Particles and Phenomena

How to Read This

Every statement carries a status.

Status	Meaning
Arithmetic	Follows from the definitions of the framework. Can be checked.
Definition	Convention of naming or notation.
Hypothesis	Working proposal. Plausible, not demonstrated.
Reading	Resonance with an external concept. Evocation, not derivation.
Open	Unresolved question.

Notation: OBS and EXP are relations (n−1 and n+1). The acronym for the observer pivot at T⁻⁴ is PIV-OBS.

Rules of method that are kept:

A level does not contain a phenomenon. Its conditions of possibility are a floor, not a location.
Acronyms are mnemonics. The operator is Tⁿ.
Within a level, behaviors are mutually exclusive (§9.6).

Indexing convention used here:

k	0	1	−1	2	−2	3	−3	4	−4	5	−5	6
n	1	2	3	4	5	6	7	8	9	10	11	12

Canonical mode: for k > 0, k valences of 2 phases. For k < 0, |k| valences of 2 phases plus one unknown (the open BC).

1. The Observation Paradox

1.1 What the Original Annex Says

Dimensions exist independently of the observer. They manifest as empty space. They do not occupy space: they are space. Observing them is an act of choice. A 3D dimension (6 phases) observed as a particle does not allow position and “age” to be fixed at the same time. The uncertainty principle is a direct consequence.

1.2 Reformulation

Complementarity is the principle of choice (§9.6), with physical content. Position and age are two behaviors of the same level. If one occurs, the other does not. No separate principle is needed. (Hypothesis)

OBS and EXP are the neighbors of a level in the sequence of n. (Arithmetic)

OBS(Tᵏ) has n−1 phases. The observer takes one phase (“steals a phase”).
EXP(Tᵏ) has n+1 phases. The open phase appears.

Level	n	OBS (n−1)	EXP (n+1)
T⁰	1	—	T¹
T¹	2	T⁰	T⁻¹
T⁻¹	3	T¹	T²
T²	4	T⁻¹	T⁻²
T⁻²	5	T²	T³
T³	6	T⁻²	T⁻³
T⁻³	7	T³	T⁴
T⁴	8	T⁻³	T⁻⁴
T⁻⁴	9	T⁴	T⁵
T⁵	10	T⁻⁴	T⁻⁵
T⁻⁵	11	T⁵	T⁶

Consequences:

OBS is a step backward along the chain (backward chaining, §8.2). EXP is a step forward (inversion, §14).
In a negative level T⁻ᵏ, the open phase is the observer’s. What remains are the 2k closed phases of Tᵏ. That is why observing T⁻ᵏ is seeing Tᵏ as properties. It is the INV-QUOT of §20.
Examples from the original annex that fall in the table: observing T⁵ gives T⁻⁴ (9 phases + 1 from the observer). The experience of T² occurs in T⁻² (5 phases). What is observed of T⁻¹ is T¹. The EXP-EX-OBJ of T² is the experience of the OBJECT of T⁻¹.

Coexistence in 6D. The original annex said that in 6D position and age are simultaneous.

Position (T²) and age (T¹) are the factors of the ARXE-ST reading of T³ (1+2). They are not the ARXE-ST/COL pair. (Arithmetic)
The canonically conjugate pairs add up to 6: (t, E) = 1+5 and (x, p) = 2+4. Both products give ħ (T⁶). (Arithmetic)
Indeterminacy would be read as the distribution of the action between two levels that add up to 6. (Hypothesis)
T⁶ = T³·T³ allows two readings of T³ at once (§21.4). In a single T³, one is given at a time. §21.4 and the principle of §21.5 were rewritten on October 29. The last sentence of the §21.5 example remains to be pinned down (see the Fundamental Particles annex, §1.4). (Author’s decision)

1.3 OBS and EXP as Operations on Modes

Mechanics. OBS removes a phase. EXP adds one. (Arithmetic)

Removing a phase from one pair of (2,2,2) gives (2,2,1). It is T³ → T⁻².
Adding a phase to (2,2) gives (2,2,1). It is T² → T⁻².
Adding a phase to the unknown of (2,2,1) closes it into a pair: (2,2,2). It is T⁻² → T³.

The alternation of sign is not imposed. It comes out on its own: adding a phase to a closed level opens a BC. Adding it to an open one closes it. This is consistent with §7.1.

With existing operators.

