from positive (wave expanding) to zero (spanning cluster locked). The 232 attosecond timescale is the formation cost of a correlated state, which TSO identifies as the fundamental process underlying rest mass. Not a TSO-dedicated test, but the most direct experimental confirmation of TSO's claim that entanglement is a physical process with finite formation time, not an instantaneous mathematical correlation.

DESI DR2 dynamical dark energy preference. Abdul-Karim et al. 2025 (Phys. Rev. D 112, 083515). BAO measurements from 14+ million galaxies show 2.8–4.2σ preference for evolving dark energy (w ≠ −1, w₀ > −1 with w_a < 0), strengthening from DR1. Principled statistical combination gives 3.1σ exclusion of ΛCDM from DESI + CMB alone. Consistent with TSO Prediction 13 (dynamical dark energy from Avrami crystallisation kinetics). Upgraded from TESTABLE to SUGGESTIVE in v11.4. Note: TSO does not yet predict a specific w(z) curve — it calls the direction (evolving, not constant) but not the shape.

JWST Little Red Dots. Kocevski et al. 2025 (ApJ) — 341 LRDs spanning z = 2–11. Number density increases sharply at z < 8, then undergoes rapid decline at z ~ 4.5. Disappearance mechanism currently unexplained in standard models. TSO: LRDs are dust-phase nodes (W < 2/7) that sublimated as surrounding universe cooled into solid phase. Prediction 25 pre-registered April 18, 2026: sigmoid shape on decline side, inflection at z ~ 2.5–3.5. Test requires z = 1–2 data currently being gathered by ground-based surveys.

Classical gravity can transmit quantum information. Aziz & Howl 2025 (Nature Communications) showed that under full quantum field theory, classical gravity can generate entanglement between masses. Consistent with TSO: gravity couples to T and ∅, always-operational functions, so gravitational "entanglement" is operational functions being shared without ever becoming non-operational.

Diósi-Penrose collapse ruled out at Gran Sasso. Donadi et al. 2021 (Nature Physics). TSO predicts this because gravity alone is not enough — collapse requires total Γ_C crossing −p_c, not just γ_gravity.

GUP decoherence. Petruzziello & Illuminati 2021 (Nature Communications). Maps onto TSO's d_eff approaching 2 at the holographic floor.

Scale-free PPI networks. Zhang 2021. Signature of networks near percolation criticality.

Quantum biology at edge of chaos. Vattay et al. converges on a "life at criticality" picture from independent reasoning.

Rydberg programmable simulators. The correct dataset citation is Kim, K., Kim, M., Park, J., Byun, A. & Ahn, J., "Quantum computing dataset of MIS problem on king lattice of over hundred Rydberg atoms," Scientific Data 11, 111 (2024), DOI: 10.1038/s41597-024-02926-9. A prior reanalysis of "Kim et al. 2024" claiming a 2.75× tanh-over-exponential preference was withdrawn in May 2026 after a citation audit revealed the work was on a misidentified dataset (a separate paper with the same lead-author name in a different field). A fresh audit on the correct Scientific Data dataset found that the data structure (45 distinct experiments with three-parameter variation per experiment) is not suited to the four-model AIC protocol as specified — the dataset does not constitute a single-axis sweep of a control variable, and no model in the protocol's set fits the points well. The "Rydberg consistency" claim is therefore in withdrawn status pending a different dataset (the Keesling 2019 / Ebadi 2021 datasets are the natural candidates).

Full list of external references with annotations: references page.

APRIL 18–19, 2026 — PROJECTION FALSIFICATION SUITE (v11.4)

The full suite ran 8 independent tests in a single notebook session. Each test was designed to be able to falsify the projection picture if it produced different numbers than standard QM. None did.

Heisenberg from partial projection. An entity is observable only if its spanning cluster has closed in all three spatial dimensions x, y, z. Wave-state objects project into 1 or 2 spatial dimensions, drawing from a finite probability fuel budget. The product Δx·Δp recovers exactly ℏ/2 at equal split because the position and momentum projections share that fuel pool. This is HUP as a consequence of partial-cluster geometry, not as an axiom.

Virtual particles as incomplete spanning clusters. A virtual particle is a crystallisation attempt that runs out of probability fuel before the spanning cluster closes. No x, y, z extension forms — unobservable by definition. Off-shell energy (E ≠ mc²) arises because rest mass requires completed topology. Observable effects (Casimir, Lamb, g-2) are W-field disturbances from the failed attempt, not from the particle itself.

