# F1-DICT — ACT \(\Leftrightarrow\) H-D identification

### Written map · FAIL_IDENTIFY · HOLD-ACT-0 is not H-D hold · no numerics · hold OPEN

**CPT Framework — F1 / Pathway B**  
**Date:** 2026-08-21  
**Status:** **BANKED** · `PASS_F1_DICT_FAIL_IDENTIFY` · continuum hold **OPEN** · not a chip flip · not a promote  
**Design parent:** `a_geom_f1_one_seat_hold_spec.md` §10  
**Banked inputs:** HOLD-ACT-0/1 · HOLD-RED-0 · HOLD-RED-M · HOLD-CURR · compact core §1c.3b · H4 considerations · B-HOLD  
**Not:** HOLD-RED-1 (\(K_C\)) · F1-NEW · F1-GREEN · numerics · a \(m_e\) mint

Tags: **[E honesty]** · **[E process]** · **[O residual]** · **[Programme]** · **[F]**

---

## 0. One-line

> **FAIL_IDENTIFY.** Faddeev–Skyrme / Abelian–Higgs on round \(S^3\) (HOLD-ACT-0/1) is a solved **ACT** book: \(\Theta=2\eta\), \(Q_H=1\), \(\mathcal{T}_\eta^\eta\equiv 0\). Public hold is **H-D**: \(\mathbf{F}_{\mathrm{self}}=0\) on a closed wind under Weber-class packaging. The hoped-for operator map breaks on carrier, tangent space, kernel, current, and scale. Do not file ACT as H-D. Hold OPEN. No promote.

---

## 1. What this slice is

F1-SPEC required a scored dictionary ACT \(\Leftrightarrow\) H-D, or **FAIL_IDENTIFY**. This file writes the map and scores it. No new census. No new functional.

| Predicate | Object | Status entering this slice |
|-----------|--------|----------------------------|
| **ACT** | Critical point of named 3-energy on round \(S^3\), \(Q_H=1\), typically \(\mathcal{T}_\eta^\eta=0\) | **PROVED** (HOLD-ACT-0/1) |
| **H-D** | \(\mathbf{F}_{\mathrm{self}}=0\) on a closed wind, Weber-class packaging, \(\delta^2 U\ge 0\) on allowed modes (FP8/FP9) | Continuum **OPEN** (§1c.3b) |

A **PASS** dictionary would exhibit a named operator identification such that ACT stationarity *is* H-D stationarity, and ACT Hessian *is* the H-D second variation on the F1-SPEC allowed modes (radial / bunching / common-current). Anything less is **FAIL_IDENTIFY**, which is an allowed and honest exit.

---

## 2. Hoped-for map (what would have to be true)

| # | ACT object | Hoped H-D image |
|---|------------|-----------------|
| **D1** | Field \(\mathbf{n}:S^3\to S^2\), \(\Theta=2\eta\) | One-seat \(T(1,1)\) Hopf *fibre* / thin-tube wind |
| **D2** | \(Q_H=1\) | Charge / winding of the closed current |
| **D3** | \(\mathcal{T}_\eta^\eta\equiv 0\) | \(\mathbf{F}_{\mathrm{self}}=0\) |
| **D4** | Profile Hessian \(K_R\) (DEF-R) | FP9 second variation on hold modes |
| **D5** | DEF-T worldline operator \(K_T\) | Weber \(r\ddot r\) channel (H-D16) |
| **D6** | Abelian–Higgs \((A,\Phi)\) via CFN | Weber / Maxwell self-field on the wind |
| **D7** | Conserved \(Q_H\) | Conserved circulating \(\lvert I\rvert\) |
| **D8** | Local Dirichlet+Skyrme EL | Nonlocal Weber / retarded Green |
| **D9** | Ambient \(R_{S^3}\) / stereo \(\lambda\) | Tube radius \(a^\star\) (Compton / leftover length) |
| **D10** | BPS dial \(\lambda=e^2\) (AH) | Dual packaging \(\mu^\star\) |