EXP of a positive level is INV-QUOT: T² → T⁻².
OBS of a negative level is the inverse of INV-QUOT: T⁻² → T².
EXP(X) = X + T⁰ by FUSION. OBS is its inverse. The stolen phase is the observer’s (T⁰). Since FUSION does not retain components, the observer disappears from what is observed. This is the observation paradox.
(Open): decide whether OBS(Tᵏ) + T⁰ = Tᵏ is a sum or a rewriting. The sum of arities is not uniquely invertible. Here T⁰ fixes the inverse.

Interface identity. (Arithmetic)

T⁻ᵏ = EXP(Tᵏ) = OBS(Tᵏ⁺¹)

n(T⁻ᵏ) = 2k + 1. n(Tᵏ⁺¹) − 1 = 2k + 2 − 1 = 2k + 1. They are the same level.

Every negative level is the interface between two consecutive positive ones. It is what the first one experiences and what the second one shows when observed. The loose phase is the point of contact with the observer. One cannot say whether it is “its phase” or “the gap it left”, because phases are indistinguishable.

Test by level. The two readings of a negative level must be compatible. (Reading)

Level	EXP(Tᵏ)	OBS(Tᵏ⁺¹)	Verdict
T⁻¹	alternation of time	motion of space	Compatible
T⁻²	position, entanglement	curvature of mass	Compatible
T⁻³	inertia, color	information as a difference in a carrier	Compatible, with a condition
T⁻⁴	BIT, entropy production	dissipation of energy	Compatible
T⁻⁵	U(1) field (Kaluza–Klein)	phase of the action: S/ħ (T⁶ as action)	Compatible with the action reading. Doubtful with the Calabi–Yau one

Case T⁴ → T⁻³. T⁴ is (2,2,2,2). The observer takes one phase from a pair. The four pairs are indistinguishable, so the choice is trivial (§9.7). What remains is (2,2,2,1) = T⁻³. (Arithmetic)

Reading: observed information shows up as the variation of something massive. Distinguishing two states requires a carrier with VAR-MASS or VAR-COL. This goes in the direction of Landauer.
Condition. A photon carries a massless bit. The rule saves it only if “observed” means detected. Detection happens in a massive absorber: PIV-OBS ⊕ REG (§21.6). In transit, the bit is not observed.
Falsifier. Any observed information without variation of a massive carrier refutes T⁴ → T⁻³.
Reading coherence (§21.5). If T⁴ chooses the 3+1 partition with COL, what is observed must be read as VAR-COL, not VAR-MASS. (Open)

Case T⁶ → T⁻⁵. T⁶ houses two terms: action (§12.1) and CAL-YAU (§19). §18 says that the single choice is demanded by the dialogical formula, not by the level. There is no need to choose one forever. The two must not be mixed within one derivation (§21.5).

T⁶ as action (ħ, §12.1). The phase of the wave function is S/ħ. Observing the action shows a continuous phase: T⁻⁵ (REG, U(1)). It is compatible, and needs no internal manifold. (Reading, with a dimensional basis: ħ = T⁶)
T⁶ as Calabi–Yau (§19). If T⁻⁵ must come out of the internal manifold, the gauge fields should come from its isometries. A compact Ricci-flat manifold with full holonomy has no continuous isometries (Bochner). This reading is doubtful.

For this test the action suffices. The Calabi–Yau reading remains a separate derivation, doubtful because of Bochner. The rule holds at T⁶ by the first one. (Hypothesis)

Scope. The identity is arithmetic. That the annex agrees with it confirms coherence between the annex and the indexing. It is not an independent proof. All the content is in the assignments. That is why the tests above are the ones that count.

1.4 “Eternal” Particles and the Isolated Particle

An ideally isolated particle lacks dimensionality, history, space, and information. It is not a historical reality. It is a consequence of the observation paradox.

Reformulation: a particle is a phenomenon of level k < 0. It has an open BC and does not exist in isolation (§7.1). Isolating it is participating in what is observed. That is why observation demands a positive action: an axiomatic act of participation (“I EXIST”), which corresponds to T⁰. (Hypothesis)

The appearance of eternity is an effect of that participation. What is observed shows no history because history (record) only exists from T³.

2. Phenomena by Level

For each level: canonical mode, reading of the BC, phenomenon, and what it lacks. Locations are readings unless stated otherwise.

T⁰ — Act (n = 1)

Tempus instans. Observer. Essential, not existent.
There is nothing to order or choose.
It has no particle. Objectivity is relative to it.

T¹ — Finite Time (n = 2, mode (2), BCC)

Phenomenon: homogeneous temporal flow. One instant after another, in one direction.
The direction is trivial: both senses are indistinguishable. No inversion is possible.
Observed from T⁻¹ (OBS), what remains is a fundamental existence without distinguishable properties.
It is the level that can be taken as universal: every choice is trivial.