Unification of five quantum vacuum phenomena. W-space bobbing — semi-crystallised electrons at W ≈ p_c held in frustrated equilibrium by Γ = Σγ_c, jostled by γ_c fluctuations, accelerating-charge Larmor radiating — accounts for: Space Roar (cosmological), zero-point energy (kinetic), Casimir force (W-field disturbance reaching plates), Lamb shift (bobbing in hydrogen orbitals), anomalous g-2 (fluctuating effective charge distribution). Five separate phenomena at different scales from one mechanism.

Projection Falsification Suite (Colab)

PRIOR RETIREMENTS

Full list of retirements and downgrades documented by the framework:

1. IBM QPU as a TSO test platform — retired mid-2025 based on single-qubit T2 analysis; un-retired April 5, 2026 based on gate-depth-sweep argument. Now prediction 19.

2. The Fire Model's specific damping coefficients — retired late 2025. Over-determined by observation; replaced with lattice-derived or empirically-measured quantities.

3. δW = 2/77 — retired v11.0. Replaced with exact δW = p_c − 2/7. The "11" in 77 has no physical meaning.

4. Γ_c = 1.5 × 10¹⁵ Hz (numerical value only) — retired April 4, 2026. See full trace above. Note: this retired a specific frequency value and its spurious derivation, not the concept of Γ_c = Σγ_c,i as aggregate closing tension, which remains foundational. See also notation section.

5. Rest mass as "topology maintenance cost" — retired April 4, 2026. Reframed as γ_o,stored — topology formation cost, paid once and locked, not a continuous expense. Protons do not eat ATP.

6. SM charge spectrum as "Z=7-specific derivation with zero free parameters" — downgraded April 5, 2026. See topology null test above. Reframed as a consequence of n_spatial = 3 plus sector combination, not Z=7-specific.

7. Spatial-cluster-shape enumeration as Z=7-specific — narrowed April 10, 2026. See Z null test above. Polycube enumeration on SC, BCC, and FCC lattices produces qualitatively similar chirality and stability patterns. The framework's TSO-specific content is relocated to the path identity space {X₁, X₂, x, y, z, T, ∅}, which is categorically distinct from spatial cluster combinatorics.

8. Chirality-based charge formula Q = χ_spatial / 3 — retired April 10, 2026. Replaced by the bidirectional X₁ bond proposal with 1/3-per-bond historical rescaling (see bidirectional bonds and 1/3 rescaling above). Different mechanism, different motivation; the two /3's are not compatible and should not be combined.

9. The Kim et al. 2024 "2.75× tanh over exponential" Rydberg reanalysis — withdrawn May 2026. A citation audit revealed the prior reanalysis was on a misidentified dataset; a fresh audit on the correct dataset (Kim et al., Sci. Data 11, 111, DOI 10.1038/s41597-024-02926-9) showed the data structure is not suited to the four-model AIC protocol as specified. The Rydberg-evidence claim is in withdrawn status pending audit on a different dataset (Keesling 2019 or Ebadi 2021).

10. Magnitude predictions on quantum-computer hardware — narrowed May 24, 2026 (v11.7). Across four pre-registered hardware tests (Fix 4 lepton scale; P40 fidelity lock σ = 0.18058; P41 three-Fano replication; P42 hexagon-with-roles, queued at allocation exhaustion), TSO's structural predictions (sign patterns, ordering relations, combinatorial Fano-line structure) replicated, but the framework's specific-magnitude predictions on quantum hardware did not. P40 came in at exactly σ/2 (ratio 0.4954 ± 0.043). P41 missed the May 14 ratio thresholds. Fix 4 was sign-right and 14σ magnitude-off. The platform constraint identified in this round: IBM superconducting QPUs are engineered to stay above p_c (classical-side); Quandela photonic QPUs are engineered to stay below p_c (wave-side); neither is allowed to cross the transition because crossing it would mean the chip isn't functioning. The decisive sigmoid magnitude prediction (Prediction 1) therefore requires a platform tunable across p_c — a Rydberg neutral-atom array is the natural candidate. Until Rydberg lab access is established, all quantum-hardware-based TSO testing should be considered structural rather than quantitative. This is not a retirement of magnitude predictions in general — Rydberg, Ω_m, the syn3.0 match, and the Goldilocks band all remain quantitatively testable on their native domains. What is narrowed is specifically "test the σ magnitude on a quantum computer." See predictions.html empirical scope section for the formal documentation.

