TNFR p-adic Dynamics — the Projective Network Tower (R4)¶

Status: transport consistency DERIVED (fiber-averaging intertwining) + MEASURED (exact rational residual 0; spectrum containment ~1e-15). The tower→REMESH identification (NT-P04) is CONJECTURAL — the REMESH operator contract is deliberately left unverified. No factoring, complexity, cryptographic, or Millennium claim; no extrapolation to the limit ℤ_p. Module: src/tnfr/mathematics/padic_tower.py · Tests: tests/mathematics/test_padic_tower.py · Benchmark: benchmarks/padic_scale_consistency.py · Depends on: R3 (multiscale composition), C4 (spectral tolerance), C5 (claim manifest).

1. The tower and its scale maps¶

The p-adic tower is the inverse system

\[\mathbb{Z}/p\mathbb{Z} \xleftarrow{\pi_1} \mathbb{Z}/p^2\mathbb{Z} \xleftarrow{\pi_2} \mathbb{Z}/p^3\mathbb{Z} \xleftarrow{\pi_3}\cdots, \qquad \pi_e(x) = x \bmod p^e.\]

Each coarse node has a fiber of p fine nodes. Two linear maps encode the scale change between level e+1 and level e:

the fiber-averaging (aggregation) map R_e (projective_scale_map), a row-stochastic p^e × p^{e+1} operator, \((R_e f)(y) = \tfrac1p \sum_{x \equiv y\,(p^e)} f(x)\);
the lift (prolongation) map Lift_e (padic_lift_map), the p^{e+1} × p^e pullback \((Lift_e\,g)(x) = g(x \bmod p^e)\), constant on fibers.

The scale map is given the neutral name projective_scale_map — not REMESH (§5).

2. The reduction-compatible family¶

Transport is projective only for connection sets that reduce uniformly over fibers. The declared family lifts any nonempty base pattern mod p by ignoring higher p-adic digits:

\[S_e = \{\, x \in \mathbb{Z}/p^e\mathbb{Z} : x \bmod p \in \text{base} \,\}, \qquad \text{base} \subseteq (\mathbb{Z}/p\mathbb{Z})\setminus\{0\}.\]

Because fiber-averaging only sees the residue mod p, every coarse connection element is represented with equal multiplicity p at the finer level — the condition that makes the diagrams below commute. Special cases: base = {1} (a lifted successor pattern) and base = (ℤ/pℤ)^* (the unit tower, connecting to R3).

3. Projective transport (DERIVED + MEASURED, exact over ℚ)¶

Writing P_e = (1/|S_e|) W_e for the random-walk transition and L_e = I − P_e, the compatible family satisfies three exact identities (fractions.Fraction, residual 0):

\[\boxed{\,R_e\,P_{e+1} = P_e\,R_e\,}\qquad \boxed{\,P_{e+1}\,Lift_e = Lift_e\,P_e\,}\qquad \boxed{\,R_e\,Lift_e = I\,}.\]

Commutation R_e P_{e+1} = P_e R_e: fiber-averaging the fine transport equals coarse-transporting the averaged state — transport is projective (projective_commutation_residual, and its Laplacian form laplacian_commutation_residual).
Intertwining P_{e+1} Lift_e = Lift_e P_e: the lift is an operator intertwiner (lift_intertwining_residual).
Right inverse R_e Lift_e = I: averaging a lifted (fiber-constant) state returns it (lift_reduction_residual).

Proof sketch (commutation). For s ∈ S_{e+1}, (x+s) mod p^e depends only on x mod p^e and s mod p^e; since each coarse shift t ∈ S_e has exactly p fine preimages in S_{e+1}, summing over the fiber of y and over S_{e+1} reproduces p·d_e copies of the coarse sum, and the two 1/p and 1/d normalisations collapse to P_e R_e. The control (test_non_uniform_fine_set_ breaks_commutation) drops one fiber element to make the multiplicity non-uniform and the residual becomes non-zero. This control shows failure for that altered set. The theorem concerns the displayed uniform reduction condition, not a classification of all possible weighted projective families.