T⁻¹ — Alternation (n = 3, mode (2,1), BCC + BCO)

Temporal particle. An objective experience relative to the temporal level. It reveals a BCO, a choice.
Object. An existence (2 phases) that is objective because a third (1 phase), a validator, takes part. It is the mode (2,1). (Arithmetic)
Variation. It is the alternation (1 phase + 2 phases). It is not objectively observable. Observed, it would only reveal void or infinity.
Precedence (SEQ). (a→a’) ≠ (a’→a). It is the only asymmetric mode of T⁻¹. It is structural. It is not yet the arrow of time.
Causality. Minimal causal order: SEQ without space. (Reading)
Directional appearance. It establishes a direction with the observer. It seems to come from infinity. It has no origin.
Choice of finitude. If one is finite, the other is infinite. They cannot both be finite or both infinite at once.
Lacks: space, memory (MEM). It is a single line, with no simultaneity.
Frequency (SI dimension T⁻¹) is its dimensional reading.

T² — Empty Space and Return (n = 4, mode (2,2), BCC)

Return. (start, end, start, end). It is the copy of §9.11.1. A T¹ that turns back on itself. (Reading)
Simultaneity. Two directions: one and its inverse. They are not orderable.
Spatial field. Virtual positions, indistinguishable from one another. It is the isolable (virtual) form of space.
Passage from T⁻¹. Temporal undecidability generates space (§14). This passage is what the original annex called “pivot”: stopping time and turning it into space. Another name is advisable (see §4), because PIVOT in §20 is a different operator.
Observed (OBS) it shows T⁻¹. Its experience (EXP) is at T⁻².
Geometric anchor: π, the first level with circle closure (§16).

T⁻² — Spatial Particle (n = 5, mode (2,2,1))

Position and entanglement. Two ends (2 + 2 phases), each without a position of its own. It is an ambivalence: a single phenomenon, pure simultaneity, undecidable. (Arithmetic on §9.10.2)
Collapse. Determining one end takes (2,2) to (3,1). The sizes become different, and the state becomes orderable. (Reading)
Correction to the original annex. What is observed of T⁻² has 4 phases, (2,2). It does not have 3. The 3 phases of (3,1) appear after one end is determined.
Origin. It is the mode (3,1): 3 phases (T⁻¹) + 1 phase (T⁰). A position that endures in time. It persists, but does not record. (Reading)
Point-likeness. A point exists even if there is nothing in it. Observed, what remains is a wavelength as a property.
Curvature, expansion, and acceleration. Acceleration is VAR·VAR. By adding exponents it is T⁻¹·T⁻¹ = T⁻². (Arithmetic) Temporal curvature also goes here.
Angularity and axis. It is a freedom to choose one direction among many equivalent ones. The mode (2,2,1) is “two identical elements plus an axis”. It is the IMP-REL (B) structure of §23, which had no location. The choice of axis is angular CHOICE. Spin and its quantization axis would be the physical case. (Hypothesis) [Updated] Spin moved to T⁶ (action) and the projection is an exclusive OBS (Fundamental Particles annex, §2.1). The (2,2,1) mode of T⁻² remains only as a candidate for IMP-REL (B).
Medium. Two opposite points lack a medium. The medium appears when multiplying by T⁻¹: T²·T⁻¹ = T¹, which is velocity. Without T⁻¹ there is no propagation: there is correlation. This is the no-signaling of entanglement. (Arithmetic + Reading)
2+1 gravity. T² ⊕ T⁻¹ (CONJ) is a triangulable space. 2+1 gravity is quantized on triangulations and has no local degrees of freedom. It is the same reason why GEO cannot be at T⁻² (§19). (Reading)

T³ — Mass and Record (n = 6, mode (2,2,2), BCC)

Mass, 3D space, extension, distance. Existing start, middle, and end appear. 2D space is a directional field (simultaneity). 3D is extensional.
3D field. It is the Higgs field in structural reading (T⁻²→T³ step of §14). (Reading)
Continuous rotation (VAR-ANGULAR). Ternary orientation without a preferred axis. π becomes undecidable (§15.2). To distinguish: at T⁻² angular freedom is choosing an axis. At T³ it is rotating continuously.
Record and memory. The mode (3,2,1) gives past, present, and future. Here MEM and irreversibility appear (§9.12).
Two exclusive readings. ARXE-ST (TIME·SPACE2) or COL (pure ternary symmetry).
Observed (OBS) shows T⁻²: curvature.
It has no temperature of its own. It is a closed, isolated level. Temperature is relational (see §3).