Additionally retired framing items: calling Γ a "fifth force" (it is aggregate coupling pressure, not a force); using bare Γ as a running sum (now Γ_C, Γ_O, Γ_net).

v11.3 language clarifications (not retirements)

Classical matter "at 2/7" — corrected April 10, 2026 to the three-regime picture. Older text described classical matter as sitting at the 2/7 floor; the corrected reading places biology and cosmic-scale matter at p_c (via metabolism and gravity respectively), and the 2/7 floor is occupied by the intermediate inert regime (rocks, planets, dust). All evidence at p_c was always being collected — only the description of where it placed classical matter was wrong. See the three-regime clarification.

δW ≈ k_BT as band-width coincidence — reframed April 10, 2026 as the thermodynamic climb energy from the 2/7 dead floor up to p_c. Same numbers, different (and sharper) interpretation.

MAY 14, 2026 — INTERMEDIATE THEORY, ∅ PATH, AND CY FOURFOLD

TSO as intermediate theory [v11.5]

The complete layered structure:

M-theory has no complete non-perturbative formulation. TSO provides what M-theory needs: a non-perturbative, background-independent description of the phase transition between classical and quantum phases. The percolation criticality at p_c is that mechanism. Intermediate theory Colab →

σ = 6/√1104 — the bulk friction ratio [CONFIRMED]

7 paths − ∅ = 6 directional paths (x, y, z, X₁, X₂, T). Anderson localization gap = p_q − p_c = 0.3440 = 1104 Pips. σ = 6/√1104 = 0.18058. Joshua Osborne (interpretation): "your numerical bracketing is staggering. You have isolated the exact noise scale, 0.1804, the bulk's friction ratio." This is a proposed reading, not an independent confirmation. The formula: (non-null path count) / √(quantum-classical separation in Pips). 6³ = 216 = Freudenthal cubic dimension on 6 non-null paths. The v11.6 holographic-projection derivation of the same σ from a different route (K3 + tetrahedron) is documented in the v11.6 section.

The ∅ path as "HERE" [NEW INSIGHT, v11.5]

In the wave state, (x, y, z) alone are insufficient to specify position. ∅ must also be functional. ∅ is "HERE." When ∅ is functional, "here" is defined — classical, localized. When ∅ is non-functional, "here" is undefined — quantum, non-local.

• Photon propagation: ∅ is non-functional. The photon propagates as a spherical shell — not because its position is uncertain, but because "here" does not exist. At absorption, ∅ becomes functional and "here" crystallises.
• Dead matter: Environmental interactions close ∅ faster than it can reopen. Always "here." The interaction rate Γ_close >> Γ_open at W_dm.
• Measurement: The act of measurement IS ∅ becoming functional through a γ_c drain hole. Wave function collapse is not mysterious — it is ∅ becoming functional. The measurement problem dissolves.
• Double slit: ∅ is non-functional at both slits simultaneously — no "here" at either slit. Interference = geometry of non-functional ∅ in x,y,z space. A detector at one slit makes ∅ functional there.

Connection to Lindblad: the anticommutator term {L†L,ρ}/2 does not close any path — it is the permanent background contribution of ∅, the zero-point floor. Gogberashvili (2022, Entropy): octonion non-associativity generates unavoidable 18.6-bit entropy between G2 and SO(7) — directly consistent with ∅ as the identity path constraining the G2 structure. ∅ path Colab →

Calabi-Yau fourfold trail (Joshua Osborne) [SUPERSEDED — see v11.6]

Joshua Osborne directed the search to: Sethi-Vafa-Witten (1996) hep-th/9606122, Kreuzer-Skarke reflexive polyhedra classification, Freudenthal determinant of J(3,O), and quasi-homogeneous hypersurfaces in weighted projective space. The G2 manifold is the compactified boundary of a CY fourfold, not enclosed by an exceptional algebra. The Freudenthal determinant Det(A) ≥ 0 is the survival inequality — its zero locus defines the unique ambient hypersurface. TSO tadpole N = 15 originally taken to imply χ(CY4) = 24×15 = 360 (candidate). Finding the specific WP(w₀,...,w₅) was identified as the next step to close the σ derivation from first principles. v11.6 update: the correct identification is χ(CY4) = −3192 at the E₇ level, with N = 15 living at the kaon WP4[1,1,1,6,9] rung. See v11.6 section.

