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This is a raw source document. Integrity tags [E] [M] [O] [F] are coloured for readability. Content is as written in the research file.

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.
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