T⁻³ — Massive Particle (n = 7, mode (2,2,2,1))

Finite particle, with mass. Mass in its objective form, the one that is experienced: VAR-MASS.
Inertia. VAR-MASS read as resistance to varying. (Reading)
Color and confinement. VAR-COL. It is a mass with one degree that does not close.
Hadron and proton. Confined structure with mass. It goes here, not at T².
It is what is observed of T⁴ (§1.3). Condition: “observed” means detected.
Geodesics (GEO). Trajectories with real physical effect. They require 3+1 dimensions (§19).
(Open): GEO and VAR-MASS are at the same level. If they are readings of the same behavior, the equivalence between inertial and gravitational mass comes for free. If they are exclusive (§9.6), there is a tension to clarify.

T⁴ — Information (n = 8, mode (2,2,2,2), BCC)

INF. Momentum and power (dimensional).
Pivot between mass and energy. 4 = (3+5)/2. Dimensionally p² ∝ m·E. (Arithmetic)
INF / SPACE2 = T²: holographic reading (§21.6).
Two partitions: 2+2 (two polarization planes) and 3+1 (corpuscular aspect).
OBS(T⁴) = T⁻³: observed information shows up as the variation of a massive carrier. Condition and falsifier in §1.3.

T⁻⁴ — Observer and Entropy (n = 9, mode (2,2,2,2,1))

PIV-OBS: observer pivot. T⁻⁴ is the midpoint of T⁻³ and T⁻⁵. (Arithmetic)
The acronym is renamed from OBS to PIV-OBS. OBS stays reserved for the n−1 relation (§1.2).
ENT-R: entropy production rate. BIT: the particle of INF.
Physical magnetism does not go here. It goes at T⁻⁵.

T⁵ — Energy (n = 10, BCC)

Energy (dimensional). It is missing from the registry of §19.
Its open counterpart is T⁻⁵. An excitation is the exchange of energy with the field.
Unruh. kT = ħa/(2πc). In the algebra of the framework: T⁶·T⁰/T¹ = T⁵. It closes. (Arithmetic)

T⁻⁵ — Electromagnetic Field (n = 11, mode (2⁵,1))

REG: U(1), continuous phase.
E and B as two faces. The lines of B return to the origin: that is the BCC face. E has sources, the charge: that is the BCO face. The absence of monopoles would be a consequence: B admits no open end. (Hypothesis)
Detection. PIV-OBS ⊕ REG: the field needs something to absorb it in order to be a discrete event.

T⁶ — Coexistence (n = 12, T³·T³)

Level with enough budget for the two readings of T³ to coexist (§21.4). See §1.2.
OBS(T⁶) = T⁻⁵ is under test (Bochner). See §1.3. (Open)

3. Tuples and Compounds

A phenomenon may not be a level. It may be a tuple (CONJ).

3.1 Photon

It is a tuple: T⁻⁵ (U(1) identity) projected onto T² (geometry), with c = T¹ (velocity). What defines it is the absence of T³: no mass, no volume.

It has position but no volume. It is point-like and directional.
Two transverse polarizations: the 2+2 of T⁴ (§21.3). [Updated] In the Fundamental Particles annex the photon is electric charge + spin, and the two polarizations are read as multiplicity (2 phases), not as geometry.
“Directional simultaneity”: an angular choice out of one. It is the freedom of the axis of T⁻².

(Hypothesis)

3.2 Wave

λ (T²) · ν (T⁻¹) = c (T¹). Sum of exponents. (Arithmetic) It is the relation among space, alternation, and velocity.

3.3 Temperature

Temperature is not a level. It is a compound. Before locating it, its condition must be fixed (§24).

kT is energy: T⁵. With ħ = 1 it is frequency: T⁻¹.
A thermal state is periodic in imaginary time, with period ħ/kT (KMS condition). That is a closed time: T¹ with BCC. Agitation would be its T⁻¹ face. “Absolute zero” is T¹ without varying. (Reading)
Unruh links temperature and acceleration (see T⁵).

3.4 Force

The annex uses two senses. They should be separated.

Dimensional force: ML/T² = T³ (same level as mass).
Ontological force: time forcing a choice. It is CHOICE applied to T⁻¹.