G2 Monge-Ampère metric and Fano = Hamming code [NEW]

Joshua Osborne: "The Fano plane only dictates the local error-correction. The geometric weights w2=3/14 and w3=1/14 are absolute — they cannot survive in a naive simulation without the Monge-Ampère metric as an error-correcting firewall. The original fitted weights (0.1487, 0.782) were artificially inflated to compensate for the missing correction." The Fano plane IS the [7,4,3] Hamming code — the world's first perfect error-correcting code. The Monge-Ampère metric encodes this code into the bond activation covariance. Geometric weights: w1=1 (loop cost), w2=3/14 (triality eigenvalue / dim G2), w3=1/14 (topological capacity / path space dim). Status: confirmed directionally (61% improvement with Itô SDE), exact implementation in progress. G2 corrected Colab →

Acknowledgment

Mathematical direction for this session from Joshua Osborne (RF engineer, G2/octonion expertise): G2 weights, Monge-Ampère metric, CY fourfold trail (Sethi-Vafa-Witten, Freudenthal, Kreuzer-Skarke), σ = 0.1804 identification, and the Fano = [7,4,3] Hamming code observation. Implementation and any errors by John Pepin.

MAY 11, 2026 — THERMODYNAMIC DYNAMICS, QUANTUM HARDWARE, AND WEINBERG ANGLE

Thermodynamic dynamics: V(p) and equation of motion [v11.5]

The key insight: TSO had geometry and phase structure but no dynamics. γ_o and γ_c provide that dynamics as forces on a potential energy landscape:

The equation of motion dp/dt = γ_o(p, stored) − γ_c(p, interactions) is the RG beta function with physical meaning: γ_o draws from stored formation cost, γ_c is consumed by interactions. The cosmological interpretation follows directly: Big Bang = γ_o releasing from maximum storage; dark energy = global γ_o not yet found a closing interaction; heat death = all γ_o spent, stuck at W_dm.

Thermodynamic dynamics Colab (May 2026) →

IBM quantum hardware: Anderson localization confirmed [v11.5]

ibm_marrakesh, 156-qubit heavy-hex, source qubit 89, sink qubit 0 (17 hops):

IBM entanglement depth sweep: Bell pair (∅⊗x), depths 1–21 (Fibonacci spacing), CHSH parameter S measured. The definitive run (500 configs, depths 4/8/12) showed monotonic S decay: 2.231 (d=4) → 2.096 (d=8) → 1.895 (d=12). Classical limit crossed between depth 8 and 12. A previous apparent "revival" at depth=8 in lower-statistics runs was not confirmed at higher statistics. The depth=8 Fibonacci/φ holonomy hypothesis remains open but unconfirmed.

Quandela Belenos QPU data retrieved [v11.5]

All four completed Belenos QPU jobs retrieved and fully decoded (May 2026) using Perceval RemoteJob.from_id():

The 3-photon 6-mode run (b54b4c67) is the most data-rich Quandela result: 57% more localized than a uniform quantum distribution, with H = −0.258 (anti-persistent mode distribution — quantum interference spreads photons more uniformly than random). The chip indistinguishability from the noise model is 0.917. Physical_perf = 0.05% limits statistics on all runs.

sin²θ_W = 3/13 from Fano geometry [PROPOSED, v11.5]

Pure Fano degree counting: X₂ (hypercharge path) lies on exactly 3 Fano lines, each containing one spatial path. Total gauge degrees (excluding Higgs) = 13 = 15 − 2 = tadpole N − d_Higgs. sin²θ_W = 3/13 = 0.23077. Measured: 0.23100. Δ = 0.09%.

The "3" appears four times: Fano degree of X₂, number of spatial dimensions, number of fermion generations, number of X₂ couplings to spatial paths. These are four views of the same geometric object.

Independent parallel: Macedonia (Kosmoplex, Nov 2025) derives sin²θ_W from octonionic geometry by a different route, getting 0.23064 — farther from measurement than TSO's 0.23077. Convergence of two independent groups on Fano/octonion geometry as the substrate is significant but does not prove either derivation is the correct mechanism. Status: PROPOSED. Colab →

MAY 3–7, 2026 — G2 / OCTONION / MANIFOLD SESSION

Summary. A LinkedIn thread on the p_c formula post attracted Joshua Osborne (RF engineer, G2/octonion expertise) and led to ten notebooks taking TSO from discrete percolation into continuous G2 geometry. Terminal results: proposed TCS G2 manifold (b₂=1, b₃=8), p1=30, Witten shift=7.5, tadpole N=15, and sin²θ_W=3/13 — all derived from the Fano degree structure and p_c alone. All results on the roof at proposed status.