4. Surviving modes (DERIVED + MEASURED)¶

The intertwining P_{e+1} Lift_e = Lift_e P_e has an immediate spectral consequence. If P_e v = λ v then

\[P_{e+1}(Lift_e\,v) = Lift_e(P_e\,v) = \lambda\,(Lift_e\,v),\]

and Lift_e v ≠ 0 (the lift is injective), so λ ∈ spec(P_{e+1}). Hence

\[\operatorname{spec}(L_e) \subseteq \operatorname{spec}(L_{e+1}):\]

every coarse mode survives the lift, and level e+1 only adds finer, fiber-varying modes (those annihilated by R_e). Numerically the coarse spectrum sits within the fine spectrum to ~1e-15 (surviving_spectrum_containment, below the derived tolerance √ε·‖L‖₂). The measured spectral gap λ₂(L_e) decreases from level 1 and then stabilises (e.g. p = 3, unit base: 1.5 → 1.0 → 1.0) — the slow-mode timescale reaches a fixed value in that finite sample. For directed operators the helper reports the minimum nonzero eigenvalue modulus, which need not be the decay rate nu_f * min Re(lambda) of the heat evolution.

5. REMESH is not claimed (the contract audit)¶

These are exact identities of declared finite linear transport models. The prime, residue carrier and connection family are inputs; neither their formation nor a joint phase/capacity/support law is derived. Projective consistency of the EPI channel does not by itself transport the full tetrad or the multichannel pressure constitutive map.

TNFR's REMESH operator (recursivity, AGENTS.md §5) has a specific contract: a recursive EPI echo across scales, NETWORK-scale generator/closure, preserved nodal identity, and U5 multiscale coherence. Projective transport consistency (§3–4) is necessary but not sufficient for that contract. RemeshContractAudit records the four conditions, all unverified, so remesh_contract_audit().realizes_remesh == False and the map keeps its neutral name. The claim "a compatible p-adic tower realises REMESH" (NT-P04) is therefore CONJECTURAL; naming the reduction REMESH is forbidden until every contract field is independently established.

5b. The R4b campaign — temporal memory and the U5 boundary (N09)¶

The campaign measures the missing temporal ingredient (remesh_audit.py) without conflating field dispersion with hierarchy coherence. The static tower candidate is the same-scale projection P = Lift_e · R_e; the temporal candidate is EPI_new = (1-α)² EPI(t) + α(1-α) EPI(t-τ_l) + α EPI(t-τ_g). Their default audits are:

candidate	temporal echo	NETWORK probe	identity probe	`1/(1+std(EPI))`	declared U5	REMESH
static `Lift·R_e`	no (`0`)	yes	yes	preserved	unverified	no
temporal recurrence	yes (`≈ 1.5`)	yes	yes	preserved	unverified	no

The dispersion score is now reported as field_uniformity_*. It reads only the sampled scalar EPI field; it does not read DeltaNFR or dEPI and declares no parent/child relation. It therefore cannot set u5_multiscale_verified. The default campaign establishes only that the recurrence responds to delayed inputs while the projection does not: temporal_echo_discriminates == True, but audit_discriminates == False because U5 has no declared evidence.

A caller can provide candidate-specific RemeshU5Evidence for a concrete graph materialized after the update. The audit then invokes assess_u5_parent_child_coherence with the declared parent, children, alpha and tolerance. A satisfying assessment can complete the temporal candidate's scoped four-condition audit. This is evidence for that hierarchy and state; it is not a universal U5-preservation theorem for later states or arbitrary recurrences.

6. Claim ledger¶

Claim	Basis	Status
`R_e P_{e+1} = P_e R_e` (compatible family)	uniform-fiber reduction	DERIVED + MEASURED (exact `0`)
`P_{e+1} Lift_e = Lift_e P_e`	operator intertwining	DERIVED + MEASURED (exact `0`)
`R_e Lift_e = I`	fiber-constant averaging	DERIVED + MEASURED (exact `0`)
`spec(L_e) ⊆ spec(L_{e+1})`	intertwining corollary	DERIVED + MEASURED (`~1e-15`)
compatibility necessary	non-uniform control	MEASURED (residual `≠ 0`)
tower realises REMESH	contract unverified	CONJECTURAL (`NT-P04`, `realizes_remesh = False`)
static `Lift·R_e` lacks temporal echo and declared U5 evidence	scoped audit (N09)	MEASURED (2/4 pass, echo `= 0`)
temporal recurrence has delayed-input sensitivity	temporal probe (N09)	MEASURED (`temporal_echo_discriminates = True`)
default campaign realises REMESH	no declared hierarchy	UNVERIFIED (`audit_discriminates = False`)
recurrence with declared hierarchy evidence	canonical U5 assessment	SCOPED to the supplied post-update state

Scope. The reduction-compatible p-adic tower carries a genuinely projective transport: coarse and fine dynamics commute with fiber-averaging, the lift intertwines the levels, and coarse modes survive exactly. This is a structural scale-consistency result on small p and low exponents. It does not identify the scale map with REMESH, and it closes no open problem.