(Open)

4. Relocations with Respect to the Original Annex

Element	Before	Now	Reason	Status
Acceleration	T²	T⁻²	VAR·VAR = T⁻¹·T⁻¹	Arithmetic
Temporal curvature	T²	T⁻²	It is variation; T² is BCC	Reading
Distinction	T²	T⁻¹ (DIFF)	It has a BCO. T² is entirely BCC	Arithmetic
Proton	T²	T⁻³	Confinement = open BC with mass	Reading
Higgs field	T²	T³ (field) and T⁻³ (boson)	T⁻²→T³ step of §14	Reading
Photon	T⁻²	Tuple T⁻⁵, T², T¹	EM is at T⁻⁵	Hypothesis
Magnetism	T⁻⁴	T⁻⁵	T⁻⁴ is PIV-OBS/ENT-R/BIT	Reading
One-way direction of time	T⁻¹	T¹ (direction), T⁻¹ (precedence), T³ (irreversibility)	§9.12	Hypothesis
Temperature and excitation	T³ and T⁻³	Compound (§3.3); excitation at T⁵/T⁻⁵	An isolated level has no temperature	Reading
“Pivot” (T⁻¹→T² step)	T²	Rename	Clashes with PIVOT of §20	Definition
Observed geodesics	T³ → T⁻²	Curvature at T⁻²; physical geodesics at T⁻³	OBS(T³) = T⁻²	Open

5. New Phenomena and Properties Proposed

Proposal	Level	What it contributes	Status
Causality	T⁻¹	Minimal asymmetric order (mode (2,1))	Reading
Wave and medium	T²·T⁻¹ = T¹	Distinguishes correlation from signal	Arithmetic + Reading
Spin and quantization axis	T⁶ (spin), exclusive OBS (axis). T⁻² (2,2,1) only for IMP-REL (B)	[Updated] See the Fundamental Particles annex, §2.1	Hypothesis
2+1 gravity	T² ⊕ T⁻¹	Explains why GEO is not at T⁻²	Reading
Inertia	T⁻³	VAR-MASS as resistance to varying	Reading
Energy	T⁵	Missing from the registry. Counterpart at T⁻⁵	Arithmetic
Unruh	T⁶·T⁰/T¹ = T⁵	Closes the algebra of temperature	Arithmetic
Mass-energy pivot	T⁴	p² ∝ m·E	Arithmetic
Observer pivot (PIV-OBS)	T⁻⁴	Midpoint of T⁻³ and T⁻⁵	Arithmetic
Interface identity	T⁻ᵏ = EXP(Tᵏ) = OBS(Tᵏ⁺¹)	Every negative level joins two positive ones. Gives tests by level	Arithmetic
Information as variation of a carrier	T⁴ → T⁻³	Falsifiable: observed information without a massive carrier	Hypothesis
B as a closed face	T⁻⁵	Absence of monopoles as a consequence	Hypothesis
Casimir	T² (boundary)	A closed boundary defines a scale (§7.2)	Reading
Persisting ≠ recording	T⁻² versus T³	Origin endures without memory	Hypothesis
Complementarity as exclusion	§9.6	Uncertainty without a separate principle	Hypothesis

6. Proposed Adjustment to §14

The narrated chain reaches T⁻³. The text continues with “T⁻³ → T⁻⁴ → T⁻⁵”. If the chain follows the order of n, it skips T⁴. The step would be:

T⁻³ → T⁴. The internal degree of color is not decided from T⁻³. It stabilizes in a closed structure that records which is which. That is information.
T⁴ → T⁻⁴. What information does not decide is who reads it. That is the observer and entropy, which §19 already places at T⁻⁴.

(Narrated hypothesis, like the previous steps)

7. Open Questions

Is “position/age” the same pair as ARXE-ST/COL? If not, how many readings does T³ have?
Are GEO and VAR-MASS one behavior or two exclusive ones?
Does B belong to T⁻² (§14) or is it the BCC face of T⁻⁵ (§3.3)? It has to be decided.
ACC appears at T⁻¹, T⁻² and as a dimensional T⁰. Fix a single use.
Fix the condition of temperature before locating it (§24).
Verify PIVOT on more triples, for example T⁻² between T⁻¹ and T⁻³.
No mnemonic in this annex has passed the reading rule (§18).
T⁶ → T⁻⁵: keep the action derivation (S/ħ phase, compatible) and the Calabi–Yau one (doubtful, Bochner) separate.
T⁴ → T⁻³: what selects the VAR-COL or VAR-MASS reading when T⁴ chooses 3+1?
Decide whether OBS(Tᵏ) + T⁰ = Tᵏ is a sum or a rewriting.
Narrate the steps T⁻³ → T⁴ → T⁻⁴ in §14 (proposal in §6).