Spontaneous G2 symmetry breaking at p_c — colour SU(3) only [PROPOSED; gauge-group claim CORRECTED July 13, 2026]

Fano line breaking sequence: both forbidden lines (x×X1=X2 and x×T=∅) break exactly at W=p_c. Both contain x. x loses 2 of 3 Fano connections below p_c and becomes purely spatial. [CORRECTED July 13, 2026 — rank obstruction.] An earlier version read the residual 5 lines as giving the full G2→SU(3)×SU(2)×U(1) = Standard Model group. That is group-theoretically impossible and is retracted: G2 has rank 2 (computed: dim Der(𝕆)=14, rank 2) and the Standard Model group has rank 4 (SU(3):2, SU(2):1, U(1):1); a subgroup cannot exceed its parent's rank, so the full SM group cannot embed in G2. (The same flaw affected the earlier "SU(4) ⊃ SU(3)×SU(2)×U(1)" statement — SU(4) is rank 3 — now corrected on foundation.) What is legitimate: the colour factor SU(3) (rank 2) is the stabilizer of ∅/HERE inside G2 — an 8-dimensional su(3) subalgebra, computed — matching the octonionic-SM literature (Furey; Dixon), where Der(𝕆)=G2 yields only SU(3). The electroweak SU(2)×U(1) requires a separate rank-≥4 source (octonion left-multiplication / triality, up to SO(8)/Spin(8)), not G2; that construction is open. Corrected claim: colour SU(3) from G2 at p_c [proposed]; electroweak elsewhere [open]. Colab →

p1=30, Witten shift=7.5, Tadpole N=15 [EXACT given K5]

5 surviving lines form complete graph K5. Exact eigenvalues: 4,−1,−1,−1,−1. Boundary flux: p1 = 2×(8×W_pc² + a²+b²) = 30. Tadpole N = p1/2 = 15 exactly. Witten anomaly shift = 7.5 (half-integer global cancellation). All determined by p_c alone. Colab →

sin²θ_W = 3/13 = 0.2308 [PROPOSED, 0.1% from measured]

Pure Fano degree counting. d_SU2/2 = 3, total gauge degree = 13 = N−d_Higgs = 15−2. Higgs decoupling encoded in degree structure. The tadpole connects directly to the mixing angle. Colab →

Caveat (July 13, 2026): sin²θ_W runs with energy scale (≈ 0.2312 at M_Z, ≈ 0.2386 as Q→0), so a fixed integer ratio can match at most one scale; the match to 0.231 implicitly selects the M_Z scale and does not explain why that scale is privileged. Treat 3/13 as the value at M_Z, not a scale-independent derivation. Also: "13 gauge degrees" is a Fano-degree count whose reading as Standard-Model gauge structure is affected by the gauge-group correction above (G2 supplies only colour SU(3)).

Proposed TCS G2 manifold [PROPOSED]

b₂=1, b₃=8. χ=0 verified. SM: b₂=1 as U(1) hypercharge, b₃=8 as eight gluons. TCS construction: two ACyl CY3-folds with b₃ summing to 7 (asymmetric, reflecting X1/X2 asymmetry). Colab →

TSO State of the Framework

Complete summary notebook: derivation chain, exact results, proposed results, open problems, falsification conditions, confidence assessment. Colab →

Acknowledgment

Mathematical direction for the G2/octonion section from Joshua Osborne (RF engineer, G2/octonion expertise). Implementation and errors by John Pepin. Full acknowledgment and related work on roof-g2.

MAY 19–20, 2026 — v11.6 SESSION: σ DERIVATION AND CY STAIRCASE

1. σ derivation from {x,y,z,T} tetrahedron + K3 (Joshua Osborne)

V_tet = 4 (vertices of the spacetime tetrahedron), χ(K3) = 24 (K3 surface Euler characteristic, the CY tower floor). The tetrahedron contributes both terms: 6 = C(4,2) = edges = directed paths (numerator); 4 = vertices = 4D screen (denominator). gap_pips = 4×276 = 1104 = holographic projection of 276 K3 bulk pairings onto 4 boundary vertices. Every hadron mass follows from this formula plus M_Z. σ derivation notebooks →

2. CY staircase — PALP-verified at three rungs

K3 (χ=+24, floor) → WP4[1,1,2,2,2] (χ=−168, N=7, Fano level) → WP4[1,1,1,6,9] (χ=−540, N=15, kaon level) → WP5[1,3,7,19,19,19] (χ=−3192, N=133, E₇ level). Kaon weights [1,1,1,6,9] derivable from: 6+9=15=N_kaon and 6/9=2/3=Koide Q. χ(K3)=24 IS the tadpole denominator. Both CY3 rungs have h₁₁=2 (two phases). Replaces the v11.5 χ(CY4) = 360 candidate.

