TWIST-J Canon Core Summary

canon basis v177, 11 July 2026
canon pin SHA-256 a2872dd586bb6ec590556a64781e08add924590331c909c9916187f098bec79f
purpose stable public entry point for humans and AI systems that cannot read the full Canon
authority authoritative public summary, subordinate to the current private Canon repository HEAD; it does not replace the full Canon or its status registry
portable source core-summary.md, a Markdown copy of the same summary for systems that can read it

full canon latest.md the complete committed body of record
canon index /canon/ · manifest /manifest/
portable copy core-summary.md the same summary as Markdown, for humans and tools that prefer it

This HTML page is the primary public reading surface. The Markdown file is a portable copy of the same summary. If they differ, treat the discrepancy as a publication error. The full Canon remains authoritative over both.

1. What TWIST-J is

TWIST-J is a program of discrete integer physics built on one closed arithmetic object.

A0. Reality is the closed integer J-Cayley plenum. It has no external boundary, no external clock, and no external reader. Its algebraic generator is

J = 1 + ζ₅², ζ₅⁵ = 1, ζ₅ ≠ 1.

The program uses zero free dimensionless parameters and one SI calibration anchor, the electron mass m_e. The structured background is called the plenum, not vacuum.

The shortest canonical reading is:

Time is a counter. Space is a commutator. J is the verb. The modulus and argument of J are the two abelian projections.

2. Status discipline

No statement may be stronger than its Canon label.

Label	Meaning
T-LOCK	Immutable theorem in the Canon
LOCK	Immutable canonical datum
T	Theorem at its declared scope
D	Derived physical or structural dictionary
C	Exact computation at finite scope
H	Hypothesis with an explicit falsifier
O	Open obligation
F	Falsified route, retained permanently

Falsification is progress. A negative result is archived, not erased. Exact arithmetic, preregistration, and independent two-platform verification are the default promotion gates.

3. The immutable algebraic core

Let j := ζ₅, K := ℚ(j), 𝒪_K := ℤ[j]. The basic exact identities are

N_K/ℚ(J) = 1,    Tr_K/ℚ(J) = 3,
J = j/φ,    φ = ‖J‖⁻¹ = (1 + √5)/2,
j = (J − 1)³,    J⁵ = φ⁻⁵.

Thus J generates the same cyclotomic field as j. The phase closes after five powers and the modulus remains.

The foundational analytic action value is

L := Re Li₂(J) = π²/100, π = −5i Li₁(J).

The canonical plenum point is

T = s + iφ = 2i(1 − J), s² = 1 + J J = 3 − φ, |T| = 2.

These objects are projections or consequences of J, not independent primitives.

4. The integer kernel

In the basis {1, j, j², j³}, multiplication by J is the exact integer matrix

        (  1   0  -1   1 )
M_J  =  (  0   1  -1   0 )   in SL(4, Z)
        (  1   0   0   0 )
        (  0   1  -1   1 )

Equivalently, (a, b, c, d) ↦ (a − c + d, b − c, a, b − c + d). Both M_J and M_J⁻¹ use only −1, 0, 1. The kernel therefore needs additions and subtractions, not floating-point arithmetic.

The canonical finite state is ψ_n = (p₁, p₄, p₁′, p₄′, q, t₅) ∈ 𝔽₅⁶, with |𝔽₅⁶| = 5⁶ = 15625. The global binary driver is the Thue-Morse bit t_n = popcount(n) mod 2.

The compact Canon form of the update is

ψ_n+1 = Π_{t_n}(M_J ψ_n)

Dimension note: the displayed M_J is the 4×4 kernel acting on the four piston coordinates (p₁, p₄, p₁′, p₄′) of ψ_n ∈ 𝔽₅⁶; the compact form abbreviates the extended finite map on 𝔽₅⁶, in which M_J updates the piston block while the fiber q and the trace register t₅ are updated by the accompanying rules of the same map.

Here Π₁ = Id for Flow and Π₀ for Snap. Snap is the source of irreversibility. In the Cayley realization the active generator is selected by i_n = Tr₆(ψ_n) + 2t_n (mod 5). The offset 2 is the exponent in J = 1 + j². The finite Cayley system is the canonical finite realization of the same J-action, not a second axiom.

5. Geometry, gauge housing, and the decoder

The Galois projection of the four piston coordinates has Gram form

G = I₄ − (1/5) 11^T, spec(G) = {1, 1, 1, 1/5}.