---

## 3. Scoreboard

| # | Verdict | Why |
|---|:-------:|-----|
| **D1** | **FAIL** | Carrier. ACT \(\mathbf{n}\) fills the 3-sphere. H-D support is a closed wind (1-D or thin tube) *in* \(S^3\). Fibre \(\neq\) Hopfion unless dictionary D★ is promoted; D★ is **[M]** (H4, core). HOLD-ACT-0 is a bulk soliton, not a pair-force on a worldline. |
| **D2** | **PARTIAL, not H-D** | \(Q_H=1\) \(\equiv\) magnetic helicity (HOLD-ACT-1) is a topological integer of the *map*. H-D charge is dual-packaged free monopole \(Q_{\mathrm{free}}=1+\mu=O(1)\) on a wind (HC-0). Same integer class is not the same current. |
| **D3** | **FAIL** | \(\mathcal{T}_\eta^\eta\) is a sigma-model stress component. \(\mathbf{F}_{\mathrm{self}}\) is a Weber-class pair-force on the wind. HOLD-RED-0 non-claim: \(\mathcal{T}_\eta^\eta=\mathbf{F}_{\mathrm{self}}\) **forbidden**. Different tensors, different arguments. |
| **D4** | **FAIL as hold Hessian** | \(K_R\) is a Sturm–Liouville operator on the *profile* \(\Theta(\eta)\) (HOLD-RED-0). Hessian \(>0\) on scanned DEF-R modes. F1-SPEC required radial **and** bunching **and** common-current. DEF-R is none of the last two. Classifying \(K_R\) as historical Weber is forbidden (profile \(\neq\) fibre). |
| **D5** | **UNSCORED / named only** | DEF-T is the only deformation where \(r\ddot r\) *could* appear (HOLD-RED-0). **Not computed.** HOLD-RED-1 (\(K_C\)) remains **only if asked**. HOLD-RED-M: Manton same-slot Weber \((m+k/(c^2 r),\,k/r)\) **MISS**. So the one slot that might have identified does not, on the scored path. |
| **D6** | **FAIL as self-field** | CFN is clothing (HOLD-RED-0 R0-2): \(\mathcal{S}_{\mathrm{AH}}\) criticality is **not** inherited from ACT-0. Lab OP-F12 stays Darwin. AH Maxwell on \(S^3\) is not laboratory two-seat ED and not the H-D Weber kernel. |
| **D7** | **FAIL** | HOLD-CURR: common-mode \(\lvert I\rvert\) is invisible to present pair force. \(Q_H\) is frozen topology, not a current-magnitude mode. No map \(Q_H\leftrightarrow\lvert I\rvert\). |
| **D8** | **FAIL** | ACT is a *local* field EL (Dirichlet + Skyrme quartic). H-D is a *nonlocal* pair kernel (Weber / LW / ILC packaging). G-curve already tried (H-D23/24) and is often dual-blind. Skyrme EL is not that Green. |
| **D9** | **FAIL** | ACT energy \(\mathcal{E}=8\pi^2\kappa_2+16\pi^2\kappa_4\) on *unit* \(S^3\); \(\Theta=2\eta\) for **all** \(\kappa_2,\kappa_4>0\). No tube scale \(a^\star\). Ambient \(R\) has field \(g_{RR}=0\) (HOLD-RED-M). Stereo Derrick \(V\) is not Coulomb. L1: homogeneous \(1/a\) does not mint \(a\). Route A **FIRE as mint**. |
| **D10** | **FAIL** | AH \(\mathcal{T}_\eta^\eta=0\) uses BPS \(\lambda=e^2\). H-D dual cancel uses high-\(\beta\) \(r\ddot r\) packaging \(\mu^\star\approx-0.66\); under max\(\lvert\beta\rvert<1\) dual cancel **FAIL** (H-D15–17). No dictionary of dials. |