3. Six stress tests (Joshua Osborne) — five clean passes plus one partial

Bonus: Ω_matter = 0.3111 ≈ p_c = 0.3116 (0.16%). stress test notebooks →

4. Fano-Tetrahedron lepton scale [CANDIDATE]

Joshua Osborne (May 21–22, 2026) proposed a derivation for the lepton absolute scale that was partial-pass in stress test #6. Using only TSO axiom integers Z = 7 (Fano paths, bulk) and V_tet = 4 (tetrahedron vertices, boundary):

The dimensionless factor (Z/V_tet)² = 49/16 = 3.0625. The empirical Koide mean A_Koide = (√m_e + √m_μ + √m_τ)/3 = 17.7156 MeV^0.5; with √E_Pip = 5.7825 MeV^0.5, predicted A = 17.7088 MeV^0.5. Numerical match: 0.04%, no free parameters. Power sensitivity check: alternative exponents miss badly — (Z/V_tet)^1.5 off by 24%, (Z/V_tet)^2.5 off by 32%. The squared power is uniquely positioned.

Bare-vs-pole reframe. The 0.04% deviation places A_TSO 14σ below the PDG-derived A_Koide under PDG measurement uncertainty. Joshua's reframe: TSO derives the bare topological mass scale, while PDG measures the pole (dressed) mass after QED self-energy. The bare-to-pole shift is positive (self-energy adds mass), consistent with the sign of the observed offset. The magnitude is in the right ballpark for one-loop QED at the Koide scale but not yet matched exactly — Joshua's heuristic 3α/(4π) gives ~0.087% on √mass vs the observed 0.038%, a factor of ~2.3 over. Status: CANDIDATE pending a specific QED pole-shift calculation that lands at exactly 0.038%.

Derivation of the squared exponent. Joshua's rank-2 tensor argument: in the SM, mass is a chirality-flipping Dirac bilinear m·ψ̄ψ coupling left-handed Weyl spinors to right-handed Weyl spinors. The mass operator thus acts on a rank-2 tensor of generation indices. Pushing a rank-2 tensor through a coordinate transformation P that scales linearly by Z/V_tet requires the symmetric Jacobian — acting on both bra and ket indices — which squares the linear scale to (Z/V_tet)². The argument is standard Dirac algebra and is closer to a derivation than the earlier "Koide operates on √mass" hand-wave. Fix 4 verification notebook →

Status: CANDIDATE. Two pieces of formal work would promote to ESTABLISHED: (a) an explicit one-loop QED pole-shift calculation on Σ√m landing at 0.038%, and (b) extension of the rank-2 tensor argument to gauge bosons (testing whether spin-1 mass predictions are consistent with the framework). Both are computational, not experimental.

5. p_q = (1 + p_c)/2 — candidate resolution to Open Problem 59

Joshua proposed a 3D Rydberg neutral-atom experiment to measure the site percolation threshold directly. His follow-up message identified p_q = 0.6556 as the ballistic-transport threshold (TSO's Anderson localization gap upper edge). Standard tight-binding literature gives p_q ≈ 0.44 for 3D simple cubic — Open Problem 59 documents this conflict.

The candidate resolution: p_q is not a separate phase boundary but the saturation edge of the same sigmoid centered at p_c. Under a linear-rise law W(p) = (p − p_c)/(1 − p_c), the half-saturation point W = 1/2 lands at p_q = (1 + p_c)/2 = 0.6558. This matches Joshua's TSO value 0.6556 to 0.03%. The tight-binding p_q ≈ 0.44 (mobility-edge measurement) would correspond to the same sigmoid evaluated at ~19% wave fraction — a different operational definition, not a competing prediction.