Therefore 𝔽₅⁴ = ker(Tr₄) ⊕ ⟨1⟩, with three unit-weight traceless directions and one trace direction suppressed by 1/5. This is the exact algebraic source of the 3+1 split. The trace direction is not zero and the gauge fiber is not a fourth spatial direction.

The attractor carries an invisible ℤ₅ fiber. The composition g = e ∘ d acts as a pure translation q ↦ q + 1, without moving the Galois base. The fiber is housing for private phase. It is not itself a measurement event or an observer.

The canonical reading split is:

Sector	What it carries	Reading mode
J and piston kernel	state and geometry	linear projection
Thue-Morse driver	the cut in ordered time	binary
gauge housing	registered amplitude	quadratic

The decoder factorizes as

D = D_clock ∘ D_geom ∘ D_matter

It is acyclic and has the structural form 1+3+0: one clock output, three spatial directions, and a gauge sector with no directly observable coordinate. The factorization is canonical. D_clock and D_matter are T-LOCK; D_geom is complete at theorem grade with some promotion labels still below T-LOCK.

6. Stable finite facts

The exact finite substrate has a theorem-grade census of 313 attractor supports. The current hosting decomposition is

313 = 2·144 + 24 + 1 = 13² + 12².

The recurrent dynamics uses the mirror algebra ⟨b, d, e⟩. Its equilibrium weights are w_b = 2/3, w_d = w_e = 1/6. The unique linear piston readout is Tr₄, with exact ensemble visibility rate 4/5.

A gyron is a decoder event, not a substrate particle. On the recurrent system,

Gyron(n) = [z_obs(ψ_n) = 4] ∧ [t_n = 0] = [t_n−1 = 0] ∧ [t_n = 0].

It is exactly the Thue-Morse pair 00. Hence

ρ = P(00) = 1/6

The fiber is uniform and independent of this event: P(q = k) = 1/5 and P(Gyron | q = k) = 1/6 for every k ∈ 𝔽₅. This separation is fundamental: geometry, gauge housing, and registered events must not be conflated.

7. Canonical physical dictionary

The following is the stable TWIST-J dictionary. Its algebraic inputs are exact; individual physical lifts retain their own Canon labels.

Canon object	Physical reading
ordered index and Thue-Morse cut	time as counting
noncommuting generator action	space and discrete curvature
\|J\| = φ⁻¹	gravity and scale projection
arg J = 2π/5	electromagnetic phase projection
ker(Tr₄)	three-dimensional spatial carrier
q-fiber	invisible gauge-phase housing
gyron event	registered matter event
Log plus Checkpoint	complete internal memory primitives

A particle name belongs to the decoder-lifted physical layer. It must not be projected backward into an unsupported substrate-level particle claim.

8. Scoped sealed theorems at v177

These results are exact and sealed at their declared scopes. They are not broader than those scopes.

8.1 General measurement coupling, v175

For finite carriers over ℚ or ℚ(i), Euclidean Gram geometry, and fine-grained canonical instruments with one Kraus operator per outcome:

every instrument has an exact square-root-free reversible coupling to system, pointer, and memory;
the coupling writes a physical record and induces the original Kraus operators exactly;
state update is conditioning on that record, not an extra postulate;
the effects E_a = K_a^†K_a are the shadow of the coupling;
effects do not determine the instrument, and distinct couplings may induce the same instrument.

The Galois-Gram lift of this theorem is sealed at v177; see 8.3.

8.2 Covariant canonical coefficient, v176

For the DeWitt form 𝒢_λ[g](h, h) = h_abh^ab − λ(tr_g h)², the value on the metric itself is independent of the metric:

𝒢_λ[g](g, g) = d(1 − λd).

At the substrate value λ = −1 and d = 3,

d(1 − λd) = d(d + 1) = 12.

At λ = −1 the form is positive definite on symmetric tensors. The GR value λ = +1 has the negative conformal direction. This fixes the covariant home of the coefficient 12 at theorem grade.

It does not prove full equivalence with general relativity, and it does not yet select the physical dressing insertion 1 + 12α². That insertion and its normalization bridge remain open.

8.3 Galois-Gram measurement coupling and the carrier no-go, v177

For the fixed rational Gram of section 5, G = I₄ − (1/5) 11^T, its integer scale G₅ = 5G, carriers over ℚ or ℚ(i), the sharp adjoint A^♯ = G⁻¹A^†G, and fine-grained canonical instruments ∑_a K_a^♯K_a = I:

1. The correct density against the Gram form is

ρ_ψ = ψψ^†G / (ψ^†Gψ)

ordinary trace one and sharp self-adjoint, and the Born value p(a) = Tr(E_aρ_ψ) is the branch G norm ‖K_aψ‖_G²/‖ψ‖_G², invariant under G versus G₅. On the sealed piston witness the pinned projective instrument gives exactly p_G = (9/34, 25/34); the three Euclidean contaminations give 1/7, 3/14, 3/17, all different.