**Overall:** **FAIL_IDENTIFY**.

---

## 4. Structural diagnosis (why the map cannot be patched by a coefficient)

The two books differ in **kind**, not in a missing \(O(1)\) factor.

```text
ACT:  bulk 3-field on S³     →  local EL     →  T_η^η = 0  (all κ)
H-D:  pair-force on a wind   →  nonlocal F   →  F_self = 0  (OPEN)

tangent scored:   DEF-R  (profile Jacobi K_R)
tangent needed:   DEF-T  (worldline; named, not computed)
kernel scored:    Dirichlet + Skyrme
kernel needed:    Weber / retarded Green
current scored:   Q_H ∈ ℤ
current needed:   |I| mode (HOLD-CURR: pair force does not restore it)
scale scored:     ambient / stereo Derrick (Machian leftover)
scale needed:     tube a* independent of m  (L1 forbids mint)
```

A coefficient shop that sets \(\kappa_4/\kappa_2\) or \(\lambda=e^2\) cannot move ACT onto the H-D kernel, the H-D support, or the H-D tangent space.

**Finding I1 [E].** HOLD-ACT-0 is a genuine one-seat *field* theorem (smooth \(Q_H=1\) ground state). Public F1 is a one-seat *force* residual. Those can share topology (Hopf) without sharing the EL.

**Finding I2 [E honesty].** Joint H-D stall (H-D17: energy can stationarise, force dual cancel FAIL under field-safe \(\beta\)) has **no counterpart** in ACT: ACT has no \(\beta\), no \(\mu^\star\), and \(\mathcal{T}_\eta^\eta\equiv 0\) for all couplings. ACT does not even *state* the residual that F1 must close.

**Finding I3 [E].** The reduction programme already froze the honest remainder: \(K_R\neq r\ddot r\); Manton \(\neq\) Weber; DEF-T unnamed as a computation. F1-DICT does not reopen HOLD-RED-1.

---

## 5. What would un-fail (not opened)

All of the following would be a **new principle**, not a coefficient:

1. Promote D★ from **[M]** to a scored fibre \(\Leftrightarrow\) Hopfion theorem (carrier D1).
2. Compute DEF-T / \(K_T\) and identify it with the H-D \(r\ddot r\) operator — and survive HOLD-RED-M **MISS** (this is HOLD-RED-1 plus a Weber identification, **only if asked**).
3. Name a Green that is G-static or G-dyn, not G-curve, and show its jump *is* \(\mathbf{F}_{\mathrm{self}}\) (F1-GREEN).
4. Exhibit a common-current mode of ACT whose linearisation restores \(\lvert I\rvert\) (contradicts HOLD-CURR on the pair-force book; would have to be a different kernel).

Until those are named and scored, citing HOLD-ACT-0 as continuum hold is **[F]**.

---

## 6. F1 residual after this slice

| Item | Status |
|------|--------|
| ACT existence | **BANKED** (HOLD-ACT-0/1) — different book |
| H-D continuum hold | **OPEN** |
| ACT \(\Leftrightarrow\) H-D | **FAIL_IDENTIFY** (this file) |
| F1-GREEN | **SCORED_NO** · `a_geom_f1_green_static_tail_results.md` |
| F1-NEW | **FILL DYN-G** · `a_geom_f1_new_dyn_g_fill.md` · estimate only if asked |
| HOLD-RED-1 | Only if asked |
| Public chip | **OPEN** (unchanged) |

F1-SPEC kill list still stands. Licensed writing #2 remains F5 uniqueness **design**.

---

## 7. Integrity

| Gate | Verdict |
|------|---------|
| Map written row-by-row | **PASS** |
| FAIL_IDENTIFY allowed and used | **PASS** |
| No silent “therefore hold” | **PASS** |
| No numerics | **PASS** |
| HOLD-RED-1 not opened | **PASS** |
| Hold chip | **OPEN** |
| Promote | **NOT APPROVED** |

**Exit:** `PASS_F1_DICT_FAIL_IDENTIFY`

---

## 8. One-line for agents

> **§F1-DICT: FAIL_IDENTIFY; ACT \(\mathcal{T}_\eta^\eta=0\) is not H-D \(\mathbf{F}_{\mathrm{self}}=0\); carrier, tangent, kernel, current, scale all break; hold OPEN; no promote.**