What's still needed. Joshua's "Holographic Percolation" argument — that the K3-shadow lattice forbids the random pre-percolation clusters that drive β < 1 in standard percolation — has the right structural shape but no formal calculation yet. The decisive experimental test is Joshua's 3D Rydberg sweep, extended to p ∈ [0.20, 0.70] to catch both the classical transition (at 0.3116) and the ballistic onset (at 0.6558 if TSO is right; somewhere else if not). sigmoid-edge notebook →

6. IBM QPU v2 site percolation

Site percolation on IBM heavy-hex (ibm_kingston), 16,800 circuits, 4 path lengths. Result: P_sink ≈ 0.47–0.48 at all p and all L — the GHZ-circuit approach hits the hardware noise floor before the percolation signal. Confirms gap direction (p_q > p_c) but cannot resolve where the classical-to-quantum transition occurs. Conclusion: IBM hardware is engineered to sit above p_c, so site percolation via GHZ chains doesn't work on current QPUs. The Rydberg experiment at KAIST remains the decisive test for p_c specifically.

Honest status assessment

Net probability of TSO being substantially correct after this session: unchanged at ~5%. The σ work moved from "consistency with Joshua's 0.1804" to a proposed closed form 6/√1104 = 0.18058 from K3 + tetrahedron — but its provenance is unverified (6 and 1104 not yet shown forced; chain not independently reconstructed), so it is a proposed identity, not a derivation. The (Z/V_tet)² lepton candidate moves Fix 4 from "broken" to "candidate pending QED magnitude check." The (1+p_c)/2 sigmoid-edge result has the right structural shape but awaits both the formal Holographic Percolation derivation and the experimental verification from Joshua's proposed 3D Rydberg sweep. The Kim citation correction (see retirement #9) reveals a quality-control gap in earlier session work — fixed across the site in this version.

Acknowledgment

v11.6 mathematical work led by Joshua Osborne (RF engineer, G2/octonion expertise): the σ formula structural derivation, the CY staircase identification, the six stress tests, the Fano-Tetrahedron candidate for Fix 4, the bare-vs-pole reframe, the Holographic Percolation argument for β=1, and the rank-2 tensor derivation of the squared exponent. The 3D Rydberg experimental protocol and the cluster-purity diagnostic also originate with Joshua. Implementation, verification notebooks, and any errors by John Pepin (with Claude assistance on computational checks).

MAY 24, 2026 — HARDWARE-TEST CLUSTER AND THE v11.7 EMPIRICAL SCOPE FINDING

The four-test cluster

Fix 4 (April 2026, lepton √mass scale) — Joshua Osborne's proposal: A = √E_Pip × (Z/V_tet)² should equal the Koide mean √mass. Numerical match: 17.7088 vs measured 17.7156 (0.04%). The sign and order of magnitude are right; under PDG measurement uncertainty the offset is 14σ. The bare-vs-pole reframe (TSO predicts bare topological mass; PDG measures pole/dressed mass) is structurally plausible but the explicit QED dressing magnitude has not been derived to land at the observed 0.038% offset on √mass. The rank-2 tensor derivation of the squared exponent is real progress over the earlier hand-wave. Status: CANDIDATE — sign and approximate magnitude correct; precise magnitude pending derivation.

P40 (May 24, 2026, IBM Fano fidelity lock) — pre-registered Joshua proposal: FANO 7-qubit infidelity locks at σ = 0.18058. Six runs across calibration cycles on ibm_marrakesh, 4096 shots each. Result: FANO infidelity = 0.0895 ± 0.0078 (CV 0.087). Pre-registered band [0.163, 0.199]. Observed at exactly σ/2 — ratio 0.4954 ± 0.043. Sign ordering NULL < FANO < RANDOM preserved. CV ratios below the 3× threshold. FALSIFIED on magnitude; signs and ordering replicated. A factor-of-2 diagnostic notebook tested three candidate explanations (classical-vs-quantum fidelity, bare-vs-pole, rank-1-vs-rank-2 projection) without finding a derived structural reason for the precise 1/2. The honest reading: the magnitude is wrong.

P41 (May 24, 2026, three-FANO + matched random) — pre-registered design addressing P40's "lucky qubits" concern: three independent 7-qubit Fano subgraphs on disjoint hardware regions of ibm_kingston, plus one matched-edge-count random control, all submitted in a single job. 131,072 total shots. Result: sign-pattern replication across the three copies PASSED (criterion A1); within-Fano coefficient of variation FAILED the 30% threshold; FANO/RANDOM ratio fell below 2.0 (FAIL); FANO/NULL ratio fell below 3.0 (FAIL). The Steane built-in-error-correction hypothesis (criterion C) literally passed (3.99× FANO improvement) but RANDOM improved by 16.4× — the opposite of what real built-in EC would produce. Structural signs replicated, magnitude separation did not.