2. Every such instrument is induced by an exact square-root-free Ĝ-unitary coupling of system, pointer, and memory, with effects E_a = K_a^♯K_a the shadow of the coupling; the construction never uses a Cholesky factor, and at G = I it reduces to the sealed Euclidean theorem of 8.1.

3. Carrier no-go: on the coordinate piston carrier (ℚ⁴, G₅) the identity

w = v₀ − v₂ = v₁ − v₃ = (−1, −1, 1, 1) ≠ 0

puts a nonzero vector in both gyron class spans, so perfect class discrimination is impossible for any instrument-realizable effect; the best sharp projector leaks exactly 7/22 and 172/187. The sealed exact class readout therefore lives on the orthogonal label carrier indexed by the piston addresses, not on the coordinate carrier. This locates the gyron readout of section 6 precisely.

Scope ceilings: the fixed G and its scale only; mixed states only as prepared finite rational mixtures; no identity is claimed between the coordinate Gram involution and the cyclotomic, Born, residue-field, or global involutions; the identification of the hybrid G ⊗ I_label carrier with the physical gyron decoder is recorded at most at D.

9. What this summary does not claim

The following are not part of the immutable core and must be read from the current full Canon with their actual labels:

complete SI metrology and the tick-to-edge unit bridge;
the quadratic decoder effects clause on data; any Gram beyond the fixed G = I₄ − (1/5) 11^T and its integer scale; the identity of the coordinate Gram involution with the cyclotomic, Born, residue-field, and global involutions;
the dressing insertion and Level-1 to Level-2 normalization inheritance;
the full inhomogeneous Einstein sector or a proof that λ = −1 is GR;
numerical tensor normalization for primordial or radiative r(k);
QCD dynamics, confinement, and running from the finite color carrier;
the observer write-port and any consciousness claim;
any numerical agreement whose Canon status is D, C, H, or P rather than T.

10. Rules for AI readers

Use the private Canon HEAD as the source of truth. This summary is only an entry point.
Preserve status labels. Never promote D, C, H, or O by paraphrase.
Use exact integer, rational, cyclotomic, or ℚ(√5) arithmetic in assertions.
Keep substrate, driver, gauge housing, decoder event, and observer layers separate.
Keep ℚ(ζ₅) and ℚ(ζ₈) distinct. Their intersection is ℚ.
Treat a fired falsifier as a permanent Canon result.
State the scope of every theorem, especially measurement and continuum claims.

Minimal machine-readable card

The card separates fixed mathematics from physical reading. Everything through the event block is exact: the existence of the Thue-Morse pair 00 and its density 1/6 are theorems of the finite carrier. The physical channels block is the strongest physical reading of the program; the full identification with the measured gyron event and the closed bridge to gravity and electromagnetism are where the decoder and measurement work continues, with its own Canon labels.

name: TWIST-J
canon_basis: v177

axiom:
  A0: closed integer J-Cayley plenum

arithmetic:
  J: 1 + zeta_5^2
  field: Q(zeta_5)
  ring: Z[zeta_5]
  unit_facts:
    norm: N(J) = 1
    trace: Tr(J) = 3
    reconstruction: zeta_5 = (J - 1)^3

finite_carrier:
  state_space: F_5^6
  state_count: 15625

internal_driver:
  sequence: t_n = popcount(n) mod 2
  name: Thue-Morse
  update: psi_(n+1) = Pi_(t_n)(M_J psi_n)

geometry:
  gram: G = I_4 - (1/5) 11^T
  spectrum: [1, 1, 1, 1/5]
  space: ker(Tr_4)
  spatial_dimension: 3

three_readings:
  kernel: linear
  driver: binary
  gauge: quadratic

gauge:
  fiber: Z_5
  invisible_coordinate: q

decoder:
  order: D_clock o D_geom o D_matter

event:
  gyron_address: Thue-Morse pair 00
  asymptotic_density: rho = 1/6

physical_channels:
  modulus_J: scale / gravity channel
  argument_J: phase / electromagnetic channel

Axiom: J = 1 + j². Zero free dimensionless parameters; one SI calibration anchor. Time is a counter. Space is a commutator.

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