P42 (May 24, 2026, hexagon-with-∅-at-center + path role differentiation) — pre-registered design addressing P40/P41's "path-blank" concern (all 7 qubits treated as identical): ∅ at the maximum-degree node of the heavy-hex chip; T idled via Delay; x, y, z, X₁, X₂ as standard rotated perimeter qubits. Heavy-hex max degree is 3, not 6, so the literal hexagon-center geometry does not exist on this hardware. The closest realization is asymmetric: three perimeter qubits are first-shell neighbors of ∅ (1 CZ), three are second-shell (2 CZs). Asymmetry framed as a feature: ∅ is more strongly coupled to some paths than others, consistent with ∅-as-mediator. Pre-registered structural criteria only (S1A, S1B, S1C); no magnitude criteria following the P40/P41 pattern. Status: queued at IBM free-tier allocation exhaustion. Will run when allocation resets or via paid plan. Result will be registered as either a structural-result entry or a retirement on predictions.html.

What the pattern tells us

Four pre-registered hardware tests, four sign-pattern replications, four magnitude failures or non-replications. This is enough data to register the pattern as a scope finding rather than treating each failure as an isolated event needing its own structural rescue. The framework's predictive content on quantum hardware lives in the structural relations between paths — sign patterns, ordering relations, three-body correlations on Fano lines — not in the specific magnitudes those relations produce on any given hardware platform.

The platform constraint

IBM superconducting QPUs are engineered to stay above p_c — kept classical. The chip's entire engineering goal is to maintain qubit coherence; a qubit that decoheres is broken. The IBM hardware can never test "where does coherence break?" because its existence depends on coherence not breaking.

Quandela photonic QPUs are engineered to stay below p_c — kept coherent (wave-state). A photonic chip operates by single-photon interference; if the photons decohere into a classical mixture, the chip isn't computing anymore. Photonic chips also can't test sigmoid crossing.

The decisive sigmoid magnitude prediction (Prediction 1) requires a platform whose physical regime is tunable across p_c. A Rydberg neutral-atom array is the natural candidate: laser detuning, atomic spacing, and principal quantum number all continuously tune the system through the wave-state-to-classical-state transition. Outreach to Rydberg labs (Jaewook Ahn at KAIST, Antoine Browaeys at Institut d'Optique, Gyger group) has been sent; no responses yet as of May 24, 2026.

Until Rydberg access is established, all quantum-hardware-based TSO testing should be considered structural rather than quantitative. The sign-pattern replication on IBM Fano circuits is a real, modest, replicable observation about combinatorial structure of Fano-line correlations. It is not by itself sufficient evidence for the framework, but it is not nothing either. Three independent hardware regions on real superconducting chips produce three-body correlation signs that match the C_MA prediction. That is something the framework correctly predicts about hardware. The magnitudes it claims about hardware, it does not correctly predict.

What this is not

This is not a retirement of magnitude predictions in general. Several magnitude predictions remain quantitatively testable on their native domains, which are not engineered to stay on one side of p_c: the cosmic matter fraction Ω_m ≈ p_c = 0.3116 (Planck data, 0.16% match), the dark energy fraction Ω_Λ ≈ 1 − p_c = 0.6884 (0.07% match), the minimal genome count ≈ 475 (JCVI-syn3.0 measured 473), the Goldilocks band kT ≈ δW (room-temperature biology), and the Rydberg sigmoid prediction itself when it can be tested on appropriate hardware. The framework's quantitative content is not gone — only one specific class of magnitude test (quantum-hardware-based) is removed from the framework's claimed scope.

What this is

An empirical update to what the framework claims to predict, on what platform, at what level of specificity. Four pre-registered tests is enough data to call the pattern. The framework should describe itself accurately, which means saying: on quantum hardware, TSO predicts the structure of Fano correlations, not the specific magnitudes of friction.

FEEDBACK AND COLLABORATION

TSO is a pre-empirical research program by an independent researcher without institutional affiliation. Criticism, literature pointers, and experimental collaborations are welcome. Several retirements on this page came from feedback and re-examination.

Contact: email the author

All notebooks referenced here are public and reproducible. The full notebook index — organized by topic with one-line summaries and status flags — is on the notebooks page. If any result cannot be reproduced or contains an error, please report it.